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Title:
Statistical environmental models hurricanes, lightning, rainfall, floods, red tide and volcanoes
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Book
Language:
English
Creator:
Wooten, Rebecca Dyanne
Publisher:
University of South Florida
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Tampa, Fla
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Subjects / Keywords:
Parametric analysis
Extreme value distribution
Linear and non-linear modeling
Prediction and forecasting
Dissertations, Academic -- Mathematics and Statistics -- Doctoral or Masters or Specialist -- USF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
ABSTRACT: This study consists of developing descriptive, parametric, linear and non-linear statistical models for such natural phenomena as hurricanes, lightning, flooding, red tide and volcanic fallout. In the present study, the focus of research is determining the stochastic nature of phenomena in the environment. These statistical models are necessary to address the variability of nature and the misgivings of the deterministic models, particularly when considering the necessity for man to estimate the occurrence and prepare for the aftermath.The relationship between statistics and physics looking at the correlation between wind speed and pressure versus wind speed and temperature play a significant role in hurricane prediction. Contrary to previous studies, this study indicates that a drop in pressure is a result of the storm and less a cause. It shows that temperature is a key indicator that a storm will form in conjunction with a drop in pressure.^ This study demonstrates a model that estimates the wind speed within a storm with a high degree of accuracy. With the verified model, we can perform surface response analysis to estimate the conditions under which the wind speed is maximized or minimized. Additional studies introduce a model that estimates the number of lightning strikes dependent on significantly contributing factors such as precipitable water, the temperatures within a column of air and the temperature range. Using extreme value distribution and historical data we can best fit flood stages, and obtain profiling estimate return periods. The natural logarithmic count of Karenia Brevis was used to homogenize the variance and create the base for an index of the magnitude of an outbreak of Red Tide. We have introduced a logistic growth model that addresses the subject behavior as a function of time and characterizes the growth rate of Red Tide.^ This information can be used to develop strategic plans with respect to the health of citizens and to minimize the economic impact. Studying the bivariate nature of tephra fallout from volcanoes, we analyze the correlation between the northern and eastern directions of a topological map to find the best possible probabilistic characterization of the subject data.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
Statement of Responsibility:
by Rebecca Dyanne Wooten.
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Title from PDF of title page.
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Document formatted into pages; contains 214 pages.
General Note:
Includes vita.

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University of South Florida Library
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University of South Florida
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Resource Identifier:
aleph - 001936628
oclc - 226377184
usfldc doi - E14-SFE0001824
usfldc handle - e14.1824
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SFS0026142:00001


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StatisticalEnvironmentalModels: Hurricanes,Lightning,Rainfall,Floods,RedTideandVolcanoes b y RebeccaDyanneWooten A dissertation submittedinpartialfulfillment oftherequirementsforthedegreeof DoctorofPhilosophy Departmentof MathematicsandStatistics CollegeofArtsandSciences UniversityofSouthFlorida Major Professor: C hristopher P.Tsokos,Ph.D. K andethody Ramachandran,Ph.D. M arcus McWaters ,Ph.D. C harles Connor,Ph.D. DateofApproval: October 26 ,2006 Keyw ords:ParametricAnalysis,ExtremeValueDistribution,LinearandNon linear Modeling,PredictionandForecasting. Copyright2006,RebeccaDyanneWooten

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NOTETOREADER Thisistoinformthereaderthatoriginaldocumentcontainscolorthatisnecessar yfor understandingthatdata.TheoriginaldissertationwillbeonfilewiththeUSFlibraryin Tampa,Florida.

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DEDICATION Toallthosewhohaveeducatedme: primarily,mymotherandfatherbutalsoall thosewhofollowed,thankyou. Aspecialdedica tiontomyfather,JerryArthurWooten,andmother,Altha AustinaWooten;bothinspiredmetofollowmydreamsofacademia.

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ACKNOWLEDGEMENTS Iwouldliketoexpressmysinceregratitudeandappreciationtomyresearch supervisoranddissertationadvisor ProfessorC.P.Tsokos mymentorandguide throughoutmyresearch. Iwouldalsoliketoacknowledgetheothermembersofmydoctorialcommittee, Dr.M.McWaters,Dr.K.Ramachandran,andDr.Connorwhomsuggestionsand guidancewasgreatlyappreciate d. Inaddition,IwouldliketoacknowledgethelateDr.A.N.V.Raoforhisteachingsand usefuldiscussiononthesubjectofstatistics. Finally,IwouldliketothankDr.G.Yanevforhishelpfulsuggestions.

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i TABLEOFCONTENTS LISTOFTABLES v L ISTOFFIGURES i x ABSTRACT xvii CHAPTER1: MOTIVATIONOFTHEPR ESENTSTUDY1 1.1 Introduction 1 1.2 TropicalStorms 1 1.3 Hurricanes 2 1.4 Lightning 3 1.5 Flooding 4 1.6 RedTide 5 1.7 Volcanoes 6 CHAPTER2: STATISTICALMODELING THECONDITIONSUNDE R WHIC H TROPICALSTORMSARE FORMED:ESTIMATING THEBIRTHOFASTORM 7 2.1 Introduction 7 2.2 DifferencingEquationsandMolecularPhysics 8 2.2.1 TestingtheConjecture 9 2.2.2 StartofHurricaneSeason2006 10 2.3 Descriptionof R esponse V ariableand C ontributing E ntitie s 12 2.3.1 WindSpeed(WSPD) 13 2.3.2 WindDirection(WD) 14 2.3.3 Gust 15 2.3.4 Pressure(BAR) 15 2.3.5 AtmosphericTemperature(ATMP) 16 2.3.6 SeaSurfaceTemperature(WTMP) 17 2.3.7 DewPoint(DEWP) 18 2.4 ParametricAnalysis 18 2.4.1 ParametricAnalysisofWindSpeed 20 2.4.1.1 TheWeibullProbabilityDistri bution 20 2.4.1.2 MaximumLikelihoodEstimates 22 2.4.2 Parametr icAnalysisofWindDirection 24 2.4.3 Pa rametricAnalysisofPressure 27 2.4.4 ParametricAnalys isofAtmospheric Temperature 30 2.4.5 Parametric AnalysisofWaterTemperature 33 2.4.6 Par ametricAnalysisofDewPoint 36

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ii 2.5 Multiva riateModelingofWindSpeeds N eartheSeasSurface 39 2.5.1 ContributingVariables 39 2.5.2 StatisticalModel 39 2.5.3 Rankinorderofsignificant( p value)and contribution 40 2.5.4 HigherOrderTerms 41 2.5.5 EffectofInteraction 43 2.5.6 EffectofDummyVariables 44 2.6 ModelValidati on 4 7 2.6.1 Line arStatisticalModel 50 2.6.2 Complete InteractiveStatisticalModel 51 2.7 SurfaceResponseAnalysis 52 2.8 Useful nessoftheStatisticalModel 55 2.9 Conclusion 55 CHAPTER3: LINEARANDNON LINEARSTATISTICALM ODELINGOF HURRICANEFORCEWIND S:HURRICANEINTENSI TY 5 7 3.1 Introdu ction 57 3.2 Descriptionof ResponseVariable(WindS peed) 58 3.2.1 DataforFiveCategory5TropicalStorms 58 3.2.2 ResponseVariables 60 3.2.3 ComparisonofLatitudeandLongitudeand theCartesiancoordinates 61 3.2.4 Whatisthedi fferenceindirectional moveme ntwithrespecttothestorm? 62 3.3 MultivariateMode lingofHurricaneForceWinds 6 3 3.3.1 ContributingVariables 63 3.3.2 Rankinorderofsignificant( p value)and contribution 64 3.3.3 TheStatisticalModel 65 3. 3.5 HigherOrderTerms 67 3.3.6 Interaction 68 3.4 CrossValidationand EstimatingHurricaneKatrina 71 3.5 ModelValidation ofCompleteInteractiveModel 71 3.6 SurfaceResponseAnalysis 72 3.7 Analysi softheSaffir SimpsonScale 74 3.7.1 Analysis o ft heSaffir Simpsonscale 74 3.7.2 HowdoestheSaffir Simpsoncomparewith thesefivestorms? 75 3.7.3 TheThermodynamicsbehindMolecular Physics 81 3.8 Useful nessoftheStatisticalModel 83 3.9 Conclusion 83 CHAPTER4: STATISTICALANALYSIS ANDMO DELINGOFLIGHTNING 86 4.1 Introduction 86 4.2 Descriptionof VariousC ontributing E ntities 88 4.3 ParametricAnalysis 89

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iii 4.4 Bootstrapping 92 4.4.1 ConvergenceofMeanandStandardError 94 4.4.2 PercentageChangeintheMeanand StandardError 96 4. 5 MultivariateModelofthe N umberofLightningStrikes 97 4.5.1 TheStatisticalModel 97 4.5.2 RankingofIndependentVariables 101 4.5.3 InteractiveStatisticalModel 102 4.6 StatisticalModelValidation 104 4.7 UsefulnessoftheSt atisticalModel 106 4.8 Conclusion 106 CHAPTER5:ANALYS ISANDMODELINGFLOO DSTAGES 108 5.1 Introduction 108 5.2 Generalize dExtremeValueDistribution 109 5.2.1 DerivationoftheGumbelDistribution 110 5.2.2 CharacteristicsoftheVariousDistributions 110 5.2.3 Maximu mLikelihoodEstimates(MLE;s) forthegivenProbabilityDistributions 110 5.2.4 PercentilesfortheFrechet,Weibulland GumbelProbabilityDistributions 113 5.3 Desc riptiveAnalysisoftheData 113 5.4 ParametricAna lysisofFloodStageHeights 11 9 5.5 Modeling F low V olumein T ermsof D urationand F low R ates 121 5.5.1 FloodStageasaFunctionofDuration 121 5.5.2 FloodStageasaFunctionofFlowRate 122 5.5.3 FloodStageasaFunctionofBothDuration andFlowRate 122 5.5.4 FloodStage asaFunctionofDuration, FloodRateandTime 123 5.6 Bes t FitDistributionProfiledbyD uration 125 5.7 Usefulness 127 5.8 Conclusion 127 CHAPTER6:ANALYSIS AND MODELINGOFREDTID EBLOOMS 128 6.1 Introduction 128 6.2 AnalysisoftheVariousOrganism sMeasuredinRedTide 130 6.3 DescriptiveStatisticsforKarenia Brevis 132 6.4 ParametricAnalysis 136 6.5 Logarithmic T ransformationandits P roperties 137 6.5.1 MaximumLikelihoodFunction 139 6.5.2 Two parameterWeibullProbability Distribution Function 140 6.6 MixedDistributions 143 6.7 RegionalAnalysis 147 6.7.1 RegionalDifferencesontheEastandWest Coasts 147

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iv 6.7.2 RegionalDifferencesNorthandSouthof TampaBayontheWestCoast 148 6.7.3 RegionalDifferencesBetweenTampaBay andOtherRegions 149 6.7.4 RegionalAnalysisBetweentheBasins AlongtheWestCoast 150 6.8 RecursionAnalysis 154 6.9 LogisticGrowth 161 6.10 StatisticalModeling 164 6.10.1 AmmoniumIon: 4 NH 164 6.10.2 NitrateIon: 3 NO 165 6.10.3 SulfateIon: 2 4 SO 6.11 Usefuln essoftheStatisticalModel 169 6.12 Conclusion 170 CHAPTER7: STATISTICALMOD ELINGOFVOLCANICAC TIVITIES 171 7.1 Introduction 171 7.2 AnalysisofVolcanicExplosivityInde xo fCerroNegro,Nigeria 173 7.3 TrendAnalysis 175 7.4 Directionof Deposition:ConicSections 177 7.5 RadialAnalysis 180 7.6 ParticleS izeProbabilityDistribution 185 7.7 Statistica lModelingofTephraFallout 187 7.8 BivariateDistribution 189 7.9 Comparisono f F our F ormsoftheB ivariateNormalProbability Distribution 197 7.10 UsefulnessoftheStatisti calProbabilityDistribution 204 7.11 Conclusion 204 REFERENCES 206 ABOUTTHEAUTH O R EndPage

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v LISTOFTABLES Table1 : Momentsforthethree parameter Weibull 20 Table2 : MaximumLikelihoodEstimatesforthethree parameterWeibull 23 Table3 : Estimatedparameters 26 Table4 : Goodness of fittestsincluding statistics 27 Table5 : ParametersandStatisticsforassociatedWeibulldistributionsfor pr essure 29 Table6 : ParametersandStatisticsforassociatedWeibulldistributions 32 Table7 : ParametersandMomentsforassociatedWeibulldistributions 32 Table8 : ParametersforassociatedWeibulldistributions;including descriptivestatistics 36 T able9: ParametersandStatisticsforassociatedWeibulldistributions 37 Table10 : ParametersforassociatedWeibulldistributions;including descriptivestatistics 38 Table11 : Variablesofinterest 40 Table12 : Rankingofcontributingvariablesinf irstordermodel 41 Table13 : Listofcontributingvariablesincludingsecondorderterms 43 Table14 : Listofcontributingvariablesincludingsecondorderterms 44 Table15 : Listofcontributingvariablesincludinginteraction,secondorder termsand thedummyvariableforlocation 45 Table16 : Listofcontributingvariablesincludinginteraction,secondorder termsandthedummyvariableforlocationincludinginteraction 47 Table17 : Evaluationstatisticsforthequalityofthemodels 50

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vi Table18 : SummaryforForwardSelection(maineffects)including Mallows ) ( p C statistics 50 Table19 : SummaryforForwardSelectionincludingMallows ) ( p C statistics 51 Table 20 : Solutionstothepartialdifferentialequations 53 Table21: Tableofmaximumhurricaneforcewindsandtheirassociated pressuresforfiveresentstormsintheAtlanticregion 60 Table22 : Variablesofinterest 64 Table23 : Rankingofindependentvariables 65 Table24 : Estimatesforcoeffici entsinfulllinearmodel 66 Table25: Multipleleast squaresregressionincludingsignificantlinearterms andasinglequadratictermforpressure 67 Table26: Multipleleast squaresregressionincludingsignificantlinear termsandasinglequadrat ictermforpressureandsignificant interactions 69 Table33 : SummaryforForwardSelectionincludingMallows ) ( p C statistics 72 Table34: Saffir SimpsonHurricaneRatingScaleascategorizedbypressure andwindspeed 74 Table35 : Comparisonofpressureaccording to theSaffir SimpsonHurricane 75 Table3 6 : Testformeanpressure 78 Table37 : DescriptivestatisticsforwindspeedasassignedbytheSaffir SimpsonscaleandtheWootenscale 80 Table 38 : Descriptivestatistic sformeansealevelpressureasassignedbythe Saffir SimpsonscaleandtheWootenscale 80 Table3 9 : ComparisonofpressureaccordingtotheWootenHurricaneRating Scaledevelopedusinghistoricaldataforthefivehurricanes outlinedinthestudy. 8 1 Table 40 : Best fitdistributionforthenumberoflightningstrikespermonth 90

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vii Table4 1 : Estimatedparametersforthenumberoflightningstrikespermonth 90 Table42 : Estimatedvaluesusingthetwo parameterWeibull 92 Table4 3 : Samplesize, mean,standarddeviation,standarderrorandthe lowerandupperboundsonthe95%confidenceintervalforthe meannumberoflightningstrikespermonth 93 Table44 : Confidenceintervalsforthemeanusingthetwo parameter Weibull 96 Table4 5 : Variab lesofinterest inestimatingthemeannumberoflightning strikespermonthintheStateofFlorida 98 Table46 : Linearregressionforthenumberoflightningstrikesinamonth withrespecttotherankedindependentvariablesincludingthe associatedp values 100 Table4 7: Rankingofindependentvariablesusingforwardselectionwitha 0.05leveltogetintothemodelanda0.05leveltoremaininthe model. 101 Table48 : Least squaresregressionforthenumberoflightningstrikesina monthwithre specttotherankedindependentvariablesincluding interaction;alsoincludedaretheassociated p values 104 Table 49 : SummaryforForwardSelectionincludingMallows ) ( p C statistics 105 Table50 : HistoricalCrestsintheSt.John sRiver near 115 Table51 : FloodEventsintheSt.JohnsRivernearDeland 118 Table52 : Testforfitoffloodstageheight 120 Table 53 : Datacompiledbynumbero ftimesrecordedinsamplings. Includestotalcountovertimeandmeancountpersampl easwell astheorganismandhowittheseorganismsarecoded. 131 Table 54 : Percentbycategory 134 Table 55 : Percentbycategory(redefined) 135 Table 56 : Statisticsbasedonthetwoandthree parameterWeibull 141 Table 57 : Goodness of FitTest forWeibull 141

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viii Table 58 : Estimat ionsforvariousreturnperiods 143 Table 59 : Estimationsofparametersandassociatedchi squaredstatistic 146 Table 60 : Descriptivestatisticsforvariousregions 152 Table61 : Comparisonofregionsand p value foradjacentregions 153 Table 62 : Descriptivestatisticsformajorregions 154 Table 63 : Comparisonbymonthand p valueforconsecutivemonths 155 Table 64 : Descriptivestatisticsbymonth 155 Table65 : Correlationsgivenvariousmaximumcapacit ies 163 Table66 : Locationofsiteswhereconcentrationsaremeasured 166 Table67 : VariablesofInterest 168 Table 68 : Rankofadditionalcontributingvariables 169 Table 69 : EruptiondataforCerroNegro 173 Table7 0 : Frequenciesandproportion sforthetwodatasources 175 Table7 1 : Databydirection(eightsectors) 179 Table7 2 : Databydirection(twenty foursectors) 180 Table7 3 : Massbydistance 183 Table7 4 : Massbydiameterandsize 185 Table7 5 : Testforbest fitdistribution 186 Table7 6 : Descriptivestatisticsandregressedslopeandcorrelation coefficient 201

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ix LISTOFFIGURES Figure1 : MapofTropicalStormAlberto;eachbuoyismarkedbyastarand numbered 9 Figure2 : Temperature(red)andPressure(blue)diffe rencesatthefirstbuoy (42056) 10 Figure3 : Temperature(red)andPressure(blue)differencesatthesecond buoy(42036) 10 Figure4 : Linegraph(byday)ofatmospherictemperaturef orthefirstbuoy (42056). 11 Fig ure 5 : Linegraph (byday)ofatmo spherictemperatureforthesecond buoy(42036) 11 Figure 6 : Linegraph(byday)ofatmosphericpressure(hPa )forthefirst buoy(42056). 12 Figure7 : Linegraph(byday)ofatmosphericpressure(hPa) forthesecond buoy(42036). 12 Figure8 : Linegra ph ofwindspeed(m/s);thefirstbuoy(42056)inburnt orangeandthesecondbuoy(42036)inaqua 14 Figure9 : Linegraph ofwinddirections(bearings);thefirstbuoy(42056)in burntorangeandthesecondbuoy(42036)inaqua 14 Figure10 : Linegrap h ofwindgust(m/s);thefirstbuoy(42056)inburnt orangeandthesecondbuoy(42036)inaqua 15 Figure11 : Linegraph ofatmosphericpressure(hPa);thefirstbuoy(42056) inburntorangeandthesecondbuoy(42036)inaqua 16 Figure12 : Linegraph ofatmospherictemperature( C );thefirstbuoy (42056)inburntorangeandthesecondbuoy(42036)inaqua 17 Figure13 : Linegraph ofwatertemperature( C );thefirstbuoy(42056)in burntorangeandthesecon dbuoy(42036)inaqua 18

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x Figure14 : Linegraph ofdewpoint;thefirstbuoy(42056)inburntorangeand thesecondbuoy(42036)inaqua 19 Figure15 : Histogramofwindspeedrecordedatbothsites;including descriptivestatisticsJune2006 21 Figure 16 : Histogramofwindspeedrecordedateachsite 22 Figure17 : Histogramandbest fitdistribution 24 Figure18 : Histogramofwinddirectionrecordedatbothsites;including descriptivestatistics 24 Figure19 : Histogramofwindspeedrecordedatea chsite 25 Figure20 : Cumulativeprobabilitydistributionforthedata,thenormalsub functionsandthemixeddistribution 26 Figure21 : Histogramofatmosphericpressurerecordedatbothsites; includingdescriptivestatistics. 27 Figure22 : Histogra mofatmosphericpressurerecordedateachsites 28 Figure23 : Histogramandpotentialbest fitdistributions 30 Figure24 : Histogramofatmospherictemperaturesrecordedatbothsites; includingdescriptivestatistics. 31 Figure25 : Histogramofatmo spherictemperaturesrecordedateachsites 31 Figure26 : Histogramofprobabilitydensityfunctionsforatmospheric temperature 32 Figure27 : Histogramandbest fitdistributionforatmospherictemperature 33 Figure28 : Histogramofwatertemperatu resrecordedatbothsites;including descriptivestatistics. 34 Figure29 : Histogramofwatertemperaturesrecordedateachsites 35 Figure30 : Histogramandbest fitdistributionsforwatertemperature 36 Figure31 : Histogramofdewpointrecordedat bothsites;including descriptivestatistics. 37 Figure32 : Histogramofdewpointrecordedateachsite. 37

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xi Figure33 : Histogramandbestfitdistributionfordewpoint 38 Figure34: Residualplotandnormalplot forlinearstatisticalmodel. 48 Fig ure35: Residualplotandnormalplotforcompleteinteractivestatistical mod el. 49 Figure36 : Contourplotofwindspee doverpressureandatmospheric temperature 54 Figure37 : Contourplotofwindspeedoverpressureandwatertemperature 54 Figure 38 : Contourplotofwindspeedoverpressureanddewpoint 54 Figure39 : Mapoffivestorms:Isabel(2003),Ivan(2004),Katrina(2005), Rita(2005)andWilma(2005). 59 Figure40 : Scatterplotoflatitudeversuslongitude 61 Figure41 : Scatterp lotofconvertedlatitudeversuslongitudeintoCartesian coordinates x and y 61 Figure42: Linegraph forlongitude 62 Figure43: Linegraph forlatitude 62 Figure44: Linegraph forconvertedlatitude andlongitudewithrespectto x 62 Figure45: Linegraph forconvertedlatitudeandlongitudewithrespectto y 62 Figure46 : Resi dualplot forlinearstatisticalmodel. 66 Figure47 : Normalprobabilitypl otfortheresidualsofthesimplelinear mode l. 66 Figure48 : ResidualplotformodeloutlinedinTable25 68 Figure49 : Normalprobabilityplot 68 Figure5 0 : Linegraph comparisonforHurricaneWilma 70 Figure5 1 : Linegraph comparisonforHur ricaneIvan 70 Figure5 2 : Linegraph comparisonforHurricaneRita 70 Figure5 3 : Linegraph comparisonforHurricaneIsabel 70

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xii Figure5 4 : Linegraph comparisonforHurricaneKatrina 70 Figure55 : Meanpressureversusrecordedwindspeedsforth elistedfive storms 76 Figure56 : Meanpressureversusrecordedwindspeedsforallstormsrecorded inthe1990s 76 Figure57 : Meanpressureversusrecordedwindspeedsforallstormsrecorded inthe1980s 76 Figure58 : Boxplotforpressurewith respecttorecordedwind speed sforthe listedfivestorms. 77 Figure59 : Barchartsforpressure,withcategoriesasassignedbytheSaffir Simpsonscale 79 Figure60 : Barchartsforpressure,withcategoriesasassignedbytheWooten scale 79 Fig ure61 : Scatterplot ofpressureversuswindspeed 82 Figure 62 : Linegraph ofpressureforeachstorm 83 Figure 63 : Histogramforthenumberoflightningstrikespermonthincluding descriptivestatisticsforthenumberoflightningstrikespermont h 89 Figure64 : Weibullcumulativeprobabilitydistribution 92 Figure65 : Re Samplingofsize10,000 94 Figure66 : Convergenceofthemean x 94 Figure67 : Convergenceofthestandarddeviation n s 95 Fi gure68 : Convergenceofthestandarderror n 95 Figure69 : Convergenceofthestandarderror n p 96 Figure70 : Convergenceofthestandarderror n q 96 Figure71 : Residualsversusthe estimatedvalueswithmaineffectsvariable precipitablewater 102 Figure72 : Normalitytestfortheres iduals. 102

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xiii Figure73 : Estimat edvaluesversusrecordedlightningstrikes 102 Figure74 : MapofdailystreamflowconditionsfortheStateofFlo rida 114 Figure75 : GaugesitesalongtheSt.JohnsRiver 114 Figure76 : MapofgaugesiteofalongtheSt.JohnsRiver nearDeland. 115 Figure 77 : Realtimes tage(waterheightinfeet)forSt.JohnsRiver (rig ht) andcolorledge(above). 116 Fig ure 78 : Linegraph (daypast01/01/1934)forstageheightintheSt.Johns RivernearDeland 117 Figure 79 : Linegraph(dayofyear)forstageheightintheSt.JohnsRiver nearDelandfortheyears1938 2004 118 Figure80 : Histogramoffloodstage heightinfeetincludingdescriptive statistics. 119 Figure 81 : Comparisonofempiricalcumulativeprobabilitywiththatofthe cumulativeWeibulldistributionandthecumulativeGumbel distribution. 120 Figure 82 : Thereturnperiodsinyearsofvarious floodstagesdependingon thedurationofthefloodevent. 127 Figure 83 : Histogramofcountof KareniaBrevis sampledovertime 132 Figure 84 : Histogramofthenaturallogarithmofthecountof KareniaBrevis sampledovertime,giventhecountwasa tleastone 133 Figure 85 : Histogramofthenaturallogarithmofthecountof KareniaBrevis sampledovertime,giventhecountwasatleasttwo. 134 Figure 86 : Histogramofthenaturallogarithmofthecountof KareniaBrevis sampledovertime,given thecountgreaterthanfive 135 Figure 87 : Probabilityplotofthedoublenaturallogarithmofthecountof KareniaBrevis sampledovertime,giventhecountwasgreater thenfive. 136 Figure 88 : Histogramwithbest fitdistributionforthenaturall ogarithmofthe countof KareniaBrevis sampledovertime,giventhecountwas greaterthenfive. 136

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xiv Figure 89 : Returnperiodsofthenaturallogarithmofthecountof Karenia Brevis 142 Figure 90 : Comparisonofthebest fitdistributionsandtheemp irical probabilitydistribution 146 Figure 91 : Thecontourplotofthenaturallogarithmofthecountof Karenia Brevis withrespecttothesamplinglocation(longitude,latitude) 147 Figure 92 : Thescatterplotofsamplinglocationsbydefinedregio ns 148 Figure 93 : ThescatterplotofsamplinglocationsbynearTampaBayand otherregions 150 Figure 94 : MapofregionbyDistrictBasins 151 Figure95 : Thescatterplotofsamplinglocationsbygroupeddistricts 151 Figure96 : Boxplotoft hemagnitudesbyregion 152 Figure 97 : Thescatterplotofsamplinglocationsbymajorregions 153 Figure 98 : Mapoflocationwheredatameasuredhourly. 156 Figure 99 : Thescatterplotofsamplinglocationsbymajorregionsfor November1957 156 Figure 100 : Linegraphofmagnitudeofbloo msinthesouthernregion. 159 Figure 101 : Linegraphofmagnitudeofbloombymonth 161 Figure 102 : Linegraphofmagnitudeofbloombylocation(latitude,longitude) 162 Figure103 : Linegraphofmagnitud eofbloom i i c x ln atasinglelocationat time i t 162 Figure 104 : Linegraphofpercentagesbasedondailymeanofthedata(blue) andtheestimatedpercentagebasedonthepreviousdata 164 Figure1 05 : Linegraph ofVEIovertheyears(firstsource) 175 Figure1 06 : LinegraphofVEIovertheyears(secondsource) 175 Figure1 07 : Linegraphofcumulativevolume 3 m sincethebirthofCerro Negro 176

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xv Figure1 08 : Linegraphofcumulative frequencies 177 Figure1 09 : Linegraphofpercentagesovertime(probabilities) 177 Figure1 10 : ScatterplotofTephraFalloutbyconicalsection 178 Figure1 11 : ScatterplotofTephraFalloutbyconicalsection 179 Figure1 12 : Histogramofthe estimateddistancefromthecenter(mainvent) includingdescriptivestatistics 181 Figure1 13 : Scatterplotofmass 2 / m kg anddistancefromthecenter(main vent) 183 Figure1 14 : Histogramoftheestimatedanglesoffthehorizoni ncluding descriptivestatistics 183 Figure1 15 : Scatterplotofmass 2 / m kg andangleoffdueeast 184 Figure1 16 : Normalprobabilityplotfordirectionoffallout 184 Figure1 17 : Boxplotfor directionoftephrafallout. 184 Fig ure1 18 : Scatterplotofmass 2 / m kg bydiameter d 186 Figure1 19 : Scatterplotofmass 2 / m kg bysize d 2 log 186 Figure1 20 : Scatterplotofmass 2 / m kg 190 Figure 121 : Empiricalprobabilitydistributionforthegivennorthernand easterncoordinates 191 Figure 122 : Non correlatedandcorrelatedbivariateGaussiandistribution 192 Figure123 : Tephrafalloutbyparticlesize 193 Figure124 : Scatter plotfortheempiricalprobabilitydist ributionanddistance north. 195 Figure125 : Scatterplotforthecorrelatedbivariatenormaldist ributionand distancenorth. 195 Figure126 : Scatterplotfortheempiricalprobabilitydistributionanddistance e ast. 195

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xvi Figure127 : Scatterplotforthecorrelatedbivariatenorma ldistributionand distance. 195 Figure1 28 : Three dimensionalscatterplotoftheempiricalprobability distributionovertheunderlyingdistancenorthanddistanceeast. 196 Figure 1 29 : Three dimensionalscatterplotofthecorrelatedbivariatenormal distributionovertheunderlyingdistancenorthanddistanceeast. 196 Figure1 30 : Three dimensionalscatterplotoftheempiricalprobability distributionovertheunderlyingdistanc enorthanddistanceeast. 197 Figure1 31 : Three dimensionalscatterplotofthecorrelatedbivariatenormal distributionovertheunderlyingd ista ncenorthanddistanceeast. 197 Figure1 32 : Scatterplotof original data 198 Figure1 33 : S catterplot oftherotateddata. 198 Figure1 34 : Contourplotofempiricalprobabilitydistribution 201 F igure 135 : Contourplotfortheestimatednon correlatedbivariatenormal d istribution 202 Figure1 36 : Contourplotfortheestimatedcorrelatedbivariate normal distribution. 202 Figure1 37 : Contourplotfortheestimatedrotatednon correlatedbivariate normaldistribution. 204 Figure1 38 : Contourplotfortheestimatedrotatednon correlatedskewed bivariatenormaldistr ibution(assumingindependence ). 204

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xvii STATISTICALENVIRONM ENTALMODELS: HURRICANES, LIGHTNING,RAINFALL, FLOODS,REDTIDEAN DVOLCANOES RebeccaDyanneWooten ABSTRACT Thisstudyconsistsofdevelopingdescriptive,parametric,linearandnon linear statisticalmodelsforsuchnat uralphenomenaashurricanes,lightning,flooding,redtide andvolcanicfallout.Inthepresentstudy,thefocusofresearchisdeterminingthe stochasticnatureofphenomenaintheenvironment.Thesestatisticalmodelsare necessarytoaddressthevariab ilityofnatureandthemisgivingsofthedeterministic models,particularlywhenconsideringthenecessityformanto estimate theoccurrence andpreparefortheaftermath. Therelationshipbetweenstatisticsandphysicslookingatthecorrelationbetwee n windspeedandpressureversuswindspeedandtemperatureplayasignificantrolein hurricaneprediction.Contrarytopreviousstudies,thisstudyindicatesthatadropin pressureisaresultofthestormandlessacause.Itshowsthattemperatureis akey indicatorthatastormwillforminconjunctionwithadropinpressure. Thisstudydemonstratesamodelthat estimates thewindspeedwithinastorm withahighdegreeofaccuracy.Withtheverifiedmodel,wecanperformsurface responseanalysist oestimatetheconditionsunderwhichthewindspeedismaximized or minimized .Additionalstudiesintroduceamodelthat estimates thenumberoflightning

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xviii strikesdependentonsignificantlycontributingfactorssuchasprecipitablewater,the temperatures withinacolumnofairandthetemperaturerange.Usingextremevalue distributionandhistoricaldatawecanbestfitfloodstages, andobtain profilingestimate returnperiods. Thenaturallogarithmiccountof KareniaBrevis wasusedtohomogenizet he varianceandcreatethebaseforanindexofthemagnitudeofanoutbreakofRedTide. Wehaveintroducedalogisticgrowthmodelthataddressesthesubjectbehaviorasa functionoftimeandcharacterize sthegrowthrateofRedTide. Thisinformationc anbe usedtodevelopstrategicplanswithrespecttothehealthofcitizensandtominimizethe economicimpact. Studyingthebivariatenatureoftephrafalloutfromvolcanoes,weanalyzethe correlationbetweenthenorthernandeasterndirectionsofat opologicalmaptofindthe bestpossibleprobabilisticcharacterizationofthesubjectdata.

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1 CHAPTER1:MOTIVATIO NOFTHEPRESENTSTU DY 1.1 Introduction Thepresentstudyconsistsofaddressingsomeveryimportantproblemsthatare inherentinthe StateofFlorida: tropicalstorms hurricanes lightning ,and flooding in additiontoastudyon volcanicactivity .Althoughallbutoneofthesubjectmodels developedusedrealdatafromtheStateofFlorida,thesemodelscaneasilybeadaptedto other statesorotherglobalregions. Thestudyconsistsofperformingparametric inferential analysisonthesubject problemusingrealdataanddevelopinglinearandnon linearstatisticalmodelsto estimatethekeyresponse s inthephenomenalistedabove. Morespecifically,weproceed toidentifyandrankthekeyattributablevariablesandtherelevantinteractionamongthe variousentitiesthatdrivethebehaviorofthesubjectresponse s .Thestatisticalmodels wehavedevelopedareofhighqualityandca nbeeasilyimplementedtoestimatethe phenomenonofinterestineachofthecaseswestudied. Theinformationobtainedfromthese statistical models ( tropicalstorms, hurricanes,lightning,floodingandvolcanicactivity ) areveryimportantintermsof strategicplanningwithrespecttopublic,environmentalandeconomicalissues Givenbelowarebriefdescriptionsofthecontentofthestudyforeachofthe chaptersthataretofollow. 1.2 TropicalStorms Webeginthestudyinthesecondchapter,St atisticalModelingtheConditions underwhichTropicalStorms are F orm ed :EstimatingtheBirthofaTropicalStorm,by introducingabasicconjecture,whichrelateswindspeedtopressureandtemperature,the keyentitiesthatcharacterizeandidentifya tropicalstorm. UtilizingrealdatafromkeystrategicallyplacedbuoysmaintainedbytheNational BuoyDataCenter(NBDC)tomakestatisticalinferencesthatprovethesubject

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2 conjecture. Firstweintroducethebasicdescriptivestatisticsandparametr icallyanalyze theresponsevariable(windspeed)andallattributingvariables. Second, wedevelopedastatisticalmodelwheretheresponseisthewindspeedas afunctionof atmospherictemperature seasurfacetemperature dewpoint and pressure .Inad dition, windgust winddirection and dayofyear consideredas explanatoryvariablesandrelevantinteractionsareincludedtodevelopthehighestquality model possible Theimportanceofthisconjectureistobeabletoestimatetherealizationofa tr opicalstorm.Furthermore,thedeveloped statistical model s enableustoestimatethe windspeedofatropicalstorm,whichplaysamajorrollinthestatisticalmodelingof hurricanes. Thequalityofthemodelwasjudgedbythefollowingfivecriteria: 2 R 2 adj R hypothesistestsofcoefficientsusingforwardselectionincluding p values, F testfor overallmodelqualityandMallows ) ( p C statistic.Allcriteriauseduniformly support thequalityofthe structured model. Keyreferencesofthepresentstudyare[5],[10],[13], [19],[32],[33],[38],[42], [43],[44],[51],[52],[62],[69],[71]and[76]. 1.3 Hurricanes Inthepresentchapterwedeveloplinearandnon linear statisticalmodelsof hurricaneforcewindsintermsofthecontributingentitiesmeasuredduringthecourseof fiverecenttropicalstormswhichreachedhurricanestatus,categoryfive: Wilma (2005), Rita (2005), Katrina (2005), Ivan (2004)and Isabel (20 03). Thedevelopmentofthesubjectsstatisticalmodelutilizedthefollowing independent(attributable)variables: pressure latitude and longitude convertedinto Cartesiancoordinates,the duration ofthestormuptothegivenpointintime,andthe d ayofyear .Otherdependentvariablesinclude:the changeinposition intheconverted coordinatesystem,the distance traveled (afunctionofthechangesinposition),andthe linearvelocity ofthestormintheconvertedcoordinatesystemalonewiththe appropriateinteractions.

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3 Theestimationofthewindspeedwithinatropicalstormisextremelyimportantin beingabletoaccuratelyidentifythelevelofhurricanethatwe willbe encounter ing .That is,stormcategoriestraditionallydefinedby:atrop icaldepression(<34knots),atropical storm(34 63knots),hurricanecategory1(64 82knots),category2(96 110knots), category3(96 112knots),category4(131 135knots)andcategory5(>135knots).In addition,itishelpfulinidentifyi ngthedirectionofthestorm. Thekeystatisticalmodelswehavedevelopedareofveryhighqualityasbeing usingthefivecriteriagivenabove .Allcriteriauniformlysupportthequalityofthe developedstatisticalmodel. Thesubjectmodelcanbeeas ilyapplieduponreceivingthe necessaryinformationthatdrivesthemodel;thatis,usingrealtimedatawecanupdated themodelandaccuratelyestimatethewindvelocity.Furthermore,thenetworkthatisin placeforcollectingthenecessarydatacanea silyfacilitatetheapplicabilityofthemodel onanhourlybasis. Inadditiontodevelopingthestatisticalmodel,wehavepreformedsurface responseanalysistoeithermaximizeorminimizetheresponsesubjecttoidentifythe valuesoftheindependentva riablesthatwillallowustoachievetheobjective;thatis, determinetheminimumormaximumwindspeedwithaspecified(acceptable)degreeof confidence. Inthischapter we alsointroducesanewscalingprocessforhurricanestatusthat referredtoas theWootenScale;thisscalecategorizeshurricanestatusinawaythatwe feelismorestablethanthecommonlyusedSaffir SimpsonScale.Usinghistoricaldata, wewereabletoestablishascalewithlessvariancewithinthe different categories of hur ricanes .Thus,applyingtheproposedscale wewereableto obtainmorestable estimatesofthewindspeed. Keyreferencesassociatedwith the presentstudyare: [4],[6],[11],[23],[26], [27],[34],[39],[40],[41],[53],[54],[57],[58],[59],[61],[ 64],[66],[67],[68],[70], and[76]. 1.4 Lightning Thefourthchapter,StatisticalAnalysisandModelingofLightningfocuseson lightningintheStateofFlorida,thelightningcapitaloftheworld.Utilizinghistorical

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4 datacollectedbytheNational Light ningDetectionNetwork(NLDN)( Fieux,Paxton, StanoandDiMarco,2005) ,wepreformedparametric inferential analysiswiththe responsevariablebeingthenumberoflightningstrikes.Wehaveidentifiedthatthebest probabilisticcharacterizationof thenumberoflightningstrikesisgivenbytheWeibull probabilitydistribution.Knowingtheprobabilitystructureofthisphenomenonwearein apositiontoprobabilisticallycharacterizethebehaviorofthenumberoflightningthat mayoccurintheSt ateofFloridagiventhesurroundingenvironmentalconditions.In addition,weareinapositiontoobtainestimatesofthemeanresponse,standarderrorand confidencelimitswithanacceptabledegreeofconfidence. Thestatisticalmodelsinvolvethe nu mberoflightningstrikes asbeingthe dependentvariable(res ponse)whichisafunctionof21 explanatoryvariables: precipitablewater tropicalstormwindtotal sealevelpressureanomaly ,t ropical stormwindsanomaly ,and Bermudahighaverage .Inaddi tiontothe relative humidity(1000mb) raininHernandocounty seasurfacetemperature temperature range precipitationanomalydistrictone relativehumidity(850mb) relative humidity(500mb) temperature(850mb) PacificDecadalOscillation( PDO) st andardanomaly raininHillsboroughcounty raininHighlandscounty solarflux standardanomaly Pacific NorthAmericaIndex( PNA ) standardanomaly precipitationanomalydistrictfour dayofyear and ArcticOscillation ( AO) standard anomaly .Thedeve lopedmodelalsoincludes appropriateinteractions.Toour knowledge,thisistheonlystatisticalmodelinexistencetoaddressthesubject phenomenonandalthoughitisdevelopedusingdatafromtheStateofFlorida,themodel canbeeasilyadjustedand updatedtobeapplicabletootherstatesintheU.S.andinother countries. Thesubjectmodelisofhighquality,beingevaluatedusingthefivecriteriastated above.Allcriteriauniformlysupportthequalityofthemodel.Thus,thismodelcanbe ea silyappliedtoestimatingthesubjectphenomenonandoncenewinformationis gathered,themodelcanbeeasilyupdatedonadailybasis. Keyreferencesassociatedwith the presentstudyare:[9],[28],and[35].

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5 1.5 Flooding InthefifthchapterAnalys isandModelingofFloodStagesweutilizerealflow ratedatacollectedintheStateofFloridastartingwiththeSt.JohnsRiver,thelargest riverintheStateofFlorida,tostatisticallyanalyzethefloodstagemeasuredinfeet.We performparametri c inferential analysisofthefloodstage (NadarajahandShiau,2005) anddeterminedthattheGumbelprobabilitydistributionfunctionoffersthebest probabilisticcharacterizationofthefloodstage.Havingidentifiedtheprobability distributionfunctio n,weareinapositiontoprobabilisticallycharacterizethebehaviorof thiskeyresponsevariable.Inaddition,wecanobtainestimatesofthesubject phenomenon,whichisveryusefulforstrategicplanning. Wedevelopedastatisticalmodeloftherespo nsevariable( floodstage )whichisa functionofthe flowrate ofthewaterandthe duration oftheflood.Thismodelis applicabletoanywaterwayandcanbeupdatedonceadditionalinformationbecomes available.Accurateestimatesofthefloodstageis importantforpublicsafety,strategic planning,amongothers. Keyreferencesassociatedwith the presentstudyare:[2],[7],[8],[9],[25],[50], and[55]. 1.6 RedTide Inchaptersix,AnalysisandModelingofRedTideBloomslooksatacritical pr oblemfacingtheStateofFloridainrecentyears;thecommonoccurrenceofRedTide (Dixon,2003) The RedTideproblemisveryimportanttotheStateofFloridawith respecttohealthissuesandeconomicimpacts,amongothers. Thepresentstudy,whichto ourknowledgeisthefirstofitskind,is presented in twoparts.First,weperformparametricanalysisonthemagnitudeofthebloomthat drivesRedTide.WeshowthattheWeibullprobabilitydistributionfunctiongivesthe bestprobabilisticcharacte rizationofthesubjectphenomenon.Thus,wecan probabilisticallycharacterizethebehaviorofthebloom andobtainusefulinformation Second ly ,wedevelopastatisticalmodeltoestimatethemagnitudeofabloom (thelogarithmictransformationofthec ountoforganismpresentinagivenlocation.)The logisticgrowthmodelwiththeinherentlogarithmictransformationofthe key parameter,

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6 effectivelyestimatesthemagnitudeofthebloomwithahighdegreeofaccuracy.Such estimatesareveryimportantp lanningpurposes ,amongothers Weplantoextendthepresentstudytothestageofbeingabletoidentifythekey contributingvariablesincludingrelevantinteractionthatdrivetheresponsevariable the magnitudeofaRedTidebloom.Inaddition,wep lantostudyregionaldifferencesand timedelaysbetweenbloomsinkeylocationsoftheStateofFlorida. Keyreferencesassociatedwith the presentstudyare:[24]and[25]. 1.7 Volcanoes IntheseventhchapterBivariateDistributionofTephraFallout westudythe problemofthedispersionofashfallortephrathataccumulatesduringavolcanic eruption.WeutilizedatathatwascollectedatCerroNegroinNicaragua (Connorand Hill,1995) bytheGeologyDepartmentattheUniversityofSouthFlorida, which includesnorthernandsoutherndirectionsandmassoftephrasievedbygrainsizealone withdatagleanedfromvariousonlinesourcesareusedtobetterunderstandthe underlyingprobabilitydistributionthatcharacterizesthesubjectdata. Wepre formedparametric inferential analysis;thatis,analysisofthebivariate dispersionoftephrafalloutbyparticlesizetodetermineiffalloutcanbeconsidered collectively.Thebivariatedistributionsconsideredtoanalyzethedataarenon correlated b ivariateGaussiandistribution,thecorrelatedbivariateGaussian,therotated (independent)non correlatedbivariatedistributionandtherotated(independent)skewed bivariatenormal. Statisticalanalysiswaspreformedtodetermineandranktheprobabi listicfitof thefourmodelsusing 2 and 2 R criteria .Thebestbivariateprobabilisticdistribution thatcharacterizessuchphenomenonisimportantinunderstandingsuchvolcanicfallout. Key referencesassocia tedwiththe presentstudyare:[15],[16],[17],[18],[20], [ 21],[22],[30],[31],[46],[61],[75],and[76 ].

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7 CHAPTER2:STATISTIC ALMODELINGTHECOND ITIONSUNDERWHICH TROPICALSTORMS ARE FORM ED :ESTIMATINGTHEBIR THOFA STORM 2.1 Introduction Inthisfirstcasestudy,weintro duceabasicconjecture: temperatureisabetter indicator when atropicalstormisbrewing anditisasignificantexplanatoryvariable whenwindvelocityistheresponsevariable First,weconsiderthehistoricalda tagatheredbeforeandduringTropicalStorm Alberto,2006.Usingdifferencing,weanalyzedchangesintemperatureandpressureas relatedtotheformationofTropicalStormAlberto.Inaddition,molecularphysicsare usedtoestablishthatthereisano n linearrelationshipbetweenvelocityandtemperature andthereforebetweenvelocityandpressure. Secondly,wedevelopastatisticalmodeltoestimatethewindvelocity( Chiu, 1994) whichenablesonetoidentifyandrankthecontributingentitieswhichd rivewind speeds.Itisacommonlyheldbeliefthatwindspeedswithinastormaredrivenbylow pressuresandwhilet herelationshipbetweenwindspeedandpressureisextremely important .Inthepresentstudywestatisticallymodelthe wind speed asafu nctionof pressure and atmospherictemperature seasurfacetemperature windgust ,and winddirection aswellas dewpoint T hequalityofthemodelisverifiedusingfivecriteria thatarestatedinChapter1 Allstatisticalcriteriauseduniformlysu pportthequalityofthedevelopedmodels.Once themodelisestablished,wepreformedsurfaceresponseanalysistodeterminethevalues

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8 oftheattributingvariablesthateithermaximizeorminimizetheresponsewithan acceptabledegreeofconfidence. Mo respecifically,inthepresentstudy,wewilladdressthefollowingquestions: 1. Whatisthephysicalrelationshipbetweenwindspeedandtemperature? 2. Whatisthephysicalrelationshipbetweenwindspeedandpressure? 3. Whataretheotherattributingentitie swhichcontributetowindspeed? 4. Whatprobabilitydistribution bestcharacterizesthebehaviorofthe response variable ,windspeed ? 5. Whatprobabilitydistribution bestcharacterizesthebehaviorofthe explanatory variables? 6. Usingdevelopedstatisticalmo dels,estimatethesubjectresponseasafunctionof significantlycontributingvariable? 7. Whatisthequalityofthedevelopedstatisticalmodel? 8. Underwhatenvironmentalconditionsistheexpect ed windspeed maximizedor minimized? 2. 2 DifferencingEquat ionsand MolecularPhysics Theconjectureposedinthischapter istwofold;first,temperatureisabetter indicatorthatastormisbrewingandcandetectpotentialformationsdaysbeforeastorm isclassifiedasatropicalstorm.Second,thattemperat ure(squared)isamoresignificant entitythanpressure(squared).Thesquaredrelationship isbaseduponconcepts introducedinstudyofmolecularphysics;namely,theBoltzmanndistributionofenergy asafunctionoftemperatureandtheIdealGasLaww hichrelatespressureandvolumeto temperature.

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9 Conjecture :Temperatureisabetterindicator when atropicalstormisbrewing andisa significantexplanatoryvariablewhenwindvelocityistheresponsevariable 2.21 TestingtheConjecture Totestthe conjecturethattemperatureisabetterindicatorwhenatropicalstorm isbrewing wegathered datagatheredbyNationalBuoyDataCenter(NBDC)forthe firstpartofJune2006.Duringthistime,TropicalStormAlbertoformedanddissipated. Thedatau tilizedtakenfromtwobuoysthatarenearthepathoftheTropicalStorm AlbertoasshowninFigure1( Zehr,1995) .InFigure1,thelocationofthefirstbuoynear Cubaisstaredandenumerated(1)andthelocationofthesecondbuoyintheGulfof Mexi coisstaredandenumerated(2).Figure1showsthatTropicalStormAlberto formednearthefirstbuoynearCubaandmovednortheastclosethesecondbuoybefore dissipatingovertheUnitedStates. Figure1 :MapofTropicalStormAlberto;eachbuoyismarkedbyastarandnumbered 1 2

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10 2.2.2 StartofHurricaneSeason200 6 Considerthefirsttropicalstormofthe2006hurricaneseason. Intermsof ind icatingatropicalstorminbrewing,datashowninFigures2and3illustratesthatthere isdropintemperaturedaysbeforetheformationofTropicalStormAlberto,whereas pressuredropsclosertothetimeoftheofformation. InFigure2andFigure3,r ed denotesthechangeintemperature ) 1 ( ) ( t T t T dT andblueisthechangeinpressure ) 1 ( ) ( t P t P dP .Thereisdropintemperatureseveraldaysbeforetherewasadrop inpressure,especiallyinthebuoynear thelocation wheretropical stormAlbertoformed onthetenth ofJune. Figure2 :Temperature(red)and Pressure(blue)differencesatthefirst buoy(42056) Figure3 :Temperature(red)and Pressure(blue)differencesatthesecond buoy(42036) 7 th &9 th 10 th &11 th 12 th D i f f e r e n c e s DayofYear D i f f e r e n c e s DayofYear

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11 Compare Figure s 4and 5,theatmospherictemperaturesrecordedatthefirstand secondbuoy,respectively.Theatmospherictemperaturerecordedatthefirstbuoy, becomeerraticdaysbeforetheformationofTropicalStormAlbertoas showninFigure 4.Theatmosp herictemperaturedropsrecordedatthesecondbuoydropsseveraldays beforethestormpassesnearthesecondbuoy. Figures6 and 7illustratethat therecanbeadropinpressurewithoutastorm forming;thisismoreobvious inthesecondbuoy Figure7.Thereisadropinpressure onthefifth (yellow) andsixth (orange)ofJuneanditis aslowasthepressuredroponthe twelfth ofJune(blue) ,whenthestormisdirectlyoverthebuoy;butduringthefirst pressuredropt hetemperaturewasrelativelystable asshowninFigures4and5onthe fifth(yellow)andsixth(orange)ofJune, andnostorm was formed.However,duringthe seconddropinpressure onthetenth(purple)ofJune ,thetemperaturewassignificantly lowert hannormalandastormformednearthefirstbuoy. A t m o s p h e r i c T e m p e r a t u r e DayofYear Day DayofYear A t m o s p h e r i c T e m p e r a t u r e Figure4 :Linegraph(byday)of atmospherictemperatureforthe firstbuoy(42056). Fig ure 5 : Linegraph (byday)of atmospherictemperatureforthe secondbuoy(42036)

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12 Moreover,thetemperatureatthefirstbuoywherethestorm was firstdeveloped becameextremelyvariableontheeighthandninth ofJune ,thedaybeforethestorm formed whereasthepressuredidnotsignificantlydropuntilthetenth ofJune .Therefore, it is reasonableto assume thattemperaturemightbeabetterindicator when astormis brewingthanpressure.Depressions(dropsinpressure)appeartobetheresultof the storm,notacause.Thereisacorrelationbetweenmaximumsustainedwindsand pressure,buttherelationshipispressureasafunctionofwindspeedandnottheother wayaround. Presently, temperaturedataisnotcollectedinhurricanes onlylatit ude, longitude,maximumsustainedwindsandpressureattheeye ifahurricanearecollected Additionally,inviewing imagesgeneratedbytheEarthSystemResearch LaboratorysDailyMeanComposites, thepressure s overtheoceansasdailyaverages for weeks priortoformationof the TropicalS torm Alberto indicatedapumpingactioninthe pressureseveraltimesbefore astormisformed;thisis illustratedin Figure7bythe drop in pressureatthesecondbuoyseveraldaysbeforethestorm is formed. P r e s s u r e ( h P a ) DayofYear P r e s s u r e ( h P a ) Figure 6 :Linegraph(byday)of atmosphericpressure(hPa)forthe firstbuoy(42056). Figure7 :Linegraph(byday)of atmosphericpressure(hPa)forthe secondbuoy(42036). DayofYear Day

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13 2.3 Descriptionofresponsevariableandcontributingentities T herelationshipbetweentheatmospherictemperature,pressure,water temperature,dewpoint,wind speedandwind direction isanalyzedindependently. Havingabetterunderstandingoftheat tributableentitiesthatdrivethesubjectresponse (windspeed),wewillbeinapositiontoprobabilisticallycharacterizethebehaviorofthe subjectphenomenonandstatisticallymodel windspeed asafunctionof atmospheric temperature pressure wate rtemperature dewpoint ,and wind direction 2.3.1 WindSpeed(WSPD) Thewindspeed, w ,is recorded bythebuoys inmeterspersecond (m/s), averagedoveraneight minuteperiod,andthenreportedhourly.Thewindspeeds measured attheselocationsarerathersmall;measuringonlyashighas35knotasshown inFigure8 w hereasthestormrotatingaboveclockedat60knots.Fromthetimethe stormformedonthetenth ofJune thruthefourteenth ofJune,whenitdissipatedthe highe stwindspeedrecordedduringthesedaysisnogreaterthanonthetwenty seventhof thesamemonth Thisisvastlydifferentthanthe60knotsrecordedasthestorms maximumsustainedwindspeed. Itisimportanttonotethatthis informationisatthe surfaceofthewaterandnot higherintheatmosphere.Hence,thesemeasurementsmadeatthebaseofthestorm wherewindsspeedsaresignificantlylowthanthestormpassingoverhigherinthe atmospherewillbeonadifferentscalethanSaffir Simpson.

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14 Figure8 : Linegraph ofwindspeed(m/s);thefirstbuoy(42056)inburntorangeandthe secondbuoy(42036)inaqua 2.3.2 WindDirection(WD) Thedirection inwhichthewindisblowing , ismeasuredinbearings(degr ee clock wiseofftruenorth)overthesameperiodusedforthewindspeed.Asillustratedin Figure9,themaximumwindspeedswhichoccurredintropicalstormArlenewhere duringthetimewhenthedirectionwasthesameatbothbuoys. Figure9 : Li negraph ofwinddirections(bearings);thefirstbuoy(42056)inburntorange andthesecondbuoy(42036)inaqua Stormisclosesttothis buoyintheGulfandis atitsstrongest Stormisatits strongest W i n d S p e e d ( m / s ) DayofYea r W i n d D i r e c t i o n DayofYear

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15 2.3.3 Gusts (GUSTS) Thegusts g aremeasuredasthepeak5or8secondgustspeeds(m/s) ( Figure 10 ) measureddurin g an eight minuteperiod.Otherthanafewsuspicionreadingsof9; thatis,inthedata 9mighthavebeenusedtodenotemissingdata,butregardlessofthis possibleincorrectentries,thereisnotasingledatewhichstandsoutasanindicationthat thestormwasformingorwasevenpresent.However,astheextrememeasureovera giventimeandnottheaverage,thismeasureshouldbehighlycorrelatedtotheresponse variablewindspeed Figure10 : Linegraph ofwindgust(m/s);thefirstbuoy (42056)inburntorangeandthe secondbuoy(42036)inaqua 2.3.4 Pressure(BAR) Thepressure P ismeasuredatsealevel.Thereisnosignificantdisparityfoundin thepressureleadinguptotheformationofthestorm,butdoeso btain the greatest disparityasthestormforms Figure11.Thedropinpressureassociatedwiththe formationoftropicalstormArleneisasignificantdrop,butonlyoccurredasthestorm is formed.Thisisalsotrueforthedropinpressureattheseco ndbuoyintheGulf.The W i n d G u s t s DayofYear

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16 lowestpressuresoccurredasthestormpassed,andwassignificantlylowerduetothefact thatthisisthesametimethestormwasatitsstrongest. Ofallthevariablesconsidered,pressureistheleasterratic;thereisanob vious dailyperiodicitybutthetransitionsove rtimeareonlyaffectedby stormconditions. Thereisnotagreatamountofvariabilityinpressure. Figure11 : Linegraph ofatmosphericpressure(hPa);thefirstbuoy(42056)inburnt orange andthesecondbuoy(42036)inaqua 2.3.5 AtmosphericTemperature(ATMP) T heatmospherictemperature a T measuredindegreesCelsiusattwobuoys recordedduringthesameperiodoftimeaswindspeedsandgustisillustratedinFigur e 12 .Thereissomewhatofadisparitybetweenthetemperaturesdaysbeforethestorm occurs,butasthestormforms,thesetemperatures approach equilibrium.However, unlikepressure,temperature hasgreatervariance.Whileatmospherictemperature does h aveadailyperiodthedeviationswithintheoscillationsaremoresporadicand apparentlymoresensitivetothesurroundingenvironment.Furthermore,therearedays Stormclosest tothisbuoy westofCuba Stormisclosesttothis buoyintheGulfasat its strongest DayofYear A t m o s p h e r i c P r e s s u r e

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17 wherethetemperatures approach equilibrium,butthetemperaturesarewarmerand therefor eabletoholdmoremass.Itisonlywhenthereisadropintemperatureata buoy,whichforcestheairtoreleaseitsgatheredmasses(rain)andresultsinashiftinthe volumesofairtobalancetheenvironmentintermsofboththetemperatureandthe pressure. Figure12 : Linegraph ofatmospherictemperature( C );thefirstbuoy(42056)inburnt orangeandthesecondbuoy(42036)inaqua 2.3.6 SeaSurfaceTemperature(WTMP) Itisinterestingtonotethedifferencesinw ater(seasurface)temperaturebetween thetwobuoys Figure13.Daysbeforethestormforms,thereisalargedifferencein watertemperature;andasthetemperatur esreachequilibrium,thestormisformed Again,thetemperaturenearCubaishigherthan thosetemperature s foundintheGulf. Additionallythetemperaturesdrivethewaterstomoveto approach equilibriumandform a storm.Aswillallthesemeteorologicalmeasurements ,therearedailyoscillations;these sinusoidalwavesaremorepronounced inthemeasureofwatertemperature.Thisisdue tothewaterabilitytoabsorbtheheatenergyforthesunduringthedayandreleasethis Birthofstorm June10 th Dropin temperature beforethe storms formation Stormis closesttothis buoyinthe Gulfasatits strongest A t m o s p h e r i c T e m p e r a t u r e DayofYear

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18 energyatnight.Thesewaters,whichrustleintheirbed,createchoppywatersand movementofair,whichultimat elyresultsinthebirthofastorm.Thisimpliesthat temperatureisabettergaugeofastormbrewingintheAtlanticbasin.Note,notonlyis thetemperatureatitslowestasthetropicalstormpasses,butthestormisatitsstrongest whenthetemper aturedifferenceisatitsgreatestbetweenthetwobuoys. Figure13 : Linegraph ofwatertemperature( C );thefirstbuoy(42056)inburntorange andthesecondbuoy(42036)inaqua 2.3.7 DewPoint(DEWP) Thedewpoint d T ismeasuredatthesameheightastheatmospherictemperature isthetemperaturetowhichaparcelofairmustbecooled(assumingaconstant barometricpressure)forthewatervaporintheairtocondenseintowater.Thedewpoint isre latedtotherelativehumidity RH Thecalculationofthedewpoint d T ,basedontheMagnus Tetensformula over C T C a 60 0 00 1 01 0 RH and C T C d 50 0 isdefinedby RH T RH T T a a d 27 17 7 237 ,where RH T T RH T a a a ln 7 237 27 17 Birthofstorm June10 th Stormisclosest tothisbuoy in theGulfandis atitsstrongest S e a S u r f a c e T e m p e r a t u r e DayofYear

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19 Whenconsideringthedaysbeforethestorm is formed,therearetwoobvious dropsindewpointatthesecondbuoy(42036) Figure14.Thisismimicked,however notaslarge,afewdayslateratthefirstbuoy(4 2056).However,moreinterestingly,two daysbeforethestorm,thedifferenceindewpointisthegreatest. Figure14 : Linegraph ofdewpoint;thefirstbuoy(42056)inburntorangeandthe secondbuoy(42036)inaqua 2.4 ParametricAnalysis Par ametricanalysisisimportantformanyreasons.First,inunderstandingthe probabilisticbehaviorofthephenomenonmeasured,thebest fit probability distribution canbeusedtotesthypothesis,determineaccurateconfidenceintervals.Furthermore,this probabilitydistributionfunctioncanbeusedtoestimatereturnperiodsanddeterminethe likelihoodofagivenwindspeed ,amongothers 2.4.1 ParametricAnalysisofWindSpeed Thewindspeeds ,Figure15,israthersymmetricandappearsasthoughit mightbe normallydistribution;however, theWeibull distributionistheonly probability Birthofstorm June10 th Largestdifferencein dewpointJune8 th D e w P o i n t DayofYear

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20 distributionthatfailstoberejectedatthe0.01significancelevel;theonlydifferenceis thatthemeanispositivelyskewed Figures15.Furthermore,weshould notethatthe highestrecordedwindspeedatthefirstbuoynearCubawasrecordedthedaythestorm formed,June10 th andJune12 th atthesecondbuoyintheGulfasthestormpassed. 2.4.1.1 The Weibull Probability Distribution Commonlyusedinreliabi litytheoryto characterize failuretimes,thisdistribution isboundedabove;thatis,thedatahasafiniteupperbounddefinedbythelocation parameter x .Thethree parameterWeibullwith , scale location and shape isgiven belowbye quation 1 x x x x x f 0 exp ) ( 1 1 Moment Estimation Mean 5.12792162 Standard Deviation 2.76173786 Skewness 0.98215256 Kurtosis 1.77329966 Table1 :Momentsfortheth ree parameterWeibull

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21 Figure15 :Histogramofwindspeedrecordedatbothsites;includingdescriptive statisticsJune2006 Theseextremevalues are,asexpectedfoundatthebuoyintheGulfillustratedin Figure16bandonlyoccurredasthestormp assed.However,thesehigherwindspeeds alsooccurredatthefirstbuoyjustnotasextremeillustratedinFigure16a,alonewith Figure16c,wecanconcludethatthedistributionissimilarinshaperegardlessofthe stormconditions.Furthermore,consi derthedatacollectedin2005asshowninFigure16. Thisthree parameterWeibullillustratesthegeneralshapeofthesedistributionare similar. Statistic Estimation Count 1429 Mean 5.128 Median 4.8 StdDev 2.762 Variance 7.627 Range 17.2 Min 0 Max 17.2 IQR 3.4 25th% 3.3 75th% 6.7 WindSpeed

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22 (16a) (16b) (16c) Figure16 :Histogramofwindspeedrecordedateachsite 2.4.1.2M axim um LikelihoodEstiamtes Themaximum likelihoodestimatesoftheparametersthatareinherentwithinthe Weibullprobabilitydistributionfunction isgivenbythefollowing: 42056WindSpeed 42036WindSpeed 42036WindSpeed(withoutstormconditions)

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23 n i i n i i n n x x x L 1 1 1 exp n i i n i i x x n x L 1 1 ln 1 ln ln ) ( ln Partials: n i i n i i x x x L 1 1 1 0 1 1 ln 0 ) ( ln 1 1 2 n i i x x L 0 ln ln ) ( ln 1 1 n i i i n i i x x x n x L Using numericaliterationtechniques ,wecanestimatethesolutionstothesystem ofpartialdifferentialequationswhichwilloptimizethelikelihoodofanestimator;thatis, themaximumlikelihoodest imators oftheparametersare giveninTable2. Parameter MLE (Location) 0.12817 (Scale) 5.926959 (Shape) 1.984727 Table2 :MaximumLikelihoodEstimatesforthethree parameterW eibull n i i n i i x x n n x L 1 1 ln 1 ln ln ) ( ln

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24 Figure17 :Histogramandbest fitdistribution 2.4.2 ParametricAnalysisofWindDirection Thewinddirectionispositivelyskewed Figure18,butthisisnotthegeneral distribution.Therearedistinctdifferencesateachbuoy. Figure1 8 :Histogramofwinddirectionrecordedatbothsites;includingdescriptive statistics Statistic Estimation Count 1429 Mean 5.128 M edian 4.8 StdDev 2.762 Variance 7.627 Range 17.2 Min 0 Max 17.2 IQR 3.4 25th% 3.3 75th% 6.7 WindDirection P e r c e n t a g e WindSpeed

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25 ThebuoywestofCubaislocatedbetweentwolandmasses,andthewindis directed,measuredclockwisefromtruenorthapproximately105showninFigure19a. However ,intheGulfthewindsblowinoneoftwodistinctdirectionswhichare approximatelyinoppositedirectionshowninFigure19b;thisisconsistentwiththe rotationofthestormwindswhichpassesnearlydirectlyoverthebuoyaseacheyewall passes. (19a) (19b) Figure19 :Histogramofwindspeedrecordedateachsite Thesepatternsareconsistentwiththedatafromthethreevariousbuoysin2005. Thethirdbuoy,likethefirstdemonstrateaconstantdirectionalflow,approximately161 cl ockwiseofftruenorth.WhereasthebuoyintheGulfshowsthatmoreoftenthannot, thewindsblowintwoopposingdirections.Hence,thetwobuoysintheAtlanticand nearCubacanbecharacterizedbysimilardistributions,butthebimodalnatureofbuo ys intheGulfneedsfurtheranalysis. Considerthemixedmodels,e quation 2 ;thatis,assumingthetherearetwo normaldirectionsforthewindtoblowatthesecondbuoy.Thefirstnormaldirection appearstobebetween0and180 ,andasecondbetween180and360.Hence, considerthesepartitionsandthestandardnormalprobabilitydistribution | x f where isthemeanand isthestandarddeviation. 42056WindDirection 42036WindDirection

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26 Cumulative Distributions WindDirection C u m u l a t i v e D i s t r i b u t i o n 2 1 1 1 2 2 1 1 | 1 | , , | x f x f x g 2 Toestimatethemeansandstandarddeviation s ,thedatawaspartitionedas outlinedaboveandthemeanandstandarddeviationswherecomputed.Usingthese meansasthebaseofourmixeddistributions,theempir icalprobabilitydistribution,and least squaresregressionwasusedtocomputethecoefficientswiththefollowingresults Table3.Withcorrelationcoefficients % 7 99 2 2 adj R R ,thismixeddistribution explainssignificantlythevariationsinthe cumulativeempiricalprobabilitydistribution ofthedataFigure20. Parameter Estimate 1 80.049 2 271.433 1 47.246 2 49.045 0.475879 Ta ble3 :Estimatedparameters Figure20 :Cumulativeprobabilitydistributionforthedata,thenormalsubfunctionsand themixeddistribution

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27 2.4.3 ParametricAnalysisofPressure Considertheatmosphericpressure Figures21and22;th edata i salsobest characterized by the Weibull probability distribution ;theWeibullprobabilitydistribution was theonlydistributionthatfailstoberejectedatthe0.01significancelevel. A comparisonofthebest fittwo parameterWeibullversusthe three parameterWeibull distribution isgiveninTable4.Thethreetestsforgoodness of fit:theKolmogorov Smirnov,Cramer vonMisesandAnderson Darling,indicatethatthethree parameter Weibullbestcharacterizesthesubjectphenomenon. Test Statistic42 036 1 p value Statistic42056 p value Kolmogorov Smirnov 0387 0 D 1073 0 D 0.003 <0.001 0429 0 D <0.001 Cramer vonMises 1439 0 2 W 6897 1 2 W 0.011 <0.001 2233 0 2 W < 0.001 Anderson Darling 8535 0 2 A 6727 9 2 A 0.010 <0.001 064 3 2 A <0.001 Table4 : Goodness of fittestsincluding statistics Figure21 :Histogramofatmosphericpressurerecordedatbothsites;including descriptivestatistics. 1 ThesecondsetofstatisticsisforthesecondbuoyintheGulfforthosedayswhenstormconditionsarenot p resent;thatis,thosedaysbeforeJune10,2006andafterJune14,2006. Statistic Estimation Cou nt 1422 Mean 1013.572 Median 1013.7 StdDev 3.341 Variance 11.16 Range 25.1 Min 995.8 Max 1020.9 IQR 4.2 25th% 1011.6 75th% 1015.8 Pressure(hPa)

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28 Wecansee thestormconditionsasextremes inourdata,itisthesepointsoflow pressurewhichmakesthedataseemslightlyskewedFigure20,butthisismoreapparent inFigure20.Recall,Arlenepassedextremelyclosetoth esecondbuoyintheGulf,andit wasonthesedaysFigure22aand22b,thatpressuresbelow1008arerecorded.However, ifweremovethesedaysandconsideronlythosedaysinwhichnotropicalstormsare present(beforeJune10,2006andafterJune14, 2006),thereisabimodal behavior tothe dataintheGulfFigure22c. (22a) (22b) (22c) Figure22 :Histogramofatmosphericpressurerecordedateachsites 42056Pressure(hPa) 42036Pressure(hPa) 42036Pressure(hPa)withoutstormconditions

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29 Otherthanwhentherearetropicalstormconditions,thepressuresaremuchless v ariant,butratherconsistentbothateachsiteandbetweeneachsite.Thatis,thepressure ismoreaffectedbythepresenceofstormconditions.Toverifythisconsidervarious estimates givenin Table5fordataobtainedfromthetwoinitialbuoys,Ju ne2006,the secondbuoywithoutstormdays,thetwobuoyscombined,June2006andfinallythree buoysforallof2005.Themeanatmosphericpressuresareextremecloseaswellasthe standarddeviations.Allarenegativelyskewedandwhenconsideredwit hthenatural occurringstormconditionsallholdpositivekurtosis. Parameter 42056 42036 42036 2 Combined 2005ThreeBuoys Theta 1000 996 1002.616 880 986.47 Shape 3.8 5.7 4.87 135 6.26 Scale 9.5 20 13.63 47 30.00 Statistic Mean 1012.9 1014.23 1015.1 1013.6 1013.6 StdDev 2.6209 3.807 3.0289 3.341 3.3407 Skewness 0.8727 1.211 0.2406 0.9085 0.9085 Kurtosis 0.5268 3.453 1.1445 2.8354 2.8254 Table5 :ParametersandSt atisticsforassociatedWeibulldistributionsforpressure Thismayseemsomewhatunrealisticsinceduringhurricanes,pressurescandrop tovaluesinthe800s,butnotatsealevel.Infact,in2005,arecordnumberofstorms occurred;27namedstorms, butthelowestrecordedpressureatanyofthesebuoys was 986.9 hPa .Amongthesestorms,theminimumrecordedatmosphericpressureonrecord is882 hPa ,measuredduringHurricaneWilma.Therefore,athigheraltitudes,wemight set 880 ,buttogetabetterideaofatmosphericpressuresatsealevelyearround, considerthedatacollectedbythreebuoys.Thesethreebuoysyieldsimilarresultstothe twocombinedsites asshownin Table5.Hence,wecanconcludethatsurfacepressures inopenwaterbestdescribedbythethree parameterWeibull probabilitydistribution, Figure23 ,givenbye quation 3 1 Withoutdaysinwhichstormconditionswherepresent;thatis,beforeJune10 th orafterJune14 th

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30 otherwise 0 47 986 00 30 47 986 exp 00 30 47 986 00 30 26 6 ) ( 26 6 26 5 x x x x f 3 Figure23 :Histogramandpotentialbest fitdistribut ions 2.4.4 ParametricAnalysisofAtmosphericTemperature Foratmospherictemperature,thereisskewinthedata;themeanislessthanthe medianandthed ataisnegativelyskewed,asshownbyFigure 24. Thestatisticsgiven aredescriptivestatistics basedontherealdata.Thesamplemeanatmospheric temperaturerecordedwas26.3Cwithasamplestandarddeviationof1.75C.Other descriptivestatisticssuchasthemedian,thevariance,therange,theminimum, maximum,inner quartilerange,andthe firstandthirdquartilesaregiveninthechartin Figure24. Pressure(hPa) P e r c e n t a g e s

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31 Figure24 :Histogramofatmospherictemperaturesrecordedatbothsites;including descriptivestatistics. (25a) (25b) Figure25 :Histogramofatmospherictemperaturesrecorde dateachsites Usingthedoublelogarithmofthesubjectresponse, wecanmorereadily determinethebesttypeofex tremevaluedistribution.Thenormalprobabilityplot appears concavedown,whichindicatesapossibleWeibull probability distribution Figure26;otherdistributionsconsidered:normal,lognormalandexponential.However, Statistic Estimation Count 1979 Mean 26.274 Median 26.8 StdDev 1.746 Variance 3.049 Range 7.7 Min 21.3 Max 29 IQR 2.7 25th% 25 75th% 27.7 AtmosphericTemperature 42056Temperature 42036Temperature

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32 thetwo parameter Weibullbe st fits thedata withparametersestimatesgiveninTable6 Whenthethree parameterWeibullisfittoeachbuoyindependently,Figure26 ,has parameterestimatesgiveninTable7. Test Statistic 42036 Statistic 42056 p value Kolmogorov Smirnov 0961 0 D 0829 0 D <0.001 Cramer vonMises 2254 1 2 W 8441 0 2 W <0.001 Anderson Darl ing 9706 6 2 A 1044 5 2 A <0.001 Table6 :ParametersandStatisticsforassociatedWeibulldistributions Figure26 :Histogramofprobabilitydensityfunctionsforatmospherictemperature Parameter 42056 42036 2005Th reeBuoys Theta 8.20 22.28 5.8 Shape 39.41 5.36 5 Scale 20.34 5.65 21 Moments Mean 27.49 25.27 24.5511 StdDev 0.7826 1.6767 4.8594 Skewness 0.8173 0.2720 0.9163 Kurtosis 0.1169 0.8189 0.1446 Table7 :ParametersandMomentsforassociatedWeibulldistributions 42056Temperature 42036Temperature P e r c e n t a g e s

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33 However,ifwewishtodiscusstheatmospherictemperatureatsealevel,then considerthedatacollectedbythreebuoys thetwopreviouslymentionedandathir deast ofFlorida intheAtlantic(41010).Thisadditionalbuoyisnecessarybecause thefirst buoywasonlycommissionedJune2005.Thenwecanfindthebest fitprobability densityfunctionoverdatacollecte datthreebuoysovertheentireyearof200 5asdefined ine quation 4 .However,notethat2005asthesmallersamplingofJune2006,hasa bimodalnatureFigure26;furtherdiscussonthiswillbeconsideroneseasonalityand modelingbetweenthesevariablesiscomplete. otherwise 0 8 5 21 8 5 exp 21 8 5 21 5 ) ( 5 4 x x x x f 4 Figure27 :Histogramandbest fitdistributionforatmospherictemperature 2.4.5 ParametricAnalysisofWaterTemperature Thewatertemperatureholdsther eversedistributionintermsofextreme values Thereareonlyafewdaysinwhichthewatertemperaturesriseabove30degreesCelsius. Thisisnotseenintheatmospherictemperature,howeverliketheatmospheric P e r c e n t a g e s AtmosphericTemperature

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34 temperature,thesurfacetemperatureisalsobest fitbytheWeibull probability distribution;itistheonlydistributionthatfailstoberejectedatthe0.01significance level. Itshouldablebenotedthatsimilarto theatmospherictemperature,thereisa bimodal effectthat existsintheseasurfacetemperatureaswell. Thetw ovariables,atmospherictemperatureandseasurfacetemperature,are highlycorrelatedwithan 2 R of92.5%. Thisismainlydue to thefactthatbothairand waterarewarmedbythesamesun,thewaterthroughtheair;butthevariati onisdueto thefactt he waterregainsheatmorereadilythatairandcoolsoffatadifferentrate. Therefore,therearedelayaffectsbetweenthetwocontributingvariables,butingeneral holdsimilardi stributioncharacterization, Figure28. Figure28 :Histogramofwatertemperaturesrecordedatbothsites;includingdescriptive statistics. Ifweviewthedistributionsbybuoy,weseethatbothhave extremehighs ,butthe bimodalnatureislessprevalentonlyslightlyinthedistribution ofthetemperaturesinthe Gulfas shownby Figure29. Statistic Estimation Count 1410 Mean 28.052 Median 28.4 StdDev 1.058 Variance 1.119 Range 6.1 Min 25.6 Max 31.7 IQR 1.7 25th% 27.2 75th% 28.9 SeaSurfaceTemperature

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35 (29a) (29b) F igure29 :Histogramofwatertemperaturesrecordedateachsites Usingthevariousdatasets gatheredfromthethreebuoysfromMay2,2006to June12,2006 ,wec ansee from Table8thatthemeantemperatureisapproximately25 degreesCelsiusandismorevariablewhenconsideredyearround.However,when consideredyearround Figure30,thebimodalnatureofthisphenomenonismore dominatingwhichreadsasafla tterdistributionwithnegativekurtosis.Betterfitscanbe obtainedbyconsideringtheconditionalprobabilitydistributionsbasedonseasonality withinthedata. Parameter 42056 42036 2005Three Buoys Theta 27.80 25.52 18 Sh ape 2.0883 2.0668 2.1 Scale 1.0870 1.9087 9.2 Statistic Mean 24.21 25.35 25.83 StdDev 1.3751 0.9980 4.0457 Skewness 0.6501 1.5157 0.6040 Kurtosis 0.4259 7.3050 1.1586 Table8 :Parametersforasso ciatedWeibulldistributions;includingdescriptivestatistics 42056Temperature 42036Temperature

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36 Figure30 : Histogramandbest fitdistributionsforwatertemperature 2.4.6 ParametricAnalysisofDewPoint Thedewpointaspressureismoreconsistentateachsiteandoverall.More over, whendeterminingthebest probability distribution,theWeibulldistribution givensthe bestresultsin eachcase Table9.Thedistributionsarerathersymmetricwithonlyafew lows asshownby Figure s 31and32. Test Statistic42036 3 p value St atistic42056 p value Kolmogorov Smirnov 0387 0 D 1073 0 D 0.003 <0.001 0429 0 D <0.001 Cramer vonMises 1439 0 2 W 6897 1 2 W 0.011 <0.001 2233 0 2 W <0.001 Anderso n Darling 8535 0 2 A 6727 9 2 A 0.010 <0.001 064 3 2 A <0.001 Table9: ParametersandStatisticsforassociatedWeibulldistributions 3 ThesecondsetofstatisticsisforthesecondbuoyintheGulfforthosedays withoutstormconditions;that is,thosedaysbeforeJune10 th andafterJune14 th P e r c e n t a g e s SeaSurfaceTemperature

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37 Figure31 :Histogramofdewpointrecordedatbothsites;includingdescri ptivestatistics. (32a) (32b) Figure32 :Histogramofdewpointrecordedateachsite. Statistical estimatesofthevariousparameters ofthe Weibull probability distributionhavebeenobtainedandareshownby Table10 .Asillustratedby Fig ures31 and32,thedistributionshowninFigure33wouldfitbetterunderseasonalconditions. Statistic Estimation Count 1424 Mean 24.345 Median 24.5 StdDev 1.098 Variance 1.205 Range 8 Min 18.9 Max 26.9 IQR 1.3 25th% 23.8 75th% 25.1 DewPoint 42056DewPoint 42036 DewPoint

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38 Parameter 42056 42036 2005Three Buoys Theta 21.68 8.17 2 Shape 4.5240 14.27 4.1 Scale 3.0641 16.64 23 Statistic Mean 24.48 24.21 19.02 StdDev 0.6867 1.375 6.2960 Skewness 0.0672 0.6981 0.9581 Kurtosis 0.1595 0.5980 0.0556 Table10 :ParametersforassociatedWeibulldistributions;includingdescriptivestatistics Figure 33 :Histogramandbestfitdistributionfordewpoint

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39 2.5 MultivariateModelingofWindSpeedsnearSeaSurface Asthebutterflyeffectsuggests,therearemanycontributingfactorswhichcanbe thecauseoflargerevents.Someofthesecontributingfa ctorsaremoreobviousthan others,somewillremainlurkingvariables,butwiththosefactors,whichvaryasmuchas theresponsevariable,wecanuseleast squaresregressiontobest fitamodelthatexplains themajorityofthevariation. 2.5.1 Contrib utingVariables Thevariablesofinterestrecordedbythebuoysintheopenwatersaroundthe StateofFloridaaswellasvariousotherregionsaroundtheworldarelistedinTable2.6. Theresponsevariablewillbetakenasthewindspeedandtheremaini ngvariableswillbe treatedastheexplanatoryvariables,whichdefinethewindspeedrecorded. 2.5.2 Statistical Model Thegeneralformofmulti lin earregressionisoftheformgivenbye quation 5 where ) ( t y i stheselectedresponsevariabletobe estimat ed, ) ( t x i arethe m different contributingentities m i ,..., 2 1 whichwhentreatedlikearandomvariable s overtime generatesthe N ind ependentlyobservedsamples: m i i i t x t y 1 0 ) ( ) ( 5 In theproposed model,thesecontributingvariables areshownin Table11willbe denoted asshowninequation 6 belo w Inequation 6 ,thecoefficients s i arethe weightsthatdrivethecontributingvariables,giveninTable11,and istherandom error.Theestimatesofthe s areth ekeyfactorusedtoidentifythesignificantly contributingvariables. d T T T P g w d w a 7 6 5 4 3 2 1 0 6

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40 Variable Description w Thewindspeed(WSPD) Thewinddirection(WD) g Thewindgusts(GST) P Theatmosphericpressure(BAR) a T Theatmospherictemperature(ATMP) w T Theseasurface(water)temperature(WTMP) d T Thedewpoi nt(DEWP) d Thedayofyear(DAY) Table11 :Variablesofinterest 2.5.3 Rankinorderofsignificant(p value)andcontribution For thefirst ordermodeline quation 6 will firstbeusedtorankthevariables whicharefoundtobesignificantbyletting i j j ; 0 andranktheindependent variableswithrespecttotheircontributiontoaccurately estimating theseresponses.This rankingwillbebasedontheimprovementinthecorrelationcoeffici ents: 2 R and 2 adj R where 2 R isgiven bye quation 7 increasesas n increases.Whereas 2 adj R ,given by e quation 8 doesnotincreasewhenadditionaldesignparametersareaddedtothe regressionmodel and p isaconstant;thisstatisticpenalizestheinclusionofinsignificant modeltermsandthereforeisabetterindicatorofhowwellthemodel explainsthe behavioroftheresponse. 2 2 2 y y y y SST SSR R i i 7 and 2 2 1 1 1 R p n n R adj 8

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41 Inthepresent study,theranking ofthecontributingvariablestotheresponseare listed inTable12.Rankingwasestablishedbymaximumimprovementinthecorrelation coefficient s 2 R and 2 adj R The statistical model givenbye quation 9 explainsonly38.9%ofthevariation inthew indspeed.With 2 adj R of38.6%, thisisan initialmodel.Thisillustratesthata depression(dropinpressure)wouldresultinhigherwindspeeds.Thenegative coefficientindicatesthatwhenallelseisconstant,anincreaseinpress urereducesthe windspeed.Infact,pressureissomuchmoresignificantlycorrelated(15.6%) and it rankshigherthanwindgusts(14.0%). d T T T P g w d w a 0640946 0 410118 0 276426 0 6609 0 379629 0 39936 0 00507486 0 31 394 9 Thus,giveninformationfor g P a T w T d T and d ,wecanobtainan estimateofthewindspeed, w Rank Variable Coefficient SEofCoeffi cient t ratio p value Constant 394.31 21.33 18.5 <0.0001 1 P 0.3796 0.022 17.7 <0.0001 2 g 0.3699 0.025 14.8 <0.0001 3 a T 0.6609 0.111 5.9 <0.0001 4 d T 0.4 101 0.063 6.5 <0.0001 5 0.0051 0.001 7.4 <0.0001 6 d 0.0641 0.009 7.1 <0.0001 7 w T 0.2764 0.098 2.8 0.0049 Table12 :Rankingofcontributingvariablesinfirstordermodel 2.5. 4 HigherOrderTerms Foranyindividualrandomvariable i x ,theremaybeinstanceswhenhigherorder termsneedtobeconsidered.Theadditionaltermscanbehintedatbylookingatthe residuals;dependingontheshape thecurvatur e canevenindicatethedegreeofthis

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42 higherorderterm. Thus,toincreasetheeffectivenessoftheresponse,weshallconsidera modeloftheformgivenby m i j i n j ij m i i i t x t x t y 1 1 1 0 ) ( ) ( ) ( 10 wherethe s i a retheweightsthatdrivetheestimatesofthesubjectresponseand is therandomerror. Itisworthnoting thatvelocity(windspeed)squareisrelatedtothetemperature; suchcurvaturemayalsobeestimatedbyrelatingtheveloc ity(windspeed)tothesquare ofthetemperatures.Similarly throughtheidealgaslaw,thiscurvaturemightextendto thepressuresquaredaswell.Hence,considerallsecond degreetermsofthepressureand temperatures ,thatis, 2 11 2 10 2 9 2 8 7 6 5 4 3 2 1 0 d w a d w a T T T P d T T T P g w 11 Usingthishigherorder statistical model,alltermsexcept 2 P aref oundto significant lycontributingtotheresponsevariable, w .Utilizingthesubjectdatawehave develop edthefollowingmodelgivenbyequation 12 2 2 2 0914943 0 299518 0 572203 0 0286254 0 7408 4 1906 16 8509 31 333777 0 365157 0 00534434 0 048 501 d w a d w a T T T d T T T P g w 12 Table13belowgivestheestimatesoftheappropriatecoefficientsofthemodel andtheirsignificantcontributiontothewindspeed.

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43 List Variable Coefficient SEofCoefficient t ratio p value Constant 501.048 44.57 11.2 <0.0001 1 P 0.3338 0.0205 16.3 <0.0001 2 g 0.3652 0.0235 15.5 <0.0001 3 a T 31.8509 2.335 0 13.6 <0.0001 4 d T 4.7408 1.2450 3.81 0.0001 5 0.0053 0.0006 8.24 <0.0001 6 d 0.0286 0.0094 3.06 0.0023 7 w T 16.1906 2.3970 6.75 <0.0001 8 2 a T 0.5722 0.0428 13.4 <0.0001 9 2 w T 0.2995 0.0430 6.97 <0.0001 10 2 d T 0.0915 0.0262 3.49 0.0005 Table13 :Listofcontributingvariablesincludingsecondorderterms 2.5.5 Effectof Interaction Additio nallyandmorelikelythereisnothigher ordertermsgeneratedbyhigher powersoftheindividualtermitself,butbyhowtheindividualentitiesinteractwitheach otherinpairsortriplets. Thus ,interactionswillbeconsidereduptosecondorder as s hownbythestatisticalmodelgivenby m j i i j i m j ij m i i i t x t x t x t y 1 1 1 0 ) ( ) ( ) ( ) ( 13 wherethe s i aretheweightsthatdrivethecontributingvariablesinestimatingofthe subjectresponseand ist herandomerror. Considering allpossibleinteractionswiththesignificantlycontributingvariables thereare21differentpairings;includingthe3higherordertermsandtheoriginalseven variables wehave 31terms thatconstitutesthemodel Thism odelresultsinthefinal formofthemodelwhichisgivenby

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44 0844883 0 0242107 0 0965926 0 489363 0 107258 0 830201 0 25742 2 43813 5 0271 85 0649 18 98363 2 68881 3 00402729 0 51 2855 2 2 a w w a d a d w a dT gd PT T T T T d T T T P g w 14 With % 6 52 2 R and % 1 52 2 adj R whichshowsthat theinteractionbetween thetemperaturesandthevariousrema iningconditions aresignificantcontributorsto windspeed Also,t hereisinteractionbetweentheatmospherictemperatureandthewater temperature,betweenthewatertemperatureandthepressure,andtheatmospheric te mperatureandthedayofyear.As ummaryoftheresultsandtherankingprocessare givenbyTable14. List Variable Coefficient SEofCoefficient t ratio p value Constant 2855.51 401.5 7.11 <0.0001 1 P 2.98363 0.4137 7.21 <0.0001 2 g 3 .68881 0.4837 7.63 <0.0001 3 a T 18.0649 2.319 7.79 <0.0001 4 d T 5.43813 1.185 4.59 <0.0001 5 0.00403 6.34E 04 6.35 <0.0001 6 d 2.25742 0.2599 8.68 <0.0001 7 w T 85.0271 15.14 5.62 <0.0001 8 gd 0.024211 0.002891 8.37 <0.0001 9 w PT 0.096593 0.01508 6.4 <0.0001 10 2 a T 0.830201 0.05694 14.6 <0.0001 11 a dT 0.08449 0.009337 9.05 <0.0001 12 w a T T 0.48936 0.068 7.2 <0.0001 13 2 d T 0.10726 0.02495 4.3 <0.0001 Table14 :Listofcontributingvariablesincludingsecondorderterms 2.5.6 Effectof DummyVariable s AsnotedinFigure12,thereisatemperaturedifferencebetweenthetwobuoys; thatis,ingeneral,thevariablesbehavedifferentlydependingonlocation.Ifwe introduceadummyvariablethatiszero and ifthedataismeasuredatthefirstbuoynear

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45 Cuba(baselevel)andoneifthedataismeasuredatthesecondbuoyintheGulf. IncludingthisdummyvariableTable15intotheabove statistical model weincrease the explanatorypercentto60.1%ofthevariation;however,causesthewinddirectionto becomeinsignificant.Thatistosay,thedisparitybetweenthetwositeswasaccounted forbythedisparityinthewinddirection.Ifweremovethisnowinsignificantterm,the remai ningmodele quation 15 explains60.0%ofthevari ationinthewindspeed. Thus, theresultingmodelisgivenby a w w a w d a d w a dT gd PT T T T T T d T T T P g B w 0526651 0 0151871 0 118028 0 80998 0 213864 0 049448 0 964969 0 42464 1 55116 2 841 110 6267 21 47287 3 28643 2 94285 2 14 3571 2 2 2 15 Thetablebelowgivesthespecificinformationwithrespecttothesignificantly contributingfactors. List Variable Coefficient SEof Coefficient t ratio p value Constant 3571.14 377.3 9.47 <0.0001 1 P 3.4729 0.3867 8.98 <0.0001 2 g 2.2864 0.4570 5 <0.0001 3 a T 21.6267 2.4700 8.75 <0.0001 4 d T 2.5512 1.1030 2.31 0.0209 5 d 1.4246 0.2435 5.85 <0.0001 6 w T 110.8410 14.4800 7.65 <0.0001 7 B 2.9429 0.1687 17.4 <0.0001 8 gd 0.0152 0.0027 5.54 <0.0 001 9 w PT 0.1180 0.0141 8.36 <0.0001 10 2 a T 0.9650 0.0609 15.9 <0.0001 11 a dT 0.0527 0.0088 6 <0.0001 12 w a T T 0.8100 0.1196 6.77 <0.0001 13 2 w T 0.2139 0.0723 2.96 0.0031 14 2 d T 0.0494 0.0232 2.13 0.0332 Table15 :Listofcontributingvariablesincludinginteraction,secondordertermsandthe dummyvariableforlocation

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46 Ifwefurtherconsidersignificantinteractionbetwe enthisdummyvariableandthe remainingsignificantlycontributingvariableslisted asgiven inTable16,theatmospheric temperaturebecauseinsignificant witha p valueof0.7768;thatisatmospheric temperaturescontributionismainlyexplainedbythe interactionwithothercontributing entitiesandatmospherictemperaturesquared Theresultingmodeltakenintoconsiderationthatadditionalinteractionsincluding termsremovedpreviouslysuchaspressuresquared isgivenby d T BT T dT B Bd PT T BT T Bg g BP T T BT T T gd P w d d d a w w a w w a a a a 660437 0 1003 0 01842 1 71995 5 02568 0 0015 0 070202 0 115468 0 424 122 26338 2 02116 1 387747 0 26837 2 01175 0 98408 1 97403 2 12009 1 90616 0 01433 0 38296 3 66 3418 2 2 2 16 Table16givesallrelevantinformationthatsupportsthestructureoftheproposed statisticalmodel.Notethat 2 R and 2 adj R haveincreasedto67.3%and66.8%, respectively,thusgiv entherecordeddatawecaninsertitintotheproposedmodeland obtainagoodestimateofthewindspeed.

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47 List Variable Coefficient SEofCoefficient t ratio p value Constant 3418.66 365.4 9.36 <0.0001 1 P 3.38296 0.3824 8 .85 <0.0001 2 gd 0.01433 0.002587 5.54 <0.0001 3 a T 0.90616 3.196 0.284 0.7768 4 2 a T 1.12009 0.06325 17.7 <0.0001 5 w BT 2.97403 0.2962 10 <0.0001 6 w a T T 1.98408 0.1848 10.7 <0.0001 7 BP 0.01175 0.007156 1.64 0.1008 8 g 2.26837 0.4324 5.25 <0.0001 9 Bg 0.387747 0.0423 9.17 <0.0001 10 2 w T 1.02116 0.1051 9.72 <0.0001 11 a BT 2.26338 0.3092 7.32 <0.0001 12 w T 122.424 13.74 8.91 <0.0001 13 w PT 0.115468 0.01387 8.32 <0.0001 14 Bd 0.070202 0.0169 4.15 <0.0001 15 B 0.0015 5.96E 04 2.51 0.012 16 a dT 0.02568 0.00963 2.67 0.0078 17 d T 5.71995 1.083 5.28 <0.0001 18 d BT 1.01842 0.1308 7.79 <0.0001 19 2 d T 0.100 43 0.02206 4.55 <0.0001 20 d 0.660437 0.2709 2.44 0.0149 Table16 :Listofcontributingvariablesincludinginteraction,secondordertermsandthe dummyvariableforlocationincludinginteraction 2.6 ModelValidation Thefollo wingcriteriawereusedtoidentifythequalityofthedevelopedmodes: the p valuesdeterminingsignificanceofeachcontributingterminconjunctionwiththe 2 R and 2 adj R statistics,andthe F statistics. Wehavealreadydiscussedthesignificanceofthe 2 R and 2 adj R statistics previously;thesestatisticsareoutlinedinTable13.Additionalstatisticsincludethesum ofsquareerrorsforeachmodelbyso urceandtheF ratio.Thislaststatisticsisusedin theF test.

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48 Considerthelinearmodel, X Y ,where X isan m n matrix, m j n i x ij ,..., 2 1 and ,.., 2 1 | areknownconstantsandfullrank;thatis, n m .The vector isavectorofunknownparameters m ,..., 1 0 and ] ,..., [ 2 1 n isa vectorofnon observableindependentnormalrandomvariables(RVs)withcommon variance 2 andmean 0 ) ( E .Thegenerallinearregressiontestfortestingthenull hypothesis 0 : 0 H H ,where H isan n m matrixoffullrank n m ,istorejectth e nullhypothesisattheconfidencelevel if F F .Thesignificancelevelis 0 | H F F P and F isgivenbye quation 17 and ) ( ~ m n m F F m m n n m i i m i i Z Z F 1 2 1 2 17 First, we considertheresiduals,theresidualsforthelinearmodelandthe developedcompletelyinteractive statistical model.Theresidualsshowadistinct disparitybetweenresiduals,andtheyarenot exactlynormalwithmeanzeroandvariance one,asshown by Figure34.Thisislessenedbythecomplete interactivestatistical model defined bye quation 15 asillustrated by Figure35. InFigure34,theresidualplothasa distinct patternandthenormalplotillustratesacrook thecurveisnotlinear.InFigure 35,theresidualplothasalessdistinctivepatternandthenormalplotismorelinear. Figure34 :Residualplotandnormal plotforlinearstatisti cal model R e s i d u a l s R e s i d u a l s Prediction NormalScore

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49 Figure35 :Residualplotandnormalscoreforcomplete interactivestatistical model When the modelwasdeveloped,thefullmodel,forwardselection,backward elimination,andstepwiseselectionswereconsidered,butther ankingofcontributing variableswasdeterminedbythemaximumincreaseinthe 2 R statistics.Additional criterionusedinmodelselectionisMallows ) ( p C statistic(1973). TheMallows ) ( p C sta tisticiscomputedusingEquation 18 where 2 s (variance) isthemeansquareerrorforthefullmodel, p SSE isthesumofsquareerrors forthemodelwith p parameters.Ifw ehave identified therightmodel,thenthestatistic estimatesthenumberofparametersrequiredinthemodel;thatis, p p C ) ( ) 2 ( ) ( 2 p N s SSE p C p 18 Table17summariesthekeystatisticsthatattes ttothequalityoftheproposed model. Prediction R e s i d u a l s R e s i d u a l s NormalScore

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50 Model 2 R 2 adj R Source Sumof Squares Degreesof Freedom Mean Square F ratio Regression 4168.25 7 595.465 Linear 38.9 38.6 Residuals 6556.86 1393 4.707 127 Regression 4968.69 10 496.869 Higher Order 46.3 45.9 Residuals 5756.42 1390 4.14131 120 Regression 5639.78 13 433.829 Interactive 52.6 52.1 Residuals 5085.33 1387 3.66643 118 Regression 5417.38 12 451.448 Buoy 50.5 50.1 Residuals 5307.74 1388 3.82402 118 Regression 7239.25 22 329.057 Buoyw/ interaction 67.5 67.0 Residuals 3486.66 1379 2.5284 130 Table17 :Evaluationstatisticsforthequalityofthemodels 2.6.1 Linear Statistical Model Forthelinearmodel ,theforwardselectionunderthecriterionof0.05toenterthe modeland0.05toremaininthemodelallseveninitialexplanatoryvariablesremainin themodel.Thisisalsothecaseforbackwardselimination.Moreover,whenconsidering the ) ( p C giveninTable18,we see thatthe 8 ) ( p C andsevenvariableslistedin Table15,wehave 7 p ;hence,thisisanindicationofthehighqualityofthe developed model. Rank Variable Partial 2 R 2 R ) ( p C F Pr>F 1 Pressure 15.29 15.29 553.126 252.54 <0.0001 2 Gust 12.78 28.07 243.888 248.43 <0.0001 3 Atmospheric Temperature 2.74 30.81 183.567 55.22 <0.0001 4 DewPoi nt 3.95 34.76 95.5348 84.55 <0.0001 5 WindDirection 1.83 36.59 55.9297 40.17 <0.0001 6 DayofYear 1.93 38.52 13.9449 43.77 <0.0001 7 SeaSurface Temperature 0.35 38.87 8.0000 7.94 0.0049 Table18 :SummaryforForwardSelection(maineffects) includingMallows ) ( p C statistics

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51 2.6.2 CompleteInteractive Statistical Model Usingthesamecriterionoutlinedabove,considerthecompleteinteractivemodel, wehave 8567 20 ) ( p C asshowninTable19andincludingthec onstantparameter, therearetwenty oneparameterstobeestimatedinthemodel,but 20 p ;andtherefore allthecriteriauseduniformlysupportthequalityofthemodel. Rank Variable Partial 2 R 2 R ) ( p C F Pr>F 1 P 15.25 15.25 2174.43 252.54 <0.0001 2 gd 13.94 29.19 1588.54 275.47 <0.0001 3 a T 3.26 32.45 1453.3 67.36 <0.0001 4 2 a T 6.63 39.08 1175.67 152.09 <0.0001 5 w BT 5.72 44.80 936.428 144.72 <0.0001 6 w a T T 8.28 53.08 589.216 246.33 <0.0001 7 BP 4.00 57.08 422.4510 130.07 <0.0001 8 g 1.77 58.85 349.663 60.08 <0.0001 9 Bg 1.43 60.28 291.436 50.09 <0.0001 10 2 w T 0.44 60.72 274.851 15.62 <0.0001 11 a BT 2.42 63.14 174.89 91.26 <0.0001 12 w T 0. 55 63.69 153.998 21.13 <0.0001 13 w PT 0.92 64.61 116.998 35.95 <0.0001 14 Bd 0.42 65.03 101.203 16.75 <0.0001 15 B 0.21 65.24 94.1642 8.56 0.0035 16 a dT 0.15 65.39 89 .9543 5.90 0.0115 17 d T 0.16 65.55 85.2396 6.40 0.0115 18 d BT 1.01 66.56 44.5568 41.91 <0.0001 19 2 d T 0.51 67.07 24.8629 21.62 <0.0001 20 d 0.14 67.21 20.8567 6.01 0.0 144 Table19 :SummaryforForwardSelectionincludingMallows ) ( p C statistics Moreover,whenconsideringtheonevariable t testwiththenullhypothesisthat themeanresidualiszero;thatis, 0 : 0 R H versusthealternativehypothesis, 0 : R a H ,wefailtorejectthenullhypothesiswitha p valueof0.986 ;thatis,the

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52 meanresidualisnotsignificantlydifferentfromzero .Therefore,allcriteriaused uniformlysupportthehigh qualityofthemodel. 2.7 SurfaceResponseAnalysis Now,withthefullydeveloped statistical model:verifiedandcalibratedusing historicaldatacanbeusedtofurtheranalyzethebehavioroftheresponsevariable.Using surfaceresponsemethodologyto obtainthenecessaryrestrictionsontheindependent entities;thatis,whatarethevaluesoftheindependentvariables n x x x ,..., 2 1 thatdrive theresponse tobeeithermaximumorminimumwith90%,95%and99%accuracy ? Surfaceresponseanalysiswouldbeabletoidentifythevaluethattheindependent variablesmusthavetominimizeormaximizetheresponse. Itsstatisticalformisgivenby m j i i j i m j ij m i i i t x t x t x 1 1 1 0 ) ( ) ( ) ( 19 wherethe s i aretheweightsestimatedusingrealdata. T operfo rmthistypeofanalysis wesearchtofind theregionw h e rewearesure the optimum configurationofthevalueoftheindependentvariables .Thenusingmodern daytechnology,wecancompute runsthatwillleadtothefullanalysisofa secondorder responsefunctione quation 20 m j i i j i m j ij m i i i t x t x b t x b b y 1 1 1 0 ) ( ) ( ) ( 20 I ntermsofthemodeline quation 16 ,we proceedto obtainthepart ials thatare needed: 0 115468 0 38296 3 w T P w 0 387747 0 01433 0 26837 2 B d g w

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53 0 387747 0 01433 0 0263965 0 660437 0 B g T d w a 0 97403 2 24018 2 02568 0 98408 1 B T d T T w a w a 0 01842 1 2006 0 71995 5 B T T w d d 0 04232 2 115468 0 98408 1 424 122 w a w T P T T w Recallthat B isadummyvariablethatiseitherzeroorone; hence,thesesix equationscanbesolvedforeachlocation asshowninTable20 These valuesofthe explanatoryvariablesareminimumsasimpliedbythevalueofpressure;thelowerthe pressure,thehigherthewindspeed.Ifweconsiderthefirstbuoy, 0 B wehavean estimatedwindspeedofapproximately6knots. List Variable 0 B 1 B 1 P 1019.57 1036.57 2 g ( )5.0530 20.1313 3 a T 27.7630 28.7804 4 d T 28.5142 23.4373 5 d 158.2952 131.2368 6 w T 29.2978 29.2978 Table 20 :Solutionstothepartialdifferentialequations T hecontourFigure36illustrates thatthe secondbuoyintheGulfofMexico (42036)bothpressureandtemperaturemustdropforextremewindspeedstooccur;that is,whenthepressureislessthanapproximately1008hPaandthetemperatureislessthan 26C,thenwindspeedsshowninredofupt o16.5knots.Figure36alsoillustratesthatat thefirstbuoynearCuba(42056)pressurecanbeashighas1015hPabutifthe temperaturebelow26Cthenthewindspeedscanreachashighas11.5knots(yellow). Inaddition,ifbothpressureandwinds peedsdrop,thenwindspeedscanreachas14.3 knotswhenpressureareashighas1006hPa,butwithalowertemperatureof25C.

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54 ThecontourinFigures37and38showsimilarpatternsfortheseasurface(water) temperaturesanddewpointversuspressure ;thatis,thelowerthetemperaturesthehigher thewindspeedsandifthereisadropinbothtemperatureandpressurethenrelatively highwindspeedswillbeexpected. BUOY=42036 16.5 14.8 13.2 11.5 9.8 8.2 6.5 4.8 3.2 1.5 WSPD 24 26 28 30 A T M P 1000 1010 1020 BAR BUOY=42056 14.6 13.0 11.5 9.9 8.4 6.9 5.3 3.8 2.2 0.7 WSPD 25 26 27 28 29 A T M P 1005 1010 1015 BAR Figure36 :Contourplotofwindspeedoverpressureandatmospherictemperature BUOY=42036 15.2 13.1 11.1 9.0 7.0 4.9 2.9 0.9 -1.2 -3.2 WSPD 26 28 30 W T M P 1000 1010 1020 BAR BUOY=42056 9.5 7.5 5.4 3.3 1.2 -0.9 -3.0 -5.1 -7.1 -9.2 WSPD 28 29 30 31 W T M P 1005 1010 1015 BAR Figure37 :Contourplotofwindspeedoverpressureandwatertemperature BUOY=42036 18.2 16.4 14.7 13.0 11.2 9.5 7.8 6.1 4.3 2.6 WSPD 20 22 24 26 D E W P 1000 1010 1020 BAR BUOY=42056 9.0 8.3 7.6 6.9 6.2 5.5 4.8 4.1 3.4 2.7 WSPD 23 24 25 26 D E W P 1005 1010 1015 BAR Figure38 :Contourplotofwindspeedoverpressureanddewpoint

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55 InallthecontourplotsaboveFigures36thru38showsthatwhentemperature dropsandpressureisbelow 1008,thereisaprofounddifferenceintheresponseofthe wind.Inthenextcasestudy,pressurewillberelatedtowindspeedsgiventhereare stormconditions. 2.8 Usefulness oftheStatisticalModel Thefindingsinthisstudyareextremelyimportanti nmanyaspects:forthefirst timeinrecordedhistorymanisable,usingrealtimedata,totestoldtheoriesanddevelop newones.Anoldtheoryrelatedwindspeedstopressurebutasthisstudywillshow temperaturecan estimate thebirthofastorm.M oreover,temperatureispotentiallya better estimator ofhurricaneforcewinds.Furthermore,therearebroaderimpactsinthe formofnationaldefense;planningandbetter simulatingastorm oncethestormisinfull fruition. 2. 9 Conclusion First,pa rametricanalysisoftheresponsevariable(windspeed)waspreformed andweshowedthattheWeibull probability distributionbestcharacterizesthebehaviorof thesubjectphenomenon.Themanyvariouseffectsoftheearthsrotations(theCoriolis Effect )aswellastheamountofenergyabsorbedbythesun(whichdependsonits relativepositiontotheearth)createnaturaloccurringextremes,whichskewthedata. Thischaracterizationoftheresponsevariableenablesustoestimatethemean,standard er rorandconfidenceintervalsbasedonaspecifieddegreeofconfidence. Presentdaystormchasersunfortunatelydonotrecordedtemperatureswhen trackingatropicalstormandthereforeadditionaldataisneed and tofullyunderstandthe truecontributions oftemperaturestowindspeed.However,asthisstudyshows,while theresponseofthewinddependsbothonpressureandontemperature(atmospheric,sea surfaceanddewpoint);allthreetemperaturesaresignificantwithorder2 whereas pressureisfoun dtobesignificantwithorder1;thatis,temperature s squaredarefoundto besignificant butpressuresquaredisfoundtobeinsignificant .Therefore,asindicated bythemeankineticenergyequationandtheidealgaslaw,thevariationinvelocity(wi nd

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56 speed)isbestexplainedbyanon linearrelationshipwithrespecttotemperature.It shouldbenoted,thatwhenallthecontributingvariablesarerankedusingforward regressionandmaximumimprovementin 2 R ,pressureisfound. T heoverall qualities of eachofthestatistical model sthathavebeendeveloped were evaluatedbasedonfivecriteriaoutlinedinChapter1;allfiveofthesecriteria uniformlysupportthequalityof the model s .Inaddition,withthenetworkestablishedv ia theinternet,thismodelcanreadilybeupdatedasadditionalinformationisgatheredonan hourlybasis. Weplantoextendthisstudytoestimatingalltheexplanatoryvariablesand developaweathergeneratortosimulatewindspeeds ( Georgiou,Davenpo rt,andVickery, 1983) ( Georgiou,1985) andthebirthofastorm aswellashurricanetracking:wind speed,directionality,durationandtiming

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57 CHAPTER3:LINEARAN DNON LINEAR STATISTICAL MODELINGOF HURRICANEFORCEWIND S:HURRICANEINTENSI TY(MA XIMUM SUSTAINEDWINDS) 3.1 Introduction Thebirthofastormwasconsideredinthepreviouschapter;inthepresentchapter weconsidertheonecharacteristicofastorminhurricanestatus;namely,thewindspeed ( Darling,1991) Utilizingrealhistor icaldatagleanedfromUNISYS (UnisysWeather) and NationalOceanic&AtmosphericAdministration ( NOAA ,2004) forHurricanes Wilma (2005), Rita (2005), Katrina (2005), Ivan (2004)and Isabel (2003). Thetracksofthese stormsareshownbyFigure39. We dev elop a statistical model that dependsonbasic informationpresentlygatheredbystormhunters;specifically, pressure latitude and longitude convertedintoCartesiancoordinates,the duration ofthestormuptothegiven pointintime,andthe dayofyea r .Otherdependentvariablesinclude:the changein position intheconvertedcoordinatesystem,the distance it travels(afunctionofthe changesinposition),andthe linearvelocity ofthestormintheconvertedcoordinate systemalonewiththeapprop riateinteractions. Thedeveloped statistical modelcanbeuse d tosimulcasttheassociatedwind speedswithahighdegreeofaccuracy.Inaddition,wi th real timedata the modelcan continuouslybeupdatedandusedto estimatethewindspeedsothatwec an provide stormwarnings.Thisissub modeltoaweathergeneratorwhichcanbeusedtosimulate thetrackofahurricaneasitprogressfromatropicalstormtohurricanecategories1thru 5. Thequalityofthis statistical modelwasdeterminedusingt hefivecriteriaoutlined inthefirsttwochapters.Thedeveloped statistical modelisthenanalyzedusingsurface responsetodetermineunderwhatconditionstheresponsevariable(windspeed)is maximizedorminimized.

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58 Inaddition,inthepresentchap terweintroduceanewscalingprocessfortropical stormsandhurricanes.ReferredtoastheWootenScale,thisredefinedscaleisshown tobemorestablewithlessvariancewithinthedefinedcategorieswhencomparedtoth e commonlyusedSaffir Simpso ns cale. Inourpresentstudywewilladdressthefollowingquestions. 1. Identifythestormsaddressedandwhy? 2. Whatisthedifferenceindirectionalmovementwithrespecttotheseason? 3. Whatarethecontributingvariables? 4. Whatisthemodelwhichmaybeus edto estimate windspeeds? 5. Furthermore,whataretheinteractions? 6. Isthebest fitmodellinearornon linear? 7. Howwelldotheobtainedmodel estimations comparewithactualdata? 8. Underwhatenvironmentalconditionsistheexpectedwindspeedinagiv en stormismaximized? 9. How stable istheSaffir Simpsonscale? 3.2 Descriptionofresponsevariable(windspeed) Thephenomenonofhurricaneforcewindsdependsonthesurroundingpressure aswellasthelatitudeatwhichthecirculations are form ed .Hu rricanescannot be form ed ontheequatorthankstotheCoriolisEffect.TheCoriolisEffectiscausedbythesuns gravitationalpullontheearth. 3.2.1 DataforF ive C ategory5 TropicalS torms DatagleanedfromUNISYSTropical Estimat ionCenter isuse dinthestudy the fivemostrecentstormsclassifi edascategory5,see Table21.Provisionincluded:charts onthetrackofthestorm,trackinginformation,positioninlatitudeandlongitude, maximumsustainedwindsinknots,andcentralpressure(hPa)

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59 Figure39 :Mapoffivestorms:Isabel(2003),Ivan(2004),Katrina(2005),Rita(2005)andWilma(2005)

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60 Year Storms MaxSustainedWind Pressure Color 2005 Wilma 150 882 Purple 2005 Rita 150 ---Red 2005 Katrina 150 902 Orange 2004 Ivan 145 910 Green 2003 Isabel 140 920 Blue Table21: Tableofmaximumhurricane forcewindsandtheirassociatedpressuresfor fiveresentstormsintheAtlanticregion Thesefivestormswillprovideaglimpseintounderstandingthetransitions betweenCategory0(tropicalstorm)toCategory1,etc.Thetracksofthesefivestorms areillustratedinthemap shownby Figure39. 3.2.2 ResponseVariable E itherwindspeedorpressurecouldbeconsideredastheresponsevariable; however,thebeliefis that thelowpressurescausehurricanestoform,thereforeinthis studywewilltrea tthewindspeedastheresponsevariableandthepressuretobea contributingorexplanatoryvariable. Themaximumsustainedwindspeedisanestimate(inmultiplesoffive), as computedbyNOAA usingtheObjectiveDvorakTechnique(ODT)( Zehr,1995) Other parametersmeasured shown inTable22includepressure,timeandlocation.Allother variablesconsideredaremanipulationsof thegiven information,estimatingpartiallinear movements,linearvelocity,duration,dayof the yearandyear. Furtherm ore,themeasurementsoflatitudeandlongitudearenotuniformlyscaled, theyexistinasphere;thereforelatitudesforvariouslongitudesarefurtherapartnearthe equatorandclosertogethernearthepoles.Totrymodelinghurricanesintotermsofit s position,thesemeasurementsfirstneedto be conver ted toaCartesiancoordinates;where linearmovementsareavalidmeasureandthereforeapproximation of linearvelocities exist.

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61 ConversionforlatitudeandlongitudeintoCartesiancoordinates Ifwe let m a 6378137 (theapproximateradiusoftheearth), 25722563 298 1 b 2 2 2 b b c m h 100 (heightabovegeoids)and ) sin 1 ( 2 2 b b a v ,then LON LAT h v x cos cos ) ( and LON LAT h v y sin cos ) ( 3. 2.3 ComparisonofLatitudeversusLongitudeandtheCartesiancoordinate Thetracksofthesefivestormsareverysimilar;fouroutofthefivemadetheir way through thestraightsbetweenCubaandFloridaasshown by Figures40and41. Itisinteresting tonotethatfouroutoffiveofthestormsmovewestthefurther norththestorms moves,butthelaststormmoved mainlyeastasthestormmoved, Furthermore,asillustratedby Figure42,thissinglestormstartedwheretheotherfour stormsendedande ndedwheretwooftheotherstormsbegan.Asforthelatitude,allof thestormsstartedclosertotheequatorasillustrated by Figure43andpossiblywithafew wobbles,movesnorth. Figure40 : Scatterplotof latitude versuslongitude Figure41 : Scatterplotofconverted latitudeversuslongitudeintoCartesian coordinates x and y

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62 3.2.4 Whatisthedifferenceindirectionalmovementwithre specttotheseason? Figure42: Linegraph forlongitude Figure43: Linegraph forlatitude NoticethatWilmaoccurredduringwinterwhentheearthsrotationwithrespectto thesunisinthesouthernhemisphere,whereastheotherfo urstormswereinthesummer monthswhentheearthsrotationwithrespecttothesunisinthenorthernhemisphere. Thiswillbesignificantwhenmodelingthedirectionalityoffuturehurricanes. Figure44: Linegraph forconverted latitudeandlongit udewithrespectto x Figure45: Linegraph forconverted latitudeandlongitudewithrespectto y L o n g i t u d e L a t i t u d e

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63 3.3 MultivariateModelingofHurricaneForceWinds Asatropicalstormstrengthenstohurricanestatus,th erearemanycontributing factorsthatdrivethesubjectresponse(windvelocity).Statisticallymodelingthesubject phenomenonasafunctionofpressure,relativepositionandmovementaswellasdayof theyear;least squaresregressionisusedtodete rminethebestestimatesofthe coefficientsinthemodel. 3.3.1 ContributingVariables Inthischapter,weconsidertheresponsevariable(windspeed)asafunctionof thevariablesoutlinedinTable22.First,weconsiderthemulti linearstatisticalmo del analyticallygivenby, Y t y x v d t y x P w 12 11 10 9 8 6 5 4 3 2 1 0 21 wherethecoefficients s i aretheweightsthatdrivetheestimateoftheresponseand istherandomerrorandthecontributing parameterwedefinedinTable22givenbelow.

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64 Variable Description w Maximumsustainedwindspeed P Pressureatcenter LAT Longitude LON Latitude x ConvertedtoCartesiancoordinates y ConvertedtoCartesiancoordinates x Thechangein x : ) ( ) ( ) ( t t x t x t x y Thechangein y ) ( ) ( ) ( t t y t y t y 2 2 y x Thedistancebetweenmovements t Thechangeintime t y x v v 2 2 Themagnitudeoftheapproximatelinearvelocity.Herethe approximatelinearvelocityis j t y i t x v t D Duration(uptothatpoint) d DayofYear Y Year Table22 :Variablesofinterest 3.3.2 Rankinorderofsignificant(p value)andcontribution First,wewillconsidertheregr essionusingallcategorieswithinthefiveselected hurricanesandallparameters thatwe ranked asshown inTable23.Thisshows that withouttemperature, the pressureisthemostcontributingvariable.Thenthenextmost contributingvariableistheloc ationofthestorm;sincebothcoordinatesaresignificant, thisimpliesboththelatitudeandlongitudearesignificant,thenthedurationofthestorm followedbythetimeofyear(dayofyear)andfinallytherelativemovementsofthe storm.Furthermor e,theseeminglyinsignificanceoftheyearmaybedueto the factthat onlythreeyearsareconsidered.Therefore,thismodeldoesnotdetectatrend being present

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65 Rank 2 R (%) Variable Description 1 90.72 P Pressure 2 93.00 x Convertedcoordinate 3 94.03 y Convertedcoordinate 4 94.34 t Duration 5 94.55 d Dayofyear 6 94.79 v Linearvelocity 7 94.88 x Changeintheconvertedcoordinate 8 95.07 y Changeintheconvertedcoordinate 9 95.17 t Changeintime 10 95.19 Distancetraveled 11 95.19 Y Year 4 Table23: Rankingofindependentvariables 3.3.3 TheStatistical Model First,wewillconsidertheregressionusingallcategorieswithinthefiveselected hurricanesandallparametersrankedinTable23.Usingleast squaresregressi on,this statistical model ,asgivenby e quation 22 and usingthedataoutlinedabove,wehave developed thefollowing statisticalmodeltoestimatethewinspeedofahurricane, w thatis, 08443 2 0 0 77545 7 10 80624 6 10 73014 7 10 58047 6 0837455 0 672378 0 10 31567 1 10 40912 9 12716 1 32 3079 5 5 6 5 6 Y t y x v d t y x P w 22 With % 2 95 2 R and % 1 95 2 adj R ,thisisaverypowerful statistical model; however,whilethismodeldoesexplain95.2%ofthevariationintheresponsevariable, someofthesevariablesareinsignif icantasshowninTable24. 4 Thisvariableyieldsnoimprovement

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66 List Variable Coefficient SEof Coefficient t ratio p value Constant 3079.32 3137 0.982 0.3269 1 P 1.1272 0.0155 72.8 <0.0001 2 x 0.0000 0.0000 8.84 <0.0001 3 y 0.0000 0.0000 8.86 <0.0001 4 t 0.6724 0.2182 3.08 0.0022 5 d 0.0837 0.0287 2.92 0.0037 6 v 0.0000 0.0000 1.55 0.1213 7 x 0.0001 0.0000 5. 48 <0.0001 8 y 0.0001 0.0000 3.64 0.0003 9 t 7.7755 12.0800 0.644 0.5201 10 0.0000 0.0000 0.196 0.8451 11 Y 2.0844 1.5670 1.33 0.1843 Table24 :Estimatesfo rcoefficientsinfulllinearmodel Therelationshipbetweendistance,velocityandchangeintimeis t v and thereforethismodelactuallycontainstheinteractionbetweenvelocitiesandthechangein timeanditisfoundtobeinsign ificant.Otherinsignificantvariablesarethechangein timeandtheyear. Moreover ,muchmoreinterestingistheresidualsthismodelproduces Figure46,thereisanobviousbowingofthedata. Hence,whilethescatterplotinFigure 47indicatestheres idualsarerandomnormal,thescatterplotinFigure46indicatesthat thereis atleastonehigherorderterm.Furthermore,sincepressureisthemostsignificant contributor,explainingupto90%ofthevariationinthemaximumsustainedwindspeed int hepresenceofstormconditions,wewillconsiderthisinquadraticform. Figure46 :Resi dualplotforlinear statisticalmodel Figure47 :Normalprobabilityplotfor theresidualsofthesimplelinearmode R e s i d u a l s R e s i d u a l s

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67 3.3.4 HigherOrderTerms Herewere developtheabovestatisticalmodeltoincludequadraticbehaviorofthe pressureasgivenby, 2 9 8 7 6 5 4 3 2 1 0 P y x v d t y x P w wherethecoefficients s i aretheweightsthatdrivetheestimateoftheresponseand isth erandomerror. Usingrealdataandleast squaresregression,theresultingmodelisgivenby 2 5 5 6 5 6 00410053 0 10 28245 6 10 80522 5 10 5205 3 0812246 0 207587 0 10 35435 1 10 12246 8 71895 6 09 2648 P y x v d t y x P w 23 Thisstatisticalmodelresultin % 1 96 2 R and % 0 96 2 adj R whichisan improvemen tofthepreviousstatisticalmodelwithout 2 P Allcontributingentitiesare significantwiththemaximum p valueof0.0509 asshownbyTable25below List Variable Coefficient SEofCoefficient t ratio p value Constant 2648.0 9 365.9 7.24 <0.0001 1 P 6.7190 0.7662 8.77 <0.0001 2 x 8.1224610 6 0.0000 14 <0.0001 3 y 1.3543510 5 0.0000 10.8 <0.0001 4 t 0.2076 0.1056 1.97 0.0500 5 d 0.0815 0.0243 3.35 0.0009 6 v 3.520510 6 0.0000 1.96 0.0509 7 x 5.8052210 5 0.0000 5.55 <0.0001 8 y 6.2824510 5 0.0000 4.27 <0.0001 9 2 P 0.0041 0.0004 10.2 <0.0001 Table25: Multipleleast squaresregressionincludingsignificantlineartermsanda singlequadratictermforpressure ComparetheresidualsinFigure46withtheresidualsinFigure48,thebowing patternislessened andtheresidualappearmorerandom.Furthermore,comparethe

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68 normalprobabilityplotsinFigure47withthenormalprobabilityplotinFigure49,the secondmodelshowsamuchstraighterlinearrelationshipindicatingtherandomerroris standardnormal Figure48 :Residualplotformodel outlinedinTable25 Figure49 :Normalprobabilityplotfor theresidualsofthemodeloutlinedin Figure48 Therefore,96.1%ofthevariationinthewindspeedisexplainedbythefivemain explanatoryvar iablesoutlinedasprimaryvariablesinthestudy;namely, pressure latitude and longitude converted, dayofyear and duration .Forthesimple transformationsusedtoobtaintheremainingcontributingentities asdiscussedpreviously inthe sectionlabel edconversionforlatitudeandlongitudeintoCartesiancoordinates. 3.3.5 Interaction Here,weshalldevelop fullyinteractivemodel givenanalyticallyby, xy Py P y x v d t y x P w 11 10 2 9 8 7 6 5 4 3 2 1 0 wherethecoefficients s i aretheweightsthatdr ivetheestimateoftheresponseand istherandomerror. R e s i d u a l s R e s i d u a l s

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69 Usingrealdataandleast squaresregression,wedevelopthefittedformofthe modelgivenby, xy Py P y x v d t y x P w 12 7 2 5 5 5 4 5 10 71 7 10 57 2 00338 0 10 45 9 10 00 6 10 19308 1 17842 0 650216 0 10 11 2 10 92 4 91341 3 895 727 24 Thedeveloped parsi moniousinteractivemodelexplains97.1%ofthev ariationin theresponsewith % 0 97 2 adj R .Inthefinalmodel giveninequation 24 all contributingentitiesarefoundsignificantwith p value<0.0001asshowninTable2 6. List Variable Coefficient SEofCoefficient t ratio p value Constant 727.895 473.1 1.54 0.1247 1 P 3.91341 0.8679 4.51 <0.0001 2 x 4.92E 05 3.77E 06 13.1 <0.0001 3 y 2.11E 0 4 3.07E 05 6.85 <0.0001 4 t 0.650216 0.1006 6.47 <0.0001 5 d 0.17842 0.02314 7.71 <0.0001 6 v 1.20E 05 1.75E 06 6.85 <0.0001 7 x 6.00E 05 9.33E 06 6.43 <0.0001 8 y 9.45E 05 1.34E 05 7.04 <0.0001 9 2 P 0.00338 4.02E 04 8.4 <0.0001 10 Py 2.57E 07 3.28E 08 7.82 <0.0001 11 xy 7.71E 12 7.00E 13 11 <0.0001 Tab le26: Multipleleast squaresregressionincludingsignificantlineartermsanda singlequadratictermforpressureandsignificantinteractions

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70 Estimat ions Figure5 0 : Linegraph comparisonfor HurricaneWilma Figure5 2 : Linegraph comparisonf or HurricaneRita Figure5 4 : Linegraph comparisonfor HurricaneKatrina Figure5 1 : Linegraph comparisonfor HurricaneIvan Figure5 3 : Linegraph comparisonfor HurricaneIsabel W i n d S p e e d Legend W i n d S p e e d W i n d S p e e d W i n d S p e e d W i n d S p e e d

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71 Figures50thru54illustratethefivestormspresentedinthe presentstudy;inblue istherecordedwindspeedsoverthedurationofthestorms:Wilma,Ivan,Rita,Isabeland Katrina;andinred,istheestimatedwindspeedusingthedevelopedstatisticalmodel. Whenthewatertemperaturesarehotterbeforethest ormreachesfullforceandremoves theheatenergyfromthewater,the estimat ionstendtobeunderestimate d whereasthe afterthestormpeaksandweakenstothepointofdissipation,the statistical modelismore accurate. 3.4 CrossValidationand Estimatin g HurricaneKatrina Theprocessof crossvalidation canbedonesystematicallybyremovingonedata atatimeandusingthebest fit statistical modelovertheremainingdatato estimate the datapointremovedorthisprocesscanbeusedbypartitioningth edatasetintotwo distinctgroups;onewhichwillbeusedtoestimatetheparametersofthemodel,then usingthebest fitregressionmodel, estimate theremainingdata.Inthisstudywe jackknifedthedatabystormusingfourstorms:namelyWilma,Ivan ,RitaandIsabelto best fitthesignificantinteractivemodel,whichyieldedamodelwith % 2 97 2 R and % 1 97 2 adj R .Then,usingthisleast squaresregressionmodelwe estimat ed HurricaneKatrina.The estimat ionexplained95.8% ofthevariationinthewindspeeds. Usinghypothesistesting,witha p valueof0.50,wemustacceptthatthemean differencecalculatedtobe 688 0 d betweentherecordedwindspeedsandthe estimat edwindspeedsisnotsignificantlydi fferentfromzero.Usingatwo tailtest,the teststatistic, 679 0 t and62degreesoffreedom, atthe0.01significancelevel,wefail torejectthatthemeandifferencebetweentherecordedwindspeedandtheestimated windspeeds given therealtimemeasurementsofthesurroundingenvironment iszero. 3.5 ModelValidation ofCompleteInteractiveModel Thefollowingcriteriawereusedtoidentifythequalityofthedeveloped statistical mode l s:the p valuesdeterminingsignificanceofeach contributingterminconjunction withthe 2 R and 2 adj R statistics,the F statisticsandMallows ) ( p C statistics.T hese

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72 results forevaluatingthestatisticalmodelgivenequ ation 24 are showninTable33 AdditionalstatisticsincludethesumofsquareerrorsforeachmodelbysourceandtheF ratio.ThislaststatisticsisusedintheF testoutlinedinchapter2. Considerthecompleteinteractive st atistical model,usingforwardselectionwitha 0.05levelinenterthemodelanda0.05leveltoremaininthemodel,thenwehave 43 11 ) ( p C asshowninTable33and with 11 p ; isan indication thatthedeveloped modelis of highquality Rank Variable Partial 2 R 2 R ) ( p C F Pr>F 1 2 P 91.11 91.11 767.76 4046.48 <0.0001 2 xy 2.78 93.89 407.43 178.80 <0.0001 3 P 0.79 94.68 306.17 58.38 <0.0001 4 Py 0.55 95.23 235.83 45.53 <0.0001 5 x 0.26 95.49 203.78 22.62 <0.0001 6 y 0.18 95.67 187.74 15.88 <0.0001 7 x 0.46 96.13 124.61 46.26 <0.0001 8 t 0.17 96.30 104.11 18.06 <0.0001 9 v 0.20 96.50 79.63 22.45 <0.0001 10 y 0.23 96.73 51.17 27.59 <0.0001 11 d 0.32 97.05 11.43 41.80 <0.0001 Table33 :SummaryforForwardSelectionincludingMallows ) ( p C statistics Moreover,whenconsideringtheonevariable t testwiththenullhypothesisthat themeanre sidualiszero;thatis, 0 : 0 R H versusthealternativehypothesis, 0 : R a H ,wemustfailtorejectthenullhypothesiswitha p valueof0.403. Thatis, weacceptthatthemeanresidualiszero. Therefore,all five criteria useduniformly supportthehighqualityofthemodel. 3.6 SurfaceResponseAnalysis Thefullydevelopedstatisticalmodelthatincludesallattributingvariableand appropriateinteractionisgivenby

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73 xy Py P dy dx v d D y x P w 12 7 2 5 5 6 4 5 10 71408 7 10 56827 2 00337569 0 10 44684 9 10 0005 6 10 9516 11 178422 0 650216 0 10 10721 2 10 91989 4 91341 3 895 727 25 We shallusethismodeltoperformsurfaceresponseanalysisaspreviously described. Undertheassumptionthat 0 ) ( x dx and 0 ) ( y dy ,thenwecan approximateconditionsunderwhichthe estimat edwindspeedismaximized.Sincet he partialwithrespecttodurationanddayofyearareconstant,andunderthestated assumptions,thepartialwithrespecttothevelocityiszero;thatis,weareassuminga storminmotionwillstayinandatthesamerelativelinearspeed.Theremaini ngpartials areasfollows: 0 10 827 256 00675138 0 91341 3 9 y P P w 0 10 71408 7 10 56827 2 10 10721 2 12 7 4 x P y w 0 10 71408 7 10 91989 4 12 5 y x w Tooptimizethisresponse,using standard calculus techniquesyieldsarelative maximumwindspeed whentheatmosphericpressure 261 822 P therelativeposition withrespecttotheconvertedcoordinates ) 26 6377805 56 59379 ( y x .Hence considercontourplotsforthewindspeedoverthec onvertedcoordinates ,thepointwhere thismodel estimat esanextreme highinthewindspeed isthelocatio nwhereallstorm migrateandareattheirstrongestintermsofwindspeed.Wealsoseethattheaccuracy ofthismodelisrestrictedtotheregionsinwhichthereareasignificant largenumberof stormstosample .Stormsdonotsurviveoverland,buti fthelandwherenotthere,storms wouldgathermorestrength.Butwiththelandasabumper,thestormsredirects,whichis

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74 overwater.Thisimpliesanadditionalcontributingvariablemightbethebathymetry theunderwaterequivalentofaltimetry,a swellaseddiesthatoccurintheopenwaters. 3.7 AnalysisoftheSaffir SimpsonScale Outlinedin1969,theSaffir SimpsonHurricaneScaleisascaleclassificationfor tropicalcycloneshavingsustainedwindsinexcesswhencomparedwithtropical depress ionsandtropicalstorms.Thefive pointscalecategorizeswesternhemispherical tropicalcyclonescalledhurricanesbasedontheintensitiesoftheestimatedsustained windspeed.Thesescalesalsogaugetheamountoflikelydamageandfloodingdueto l andfall. HerbertS.Saffirdevelopedascale usingpressureandwindvelocityto characterizethestageofahurricane .RobertSimpson,ameteorologistaddedinthe damagecausedbystormsurgeandresultingfloods ,creatingthe Saffir Simpsonscale which hasbeenusedtogaugestormsforthepasttwenty sevenyears. 3.7.1 Analysis oftheSaffir Simpsonscale Tropicalcyclonesareclassifiedintothreemaingroups: tropicaldepressions tropicalstorms ,and hurricanes .Hurricanesarerankedaccordingto theirmaximum windsusingthefive pointSaffir SimpsonHurricaneScaleaslistedinTable34. Type Category Pressure (hPa) Winds (knots) Winds (mph) Surge (ft) Depression TD ----<34 <39 TropicalStorm TS ----34 63 39 73 Hurricane 1 >980 64 82 74 95 4 5 Hurricane 2 965 980 83 95 96 110 6 8 Hurricane 3 945 965 96 112 111 130 9 12 Hurricane 4 920 945 113 135 131 155 13 18 Hurricane 5 <920 >135 >155 >18 Table34: Saffir SimpsonHurricaneRatingScaleascategorizedbypressureandwind sp eed

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75 3.7.2 HowdoestheSaffirSimpsonScalecomparewiththesefivestorms? Usingthesamefivestormsusedtodevelopthestatisticalmodelsinprevious sections,C omparethemidpointsofthevariousintervalsoutlineforpressurebythe Saffir Simpson scale ,givenin Table35 ,weseetherearenomidpointavailablefor tropicaldepressions,tropicalstormsandhurricanecategoryone Forhurricanecategory two,theSaffir Simpsonscalehasamidpointof972.5;however,thedataindicatesthat thisiss ignificantlylowerwithameanof960.0andmedianof959.Forhurricane categorythree,theSaffir Simpsonscaleis950whichisthesameasthedatasmedianbut slightlygreaterthanthemeanof948.7.Forhurricanecategoryfour,theSaffir Simpson s caleis932.5whichisrelativelyclosetothemeanofthedata,930.5,andthemedian, 931.Forhurricanecategoryfive,theSaffir Simpsonscalegivesnolowerboundand thereforenomidpoint;thedataindicatesthatthelowestpressureisapproximately 882 andthemeanpressureforstormswithhurricanestatuscategoryfiveis911.9andthe medianpressureof914. Category Saffir Simpson Midpoint Data: Min Max Data: Mean Data:Median TD --------983 1013 1001.6 1003 TS --------965 1007 993. 8 997 1 >980 > 955 987 975.1 979 2 965 980 972.5 940 978 960.0 959 3 945 965 950 927 969 948.7 950 4 920 945 932.5 894 955 930.5 931 5 <920 < 882 938 911.3 914 Table35 :ComparisonofpressureaccordingtotheSaffir SimpsonHurricane RatingScale versusassociatedpressuresinhistoricaldata AsshowninTable35,thepressuresatwhichtheSaffir SimpsonScalesetdonot agreewithourfindings.Thereareoverlappingpressuresatwhichtimethestormisin transition. Tofurtheranalyzethisdis parity,wecomparedthemeanpressuresversuswind speedsfortheoriginalfivestorms,Figure55,tothemeanpressureversuswindspeeds forallstormsrecordedinthe1990s,Figure56andthe1980s,Figure57.Thescatter plotsinFigures55thru57il lustratethatthereisawelldefinedrelationshipbetween atmosphericpressureandwindspeed.Inaddition,thisrelationhascurvature.Henceto

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76 beingtoanalysistherelationshipbetweenwindspeedsandatmosphericpressurealone, considertheboxplot sformeanpressurebyrecordedwindspeedasshownbyFigure58. Thereareseveralwindspeedswherethepressuresaremorevariable. Figure55 : Meanpressureversusrecordedwindspeedsfor thelistedfivestorms Category P r e s s u r e ( h P a ) P r e s s u r e ( h P a ) P r e s s u r e ( h P a ) Figure56 : Meanpressureversus recordedwindspeedsforallstorms recordedinthe1990s Figure57 : Meanpressur eversus recordedwindspeedsforallstorms recordedinthe1980s

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77 Figure58 : Boxplotforpressurewithrespecttorecordedwindspeedsforthelistedfive storms Ifweconsiderpressurescategorizedbywindspeed;first,usinghypothesistesting todetermineifthemeanpressureisdifferentasthewindspeedincreases,wehave the following:given w p themeanrecordedpressure forgiven windspeed w wecantest thenullhypothesisthemeanpressureforagivenwindspeed w isequaltothemean pressureforthe windspeed 5 w .The p valuesforthevarioushypothesistestsaregiven in Table 36;significantlydifferentmeanpressureareshowninbold P r e s s u r e ( h P a )

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78 5 0 : w w p p H w 5 w value P 10 15 0.388 15 20 20 25 25 30 0.291 30 35 0.118 35 40 0.431 40 45 0.00672 45 50 0.312 50 55 0.0595 55 60 0.338 60 65 0.134 65 70 0.302 70 75 0.211 75 80 0.0651 80 85 0.831 85 90 0.336 90 95 0.258 95 100 0.12 100 105 0.0676 1 05 110 0.684 110 115 0.408 115 120 0.0221 120 125 0.00626 125 130 0.0493 130 135 0.641 135 140 0.711 140 145 <0.001 145 150 0.00831 Table3 6 : Testformeanpressure Hence,atthealphalevelof0.01,therearefivedistinctgroups,buttwoofthe se categorieswouldcontainonlyasinglerecordedwindspeedeach;namely,145knotsand 150knots.Ifwedefinedthesetwowindspeedscombinedashurricaneforcewinds category5.Second,weneedtoaddresstheunevenlydistributionofthefirstthree categories:10thru40,45thru120and125thru140.Ifwedefinewindspeedsbetween 10and40knotsascategory0(bothtropicaldepressionsandtropicalstorms),anddefine windspeedsbetween125and140ascategory4,thenweneedtopartitionwind speeds

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79 between45and120knotsintothethreeremainingcategories;namely,category1, category2andcat egory3.AsshowninTable36 ,thereareseveraldifferentbreaking pointsatthealphalevelof0.10,twoofwhichfallreasonablywithourinterv al,therefore wewilldefinewindspeedsbetween45at70ascategory1,between75and100tobe category2andfinallywindspeedsbetween105and120ascategory3. Comparethemeanpressuresbycategoryforthefivestorms,firstbytheSaffir Simpson scale,Figure59,andtheWootenscale,Figure60.TheWootenscaleisless varianceacrossthescalewhereastheSaffir Simpsonscaleislessstable. ComparisonsofcategoricalwindspeedsbetweentheSaffir SimpsonScaleand theWootenSc ale showthatthe proposed scalehasamuchmorebalanceddistributionin mostmeasures:count,totalvariance,andindividualranges,seeTable3 7 IntheSaffir Simpsonscale,thevarianceamongthevariouscategoriesrangebetween14.60and 190.913whe reasintheWootenscale,thevariancerangesbetween6.536and82.887. ThestandarddeviationsarealsoanindicationthattheWootenscaleismorestable. Figure59 : Barchartsforpressure,with categoriesasassignedbytheSaffir S impsonscale Figure60 : Barchartsforpressure, withcategoriesasassignedbythe Wootenscale

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80 Saffir Simpson Count Mean Median Standard Dev iation Variance C0 121 39.421 40 13.817 190.913 C1 27 70 70 5.371 28.846 C2 52 89.327 90 3.839 14.734 C3 47 104.362 105 4.25 18.062 C4 106 123.679 125 6.701 44.906 C5 49 142.653 140 3.833 14.69 Wooten Count Mean Median StdDev Variance C0 65 28.385 30 7.711 59.459 C1 79 56.899 55 9.104 82.887 C2 76 91.645 90 6.292 39.592 C3 77 114.026 115 5.383 28.973 C4 87 133.046 130 6.021 36.254 C5 18 147.222 145 2.557 6.536 Table37 : DescriptivestatisticsforwindspeedasassignedbytheSaffir Simpsonscale andtheWootenscale Comparisonsofcategori calpressurebetweentheSaffir SimpsonScaleandthe WootenScalesh ow the proposed scalehasamuchmorebalanceddistributioninmost measures:count,totalvariance,andindividualranges ,seeTable38 IntheSaffir Simpsonscale,thevarianceamong thevariouscategoriesrangebetween45.686and 170.441whereasintheWootenscale,thevariancerangesbetween40.416and123.125. ThestandarddeviationsarealsoanindicationthattheWootenscaleismorestable. Saffir Simpson Count Mean Median Std Dev Variance C0 116 996.914 1000 9.914 98.288 C1 27 975.148 979 11.124 123.746 C2 52 960.000 959 6.759 45.686 C3 47 948.702 950 10.558 111.475 C4 106 930.462 931 13.055 170.441 C5 49 911.612 914 12.757 162.742 Wooten Count Mean Median StdDev Varian ce C0 65 1002.077 1003 6.357 40.416 C1 74 986.257 987 11.533 133.015 C2 76 958.329 958.5 8.284 68.624 C3 77 941.143 943 10.313 106.361 C4 87 921.057 921 11.096 123.125 C5 18 901.500 904 10.326 106.618 Table 38 : Descriptivestatisticsformeanseale velpressureasassignedbytheSaffir SimpsonscaleandtheWootenscale

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81 3.7.3 The T hermodynamics BehindM olecular V elocity Considerthethermodynamicformulaforkinetic t emperature givenby kT mv average 2 3 2 1 2 where m isth emassoftheparticlesinmotion, v isthelinearvelocityoftheparticle, k isBoltzmannconstant 1 2 2 23 10 3806503 1 K kgs m k and T isthetemperature.(See AverageMolecularKineticEnergya ndMaxwell Boltzmanndistribution.)Ifwefurther considertheidealgaslaw P T k V ,assumingthevolume V andthemass m are constant,wehavethefollowingrelationshipbetweentherelativewi ndspeedandthe depression(changeinpressure): max 2 min P P w w .Usingthisasthebasefor analyzingaquadraticrelationshipbetweenwindspeedandpressure,98.6%ofthe variationinthepressurecanbeexplainedbytheleastsquareregression ofpressureonto windspeedusingthemodel 96 1012 312669 0 00285818 0 2 w w P .Thismodelis consistentwiththefactthatnormalatmosphericpressure(meansealevelpressure),when littlewindispresent 0 w ,is1013.25whichisonlyslightly higherthan1012.96. Invertingthisregressionwehavethefollowingmodelto estimat edwindspeed basedonthisrelationstatedaboveand using historicallydata wehave 69722 54 00285818 0 511 1021 P w .Thismodelestimatesthemaximummeanlevelsea pressurei s1021.511.Thismodelsupportsthefollowingrelationships betweenwindspeed andpressure,asshownbyTable39. Type Category Pressure(hPa) Wind(knots) TropicalDepression/TropicalStorm 0 995 1010 10 42 Hurricane 1 972 994 43 77 Hurricane 2 951 971 78 102 Hurricane 3 932 950 103 122 Hurricane 4 911 931 123 142 Hurricane 5 <911 >143 Table3 9 :ComparisonofpressureaccordingtotheWootenHurricaneRatingScale developedusinghistoricaldataforthefivehurricanesoutlinedinthestudy.

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82 Rec all,undertheassumptionthatthevolumeisconstant,wehaveformtheideal gaslaw P T k V whichbringsintothepicturetemperature: dt dT T P dt dP andunderthe assumptionsthat thepressureasafunctionwindspeedisgivenb y c bw aw P 2 ; therefore, wehave dt dw b aw dt dP 2 ,hence dt dT T P b aw dt dw 2 1 Ifweassumethatthetemperatureafterthepeakofthestormiscoolerthanbefore thestorm,wewouldexpectthelowerthetemperaturethelowerthepr essure.Classify thestormas BeforeMaximumPressure ( BP ) forpressuresrecordedbeforethepeakwind speedand AfterMaximumPressure ( AP ) forpressuresrecordedafterthepeakwind speed.Asillustratedbythegreendots,thepressuresBParehigherth anthepressuresAP, thebluedots. Thenweseethatthereisadifferenceintherelationshipbetweenwind speedandpressurebeforepeakwindspeedandafterpeakwindspeed.Thisdifferenceis mostlikelyduetothetemperaturechangesasastormremo vesheatfromtheocean waters.Furthermore,therateatwhichthepressuredropsisrelatedtothedurationofthe storm.ThisisillustratedinFigure62.HurricanesWilma,RitaandKatrinaweremore intensestorms,withwindspeedsreaching150knots dissipatedmorequicklythan HurricaneIvanandIsabel.HurricaneIvanandIsabelreachedwindspeedsof145and 140,respectively. Figure61 :Scatterplo t ofpressureversuswindspeed Classification

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83 Figure 62 : Linegraph ofpressureforeachstorm He reweseethattherateatwhichthepressuredropsandthewindspeedincreases isproportionalorpossiblyexponentiallyrelatedtothetimeorduration. Thefasterthe pressuredropsthemoreintensethestorm,butwithshorterduration.Thustheshor terthe stormthemoreintensethewindsproduced. 3.8 Usefulness oftheStatisticalModel Thedevelopedstatisticalmodelcanbeusedtoaccuratelyestimatethewindspeed ofatropicalstormasitmovesandtransitionsintoahurricane.Theestimation produced canbeusedforpublicsafelyandadvisories.Furthermore,inconjunctionwithother developedstatisticalmodelsthedevelopedstatisticalmodelcanbeimprovedhurricane trackingandforecasting. 3.8 Conclusion Inthepresentstudywewereableto developanon linearstatisticalmodelusing fiveresenttropicalstormswhichreachedcategoryfivestatus: Wilma (2005), Rita (2005), Katrina (2005), Ivan (2004)and Isabel (2003).Thedifferencebetween directionswithrespecttoseasonisinthecolde rmonths,thestormsformatahigher latitudeandlowerlongitude(furthereast).Thedirectionalityofthestormwillbefurther studiedaswellasthetiminganddurationoftropicalstormsandhurricanes. Storm

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84 Thedevelopedstatisticalmodelexplains97% ofthevariationinthe subject response( windspeed ) Usingthedevelopedmodel wecan estimate thewindspeed withinastormwithoutweatherballoonsandreconnaissance.Theageofinformation enablesus,usingradar,satellitesandbuoystogathereno rmousamountsofdata, computerandtheInternetallowustostoreandsharetheinformation.Inaddition,the wayinwhichthenetworkissetup,thismodelcanbeupdatedonanhourlybasisand usedtogeneratewarningforpublicsafely. Onceadditional informations uchastemperaturesarerecordedintheeyeofa hurricane,theextended modelcouldexplainupwardsof99.9%ofthevariationinthe windspeed.Atpresent,inthis study wehaveshownthatpressureis themost significantlycontributingva riable thatisnottosaythatitisthecause.Furthermore, whentemperatureisnotincluded,thecurvatureinthe windspeed datarequiresa quadraticterm and whenthevarioustemperaturearenotavailable,pressureoforder2is foundtobeasignif icantlycontributingvariable. Themostexplanatorystatisticalmodel isnon linear.Thesignificantlycontributingvariables: pressure pressuresquared ,the location inCartesiancoordinates,the changeinlocation ,the linearvelocity ofthe storm,the time(inhours)of duration ofthestorm,andinteractionbetweentheCartesian coordinates xy andthe y coordinateandpressure, Py Analyzingthesurfaceofthisnon linearstatisticalm odel,weestimatethatthe windspeedwillbemaximizedwhenthepressureislow,about882.Thisisthelowest recordedpressureinourstudy. Thedevelopedstatisticalmodel forhurricaneintensity isas ub model which will eventuallybeusedinconju nctionwithothersub modelst hat estimatethepressure,the durationofthestorm,thedirectionalityandthetimingto createaweathergenerator whichyieldaccuratesimulationofhurricanetrackingwhichwouldenable ustoreplace thespaghettistring modelspresentlyusedtosimulatethetrackofatropicalstorm. Furthermore,theproposedmodelcanbeupdatedandappliedtootherregionswheresuch cyclonesoccur.Moreover,easilyupdatedasnewinformationisgatheredandestimations madecanbeu tilizedforadvisoriesforpublicsafely.

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85 Furthermore,wedevelopedascale,theWootenscale,bywhichhurricanestatus canbemoreaccuratebedescribed.Thatis,thereislessvariancewithinthedefined categoriesbothwithrespecttowind speedand pressure.Usingthisproposedscalewill makeestimatingtheintensityofastormasittransitionsbetweenthevariouscategories. ItwillsettheinitialsettingsoftheMarkovchainthatwilldrivetheweathergenerated discussedabove.

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86 CHAPTER 4: STATISTICALANALYSIS ANDMODELINGOFLIG HTNING 4.1 Introduction Floridaisthelightningcapitaloftheworld.Lightning strikesoccurwhenelectrostaticenergywithinstorm conditionsisunbalancedandephemeraldischargesof staticelectricitya resetofftohelpthesystemfind equilibrium.Therehavebeen63lightningevents reportedto theNationalEnvironmentalSatellite,Data, andInformationService( NESDIS ) inHillsborough Coun ty between01/01/1950and12/31/2005that includesevendeaths ,forty nineinjuries,and1.420 milliondollarsinpropertydamage.Therewere919 casesreportedinFlorida. Hence,i tisimportantto considerthiswithrespectto publicsafetyand economic impacts .Weneedtoobtainabetterunderstandingofthe sub ject phenomenon. Inthepresentstudy,wefirstperformparametric inferential analysisonthe subjectresponse,namelylightning;thatis,thenumberoflightningstrikespermonth. Usingmaximumlikelihoodestimates(MLE) wedeterminethattheWeibull pr obability distributionbestcharacterizestheprobabilisticbehaviorofthesubjectphenomenon. Havingknowledgeoftheprobabilisticnatureoflightning,enablesustoestimatereturn periodforthepeaknumberoflightningstrikesinagivenmonthaswe llasestimatesof themean,standarderrorandconfidenceintervalsatanacceptablelevelofconfidence. Suchinformationcouldbeusedasmeasuresoflightningdetectionandprotection. Inintroducingthesubjectanalysis,wewillusethetechniqueo fbootstrapping; implementingthistechnique,wewillobtainamorereliableestimateofthetrue (unknown)averagenumberoflightningstrikes.Thistechniquealsoallowsustomonitor Image1 :PhotographbyC. Clark,courtesyNOAAPhoto Library,NOAACent ral Library;OAR/ERL/National SevereStormsLaboratory (NSSL)

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87 themeanestimateasthesamesizeincreases;thatis,determinethe convergenceofthe estimate x as n andsimultaneouslyreducethestandarderror n s Second ly ,weuserealdatagatheredbyNationalLightningDetectionNetwork (NLDN)todeveloplinearandno n linearstatisticalmodelsforthe numberoflightning (response)asafunctionofthe contributable variables: precipitablewater tropical stormwindtotal sealevelpressureanomaly t ropicalstormwindsanomaly ,and Bermudahighaverage .Inaddition tothe relativehumidity atvariouslevels rain in variouscounties seasurfacetemperature precipitationanomalydistrictone temperaturesatvariouslevels temperaturerange PacificDecadalOscillation ( PDO)standardanomaly SolarFluxstandardan omaly Pacific NorthAmerica Index( PNA)standardanomaly precipitationanomalydistrictfour dayofyear ArcticOscillation ( AO)standardanomaly and O utgoingLongWaveRadiation (OLR) Toourknowledge,thisisthefirststatisticalmodelofitskind Thedevelopedstatisticalmodelenablesusto identifyandrankthecontributing entities ( explanatoryvariables / independentvariables)thatexplainunderwhatconditions lightningstrikesoccur(Fieux,Paxton,Stano,andDiMarco,2005)andestimatethe responsevariable;namely,thenumberoflightningstrikesinagivenmonth.Thequality ofthismodel wasdetermined usingthefivecriteriaoutlinedinChapter1andusedin Chapters2and3.Allcriteriauniformlysupportthequalityofthedeveloped statistical model.Finally,usingthedevelopedmodelweperformedsurfaceresponseanalysis;that is,wedeterminedthevaluesofthecontributingvariablesthateithermaximizeor minimizetheresponsewithanacceptablelevelofconfidence. Lightningi nthestateofFloridaisasignificantevent ( phenomenon ) thatwemust makeeveryefforttomonitorandunderstand.Althoughthis study concentratesonthe stateofFlorida,similarmethodology,andproceduresareapplicabletootherstatesand furtherg eneralizedtootherregionswherelightningisafactor.Moreover,withthe networksthathavebeenestablishedtomaintainthisinformationcanbeutilizedto continuallyupdatethis statistical modelandfurthermore,thismodelcanbe easily updated and a ppli ed tootherstatesaswellasotherglobalregions.

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88 Inourpresentstudywewill addressthefollowingquestions: 1. Identifytheregiontobeaddressedandwhy? 2. WhatconstitutesLightning? 3. Whatisthebest fitprobabilitydistribution? 4. Whatisthe maxi mumprobablenumberofl ightning s trikes? 5. Whatistheprimarycontributortothenumberof lightning strikes ? 6. Whatarethesignificantinteractionsthatcontributetolightning? 4.2 Descriptionof V arious ContributingE ntities Thedatausedinthepresent studywereobtainedfromseveralsourceslistedbelow. Themaindatasetconsistofmonthlytotallightningstrikesforaperiodof16years. Monthlyrecordspreviouslycompiledwithrelativehumidity,temperatureincluding Bermudahighs,tornadoes,wate rspouts,hail, amongothers. 1. NOAA,NationalEnvironmentalSatellite,Data,andInformationService (NESDIS). 2. MonthlycloudtogroundlightningdataoverFloridafrom1989to2004forMay throughSeptember,NLDN. 3. Totalmonthlyrainfallcollectedbysixt eencountiesinthestateofFlorida1989to 2004fromSouthwestFloridaWaterManagementDistrictHydrologicData. Aswese einthehistograminFigure63alongtheachartwhichincludesthebasic descriptivestatistics.T hisdataisextremelyskewedto therightwithamonthlymean numberoflightningstrikespermonth 33 691 93 x comparedtothemediannumber oflightningstrikespermonth 256 34 M .Moreover,the sample deviation 21 871 120 s dominatesthemean withacoefficientofvariation 29 1 x s CV ;thatis 129%ofthemean,andstandarderror 00 9137 n s

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89 Figure 63 :Histogramforthenumberoflightningstrikespermonthincludingdescriptive statisticsforthenumberofl ightningstrikespermonth Withsuchlargecoefficientsofvariationandstandarderrors,theinterpretationof themean couldbe misleading.Wewilladdressthisissueinthenextsection,whichwill introducethebootstrappingtechniquetoreducethes tandarderror.Inthissection,weare concernedwiththebest fitprobabilitydistribution.Themostcommonlyused distributionsare:normal,lognormal,exponential,two parameterWeibull,gammaand betadistribution. 4.3 ParametricAnalysis Asillustr ated inthehistogramsinFigure63 ,thedataisextremelyskewed.This datadoesnothaveanormaldistribution;theskewnessismeasuredtobe1.485forthe monthlydistributionindicatingthatthedataisasymmetric. Itisclearthat wehaveapeaki nourd istribution,which ismorecurve dthanthe symmetricnormaldistribution.Furthermore, kurtosismeasuringzeroisnormaland negativekurtosisindicatesaflatdistribution,thedatainthisstudyhasapositivekurtosis; namely,1.35forthemont hlydistributionandevengreater3.04forthedailydistribution. Usingstandard goodness of fitmethods ,firstforthemonthlynumberoflightningstrikes, theonly probability distributionthatfailedtoberejec tedatthe0.01level,Table40 ,isthe t wo parameterWeibull. Statistic EstimatedMonthly Count 175 Mean 93,691.33 Median 34,256 StandardDeviation 120,871.21 StandardError 9,137. 00 Minimum 102 Maximum 529,981 Range 529,879 Skewness 1.48 Kurtosis 1.35 MonthlyLightningStrikes

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90 Test Weibull Kolmogorov Smirnov <0.001 Cramer vonMises <0.001 Anderson Darling <0.001 Chi Squared 0.015 Table 40 :Best fitdistributionforthe numberoflightningstrikespermonth Furthermore,whenconsidering theempiricalcumulativeprobabilitydistribution comparedtoeachofthe individualbest fitdistribution s asshowninTable41, thethree parameterWeibullhasthehighest 2 R and 2 adj R indicatingthethree paramete rWeibullis thebest fitdistributionamongtheindividualdistributionstested. Distribution Parameters 2 R 2 adj R Weibull(2) 0 67086 6229 0 99.0% 99.0% Weibull(3) 464 0 847 187 102 99.5% 99.5% Ta ble4 1 :Estimatedparametersforthenumberoflightningstrikespermonth However,sincethesemodelsareforthemostpart are thesameandtheargument canbemadethatthenumberoflightningstrikesasshowninthe sample datacouldbeas fewasone wewillemploythelawofparsimonyandcontinueourstudyusingthetwo parameterWeibull. Thus,theprobabilitydensityfunctionthatcharacterizetheprobabilisticbehavior ofthenumberoflightninginagivenmonthisgivenbythetwo parameterWeibul lwith maximumlikelihoodestimates(MLEs)asfollows:theshapeparameter 6229 0 and scaleparameter 086 67 andthethresholdsettozero ) 0 ( ;andtherefore,the associatedprobabilit ydensityfunction isgivenbye quation 26 andcumulative probabilit ydistributionfunctiongivenbye quation 27 otherwise 0 0 67086 exp 67086 67086 6229 0 ) ( 6229 0 1 6229 0 x x x x f 26

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91 and otherwise. 0 0 67086 exp 1 ) ( ) ( 6229 0 x x x X P x F 27 Usingth emaximumlikelihoodestimatesfortheremainingtwo parameterand settingthethresholdatthemini mumnumberoflightningstrikes,zerowehavethe expectednumberoflightningthatonewillexpectwilloccurisgivenby 78 297 92 ) | ( ) ( 0 dx x xf x E andt he median 2341 247 37 M ;thatisthevalue x suchthat x dx t f 0 50 0 ) | ( Thisresultsinahighvariance,thatis, 10 0 2 10 63389094 2 ) | ( ) ( ) ( dx x f x E x x V andastandarddeviationof 8 659 161 s and 33 220 12 n s Table42givesthepercentilesforboththeobserveddataandtheestimatedvalues. Accordingtotheobserveddata,giventhatlightninghasoccurred,thereisa1%chance thattherewerelessthan213strikes.Thereisa50%chancethatgivenlightning o ccurred,therecouldbeupto34,256strikes.Thereisalsoa1%chancethatgiven lightningoccursthattherecouldbemorethan451,786strikes.Accordingtothe estimation,thereisa1%chancethatgivenlightningoccursthattherecouldbemorethan 778,687strikes. Additionalestimatescanbemadeusingthecumulativeprobability distributionisshowninFigure63.Inexample,toestimatethe80%percentileusingthe Weibullprobabilitydistributiongraph,weprojectbackwardstofindthatapproxima tely 150,000strikes;thatis,thereisa20%chancethatgiventhatlightningoccursthatthere willbeatleast150,000strikes.

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92 Percent Observed Estimated 1 213 41.64 5 457 569.98 10 1,245 1,810.13 25 7,042 9,078.44 50 34,256 37,248.38 75 143, 732 113,333.89 90 284,311 255,923.76 95 362,897 390,462.58 99 451,786 778,687.13 Table42 : Estimatedvaluesusingthetwo parameterWeibull WeibullCurve: Thresh=0Shape=0.6229Scale=67086 C u m u l a t i v e P e r c e n t 0 20 40 60 80 100 Lightning 0 100000 200000 300000 400000 500000 600000 Figure64 : Weibullcumulativeprobabilitydistribution 4.4 Bootstrapping Thetechniqueofbootstrappin gisare samplingtechniqueusedwhenthe descriptivestatisticsaresuchthatthesamplestandarddeviationissignificantlylarger withrespecttothemean, x s .Sincethesamplingdistributionhasastandard C u m u l a t i v e D i s t r i b u t i o n NumberofLightningStakes

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93 deviationof n s ,ifwecangeneratealargeenoughsamplesizeandthesamplemeanis comparabletotheoriginalestimate x thenwearemoreconfidentthatthesamplemean isanaccurateestimateofthepopulationmean Bootstrappingis aprocedurethatinvolveschoosingrandomsamples with replacement fromadatasetandanalyzingeachsamplethesameway.Thistechniqueof resamplingasameansofacquiringmoreinformationabouttheuncertaintyofstatistica l estimators; it allowsustotestthereliabilityofthe estimates andassesswhether stochasticeffectshaveaninfluenceontheprobabilisticdistributionwhichcharacterizes thephenomenonunderstudy byreducingthestandarderror T heoriginaldatas etisconsideredandthefollowingstatisticscomputed:the mean,standardsampledeviationandthestandarderror.Recallthemeannumberof lightningstrikespermonthis 33 691 93 x ,thestandarddeviationis 21 871 120 s andthe standarderrorof 00 9137 n s B ootstrappinggeneratedadatasetofsize fivehundred;andtheabovestatisticswerecalculated again .Then,independentofthe aboveset,bootstrappinggeneratedadatasetof onethousand,thenagainwe g enerate d an independentdatasetoffifteenhundredandfinally weincreasedthe dataset to ten thousand.Seeas sociatedstatisticsinTable43 Data N Mean Std. Dev. Standard Error Lower Bound Upper Bound Original 175 93691.33 120871.2 9137.00 757820. 80239 111599.8576 BS1 500 91832.02 116924.7 5229.03 81583.11816 102080.9218 BS2 1000 93197.93 120531.9 3811.55 85727.28544 100668.5746 BS3 1500 89095.06 113873.5 2940.20 83332.26857 94857.85421 BS4 10000 91138.73 118640.2 1186.40 88813.38208 93464. 07792 Table43 :Samplesize,mean,standarddeviation,standarderrorandthelowerandupper boundsonthe95%confidenceintervalforthemeannumberoflightningstrikesper month

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94 4.4.1 Convergence Thisisconsistentwiththesequenceofmeans 10000 ,..., 2 1 ; 1 n n x x n i i n ,the sequenceofstandarddeviations 10000 ,..., 2 1 ; 1 ) ( 1 2 n n x x s n i n i n ,andsequenceof standarderror: 10000 ,..., 2 1 ; n n s n n generatedusingthefifthdatasetdescribed above. There sampleddata,theconvergenceofthemean,theconvergenceo fthe standarddeviationandstandarderrorareillustratedinFigures65thru68. Notethat,as s s n as n thereforefor 1 n ,thisimplies 2 s n .In general,toreducethestandar derrorbyafactorofnine,wewouldneedeight onetimes asmuchdata;thatis,increasethedatasizefrom175to14,175.Toreducethestandard errorbyafactorof9,000wouldrequire14,175,000,000data. However,atsuchaslowrateofconvergence ,whatisthepointoflessening return;thatis,atwhatpoint itis necessary toincrease n significantlylargecomparedto thereductioninthestandarderror. N u m b e r o f L i g h t n i n g S t r i k e s M e a n L i g h t n i n g S t r i k e s Sample Figure65 :Re Samplingofsize10,000 Figure66 :Convergenceof themean x Sample

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95 Consider thepercentchangeinthemean, 10000 ,..., 4 3 ; 1 1 i x x x p i i i i andthe percentchangeinthestandarderror 10000 ,..., 4 3 ; 1 1 i q i i i i ,illustratedinFigures 69and70. Asweseetherateatwhichboththemeanandthestandarderrorconvergesis onlysignificantwhen 500 n Thus ,wecanacceptthatthecalculatedmeanis acceptable,becauseoftheconvergencewithincreasingthesamplesizeandreducingthe dominanceofthestandarderrorwecanproceedtoobtainconfidencelimits.Table44 given90%,95%,and99%confidencelimi tsofthetruenumberoflightningbasedonour sampleddata. Sample Sample S t a n d a r d D e v i a t i o n S t a n d a r d E r r o r Figure67 :Convergenceofthestandard deviation n s Figure68 :Convergenceofthestandard error n

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96 4.4.2 PercentageChangeintheMeanandStandardError: Confidence Level Lower Upper 99% 69,895 117,488 95% 75,658 111,725 90% 78,582 108,801 Table44 : Confidence intervalsforthemeanusingthetwo parameterWeibull Thatis,weareatleast 99%confiden t thatthetruemeannumberoflightning strikespermonthisbetween69,895and117,488 ,similarlyweareatleast 95%confiden t thatthetruemeannumberoflig htningstrikespermonthisbetween75,658and111,725 Similarlyweareatleast90%confidentthatthetruemeannumberoflightningstrikesper monthisbetween78,582and108,801. Sample Figure69 :Convergenceofthestandard error n p Sample Figure70 :Convergenceofthe standarderror n q P e r c e n t a g e C h a n g e i n M e a n P e r c e n t a g e C h a n g e i n E r r o r

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97 4.5 Multivariate Statistical Modelofthe N umberofLightningStrikes Inthissection,wewillusehistoricaldatacollectedbytheNationalLightning DetectionNetwork(NLDN)inconjunctionwith othermeteorologicalphenomena to identif y thecontributingentitiesorexplanatoryvariables(independentvariables)that causel ightningtooccur. Severaladditionalvariablesweremadeavailablebutwherenotfoundsignificant suchvariablesastornadosbycategory,waterspouts,hailbysize,NorthAtlantic Oscillation(NAO),Madden/JulianOscillation,etc.Foracompletelisto fthemorethan 100variablesini tiallyconsideredseeAppendixA .Torankthecontribution ofthe numberoflightning oftheseexplanatoryvariables,firstamodelwasgeneratedusing forwardregressionandthenthevariablesfo undtobesignificantatth e0. 0 1 levelwere runina s modelgeneratedbyaddingvariablesintheorderthatmaximizestheincrease to 2 R 4.5.1 StatisticalModel Theinitialstatisticalmodelwedevelopedfortheresponsevariable( numberof lightningstrikes ) asafunctionof month perceptiblewater precipitation bydistrict, sealevelpressure Bermudahighs relativehumidity atvariouslevelsinthe atmosphere, temperatures atvariouslevelsintheatmosphere, seasurface(water) temperature ,the Pacific DecadalOscillation (PDO),the Pacific/NorthAmerica Oscillation (PNA),the ArticOscillation (AO),the OutgoingLong waveRadiation (OLR), SolarF lux asanaverageorananomaly;and rainfall insixteencounties enumeratedasfollows: EnumerationofCou nties :1Levy,2Marion,3Citrus,4Sumter,5Hernando,6Lake,7 Pasco,8Polk,9Pinellas,10Hillsborough,11Manatee,12Hardee,13Highlands,14 Sarasota,15Desotoand16Charlotte Table46givesthemathematicalnotationsthatareusedinther emainderofthe studyandabriefdescription.

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98 ContributingVariables Variable Description m Monthofyear pw Perceptiblewater d p Precipitation a nomalybydistrict; 7 6 5 4 3 2 1 d w P Sealevelpressure: a verage w P Sealevelpressure: anomaly w Tropicalstormwinds: t otal w Tropicalstormwinds: a nomaly B T Bermudahigh : a verage B T Bermudahigh: a nomaly Relativehumidity: mb rh Levels:1000mb,850mb,700mb,600mb,500mb,400mb300mb,200mb,100mb Temperature: mb T Levels:1000mb,850mb,700mb,600mb,5 00mb,400mb300mb,200mb,100mb range T The maximum range between temperatures at various levels w T Seasurfacetemperature PDO PacificDecadalOscillationindex standarda nomaly PNA Pacific/NorthAmer icaOscillationindexstandarda nomaly AO ArticOscillation orl OutgoingLong waveRadiation sf Solarfluxstda nomaly i r Rainfall:totalmo nthlyrainfallcollectedbysixteencountiesinthestateofFlorida Table4 5 : Variablesofinterest inestimatingthemeannumberoflightningstrikes permonthintheStateofFlorida NotallofthevariableslistedinTable46,werefoundsignificant whenconsider thenullhypothesis 0 : 0 i H ,seeTable47.Inaddition,whenu singbackwards elimination,temperatureat the850 level ,dayinyear,seasurfacetemperature, precipitationindistrictone,sealevelpressure,AOstandardAno maly,therelative humi dityatthe500 level, andraininHillsboroughCountyareinsignificantatthe0.10 significance level.Therefore,ourmodelwillincludethefourteenvariablesfoundtobe significantinbothmodelselectionmethods .Theanalytica lmodelthatwewishto structureusingrealdataisgivenby

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99 4 14 13 12 13 11 10 850 9 8 5 7 1000 6 5 4 3 2 1 0 p PNA sf r PDO rh T r rh T P w w pw N range B w 28 wherethecontributingvariableshavebeenidentifiedinTable46givenbelow, andthe s aretheweightsthatdri vetheestimateofthenumberoflightningstrikes N and istherandomerror. Theestimatedstatisticalformoftheabovemodelisgivenby 4 13 850 5 1000 4 7 44 6306 3 3193 54 2272 62 2927 8 1823 43 1346 79 2862 69 3647 78 7376 7 24246 881055 0 21 1350 38 1119 5 13100 10 42 2 p PNA sf r PDO rh T r rh T P w w pw N range B w Thestatisticalmodelisofhighquality becauseitresultsin % 2 97 2 R and % 9 96 2 adj R .Table48liststhecontributingvariables,theestimatedcoefficients includingtheassociatedstandarderror,andtheteststatisticand p valueforthe hypothesistesting.Theh ypothesistestdeterminesifthecoefficients i issignificantly differentfromzero;thatisthenullhypothesisis 0 : 0 i H andthealternative hypothesisthatthereisasignificantdifference.

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100 Variable Coefficie nt SEofCoefficient t ratio p value Constant 2.42E+07 2.17E+06 11.1 <0.0001 pw 13100.5 790.3 16.6 <0.0001 w 1119.38 183 6.12 <0.0001 w 1350.21 160.9 8.39 <0.0001 w P 0.881055 0.03293 26.8 <0.0001 B T 24246.7 2125 11.4 <0.0001 1000 rh 7376.78 875.4 8.43 <0.0001 5 r 3647.69 1001 3.64 0.0004 range T 2862.79 905.7 3.16 0.0019 850 rh 1346.43 558.9 2.41 0.0171 PDO 1823.8 2073 0.88 0.3803 13 r 2927.62 1218 2.4 0.0173 sf 2272.54 1964 1.16 0.2489 PNA 3193.3 1535 2.08 0.0391 4 p 6306.44 1542 4.09 <0.0001 Table46 : Linearregressionforthenumberoflightningstrikesinamonthwith respecttotherankedindependentvariablesincludingtheassociatedp values Forthe completeinteractivestatisticalmodel, using theforwardselectionunder thecriterionof0.05toenterthemodeland0.05toremaininthemodelall variableslisted inTable47 remaininthemodel.Moreover,whenconsideringthe ) ( p C giveninTable 47 ,we see thatthe 59 9 ) ( p C but 21 p ;hence,this thereisabettermodel,possible withfewervariablesorinteraction.However,theanalysisofthestatisticalmodel providesarankingofthecontributingvariablesandabasisforaninteract ivemodel.

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1 01 4.5.2 RankingofIndependentVariables: Rank Variable 2 R ) ( p C F Pr>F 1 Precipitablewater 61.00 2278.48 187.72 <.0001 2 Tropicalstormwindtotal 71.44 1639.00 43.50 <.0001 3 Sealev elpressureAnomaly 78.94 1180.07 42.03 <.0001 4 TropicalstormwindsAnomaly 89.99 503.250 129.09 <.0001 5 Bermudahighaverage 92.34 360.465 35.70 <.0001 6 Relativehumidity(1000mb) 94.76 214.068 52.99 <.0001 7 RaininHernandocounty 95.53 168.810 19.60 <.0001 8 Seasurfacetemperature 96.10 135.799 16.50 <.0001 9 Temperaturerange 96.61 106.309 16.93 <.0001 10 PrecipitationAnomalydistrictone 96.90 87.763 12.15 0.0007 11 Relativehumidity(850mb) 97.10 75.090 9.33 0.0028 12 Relativehumidity (500mb) 97.40 59.078 12.66 0.0006 13 Temperature(850mb) 97.80 40.879 16.17 0.0001 14 PDOstandardAnomaly 98.00 29.026 12.25 0.0007 15 RaininHillsboroughcounty 98.10 25.633 4.94 0.0283 16 RaininHighlandscounty 98.20 22.106 5.27 0.0237 17 Solar FluxstandardAnomaly 98.30 16.945 7.23 0.0083 18 PNAstandardAnomaly 98.40 14.620 4.52 0.0359 19 PrecipitationAnomalydistrictfour 98.40 12.273 4.70 0.0324 20 Dayofyear 98.50 9.5954 5.27 0.0237 21 AOstandardAnomaly 98.56 Table4 7: Ranking ofindependentvariablesusingforwardselectionwitha0.05 leveltogetintothemodelanda0.05leveltoremaininthemodel. IllustratedinFigure 71 theresidualsarerandom;theresidualplotofthe estimat ed valuesversustheresidualsofthate stimationarescatteredwithnodiscernablepatternsas wellasthenormal probabilityplotinFigure72 oftheresidualsisapproximatelya straightlineandhencetheresidualsareapproximatelynormal.

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102 Comparetheactualnumbero flightningstrikesrecordedbyNLDNtothe estimat edvaluesaccordingtothemodeloutli nedinequation 29 andTable4 8; as showninFigure73, weseethatthismodelaccountsforseasonality.Furthermore,wesee thatthereareasig nificantlylargernumberoflightningstrikestowardtheendofMay thruSeptember. Figure73 : Estimat edvaluesversusrecordedlightningstrikes 4.5.3 Interactive Statistical Model Hereweextendthedevelopedstatisticalmodeltoincludeall possibleinteractions ofthecontributablevariables.Theanalyticalformofthemodelisgivenby Month R e s i d u a l s R e s i d u a l s Estimation NormalScores Estimation N u m b e r o f L i g h t n i n g S t r i k e s Figure71 :Residualsversusthe estimatedvalueswithmaineffects variableprecipitablewater Figure72 :Normalitytestfortheresiduals

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103 w w B w range w w P m rh T rh pw r r p T sf PNA rh rh rh T T T P P w w pw N 20 500 19 500 18 6 17 5 16 4 15 14 13 12 500 11 850 10 1000 9 8 850 7 6 5 4 3 2 1 0 29 wherethecoefficients s i aretheweightsthatdrivetheestimateoftherespon seand istherandomerror. Notethattheinteractionrelevantinthemodelareprecipitablewaterandrelative humidityatthe500mblevel,theseasurface(water)temperatureandtherelative humidityatthe500mblevel,andthes easurfacepressureandthemonthofyear. Usingtheinformationavailablewehavestructuredthefollowingstatisticalmodel toestimatethenumberoflightningstrikespermonthwiththecontributionofinteraction givenby w w B w range w w P m rh T rh pw r r p T sf PNA rh rh rh T T T P P w w pw N 587 559 607 634 526 229 25 2310 99 3628 7 4075 8 24949 69 3194 10 82 2 5 10134 72 2743 25 4481 10 91 4 10 20 1 26 5815 934669 0 6 5534 12 1315 85 1491 9 21913 10 190 500 500 13 5 4 3 500 850 1000 3 850 4 7 30 Thisstatisticalmodelresultin % 4 98 2 R whichisanimprovementofthe previousstatisticalmodel Allcontributingentities exceptseasurfacetemperature are significantwiththe p value<0.05asshownby Table48below

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104 Variable Coefficient SEofCoefficient t ratio p value Constant 1.90E+07 2400000.00 7.93 <0.0001 pw 21913.9 2226.00 9.84 <0.0001 w 1491.85 156.10 9.56 <0.0001 w P 0.934669 0.02599 36 <0.0001 w 1315.12 165.80 7.93 <0.0001 range T 5815.26 909.30 6.4 <0.0001 1000 rh 4481.25 825.90 5.43 <0.0001 850 rh 2743.72 553.90 4.95 <0.0001 500 rh 10134.5 2853.00 3.55 0.0005 B T 24949.8 1947.00 12.8 <0.0001 5 r 3628.99 779.40 4.66 <0.0001 4 p 4075.7 1205.00 3.38 0.0009 w T 4.91E+03 5893. 00 0.833 0.406 sf 3194.69 1281.00 2.49 0.0137 PNA 2.82E+03 1183.00 2.38 0.0185 13 r 2310.25 941.80 2.45 0.0153 850 T 1.20E+04 3026.00 3.96 0.0001 m 569230 272400.00 2.09 0.0383 500 rh pw 229.526 56.17 4.09 <0.0001 500 rh T w 634.607 183.50 3.46 0.0007 w P 5534.6 1751.00 3.16 0.0019 w P m 559.587 267.20 2.09 0.0379 Table48 : Least squaresregressionforthenumberoflightningstrikesinamonthwith respecttotherankedindependentvariablesincludinginteraction;alsoincludedarethe associated p values 4. 6 Statistical ModelValidation Thefollowingcriteriawereusedt oidentifythequalityofthedeveloped statistical mode l s:the p valuesdeterminingsignificanceofeachcontributingterminconjunction withthe 2 R and 2 adj R statistics,the F statisticsand Mallows ) ( p C statistics.T hese statis ticsforthemodeloutlinedine quation 30 areshowninTable5 0.Additional

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105 statisticsincludethesumofsquareerrorsforeachmodelbysourceandtheF ratio.This laststa tisticsis usedintheF testoutlinedinC hapter2. U singforwardselectionwitha0.10level ofsignificance inenterthemodelanda 0.10level ofsignificance toremaininthemodel,thenwehaveasshowninTable 49 22 ) ( p C where 21 p ;andthisisaindicationofthehighqualityofthedeveloped model. Inaddition,the F statisticalindicatethatallcontributingvariablesare significantatthe0.10leveland % 41 98 2 R ,thati s98.41%ofthevariationinthe subjectresponse(numberoflightningstrikes)isexplainedbytheleast squaresregression model. Rank Variable Partial 2 R 2 R ) ( p C F Pr>F 1 500 rh pw 60.59 60.59 3627.57 265.93 <0.0001 2 w P 13.61 74.2 2318.1 90.7 <0.0001 3 w 6.79 80.99 1665.27 61.11 <0.0001 4 w 8.77 89.76 822.406 145.46 <0.0001 5 500 rh 2.55 92.31 578.423 56.07 <0.0001 6 B T 1.58 93.89 428.4 43.33 <0.0001 7 500 rh T w 1.68 95.57 268.963 63 <0.0001 8 range T 0.6 96.17 212.804 26.11 <0.0001 9 5 r 0.46 96.63 170.474 22.47 <0.0001 10 PDO 0.26 96.89 146.971 13.94 0.0003 11 1000 rh 0.2 97.09 129.976 11.02 0.0011 12 pw 0.51 97.6 82.8983 34.29 <0.0001 13 850 rh 0.23 97. 83 62.4231 17.28 <0.0001 14 850 T 0.24 98.07 41.2136 19.94 <0.0001 15 PNA 0.07 98.14 36.7854 5.69 0.0183 16 4 p 0.03 98.17 35.5194 2.92 0.0893 17 13 r 0.07 98.24 30.2948 6.7 0.0105 18 sf 0.04 98.28 28.2024 3.86 0.0511 19 m 0.05 98.33 25.8096 4.23 0.0413 20 w P 0.04 98.37 23.8977 3.84 0.0519 21 w P m 0.04 98.41 22 3.9 0.0502 Table 49 :Summary forForwardSelectionincludingMallows ) ( p C statistics

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106 Moreover,whenconsideringtheonevariable t testwiththenullhypothesisthat themeanresidualiszero;thatis, 0 : 0 R H versustheal ternativehypothesis, 0 : R a H ,wemustfailtorejectthenullhypothesiswitha p valueof0.9999. Therefore,allcriteriauniformlysupportthehighqualityofthe statistical model. 4.7 Usefulness oftheStatisticalModel Light ningaff ectsusinseveralways;onelightningcasualtyoccurredforevery 86,000strikes(overtheUnitedStates)andonedeathoccurredforevery345,000flashes (NOAA). Second,lightningcauses poweroutages.Itwouldbeusefulfor theenergy supplycompany to havea statistical modelwhichwould estimate lightningstormsin orderforthemtobetterservecustomers and whichminimizingexpense.Notjusttobe ableto estimate thenumberoflightningbasedonthesurroundingenvironmentaldatabut appropriately allocateresources whenshouldadditionalworkersbescheduledtomake repairstothesystemby estimat ingpotentialoccurrenceIngeneral,tobeableto develop strategiesforthesafetyofourcitizens,amongothers.Thedevelopedmodelcanbeused e ffectivelytoaddresstheseissues. 4.8 Conclusion Whentheelectrostaticenergywithinstormconditionsisunbalancedand ephemeraldischargesofstaticelectricity,thisdischargeisseenaslightorlightning.The phenomenonisacommonoccurrenceint heStateofFlorida.B asicdescriptivestatistics indicatethatthemeannumberoflightningstrikesisapproximately93,961strikesin givenmonth.Toverifythisestimationsaccuracy,parametricanalysisisusedtoshow thatthenumberoflightningstri kespermonthisnotnormallydistributed,butisskewed insuchawaythatgivenlightningstrikesoccurtherearemorelikelytobealargenumber ofstrikes. TheWeibullprobabilitydistributionbest characterizethebehaviorofthesubject response( numberoflightningstrikes).Thetwoandthree parameterWeibull relatedin thatthethree parameterWeibullwithathreshold 0 isthetwo parameterWeibull

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107 distribution .Employingthelawofparsimony,aswellasthefactthetrue minimum numberoflightningstrikesiszero,thetwo parameterWeibullusedwhen estimat ingthe returnperiodforvariousnumbersoflightningstrikes.Furtheranalysisutilizingthe techniqueofbootstrappingtogenerate500countshowsthatthetruenu mberoflightning strikespermonthatthe95%confidencelevelisbetween81,583and102,080strikesina givenmonth. Second,non linearmodelingofthenumberoflightningstrikespermonth ( Fieux, Paxton,StanoandDiMarco,2005) withrespecttothe amountof perceptiblewater windshear anomalies in sealevelpressure ,various temperatures relativehumidity atdifferentlevelsintheatmosphereandseveralothersignificantlycontributingvariables, whichexplainsapproximately98.4%ofthevaria tioninthesubjectresponse.Thatis,in the numberoflightningstrikes permonthcanbeestimatedbasedonthesurrounding environmentandtheirinteractionwith % 4 98 2 R and % 2 98 2 adj R Furthermore, explaining61.0%ofthev ariationinthesubjectresponse(numberoflightningstrikes)is explainedbytheamountofperceptiblewaterintheair.Significantinteractionsinclude thefollowinginteractions:perceptiblewaterandrelativehumidityatthe500level, relativehumid ityatthe500levelandseasurface(water)temperature,theseasurface pressureandmonthofyear.

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108 CHAPTER5: STATISTICAL ANALYSISAND MODELINGOFFLOOD STAGES 5.1 Introduction Inourrecordedhistoryandasrecentlyas2005,damageinitiatedbyori nfact causedbyextremehydrologicaleventshaveplaguedmanyanddevastatedcivilized society.Inordertomitigateabalancebetweenthebeneficialimpactsoffloodwaters withnegativeimpactsofsuchphenomena as rainfallandflooding wemusthaveab etter understandingofthesubjectresponse;thatis,thefloodstagemeasuredinfeet .However, itshouldbenotedthatfloodingisnotsolelycausedberainfall.In1993,duringthegreat floodinMississippi,therewereonlyeightnamedstorms ofwhic h onereaching measures ashigh as hurricanestatus categorythree.Thesefloodwaters were causedbymelting snow,affectedcoastalcountiesinFlorida .H owever,flooding theStateofFloridais mainlyc ausedbyrainfall Inthepresentstudy,weshallst udy thefloodingintheStateofFlorida ourstudy willconsists of twoparts.First we will performparametricanalysison thefloodstages intheSt.JohnsRiver,thelarges t riverintheStateofFlorida .Determining the probability distributionwhich best characterizesthebehaviorofthesubjectresponse(floodstages), wewillbeable toestimatethemean,thestandarderrorandgenerateconfidenceintervals basedonaspecifiedlevelofconfidence. Second,wewilldevelopastatisticalmodelofth e floodstage oftheSt.Johns Riverasafunction of flowrates and estimated duration ofthefloodevent.Usingthe developedstatisticalmodelofthemeanresponse(floodstage),weareinapositionto profilevariousfloodstages.Thatis,giventhe presentflowrates,dependingonthe probabledurationofthefloodwecanaccuratelyestimatetheheightoftheflooding (floodstage).Wethenextendthedevelopedstatisticalmodeltoinclude time (dayof year)uptoorder3toestimatetheseasonalhi ghsandlows.

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109 Suchestimationsofthefloodstagesareextremelyimportantforpublicwarnings andstrategicplanning.Furthermore,thenetworkinplacetorecordandgather informationbysuchagenciesasSouthwestFloridaWaterManagementDistrict, (SF WMD),easilydevelopedstatisticalmodelcanappliedtoobtaingoodestimatesof upcomingfloodstages.Inaddition,asnewdatabecomesavailablethestatisticalmodel canbeupdatedtoimprovetheestimatesofthecoefficientsthatdrivetheattributing variables.AlthoughthepresentstatisticalmodelwasdevelopedusingdatafromtheState ofFlorida,itcanbeeasilyrestructuredforotherstatesorglobalregions( Nadarajahand Shiau,2005) Inourpresentstudywewilladdressthefollowingquest ions: 1. DiscusstheGeneralizedExtremeValueDistribution. 2. Identifythewaterwayaddressedandwhy? 3. Whatconstitutesaflood? 4. Whatisthebest fitprobabilitydistributionforthefloodstage? 5. Whatarethecontributingvariables? 5. 2 Generalized ExtremeVa lue Probability Distribution T hreeextremevalue probability distributions wereusedtodeterminewhichof theseaccuratelycharacterizesthebehaviorofthesubjectresponse(floodstage).These probabilitydistributions aretheGumbel,theFrechetand negativeWe ibull(1987).The cumulativegeneralizedextremevalue(GEV) functiondevelopedbyJenkinson(1955); alsoseeHoskingetal. (1985)andGalamboscontainsthesethreeextremevalue probabilitydistributions.ThecumulativeGEVdistributionisdef inedby, 1 1 exp ) ( x x F 31 where isthelocation, isthescaleand istheshapeparameter.

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110 For 0 ,ormoreaccurately,th elimitingfunctiongivenby ) ( lim ) ( 0 x F x G resultin theGumbeldistribution (doubleexponentialdistribution) andisgivenby x x G exp exp ) ( 32 5.2.1 DerivationoftheGu mbelDistribution Letting x n ,as 0 then n and x x n n e n x 1 1 lim 1 lim 1 0 33 However,for 0 wehave 0 1 x ,whichimpliesif 0 ,thenwe havetheFrechet probability distributionwhere x andistruncatedbelowwhereas if 0 ,thenwehavethenegativeWeibull probability distributionwhere x and istruncatedabove. 5.2.2 CharacteristicsoftheVariousDistributions Ifthe probability distributionisGumbelthenthenormalprobabilityplotfor x ln ln isastraightline.IfthedistributionisFrechet,thennormalprobabilityplot for x ln ln curvesupandisindicativeofdata possessesa heavytail.Ifthedistributionisa negativeWeibullthennormalprobabilityplotfor x ln ln hasahorizontalasymptote. 5.2.3 MaximumLikelihoodEstimates (MLE s) forthe G iven P robability D istributions Oncetheappropriatedistributionisdetermined,thentheMLEsforthe parameterscanbeestimated.FortheFrechetcumulativeprobabilitydistribution( 0 ), thealternativeformisgiven by

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111 otherwise x x x F 0 exp ) ( wheretheMLEsaregivenbythesolutionstotheequationsgivenbelow n i i n i i n i i i x x x x n n 1 1 1 ln ln n i i n i i n i i x x x n 1 1 1 ) 1 ( 1 1 and 1 1 1 n i i x n FortheWeibullcumulativeprobabilitydistribution( 0 ),thealternativeformisgiven by otherwise x x x F 0 exp 1 ) ( wheretheMLEsaregivenbythesolutionstothegivenequations, n i n i i n i i i i t t t n t n 1 1 1 ln ln

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112 n i i n i i n i i x x x n 1 1 1 1 1 1 and 1 1 1 n i i x n Inaddition,th eGumbel cumu lativeprobability distribution 0 ,theMLEs aregivenbythesolutionstothegivenequations, x x x x n i i n i i i 1 1 exp exp and n i i x n 1 1 ln However,solvingthissystemofequationsisnoteasilydone.Ifinstead, we con sidertheempiricalcumulativeprobabilitygivenby 1 n i p i ,sinceGumbelisa doubleexponential,we havetherelationshipgivenbye quation 34 where 0 and 1 1 .Usingleastsqu aresregression andtheanalyticalformoftheGumbel probabilitydistributionisgivenby

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113 i i x x p 1 0 1 ln ln 34 Additionalestimatescanbemadeusing n i p i 2 1 2 oranysequence n i i p 1 of n probabilitiessuchthat 1 i i p p .Inthepresentstudy, 1 n i p i isused;thispartitionof theopeninterval ) 1 0 ( isuniformlyspacedandeasytocalculate. 5.2.4 Percentilesfor theFrechet,WeibullandGumbel ProbabilityDistributions Using MLE stoestimatethep ercentilesfortheFreche tandtheWeibullaregiven bye quation 35 ande quation 36 ,respectively, ) 1 ln( 1 p x p 0 35 and ) 1 ln( 1 p x p 0 36 However,fortheGumbelcumulativeprobabilitydistribution,thepercentilesare givenby, p x p 1 ln ln 37 Theseestimateswillbeusedtodeterminehowwellthefloodstagedatafitsone oftheextremevaluedistributionstatedabove. 5.3 DescriptiveAnalysisoftheData TherearemanygaugesitesintheStateofFlorida withvaryinglevelsofflooding; thesesitesarehighlightedintheUnitedStatesGeologicalSurvey(USGS)mapshownby

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114 Figure75 :Gaugesitesalong theSt.JohnsRiver Figure74.AsshownbyFigure74,therearenostationsintheStateofFloridathatarein theHighFlood(black)intheStateofFlo ridaonSeptember12,2006.Thereweresix siteswheregaugereadingsaremuchabovenormal(blue);thatis,siteswereflood watersarehigherthanthe90 th percentile.Mostsitesareatorbelownormal. Figure74 : MapofdailystreamflowconditionsfortheStateofFlorida Thesiteusedforanalysisinthisstudy is one ofthree sites alongFloridaslargestriver,theSt. JohnsRiver .Data glea nedfromAdvanced HydrologicPredictiveService(AHPS)maintainedby NOAA.Thesiteofinterests ,shownbyFigure75, is locatedat29'29"latitudeand81'58"longitude inlandgrant38,T.17S.,R.29E.whichisinLake County.Thehydrologicunit isneartherightbank 142milesupstreamfromthemouthoftheriver. The St.JohnsRiverhasadrainageareaof3,066square miles(8.5751744 10 10 squarefeet).TherecordsextendfromOctoberof1933uptothe

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115 present;infact,realtimedataisava ilableforupto31 daysupdateddailythegaugeused water stagerecorder,acousticvelocitymeter,water qualitymonitoranddata collection platform. ThereadingsusedinthisstudyweretakenfromthesiteontheSt.JohnsRiver shownbyFigure76 .The historicalcrestsmeasured atthissitelistedinTable50 below. DailyreportscanbegleanedofflineatAHPS,whichshowtheriversrealtime conditionsasillustratedfor September12,2006byFigure77 Thechartontherightis there altimefloodstagefortheSt.JohnsRiver,September12,2006.Asshownby Figure74aswellaswellasFigure77,thewaterheightwhichdeterminesfloodstageis notinfloodstage.Minimumwaterheight(stage),the actionstage ,atwhichactionmust betakeninpreparationofpotentialflooding,is3.7feet.Oncethewaterheightisgreater than4.2feet,theriverisin floodstage .A moderatefloodstage is5feetanda major flood stageis5.5feet. Figure76 :Mapofgaugesiteofalong theSt.JohnsRivernearDeland (1) 6.14fton1964/09/23 (2) 6.06fton1953/10/11 (3) 5.84ft on1964/10/01 (3) 5.84fton1960/10/03 (5) 5.77fton1960/09/30 (6) 5.23fton2004/10/01 (6) 5.23fton2004/09/30 (6) 5.23fton2004/09/29 (6) 5.23fton1969/11/01 (10) 5.03fton1964/09/18 Table50 :HistoricalCrestsin theSt.JohnsRiver near Deland

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116 Figure 77 :Realtimes tage(waterheightinfeet)forSt.JohnsRiver (right)andcolor ledge(above) Therefore,weneedtounderstandthethresholdbywhichfloodstagesmeasuredat thissitear eset.Thelowestfloodstage isillustratedbyFigure77 istheActionStage whichissetat3.7feet,thenextFloodStageissetat4.2feet,theModerateFloodStateis setat5feetandtheMajorFloodStageissetat5.5ft.Therecordfloodheight is6.14feet andoccurredmorethanfortyyearsago. Therefore,considersurfacewater,fieldmeasurementsgatherbytheUSGSover thepast23yearswhichincludessuchmeasuresasgaugeheight(thesurfaceofthewater werethegagemakesitsreadings), width(ft),area(ft 2 ),meanflowvelocity(ft/s),inside gageheight,anoutsidegaugeheight,streamflow(ft 3 /s),shiftadjustment(ft).Thesewill betheonlyentitiesconsideredascontributingvariables. Thedatausedtoanalyzethesubjectrespon se(floodstage)areshownbyFigure 78asdailygaugeheight(feet)overtime.Thetimeframeforthisdatausedinthepresent studyisfromJanuary1,1934topresent. FloodCategories(infeet) MajorFloodStage: 5.5 ModerateFloodStage: 5 FloodStage: 4.2 ActionStage: 3.7

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117 Figure 78 : Linegraph (daypast01/01/1934)forstageheightintheSt.JohnsRivernear Deland Giventheflowratesatthissitebedenoted ) ( i i t f x bethestochasticsequ ences offlowratesrecordedattime i t ,thenourdatasetconsistsoftimes n i i t T 1 } { with i i t t 1 wheretimeismeasuredindayspastJanuary1 st ,1938.Thenwecanuse parametricanalysistobest fitth edistributionofthefloodstage.First,wewillfitoneof thethreeextremevaluedistributionstothefloodrates overtimeasshownbyFigure79 andestimatethelengthoftimebetweenfloodpeaks ) ( L E .Note:thereareonly13floo d eventsrecordedbetweenJanuary1,1934andDecember31 ,2004,seeTable51 .From thesetwoestimateswecancomputethereturnperiods ) ( 1 ) ( x F L E T 4.2ft

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118 ConsiderthespooleddatashownbyFigure79;thatis,thelinegraphofdataby dayofye ar.Figure79showsthatthefloodstagesarehighesttowardtheendoftheyear. Italsoshowsthattheriverdoesnotreachamajorflood(greaterthan5.5feet). Figure 79 :Linegraph(dayofyear)forstageheightintheSt.JohnsRivernear Deland fortheyears1938 2004 Event Beginning Date Ending Date Duration PeakDay Height (ft) Time between peaks 1 26 Jun 34 28 Jun 34 2 27 Jun 34 4.21 2 4 Jul 34 5 Jul 34 1 4 Jul 34 4.24 7.5 3 21 Sep 45 18 Oct 45 27 30 Sep 45 4.99 4,105.5 4 1 Oct 47 14 Nov 47 44 18 Oct 47 5.00 748.5 5 3 Oct 48 3 Nov 48 31 13 Oct 48 5.20 360.5 6 15 Sep 53 17 Nov 53 63 12 Oct 53 6.06 1,825.0 7 29 Mar 60 31 Mar 60 2 29 Mar 60 4.23 2,360.5 8 3 Apr 60 3 Apr 60 3 Apr 60 4.20 4.5 9 13 Sep 60 8 Nov 60 56 4 Oct 60 5.83 184.0 10 13 Sep 64 5 Oct 64 22 23 Sep 64 5.01 1,450.0 11 17 Nov 94 7 Dec 94 20 23 Nov 94 4.61 11,018.5 12 17 Sep 01 2 3 Sep 01 6 19 Sep 01 4.31 2 ,491.5 13 11 Sep 01 2 Nov 01 52 30 Sep 01 5.23 11.0 Means 25 4.86 2047.25 Table51 :FloodEventsintheSt.JohnsRivernearDeland FloodStage

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119 5.4 Parametric Statistical AnalysisofFloodStageHe ights Firstconsiderthedistributionofthevariousfloodstagesasillustrated byFigure 80alongwiththebasicdescriptivestatisticsofthedatathatweshallanalyze .The distributionofthisdataisnotsymmetric;themeanisskewedtotheright. Thatis,the meanfloodstageisgreaterthanthemajority(ormedian)oftheprobableheights. Figure80 :Histogramoffloodstageheightinfeetincludingdescriptivestatistics. First,considertheGumbeldistribution,usingthedescribedstatistic almethodsw e canestimatetheparameters fortheGumbelprobabilitydistributionare 86355 0 and 769598 0 Second,considertheWeibull probability distribution,using realdata,weestimate usingMLEs wehave 54039 0 093473 2 and 999629 1 .Both distribution s areacceptedatthesignifican celevelof0.01usingstandardgoodness of fit test:theKolmogorov Smirnov,Anderson DarlingandCramer vonMiser.Thegoodness of fitcan alsobeseeninthecomparisonillustratedbyFigure81,theGumbelprobability distribution(centercurve)isabetterestimateoftheempiricalprobability(uppercurve) thantheWeibullprobabilitydistribution(lowercurve).H oweverwhenconsidering 2 as showninTable5 2 ,theGumbeldistributionshowsabetterfit. Statistic Estimation Count 25385 Mean 1.308 Median 1.09 StdDev 0.984 Variance 0.968 Range 6.6 Min 0.54 Max 6.06 IQR 1.25 25th% 0.59 75th% 1.84

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120 0 0.2 0.4 0.6 0.8 1 1.2 Weibull Empirical Gumbel Figure 81 :Comparisonofempiricalcumulativeprobabilitywiththatofthecumulative WeibulldistributionandthecumulativeGumbeldistribution. FittedDistributio n Chi square Gumbel 26.55 Weibull 92.46 Table52 :Testforfitoffloodstageheight ThebestfitteddistributionistheGumbeldistributiongivenby 769598 0 86355 0 exp exp ) ( x x G 38 Knowingthiscumulativeprobabilitydistribution, wecancreateconfidence intervalsforanyspecificlevelofconfidence.Inexample,atthe95%confidencelevel, thetruemeanwaterheightisbetween 0.14feetand3.69feet.Weare99%confidentthe truemeanwaterheightisbetween 0.42and4.94. Ifweconsiderthe99%confidence interval,theupperboundisactionstageoutlinedinFigure77;theActionStageforthe St.JohnsRivernearDelandis3.7feet.

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121 5.5 StatisticallyModelingFloodStages Assuming ,thestandardd eviationoftheflowrates, isconstantthenwecan estimatethe meanflowrate f ,bydevelopingstatisticalmodelsandidentifyingthe contributingvariables: duration flowrate and timeofyear 5.5.1 Flood S tageasa F unctionof D uration Considerthestatisticalmodeloutlinedbye quation 39 ,where j d isthe duration ofthe th j floodevent.Thedataforthesevariablesisgivenby Table5 1 The analyticalformofthest atisticalmodelisgivenby, j j j d x 1 0 39 wherethecoefficients s i aretheweightsthatdrivetheestimateofthesubjectresponse (floodstage)and s i aretherando merror. Usingrealdataandleast squaresregression,thedevelopedstatisticalmodelis givenbye quation 39 .T hisis giventhat floodcondition sareinfactpresent .The minimumdurationneededtoqualifyasafloodisthreeanda halfhours.Theshortest durationwas0 + days;settingtheflowrate 2 4 x ,weget 146 0 d days ,approximately 3.5hours. j j d x 026288 0 19616 4 40 Thedevelopedleast squaresregression modelexplains 91.1%ofthevariationin thefloodlevelhigh.Moreover,thedurationofthefloodissignificantwith p value <0.0001.

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122 5.5.2 Flood S tageasa F unctionof F low R ate Furtherconsideralltherecorded floodstages overtimewithrespectto the flow rate incubicfeetpersecond.Thedevelopedstatisticalmodelgivenbye quation 41 relatest hefloodstateattime i t denoted i x withrespecttotheassociatedflowrate i f where 0 isthemeanwaterheightiftheflowratewerezero, 1 istheweightthatdrives theestimateofthesubjectresponse(floodstage)dependingontheflowrateofthewater. i i i f x 1 0 41 Usingrealdataandleast squaresregression,wedevelopthestatisticalmodel givenby i i f x 4 10 52238 3 229459 0 42 Thedevelopedstatisticalmodelusedtoestimate flowratesexplain 72 .9%ofthe variationinthewater height (floodstage) Accordingtothis statistical model iftheflow werestopped,the estimat edwaterlevelwouldbelessthanafourthofafoot.Moreover, toreachthefloodstageofhighestmagnitude,theflowrate swouldhavetobeupwardsof 11272.32cubicfeetpersecond;thisisfoundbysetting 2 4 x andsolvingfortheflow rate f .Thisisconsistentwiththemeanflowratesgivenfloodinginprogress;themean flowrate, giventhefloodstageisatleast4.2feetis12069.78cubicfeetpersecond. Whenconsideringflowratesingeneral,withandwithoutflooding,themeanis 3036.675cubicfeetpersecond ;theminimumrecordedwas3510cubicfeetpersecond Usingthe developedstatisticalmodel,wecanestimateamorerealisticminimumwater height(stage)is1.47whenwesettheflowratetotheminimumrecorded. 5.5.3 FloodStageasaFunctionofBothDurationandFlowRate Expandingthedefinitionofdurationof afloodtotheentiredataset,wehave

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123 2 4 1 2 4 0 1 i i i i x d x d whichmeanthatthedurationwillbeheldatzerowhentheriverisnotinfloodstate, otherwise, thedayoftheflooding .Thenwecanextende quation 41 toe quati on 43 whichincludesthedurationintermsofthedaywithin thefloodingperiod, i d ,thatis, i i i i d f x 2 1 0 43 Using realdataand least squaresregression theparame terestimate sgivenby i i i d f x 03173 0 10 43851 3 249643 0 4 44 Theestimatedcoefficientsine quation 44 a reextremelyclosetotherelationshipgiven bye quation 42 Ad dition ally,withtwo contributingvariable ,theleast squaresregressionmodel correlationcoefficient 2 R increasesto 73.1% ;thatis,73.1%ofthevariationinthe subjectresponse(floodstageinfeet)isexplainedbythedevelopedstatisticalmodel .Th e durationwithinthefloodeventisfoundtobesignificantwith p value< 0.001. 5.5.4 Flood S tageasa F unctionof D uration, F lood RateandTime Anothercontributingvariableisthetimeofyear i t (inda ys),asillustratedby Fig ure79 ,weseethatfloodingismorelikelytooccurinthefall.Usingleast squares regression,wefindthatthetimeofyearindaysshouldsignificantlycontributein explainingthevariationinthefloodstage.Inclu dingthisvariablesgivenby, i i i i i t d f x 3 2 1 0 45

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124 wherethecoefficients s i aretheweightsthatdrivetheestimateofthesubjectresponse (floodstage)and s i aretherandomerror. Usingrealdata,thede velopedstatisticalmodelisgivenby i i i i t d f x 00176558 0 433618 0 10 21587 3 00608185 0 4 46 Theleast squaresregressionmodelgivenbyequation 46 explains 76.4% ofthe variationinthesubjectresponse;all variable sare fou ndsignificantwith p value<0.0001; moreover 2 2 adj R R Ifweincludeahigherordertermfortimeofyear,wetakeintoaccountthe relativecurvatureintheflowratesasitrisesinthesummerreachingitspeakinthefall andthenretur ningtothenaturalflood stagewehavethestatisticalmodelgivenby, i i i i i i t t d f x 2 4 3 2 1 0 47 wherethecoefficients s i aretheweightsthatdrivetheestimateofthesubjectresponse (floodstage) and s i aretherandomerror. Usingrealdataandleast squaresregression,weobtainthestatisticalmodelwhich estimatesthefloodstageasafunctionofflowrate,durationandtimeofyear(order2), shownbyequation 48 2 6 6 4 10 29437 7 10 9098 8 468619 0 10 17626 3 166468 0 i i i i i t t d f x 48 Usingstandardhypothesistesting,thesecondordertermfortimeofyear is a significant lycontributingvariable with p value<0.001.The least squaresregression

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125 modelgivenby e quation 48 impliesthatthehigherfloodstageoccursattheendofthe yearandthatthelowestfloodstagesoccurwhen 61 0 t ,sometimeduringthefirstday oftheyearbutdoesnotlocatethehigherfloodstageacc urately. Hence,considerthe thirddegreetermfortime whichallowsforbothaminim um andamaximumtobedetectedisgivenby, i i i i i i i t t t d f x 3 5 2 4 3 2 1 0 49 wherethecoefficients s i aretheweightstha tdrivetheestimateofthesubjectresponse (floodstage)and s i aretherandomerror. Using realdataand least squaresregr essionweobtainthestatisticalmodelgiven by, 3 7 2 6 6 9 4 10 40132 1 10 45673 78 10 0122229 0 10 88571 1 10 05014 3 547803 0 i i i i i i t t t d f x 50 U s ing analyticalmethods andsetting 0 t x ,wefindthattheminimumflood stageonday94 43 94 t ,earlyMay, andamaximumfloodstageonday307 89 307 t ,earlyNovember.Furthermore,thedeveloped statisti cal modelexplains 78.6%ofthevariationinthefloodstage. 5.6 ParametricAnalysisandStatisticalModelingtoCreateProfiles Giventhatafloodev entoccurs,thenwehaveidentifiedthebestprobability distributionthatcharacterizesthesubjectres ponse(floodstage);namely, theGumbel distribution giveninequation 38 .Theconditionalprobability underthe assum ption the averagefloodstagedependsonthedurati onofthefloodeventgivenbye quation 40 w egetthatthereturnperiodinyearsis

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126 ) 2 4 | ( 1 ) ( x x G L E T where yr L E 6 5 ) ( Here,theconditionalprobabilitydistributionfunction, ) 2 4 | ( ) ( x x G x F ,theinterms of ) ( x G ,thecumulativeprobabilitydist ributiongiveninequation 36 isgivenby otherwise 0 2 4 ) 2 4 ( 1 ) 2 4 ( ) ( ) ( x G G x G x F andhence otherwise. 0 2 4 ) ( 1 ) 2 4 ( 1 4 5 ) ( x x G G x T Extendingthisformulausingthedevelopedstatisticalmodelgiveninequation 40 ,wehavethatthereturn periodscanbeprofiledisgivenby, ) ( 1 ) 026288 0 19616 4 ( 1 4 5 ) | ( x G d G d x T Knowingthisconditionalprobabilityweareabletoestimatethereturnperiodforagiven floodstageprofiledbythefloodsduration. Thenwehavethefollowingreturnpe riodsillustrate dbyFigure82 Considerthe averagefloodduration,25day,accordingtoFigure82,giventhefloodenduredfor25 days,wewouldestimatethemeanfloodheightisapproximately4.85feetandthereforea majorflood(5.5feet)wouldbeexpecttooccurev ery12.5years.Whereas,iftheflood onlyenduredfor10days,themeanfloodheightisapproximately4.46feetandtherefore amajorflood(5.5feet)wouldbeexpectedtooccurevery20.8years.

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127 Figure 82 :Thereturnperiodsinyearsofvario usfloodstagesdependingontheduration ofthefloodevent. 5.7 Usefulness oftheStatisticalModel Thistypeof statistical analysisisusefulinunderstandingthecomplexityofthe waterwaysinflood.Tobeableto estimate andcontrolfloodingin theStateofFloridais necessaryformitigatingandplanningforsuchevents. Moreover, thedevelopedstatistical modelusedtoprofilecanbeupdatedasnewinformationisgathered, theprobableflood stagethatwilloccur canbeestimatedenabling offic ialstocreateaccurate andtimely publicwarnings. Thestatisticalmodeldevelopedinthis study, alongwiththeestablished probabilitydistributionwhichcharacterizesthebehaviorofthesubjectresponse(flood stage),weareinapositiontonotonly profilevariousfloodevents,butestimatethe mean,standarderrorandconfidenceintervalforapredeterminelevelofconfidence. 5.8 Conclusion Inthepresentstudy,wedeterminedthattheGumbelprobabilitydistributionbest characterizesthebeha viorofthesubjectresponse(floodstage)intheSt.JohnsRivernear DelandintheStateofFlorida.TheSt.JohnsRiveristhelargestriverintheStateof R e t u r n P e r i o d ( i n y e a r s ) Duration (indays)

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128 Florida.Atthissite,afloodisconsideredmajorifthefloodstageisabove5.5feet,a mod eratefloodis5feet,andminimumfloodstageis4.2feet.Theactionstagehasbeen thereforesetat3.7feet. Usinghistoricalfloodsrecords,themeanfloodstageis4.86feet.Onaverage thesefloodsenduredfor25daysandoccurredwithameanint ervalbetweenfloodsof5.6 years.I fitweretofloodforaperiodof75days,thenthefloodstagewouldreachashigh as7.2feetandisprobabletobeseen withinahundredyears.T helongestduration recordedwas63daysandthemeanwascloserto25 days.Assumingthedurationofthe floodwas25days,wewouldexpecttohavefloodstagesashighas5.8to5.9every hundredyears. Duration isacontributingvariabletofloodstage,thatis,thelongertheflooding persist,thehighertherelative waterheight.The flowrate ofthewaterisalsoa significantlycontributingvariable;thefasterthewaterflows,thehigherthevolumeof waterandthereforethehigheraverageheightofthewater.Inaddition,thetimeofyear includeduptoorder3, estimatesthattheseasonallowisnearthe94dayoftheyear (earlyMay)andaseasonalhighnearthe308dayoftheyear(earlyNovember).

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129 CHAPTER6: STATISTICAL ANALYSISANDMODELIN GOFREDTIDE BLOOMS 6.1 Introduction RedTideisbecomingah ottopicintheStateofFlorida;killingfishand negativelyaffectingtourismintheState.The rearemanydifferent microorganisms responsibleforredtidereleasetoxinsinto both thewaterandair.Theseneurotoxinscan affecttherespiratoryandcar diacsystem,reducingbloodflowandslowingdownthe heart.Largebloomskillingthousandsoffishhavebeendocumentedasearlyasthe 1800s. Inthepre sentstudy,webeginwithavailabletoperform descriptivestatistics which aresimplestatisticswhi ch describethenumberoforganismwithinasampleand establishedthatthedataisbestanalyzedthroughalogarithmicfilterwhichreducesthe scaleandhomogenizesthevariance.Utilizinghistoricaldatagatheredsporadicallyover thepastseveraldeca des,wedeterminedthattheWeibullprobabilitydistributionbest characterizethemagnitudeofabloom;thatis,theprobabilisticbehaviorofthe logarithmictransformationofthecountoftheorganism KareniaBrevis ,theprimary organismfoundinRedTi de ( Duke,GivenandTinoco,2004) Second ly weuseinferentialstatistictodetermineregionaldifferences.Next,we used recursiontoestimatelogistically according to thelogisticgrowthmodel therate atwhenabloomgrows.Thatis,estimatingt hesubjectresponse(magnitudeofabloom) dependingonthepercentofthetotalcapacitytakenupbythecurrentbloomandthe remainingpercentofthetotalcapacityavailable. Furthermore, we proceed toestablisharelationshipbetweennutrientrunoff from theStateofFloridaandthemagnitudeofaRed TideBloom;thatis,wedevelopeda statisticalmodelofthesubjectresponse(magnitudeofbloom)asafunctionof soil nutrientsthatwashintotheoceans: Sulfat e 4 SO NitrateIon 3 NO and Ammonium

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130 Ion 4 NH ;t hesemineralsarecommoninfe rtilizersusedinagriculture. Thepresent study can beextendedtoinclude moreprecisestatistical models onthesubjectresponse once consistentconcurrent dataisgathered;thatis,weneedtoestablishadatabankwherenot justtheorganismcountanddatearerecordedbutsalinity,watertemperature,andother contributingvariablesonthesametemporalscale. Thepresent analysisisimportanttotheSta teofFloridaonbothanenvironmental andeconomicalpointofview.Accuratelyestimatingthesizeofabloomenableusto accuratepostwarningsinarea s affectbyandoutbreak,andabetterunderstandingofthe contributingentitiesthatthesubjectre sponse;namely,thesizeofabloomwillleadto understandingofthecauseandeffectofRedTide. Inourpresentstudywewilladdressthefollowingquestions: 1. IdentifythemaincontributortoRedTide? 2. Whatistheprobabilitydistributionofthemagnit udeofanoutbreak? 3. Whataretheregionaldifferencesintermsofmagnitudeofthebloom? 4. Whatistherelativegrowthrateofabloomintermsofmagnitude? 5. WhatarethecontributingentitiesthatfuelRedTide? 6.2 AnalysisoftheVariousOrganismsMeasu redinRedTide Thereare57variousgeneralizedorganismsfoundinover56,000samplestaken overaforty eightyeartimespanning 1954thru2002;onlythirty oneorganismsare recordedinatleasttensamplesandtwenty oneorganisms arerecordedina tleastone hundredsamples Table 53, in cludes ageneralgroupofotherplankton. O nlyten organismsfoundina tleastonethousandsamplesthatweare co nsidering : Karenia (morespecifically KareniaBrevis ), Diatom OtherPlankton Gymnodinium Dinofla gellates Micro flagellates Gyrodinium Ciliates Gonyaulax and Peridinium Image: KareniaBrevis FloridaFishandWildlife ConservationCommission

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131 Inthepresentstudy,we willconcentrateon Karenia Brevis (formerly Gymnodiniumbreve ).Thisorganism when presentinsufficientnumbers(thousandsor millionsofcellspermi lliliter)turnthewaterredinvokingthenameRedTide.Littleis knowaboutRedTideanditscauseandeffects.Onequestiontobeaddressedinthis study iswhetherRedTidebloomsaroundtheStateofFloridaarecorrelated,possibly withatimedelay ,totherun offfromtheStateofFlorida. First,wemustanalyzethe mainorganismassociatedwithRedTide:namely Karenia Brevis Code TotalCount Samples MeanCountperSample Organism KARE 8772810000 56272 155900.0924 Karenia(1) DIAT 152712300 2557 59723.23035 Diatom(2) OTHE 1043334 2424 430.4183168 OtherPlankton(3) GYMN 108634500 2088 52028.01724 Gymnodinium(4) DINO 42049600 1883 22331.17366 Dinoflagellates(5) MICR 3287360000 1865 1762659.5 17 Micro flagellates(6) GYRO 8020570 1791 4478.26354 Gyrodinium(7) CILI 22979410 1449 15858.80607 Ciliates(8) GONY 47679000 1445 32995.84775 Gonyaulax(9) PERI 145862172 1081 134932.629 Peridinium(10) CERA 7063375 857 8241.97783 Ceratium PROR 5692300 755 7539.470199 Prorocentrum NAUP 125476.9 439 285.8243736 Nauplii OSCI 12907406 348 37090.24713 Oscillatoria TRIC 688423720 347 1983930.029 Trichodesmium FLAG 267044 296 902.1756757 Flagellates BLUE 75459800 268 281566.4179 BlueGreenAlgae COPE 42321 201 210.5522388 Copepods POLY 268663 190 1414.015789 Polykrikos COCH 7443913 134 55551.58955 Cochlodinium RHIZ 95.992 114 0.842035088 Rhizosolenia Table 53 :Datacompiledbynumberoftimesrecord edinsamplings.Includestotalcount overtimeandmeancountpersampleaswellastheorganismandhowittheseorganisms arecoded.

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132 6. 3 DescriptiveStatisticsforKarenia Brevis Let ) ( t c betheconcentrationof Karenia Brevis att ime t ,where t ismeasuredin dayssinceJanuary1 st ,1954.Wehaveasampleoftheseconcentrations;namely ) ( i i t c x forthevarioussamplestakenatvarioustimes i t .Ifwec onsidertherawcount ofthisdata,thenthereisane xtremeskewinthedataasshowninFigure83.Figure83is importantbecauseitillustratesthatno KareniaBreve isacommonconditionfoundin mostofthesamples.Therearenumeroussampleswithzer ocount;thatis, 0 x and therefore weconsider 0 x and thenaturallogarithmofthecount(concentration 5 ), ) ( ln t c ,willbeconsideredtoadjustthescaleandbringtheunderlyingdistributioninto focus. Hence,thisstudyanalyzestheconditionalprobabilitydistribution ofthe magnitudeofabloom,giventhereisa countof Karenia Brevis isgreaterthanzero;that is, thisorganismis infactpresent,seeFigure84 Figure 83 :Histogramof countof KareniaBrevis sampledovertime 5 Countpersampledliter OrganismCount x Statistic Estimate Count 56272 Mean 155900 Median 0 StdDev 2594849 Variance 6733242184185 Range 358000000 Min 0 Max 358000000 IQR 333 25th% 0 75th% 333

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133 Figure 84 :Histogramofthenaturallogarithmofthecountof KareniaBrevis sampledovertime,giventhecountwasatleastone Figures84 givemoreinsightintothetruenatureof KareniaBrevis ,butthe reare manytimeswhenasingleorganismisdetected .ThiscanbeseeinFigure84,thereisa largenumberofsampleswithexactlyoneorganize.H ence,considerwhenthereisa bloom meaningthecountismore thanoneasshowninFigure85 InFigure8 4,the samplemeanmagnitudeofRedTidebloomiscalculatedas9.097,whereasinFigure85, giventhatthereisabloom(morethanoneorganismrecorded)thesamplemean magnitudeofRedTidebloomiscalculatedas10.118.Furthermore,thevarianceis si gnificantlyreducedallowingforabetterestimateofthemeanmagnitudeofabloom. 0 ; ln x x y Statistic Estimate Count 16731 Mean 9.097 Median 9.798 StdDev 4.006 Variance 16.046 Range 19.696 Min 0 Max 19.696 IQR 5. 124 25th% 6.908 75th% 12.032

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134 Figure 85 :Histogramofthenaturallogarithmofthecountof KareniaBrevis sampled overtime,giventhecountwasatleasttwo. Considerthethreesubgroups: (a) noorganismfound(N), (b) exactlyone(a single)organismpresent(P)and (c) finally whentheorganismisinbloom(B).Rarelyis thereasingleorganismpresent(P);barely3%ofthesamplesre cordedacountofone, seeTable54 .Normally,thati sinthemajority(approximately70%)ofthesamples,there arenotevenasingle KareniaBrevis present.Onlyanestimated27%ofthesamples recorded isorcontains abloom;thatis,morethanoneorganism. Table54belowgives estimatesofthepercent ofthedatainthedefinedsubgroups. Group Count % B 15042 26.731 N 39541 70.268 P 1689 3.001 Table 54 :Percentbycategory When we furtherconsiderthecount,thereisadisparitybetweentheminimum statistic, 693 0 ln min i i x andthebul koftheremainingvaluesasillustratedbythega p inthehistogramshownbyFigure84 .ConsideroutliersdefinedbyChebyshevs inequality: 2 2 t t x P whichfor 667 8 t yields 1 0 667 8 118 10 x P 1 ; ln x x y Statistic Estimate Count 15042 Mean 10.118 Median 10.309 StdDev 2.741 Variance 7.512 Range 19.003 Min 0.693 Max 19.696 IQR 4.215 25th% 8.006 75th% 12.221 outliers gap

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135 whichimpliesthat twen ty oneoutliers arepresent ,ofwhichonlytwoareextremehighs leavingnineteenlower level outliers.Allnineteenoftheseextremeoutliersare 4 3 2 x whereastheupperoutliersare 358000000 197656000 x .Empirically, theseoutliers constitute approximately0.14%ofthedata countinbloom. Hence,we willredefinethecategories:first, NOorganismfound(N) 0 x organismsPresent(P),butfew, 5 0 x andfinallywhentheorganismisinfullbloom (B) 5 x .Thisredefiningdoesnotsignificantlyaffectthepercen tagesineachcategory asgiveninTable55 ,butdoesremovethe gapinthehistogramFigure85andillustrated byFigure86 Table55givesmorepreciseestimatesofth epercentofthedatainthe redefinedsubgroups.Theredefinitionofabloomtobe 5 x insteadof 1 x ,yieldsthe descriptivestatisticsgiventhechartalongwiththehistogramillustratedbyFigure86. Whilethe sestatistics,themean,themedian,thestandarddeviationandthevarianceare allextremelyclose,themaindifferenceisillustratedinthatthereisnogapinthedatain Figure86. Group Count % B 15022 26.695 N 39541 70.268 P 1709 3.037 Table 5 5 :Percentbycategory(redefined) Figure 86 :Histogramofthenaturallogarithmofthecountof KareniaBrevis sampled overtime,giventhecountgreaterthanfive 5 ; ln x x y Statistic Estimate Count 15022 Mean 10.131 Median 10.309 StdDev 2.722 Variance 7.409 Range 17.904 Min 1.792 Max 19.696 IQR 4.22 25th% 8.006 75th% 12.226

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136 6.4 ParametricInferentialAnalysis Consider ) ln( ln x forthesampl eswhere KareniaBrevis isinfullbloom and x istheconcentrationof KareniaBrevis inagivensample .Thistransform ationofthedata helpsindicates someformofanextremevaluedistribution.Thecurvatureoftheno rmal probabi lityplotgiveninFigure87 indicates thatthe Weibull probabilitydistribution wouldbeagoodfit .Infact, that theWeibull probability distribution asshownby Figure 88 istheonlydistributionatthe0.01levelsthatcannotberejected. Figure 87 :Probabilityplotofthedoublenaturallogarithmofthecountof Karenia Brevis sampledovertime,giventhecountwasgreaterthenfive. Curve: Weibull(Theta=0Shape=4.2Scale=11) P e r c e n t 0 1 2 3 4 5 6 7 8 Ln(X);X>5 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 16.5 18.0 19.5 Figure 88 :Histogramwithbest fitdistributionforthenaturallogarithmofthecountof KareniaBrevis samp ledovertime,giventhecountwasgreaterthenfive. y = l n ( l n ( x ) ) ; x > 1 NormalScore P e r c e n t a g e MagnitudeofBloom 5 ; ln x x y

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137 Thus,theexistingdatadesignatedby x canbebestcharacterizedprobabilistically bytheWeibullprobabilitydistributionfunction. 6.5 Logarithmictransformationanditsprop erties The data thatcharacterizes astheorganismcountin KareniaBrevis havesucha largescalethatalogarithmictransformationmustfirstbetakento consider the probabilitydistribution tobeuseful .Thistransformeddatawillbereferredtoast he magnitudeofthedata;thatis, N i i x 1 ln which neednotbebasedonthenaturallogarithm thiscanbeadjustedasneededtoageneralbase N i i b x 1 log .Inthis study ,wewilluse thenaturallogarithm. Assuming 1 i x ,defined N i i y 1 asthemagnitudeoftheoriginaldataset,where i i x y ln .Further assumethistransformeddataisbestfitbythetwo parameter Weibull;thatis ) , 0 ( ~ W y ,thanconsiderthecumul ativep robabilitydensity functiongivenby otherwise y y y F Y 0 0 exp 1 ) ( 51 where istheshapeparameterand isthescaleparameter, Thenintermsoftheoriginaldata,thetransf ormedcumulativeprobability distribution functionisgiven bye quation 52 ,whichyieldthetransforme dprobability distributionfunctiongivenbye quation 53 Thatis,wehave ln exp 1 ) (ln ln ln ln ) ( x x F x Y P x X P x X P x F Y X and

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138 x x F X ln exp 1 ) ( 52 Theprobabilitydensityfunctionof x isgivenby x x x x f X ln exp ln ) ( 1 53 w hich satisfiestheconditionfortheprobabilitydensityfunction, 1 ) ( 0 x f X ,for 1 x and 1 1 ) ( dx x f X ProofofProperty1: Since ) ( y f Y isaprobabilitydensityfunctionsuch that 1 ) ( 0 y f Y for 0 y .Usingthedefinedtransformation, wehave 1 ) (ln 0 x f Y for 0 ln x ;thatis 1 x .Hence,dividingby x 1 1 max ) (ln 0 1 x x x f x Y ;thatis, 1 ) ( 0 x f X ,for 1 x ProofofProperty2 : Thistrans formeddistribution ) ( x f X isaprobabilitydensity function.Sincethetransformationiscontinuous x x f x f Y X ) (ln ) ( wecanwrite 1 ) ( ) (ln ) ( 0 1 1 dy y f dx x x f dx x f Y Y X

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139 6.5.1 MaximumLikelihoodFunction n i n i i n i i n n x x x x L 1 1 1 1 ln exp ln ) ( or TheMLEs of and areobtainedbysolvingthefollowingsystemoftwoequation. 0 ln 1 ln 1 ) ( ln 1 1 1 2 n i i i n i i x x x n x L and 0 ln ln ln ln ) ( ln 1 1 n i i i n i i x x x n x L Unfortunately,solvingthissystemofequationstooptimizethislikelihood functi onisnoteasilydoneanalytically.However,withtheadventofrecenttechnologies wecanaccurately obtain estimate softhe solution oftheaboveequation usingiterative proced ure(QiaoandTsokos,1998). The th j momentoftheran domvariable x isgivenby ) ( ) (ln ) ( ) ( ) ( 1 0 ln 0 ln ln jy Y Y jy Y x j X x j x j X j X e E dy y f e dx x x f e dx x f e e E x E n i i n i i n i i x x x n n x L 1 1 1 ln ln 1 ln ln ln ) ( ln

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140 Thus,wecanusetheaboveexpressiontoobtainestimatesofthebasicstatisticsofthe phenomenonofinterest. Let yt Y e E t MGF ) ( bethemomentgeneratingfunctionfort hestandardtwo parameterWeibullintermsof y .Underthegiventransformation,themoment generatingfunctionforthetransformedprobabilitydensityfunctioncanbeexpressedinto termsoftheoriginaldistribution. Thatis,wehav e y Y te Y Y te Y tx X tx tx X X e MGF e E dy y f e dx x x f e dx x f e e E t MGF y y 0 1 1 ) ( ) (ln ) ( ) ( 6.5.2 Two ParameterWeibull ProbabilityDistributionFunction Using numericalschemes ,wecanestimatethetwo parameterWeibull. Applying theMLE yieldsascaleparameter estimate of 11 andshapeparamete r estimate of 2 4 asshownbelow, otherwise. 0 5 11 exp 1 ) ( 2 4 x x x F 54 Fromth emomentgeneratingfunction, k n n n x E 1 ) ( wecanestimatethe sample mean, sample standarddeviation,skewnessandkurt osis foreachofthe probabilitydistributionfunctionandareshowninTable56 T hethree parameterWeibull probabilitydistribution whichgivesa best fit probability distributionusingMLEyieldsathreshold 693114 1 scaleparameter

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141 395226 9 ,andshapeparameter 484966 3 Thisyieldsthecumulative probabilit ydistributionfunctiongivenby otherwise. 0 5 395226 9 693114 1 exp 1 ) ( 484966 3 x x x F 55 Statistic Estimate2Parameter Estimate3Parameter NumberofData 15022 15022 Mean 10.13 10.14 StandardDeviation 2.72 2.68 Variance 7.40 7.18 Skewness 0.0359 0.0359 Kurtosis 0.6884 0.6884 Table 56 :Statisticsbasedonthetwoandthree parameterWeibull Thedifferencebetween the two andthree parameterWeib ullprobability distributionsis insignificant.Whenconsideringthesimpleregressbetweentheempirical probabilitydistributionandthetwoandthree parameterWeibull,99.4%ofthevariation intheempiricalprobabilityisexplainedbytheestimated two parameterWeibull probabilitydistribution whereas99.5%isexplainedbythethree parameterWeibull probabilitydistribution .Bothofthese probability distributionsareacceptedatthesa me levelofsignificanceresultsareshowninTable57below Test Statistics p value Kolmogorov Smirnov D 0.0602309 Pr>D <0.001 Cramer vonMiser W sq 12.5414661 Pr>W sq <0.001 Anderson Darling A sq 71.3164327 Pr>A sq <0.001 Table 57 :Goodness of FitTestforWeibull

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142 S incethesetwo probability distributionsa reextremelycloseintheirestimate,we willinvokethelawofparsimonyandcontinuewiththetwo p a rameterWeibull probabilitydistributionfunction Thus, wecan proceedto estimate, given Karenia Brevis ispresent,theprobability of exceedingagive ncountinagiv ensampleasshown inFigure89 Inanygivensampleinwhich KareniaBrevis ispresent,asfewas22organisms couldbepresent.However,ineverytensamplesinwhich KareniaBrevis ispresent,this numberjumpsto660,000incount or 4 13 ln x .Ineveryhundredsamplesinwhich KareniaBrevis ispresent,thisnumberjumpsagainto7,452,052,whichimplies that 8 15 ln x ;thisisanincreaseofover11fold. Recall ,that therewereover56,000samplestaken overa48 yearperiod,atthis rateupto1200samplesmaybetakeninayear.Therefore,considerthereturnperiods assuming1200samplesperyear,theninanygivenyearasamplecouldcontainupwards of41,243,332countof KareniaBrevis : thatis, 5 17 ln x .Uptosixtimeswhatisfound ineveryhundredsamples;thereforeconsiderthereturnperiodsfor 17 ln x ;that is, 784 824 39 x ,seeFigure90 andTab le58 Figure 89 :Returnperiodsofthenatural logarithmofthecountof KareniaBrevis Time(Years 1 ) 5 ; ln x x y R e t u r n P e r i o d ( S a m p l i n g )

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143 x ln x ReturnPeriod (Years 6 ) Min Max Min Max 0to10 17 18.75 24,154,952 139,002,155 10to20 18.755 19.07 139,698,906 191,423,727 20to30 19.075 19.25 192,383,243 229,175,810 30to40 19.255 19.375 230,324,559 259,690,215 40to50 19.38 19.465 260,991,918 284,146,355 50to60 19.47 19.545 285,570,645 307,812,072 60to70 19.55 19.61 309,354,986 328,484,430 70to80 19.615 19.66 330,130,965 345,326,187 80to9 0 19.665 19.71 347,057,142 363,031,439 90to100 19.715 19.755 364,851,142 379,741,000 Table 58 : Estimat ionsforvariousreturnperiods Themaximum x ln asshowninTable58above,a high ofmagnitude19.755or 379,741,000 count ina givensample hasanestimatedreturnperiodofbetween90and 100years. 6.6 Mixed Probability Distributions Fu rtherstudyofthehistogram,shownbyFigure91 indicatesa bimodal behavior andtherefore,amixtureof the twonormal probability distrib utionsmightyieldaneven betterfit. Themixtureoftwo normal probability distributions isgivenby 2 1 1 1 2 2 1 1 | 1 | , , | x f x f x g 56 Theexpectedvalueandvarianceofthemixedprobabilitydistributionfuncti onsare given,respectively,by ) ( 1 ) ( ) ( 2 1 x E x E x E f f g and ) ( 1 ) ( ) ( 2 1 x V x V x V f f g 6 Assuming1200samplesperyear

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144 Bothofthesepropertiesfollowfromthefollowingrelationship withinthemoments,that is, ) ( 1 ) ( ) | ( 1 ) | ( ) , , | ( ) ( 2 1 1 1 1 1 2 2 1 1 p f p f R p R p R p p g x E x E dx x f x dx x f x dx x g x x E Hence,wecancompute anestimateoftheexpe ctedvalue 13 10 ) ( ) ( x E x E g and variance 72 2 ) ( ) ( x V x V g .Firstwecanestimateon e ofthepeaksbyconsidering themodeofthemagnitudes 9 6 M ,whichinanormaldistributiongivensan indicationtothepotentialfirstm eanandsincethesecondpeakismore certain andcanbe estimatedas 5 12 2 thiswillbetheinitialmean.Ifwefurtherassumethatthe sample standarddeviationsarethesame ;thatis, 2 1 ,thenwecanuseleastsqua res regressiontoestimatethemixingfactor Thatis,considerthemixedmodelgivenby e quation 56 where i p isthecumulativeemp iricalprobabilitydistributiongivenby ) | ( 1 ) | ( 2 1 1 1 i i i x F x F p 57 Ifwelet 1 ,theleast squaresregressionyields 542792 0 and 73584 0 .Usingthesetwoestimatesofthemixingfactor wehave 542792 0 1 and 264155 0 2 ,simplytaketheaverageofthesetwoestimates resultsin 4034735 0 Thisestimateinabetter fit oftheinitialdatatoa mixed probability distribution with 34 187 2 ,whichis verycloseto theWeibull pr obability distribution, with 36 186 2 Therefore, we furtherconsiderthefirstestimatedmixturefactor 4034735 0 inconjunctionwith therelationshipgivenby 2 1 1 58

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145 toe itherre estimateofth elowerpeak 1 orupperpeak 2 .Ifwefixtheupperpeakand re estimatedthelowerpeak,wehave 6 6 1 .Thisyieldsaworsefitwith 09 225 2 However,ifw efixthelowerboundandestimatetheupperpeak,thisyields 3 12 2 and 4764 167 2 Byconsideringvariousvaluesof andcontinuouslyre estimating 2 ,wecan reducethechi squa redstatis ticasshowninTable59 .Furthermo re,oncewehave establishedthebestsamplemeansthatgive usthe additional adjust mentofthesample standarddeviationsusingtherelationshipgivenby 2 2 2 1 2 1 59 inordertofurtherreducethechi squaredsta tistics,alsoshowninTable59 Notethattheseestimatesforthemixingfactor andthefirststandarddeviation 1 areonlyaccuratetotheseconddecima l,butthisreducesthechi squaredstatistic to 293 15 2 ,whichindicatesabetterfitusingthemixedstatisticalmodel Thisis asignificant improvementoverboththetwoandthree parameterWeibull probabilitydistributionfunctions wi ththe chi squaredstatistics, 2782 30 2 and 7121 31 2 ,respectively.However,asillustratedinFigure 90 ,allthreeofthese distributionsarehighlycorrelatedtotheempiricalprobabilitydistribution.

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146 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 MixedModel EmpiricalCumulativeProbability Weibull(4.2,11) Weibull(1.693114,9.395226,3.484966) Figure 90 :Comparis onofthebest fitdistributionsandtheempiricalprobability distribution Trial 1 2 1 2 2 1 6.9 12.5 2.72 2.72 0.403 4735 186.359 2 6.9 12.3 2.72 2.72 0.4 167.476 3 6.9 11.5 2.72 2.72 0.3 76.650 4 6.9 10.9 2.72 2.72 0.2 34.596 5 6.9 10.7 2.72 2.72 0.15 26.500 30 6.9 10.5 1.27 2.90 0.11 15.328 31 6.9 10.5 1.26 2.90 0.11 15.318 32 6.9 10.5 1 .25 2.90 0.11 15.309 33 6.9 10.5 1.24 2.90 0.11 15.302 34 6.9 10.5 1.23 2.91 0.11 15.298 35 6.9 10.5 1.22 2.91 0.11 15.296 36 6.9 10.5 1.21 2.91 0.11 15.296 37 6.9 10.5 1.20 2.91 0.11 15.298 38 6.9 10.5 1.19 2.91 0.11 15.303 39 6.9 10.5 1.18 2.91 0. 11 15.310 40 6.9 10.5 1.17 2.91 0.11 15.320 Table 59 :Estimationsofparametersandassociatedchi squaredstatistic

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147 6. 7 RegionalAnalysis Therearehundredsofvariouslatitudeandlongitudelocationsrecorded,these pointscanbeusedtogenerateac ontourplotofthe x ln overthevariouslocationsgiven inFigu re92 ,buttherearetwodistinctregionswherethecountshavebeeninexcessofa halfmillioncountof KareniaBrevis Consider themagnitudeofthebloomasdefined by t he greatestintegerfunction ofthemagnitudeofabloom;thatis,theleastintegerbelow thevalue, ) ln( int x m ,shown i nthecontourplotinFigure93. Figure 91 :Thecontourplotofthenaturallogarithmofthecountof KareniaB revis with respecttothesamplinglocation(longitude,latitude) 6.7.1 Regional D ifferencesontheEastandWestCoasts First,considerthemean of x ln fortheregionabovethe26 latitudenorth; dividedintothenaturalregionsdef inedbytheEastandWestcoastsofFlorida partitionedat 81.75 longitudewest.Testingthehypothesisthemean x ln intheEastis thesameasthemean x ln intheWestfailstorejectthatthesemeansarediffer entwitha p valueof0.0839.Themean x ln fortheEastcoastis10.416withastandarddeviation Longitude Bloom Magnitude L a t i t u d e

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148 of2.686andthemean x ln fortheWestcoastis10.155withastandarddeviationof 2.719.Thereisadisparityint henumberofsamplestakenontheEastandWestcoastof Floridawithsamplingcountsof328and14088,respectively. However,theyarelarge enoughtoacceptthefactthatthetowregionsgivethesameresults. 6.7.2 Regional D ifferencesNorthandSouth ofTampaBayontheWestCoast Second ly ,thereisanobviousincreaseinthenumberof KareniaBrevis recorded nearandwithintheTampaBay.Thenasecond,smallregionnearTallahasseewith countsofmagnitude11;however,thelargeareanearandwithin TampaBayshows countsofmagnitudesbetween2and19.Themajority,nearly80%ofthesamplesis takenfromSouthofTampa(latitude 1 1 8 5 27 ,longitude 7 4 1 3 82 ),above26 la titudenorth;shownbyFigure92 asSW.How doesthesamplescollectedsouthof Tampa statistically comparewiththosesamplesgatherednorthofTampaBay(NW)? Figure 92 :Thescatterplotofsamplinglocationsbydefinedregions Thereisasignificantdifferencebetweenthemagnitudesof thebloomsNorthof TampaandthemagnitudesofthebloomsSouthofTampa.The sample meanmagnitude Regions Longitude L a t i t u d e

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149 intheNorthis9.869withastandarddeviationof2.532withsamplesize2109.The meanmagnitudeintheSouthis10.206withastandarddeviationof2 .748withsample size11979.With3049degreesoffreedomanda p valuelessthan0.001,thereis significantevidencetoshowthatthemagnitudeofbloomsintheSouthis somewhat greaterthanthemagnitudeofbloomsintheNorth.Thisismostlikelydu etorunofffrom TampaBay. 6.7.3 Regional D ifferences B etweenTampaBayand A ll O ther R egions C onsidered TampaBayisonthewestcentralcoastofbetween27.5 and28 latitude,north. ThemajorsourceofpollutiontoTampaBayisnitrogen. Nitrogeni s the essentialplant nutrientinexcessfuelthegrowthofalgaebloomsinthebay.Morethanhalfthe nitrogen enteringTampaBaycomesfromstormwaterrunofffromurbanandr esidential areas.Thisrunoffof stormwaterrunsoffthelandwithrainfall, whichcarriesfertilizers andpesticideresidues,andeventrashintothefourmajorriverswhichfeedtheTampa Bay:theAlafia,Hillsborough,LittleManateeandManateeRivers.Thereisasignificant differencebetweensamplestakennearTampaBay(see the variousmagnitudesinFigure 91 )andthosesamplestakenfurtheraway,denotedas(X)?Thosepointsconsidered nearTampaarelocations(latitude y andlongitude x )suchthatthequasi distance 7 is atmos tone;thatis, 1 ) 9697 27 ( 5297 82 2 2 y x ,seeFigure93 ThereisasignificantincreaseinmagnitudebetweentheregionnearTampaBay andthosemeasuredfurtheraway.ThemeanmagnitudeinnearTampais10.283witha standarddeviationof2.871withsa mplesize7931.Themeanmagnitudefurtherawayis 9.962withastandarddeviationof2.533withsamplesize7090.With15016degreesof freedomanda p valuelessthan0.001,thereissignificantevidencetoshowthatthe magnitudeofbloomsnearTampa Bayisgreaterthanthemagnitudeofbloomselsewhere. Again,thisismostlikelyduetostormwaterrunoff. 7 Thisisnotanexactdistancesincethelatitude longitudelocationnotf irstconvertedintoCartesian coordinates.

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150 Figure 93 :ThescatterplotofsamplinglocationsbynearTampaBayandotherregions 6.7.4 RegionalAnalysis B etweenthe B asins A longthe W est C oast ThereareninebasinregionsoutlinedintheStateofFlorida seeFigure94 .Of theseninebasins,seventouchthewestcoast;fiveinfactareattachedtotheTampaBay. Further, weconsiderthe partitiontheregionbysevenbands b y partitionin gthelatitudeas shownbyFigure94,creating16regionswheresamplesweretaken:NE4,NE5,NE6, NE7,NW4,NW5,NW6,NW7,SE2,SE3,SE4,SS1,SS2,SW2,SW3andSW4.The basicdescriptivestatisticsfortheseregionsaregiveninTable60. Thelowestmean magnitudeis6.485inthesoutheastregion(SE2)andhighestmeanmagnitudeis11.733 inthenortheastregion(NE4).Thedifferencesbetweenregionscanalsobeseeninthe boxplotillustratedinFigure96.Figure96shows,considerth eregionscounter clockwisearoundtheStateofFlorida,weseethatthevariousregionswherethereare largersamplesizes,thevarianceissmallerthantheregionsthesouthwestwherethe samplessizesaresmaller. Magnitude Longitude L a t i t u d e

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151 Figure 94 :MapofregionbyDistrictBasins Figure95 :Thescatterplotofsamplinglocationsbygroupeddistricts DistrictBasins 1. WithlacoocheeRiver 2. CoastalRivers 3. GreenSwamp 4. HillsboroughRiver 5. Pinellas AncloteRiver 6. Northwest 7. Hillsborough 8. AlafiaRiver 9. PeaceRiver 10. Manasota Regions L a t i t u d e Longitude

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152 Region Count Mean StdDev NE4 38 11.733 2.626 NE5 87 10.864 2.5 01 NE6 43 10.262 2.930 NE7 11 10.344 2.982 NW4 336 9.968 2.679 NW5 158 8.521 2.427 NW6 1291 9.968 2.477 NW7 324 10.028 2.461 SE2 8 6.485 1.764 SE3 124 9.99 2.592 SE4 17 10.563 1.535 SS1 222 8.724 2.692 SS2 384 9.789 2.624 SW2 502 9.952 2.520 S W3 5058 10.095 2.594 SW4 6419 10.313 2.876 Table 60 :Descriptivestatisticsforvariousregions Figure96 :Boxplotofthemagnitudesbyregion Figure96alsoshowsthat thereisasignificantamountofoverlapforthe majorityoftheseregions.H owever, theregion SE2ontheeastcoasthasan extremelowmagnitudeofbloom.Usinghypothesistestingwiththenull hypothesis j i H : 0 versusthealternativethattheyaredifferent j i a H : for j i an dboth i and j areadjacent regionsasdefinedinFigure97 Regions M a g n i t u d e i n F u l l B l o o m

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153 Thereareseveralregionswherethereisasignificantdifferencebetween thesamplesmeanmagnitudes,seeTable61.Thesearetheregionswith p v alue< 0.1.Hypothesistestingalsoindicatesthatthereisnosignificantdifferencein betweenthetworegionsintheuppernorthwest(NW7andNW6)aswellas betweenthetworegionsintheuppernortheast(NE6andNE7). Comparison p value % NW7 NW6 69.8 NW6 NW5 < 0 .1 NW5 NW4 < 0 .1 NW4 SW4 2.24 SW4 SW3 < 0 .1 SW3 SW2 12.4 SW2 SS2 35 SS2 SS1 < 0 .1 SS2 SE2 0.128 SS1 SE2 < 0 .1 SE2 SE3 < 0 .1 SE3 SE4 20.2 SE4 NE4 < 0 .1 NE4 NE5 1.57 NE5 NE6 25.2 NE6 NE7 93.6 Table61 :Comparisonofregionsa nd p valueforadjacentregions Figure 97 :Thescatterplotofsamplinglocationsbymajorregions Major Region L a t i t u d e Longitude

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154 Whatissurprisingisthatthehighestmagnitudeofbloomisinregion8,thisisthe northeastsection ofthestate,seerankinginTable62 .Wit hameanmagnitudeof10.872, thisisapproximately23,000moreorganismcountthanthesecondhighestregion,the TampaBay.Thismighthavetodowiththefactthattherearesignificantlyfewersamples takenatinthefirstregion;moreover,isthefact thatthesesamplesmayhavebeentaken whenRedTidewasinfullbloom andveryfewsamplesweretaken However,thereissignificantinformationtoshowthatoneofthehighestranking regionswithrespecttomagnitudeofbloomisthe regionjustsouth oftheTampaBay. Thenfourth,theregionjustsouthofthefirstregion,whichmightbeanindicatorofthe bloomsmove mentandgrowth .Furthermore,theregionwiththelowestmeanmagnitude onlyhad16samples.Thisisnotenoughinformationtomeasur ethemagnitudeofa bloom,butthislackofsamplingmightbeduetothefactthatnoeventofRedTidehas everbeenrecordedinthisregionandthereforeexc ludedfromthesamplingspace. Rank MajorRegion Count Mean StdDev 1 8 179 10.872 2.692 2 3 6755 10.296 2.867 3 4 5937 10.067 2.589 4 7 141 10.059 2.492 5 1 1615 9.98 2.473 6 5 222 8.724 2.692 7 2 158 8.521 2.427 8 6 15 6.709 1.781 Table 62 :Descriptivestatisticsformajorregions 6.8 Recursion Analysis Theremainderofthis study wil lconcentrateonregions3and4outlinedinFi gure 99 .Comparingsa mplesmonthlyshowninTable63 ,therearetwodistinctperiodsof lullsinthemagnitudesofbloom;namely,April,MayandJune(spring)andNovember, DecemberandJanuary(winter).How ever,inthetransitionalmonthsofFebruaryand March,andtheJulythruOctober(summer),thereisasignificantincreaseinthe magnitudesoftheblooms.During thetwodistinctlullsTable64 ,themeanmagnitudes

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155 fallat(approximately)orbelow10,w hereasinduringtheoffmonthsthemagnitudesare onaverageabove10. ComparisonofMonths p value 1 2 <0.1 2 3 0.19 3 4 <0.1 4 5 14.3 5 6 78.1 6 7 <0.1 7 8 0.311 8 9 2.41 9 10 0.84 10 11 <0.1 11 12 52.1 12 1 28.6 Table 63 :Comparisonby monthand p valuefor consecutivemonths Month Count Mean StdDev 1 991 9.801 2.522 2 969 10.322 2.736 3 872 10.711 2.625 4 659 9.782 2.551 5 587 10.002 2.737 6 440 9.957 2.51 7 431 10.8 2.889 8 904 10.296 2.926 9 1782 10.574 3.178 10 1847 10.3 16 2.697 11 1773 9.854 2.569 12 1434 9.911 2.463 Table 64 :Descriptivestatisticsbymonth Intheseregions,therehasbeenanincreaseinthemeanmagnitudesovertime. However,thisincreaseisbetterexplainedbytheincreasinginthenumberofs amples takenmorethantheyearinwhichthesamplesaretaken.Inaddition,inrecentyears, KareniaBrevis beingthemajororganismassociatedwithRedTidehasbecomeafocus ofinterest,inandofitselfgeneratesabias.Manyoftheoriginaldataon lyrecorded KareniaBrevis andthemajorityofthesamplesweretakenbecauseofthepresenceof RedTide.

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156 Considerthemeanmagnitudebydayofyearandbyregion.Thereisaslight delay,butnotgreatlyresolvedonadailybases.Consideryear1957f ortworeasons: first,thereare116sampleshourlysamplestakenoveratwenty ninedayperiodatthe samelocationnearabridge(GulfBlvd)thatseparatestheGulfofMexicoandBoca CiegaBay showninthemapgiveninFigure98 .Second,duringthissam eperiodoftime, November1957,datawasrecordedinbothregionsasshowninFigure 99. Figure 98 :Mapoflocationwheredatameasuredhourly. Figure 99 :Thescatterplotofsamplinglocationsbymajor regionsforNovember1957 DayinNovember M a g n i t u d e i n F u l l B l o o m

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157 Whenthese tworegionsconsideredovertime,comparethemeandailymagnitude ineachregion,denotedby j x 3 and j x 4 ,respectively where j isthedayinNovember 1957.Testingthehypothesisthatthemean differenceiszero;thatis,thenullhypothesis is 0 : 0 d H versusthealternativehypothesisthattheyaredifferent, 0 : d a H for j j j x x d 4 3 .Witha p valueof0.82,wefailtorejectthatthereisasignific ant differencebetweentheregionsonadailybasis.Evenifyouconsiderthedifferences betweentheminimum,medianandmaximum,thereisinsignificantevidencetoshowthat thereisadifferencebetweenthesestatisticsonadailybasis.Furthermore, forthisone month,thereisnotsignificantdifferencebetweenthemeanmonthlymagnitudesineach region;thisis,intestingthenullhypothesis 4 3 0 : H versusthealternative hypothesis 4 3 : a H wehavea p valueof0.846 .Thisiscontrarytoourfindingsthat overtime,thereisasignificantdifferenceinthesemeans.Thisisactuallytrueforthese newlydefinedregions.Ifwereturnedtotheoriginal,smalle rregionsdefinedinFigure 95 ,thesedifferencesmaybemo reprevalent. Let ) ( t A and ) ( t B bethemeandailymagnitudeinregions3and4 respectively overtime t (dayofyear.)Considerthedelayequationdefinedby ) ( ) ( t bA a t B where isthetimedelayindaysand istherandomerrorpresentintheseries.Using least squaresregressionand 10 9 ,..., 9 10 ,themostsignificanttimedelaybetween thetworegionsisfourdays.Thisfour d aytimedelayexplainsonly52%ofthevariation inthemagnitudeofbetweenthetworegions.However,thisisthereverseofwhatwas anticipated;thatis,themostsignif icanttimedelayisstatisticallymodeledbythe followingequation, ) 4 ( 3697 0 1717 6 ) ( t A t B for 365 ,..., 3 2 1 t 60

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158 Considerthemeandailymagnitudesrecordedovertheentiretimeinthetwo regions.Therankingofsignificanttimedelaysshowatfirstathree hourtimedelayfrom thenorthern regiontothesouthernregion,butthesecondmostsignificanttimedelayas aboveisafour daytimedelayfromthesouthernregiontothenorthernregion.These variationsmaybeduetotheebbandflowoftheGulfSea.Asthelunarcycleaffects tides ,weneedtocheckdelaysupto28days. Extendthedelaytimesto 30 29 ,..., 29 30 ;asbeforethetwomost significanttimedelaysbasedonthecorrelationcoefficientareathree hourtimedelay fromthesouthern regiontothenorthernregione qu ation 61 andthenafour hourdelay timefromthenorthern regiontothesouthernregionse quation 62 ) 3 ( 107 2 869 38 ) ( t A t B 61 and ) 4 ( 805 1 749 37 ) ( t A t B 62 Considerthemultipledelayeffectscausedbythesloshingoftheopenwater.We cangeneralizethedelayequationdefinedby ) ( ) ( t bA a t B to j j j t A b a t B ) ( ) ( .Usingthisgeneralize statistical modelgivenin e quation 63 wherethe s b i aretheweightsthatdrivetheestimateofthesubjectresponse(the magnitudeofabloomintheregion B ), ) 4 ( ) 3 ( ) ( 2 1 t A b t A b a t B 63 where a and b aretheweightsthatdrivetheestimateofthesubjectresponseand is therandomerror. Using least squareregressionwedeterminewhichofthetwodelaysmentioned previouslyismoresignificant.Witha p va lueof0.0076,thefour hourtimedelayfrom thenorthernregiontothesouthernregionismoresignificantthanthethree hourdelayin

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159 reverse,whichshowsa p valueof0.0547.However,thisfour hourdelayfromthe northernregiononlyexplains3.4%of thevariationinthemagnitudeofthebloominthe southernregion. Wewouldexpectthatthetimeofyearwouldhelpexplainthevariations. Considerthemodel ct t bA a t B ) 4 ( ) ( where a b ,and c aretheweightsthatdrivetheestimateofthesubjectresponseand istherandomerror. U singthesameleast squaresregression,boththefour hourdelay( p value= 0.0004)andthedayofyear( p value<0.0001)a resignificantcontributorsandexplain 10.7%ofthevariationbetweenthemagnitudesofbloomsinthetworegions.The majority(7.5%)oftheexplanationcomesfromthetimeofyear,howeverthereisan obviouslull inthesummermonthsFigure100 Ther eisprobablyatrigonometricrepresentation of thisoscillation;however,the simplificationistoconsideraquadratictermintime.Thismodelexplains28.1%ofthe variationinthemagnitudeofthebloominthesouthernregion.Whileasimilarmodel for thenorthernregionsshowsthatbothtermsaresignificant( p value<0.0001),itonly explains7.2%ofthevariationinthesouthernregion. Figure 100 :Linegraphofmagnitudeofbloomsinthesouthernregion DayofYear M a g n i t u d e i n F u l l B l o o m

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160 Considerthemodelwiththetimede lay,timeandtimesquared.Thismodel explains29.3%ofthevariationbetweenthetworegions;however,whiletimeandtime squaredaresignificantcontributors( p value<0.0001),thedelayeffectfoundtobe insignificant( p value0.2391). Therefore, themagnitudeofthebloomismoredependentonenvironmental factorsintheGulfandAtlanticandtheconditionsthatcomewiththeseasons.This includespreviousconditionsandtime.Considerthemodel 2 ) 1 ( ) ( dt ct t bB a t B where a b c and d aretheweightsthatdrivetheestimateofthesubjectresponseand istherandomerror. Thismodelexplains32.6%ofthevariationinthesout hernregion.Asimilarmodelfor thenorthernregionagainfindsthesevariouscontributingentitiessignificant( p value< 0.0001)butonlyexplainsasmallproportionofthevariation. Whenconsideringthegeneralizedmodelgivenbythefollowing: 2 ) ( ) ( ) ( et dt t A c t B b a t B k Ak k j Bj j with 4 1 A 1 B neitherofthetimedelaysfromthenorthernissignificant.Onlythe dayofyear(time),timesquaredandthemagnitudeofthebloompreviouslyintheregion. Basedontheanalys isabove,thebest fit timeandtimedelaymodelaregivenbelowby e quation 64 ande quation 65 foreachregionrespectively.Theresidualsforthe northernregionaremorerandomthantheresidualsforthesouthe rnregion.This indicatesthatwhilethemodelforthenorthernregionexplainslessofthevariation,the remainingseemsdependentonMotherNature.Whereasthemodelforthesouthern regionexplainsmoreofthevariation,buttherearestillotherfac torsthatcontributeto thesephenomena. Thestatisticalmodelsaregivenby,

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161 2 5 10 52516 3 0130446 0 ) 1 ( 234395 0 93214 6 ) ( t t t A t A 64 and 2 4 10 4892 6 199236 0 ) 1 ( 185013 0 022 22 ) ( t t t B t B 65 6.9 LogisticGrowth Model Inalltheyearsofdatacollecti ng,themostsampledyearis1957inwhich4138 samplesweretaken;however,52.3%(2165)oftheseweretaken inonlyfourmonths Figure101 .Moreover,approximately11%(233)ofthesesamplingswheretakenatthe samelocation,denotedinblueinFigure 102 ,aregatheredonaconsistenttemporalscale. Hence,wewillrestrictthefollowinganalysistothistimeandplace. Figure 101 : Linegraphofmagnitudeofbloombymonth Month DayofYear M a g n i t u d e i n F u l l B l o o m

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162 Figure 102 : Linegraphofmagnitudeofbloombylocation( latitude,longitude) Fewadditionalsamplesweretakentowardthebeginningoftheyear,butthe countwaszero;however,oncetheoutbreakwasinfullbloom,manysamplesweretaken. Therefore,unfortunately,wedonothavethesamplesfromthissiteu ntilOctober .As illustratedinFigure105andFigure106 ,oncethisbloomispresentthemagnitude increasequitequickly.However,itisinterestingtonotethatforthedatacollectedona consistentbasisatasinglesiteFigure 107 ,thereisanosci llationinthemeandaily magnitudeswhichmightbeexplainedbyalogisticgrowthpattern. Figure103 :Linegraphofmagnitudeofbloom i i c x ln atasinglelocationattime i t DayofYear M a g n i t u d e i n F u l l B l o o m DayofYear M a g n i t u d e i n F u l l B l o o m

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163 Thelogisticalmodeldefinedbyagr owthconstant r ,theproportionofspace taken(assumingthemaximumcapacityis i i x C max and C relativelylarge)by 1 1 n n n P C rP P Alternatively,ifwedefinethepresentproportiona s C x p n n where n x isthe daily sample meanmagnitudeforthe th n day,thenthislogisticalmodelbecomes 1 1 1 n n n p rp p and givenasatimeseriesyields 1 1 ) 1 ( ) ( 0 t p t rp p t p where i i p t p ) ( 66 C 2 R GrowthRate r Error 20 15.43 3.3277 1.3025 30 67.28 2.6848 0.2507 40 68.66 1.7204 0.0679 50 68.72 1. 4077 0.0227 60 68.68 1.2543 0.0059 70 68.64 1.1635 0.0016 80 68.61 1.1034 0.0052 90 68.58 1.0607 0.0070 100 68.56 1.0288 0.0080 150 68.49 0.9437 0.0082 200 68.45 0.9061 0.0071 1000 68.37 0.8270 0.0017 1E+14 68.35 0.8093 0.0000 Table65 : Correla tionsgivenvariousmaximumcapacities Thetheoreticalmodelwepropose,whereestimateisgivenbelowbye quation 66 with 50 C explains68.72%ofthevariationintheperc entisshownFigure104 Thisisequivalen ttosayingthatthemaximumcapacityinaliterisabloomofmagnitude 50;or 21 10 18 5 countisplausible. Thedevelopedstatisticalmodelisgivenby

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164 0227 0 1 1 ) 1 ( 4077 1 ) ( t p t p t p 67 Figure 104 :Linegra phofpercentagesbasedondailymeanofthedata(blue)andthe estimatedpercentagebasedonthepreviousdata Finally,theproposedmodelgivenbyequation 67 givesgoodestimates ofthegrowthasafunctionofthetimeandshould beusedtoobtainusefulinformationon thesubjectmatter. 6.10 Statistical Modeling Othercontributingentitiescanbeincludedsuchasammoniumions,nitrateions andsulfateions. 6.10.1 AmmoniumIon : 4 NH Ammoniumhydroxide is anameusedtodescribetheprocessofmixingan ammoniaandwater. Ammonia isthecompoundNH 3 ,butwhendissolvedinwater,the ammonia takesaprotonfromthesurroundingwater,producinga hydroxide anion and an ammonium cation NH 4 ,e quation 68 .Molecularmass:18.04Daltons. Ammonium ions areatoxicwasteproductproducedbythemetabolicsysteminanimalsandare excretednaturallyintheenvironmentnatural. OH NH O H NH 4 2 3 68 Information DayofYear P e r c e n t a g e o f T o t a l C a p a c i t y

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165 6.10.2 Nitr ateIon : 3 NO Nitrate isasaltof nitricacid ; Nitrateion isapolyatomicanionwithmolecular massof62.01Daltons.Thetwocompoundsoutlinedabovecomprisethechemical compound ammoniumnitrate 3 4 NO NH .Molecular mass:80.04Dalton.Thislarger compoundiscommoninfertilizerandhasbeenusedasanoxidizingagentinexplosives. 6.10.3 SulfateIon : 2 4 SO Sulfate isasaltof sulfuricacid ; Sulfateion isapolyatomicanionwithmolecular mass of96.06Daltons. Let 3 2 1 ); ( i t C i beconcentrationswhere t istime(weekof year),letthemagnitudeofbloomatagivenpointbemodeledby e quation 69 andlet j bethetime delaybetweenthegivenpointandthesites 7 ,..., 2 1 j listedinTable66 wherethemeasurementsweremade.Thereareseveralassumptions.Firstthattherateof eachoftheconcentratesisequal;thatis,thismodeldoesnottakeintoaccou ntthe interactionbetweenconstitutesandotherorganismsandnutrientsonitspath.Second, theflowratethroughthevarioussoilsastheymaketheirwaytoawatersourceandinto theopenwaterisalsoconstant.Furthermore, theproposedmodelis a logisticregression modelsince ) ( ln ) ( t x t M ,givenby 3 1 7 1 ) ( ) ( i j j i ij t C b a t M 69 where a andthe s b ij aretheweightsthatdrivetheestimateofthesubjectresponse (magni tudeofbloom), ) ( j i t C istheconcentrationofthe th i nutrientwithatime delayof j dependingonwhichof i sitesand istherandomerror.Thesite informationisgivenTable66. Thetimedelays( j )canbeestimatedtwoways.First,usingleast squares regression,determinesignificanttimedelays.Second,usingthedistancebetweenthe givenpointandthesite j d ,andwiththeassumedconstantflowratewhichcanbe

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166 estimatedusingwindspeedanddirectionfromvariousbuoysintheGulfandonland,or flowratesintheriverswhichfeedintotheGulf,denoted k Site Name Date Active Elevation Latitude Longitude FL03 BradfordForest 10/10/1978 44 29.9747 82.1981 FL05 ChassahowitzkaNationalWildlifeRefuge 8/27/1996 3 28.7494 82.5542 FL11 EvergladesNationalPark ResearchCenter 6/17/1980 2 25.39 80.36 FL14 Quincy 3/13/1984 60 30.5481 84.6008 FL23 Sumatra 1/26/1999 14 30.1111 84.9919 FL41 VernaWellField 8/25/1983 25 27.38 82.2839 FL99 KennedySpaceCenter 8/2/1983 2 28.5428 80.6444 Table66 :Locationofsiteswhereconcentrationsaremeasured EstimatedTimeDelayUsin gLeast SquaresRegression Sinceallsiteswereoperationalintheyear2000and KareniaBrevis was significantlysampledthatyear,wewillstartwiththisspecificyear.Duringthisyear,if weconsiderthethreeammoniasmeasuredatthevarioussites, theyarecorrelatedas expectedbythechemistry.Chemicalreactionsarenotaccountedforinthispreliminary model,howeverifwecorrelatethemeasureoftheconcentrationofeachconstituencyat eachsitewithuptoamonthtimedelay;recallourtem poralscaleisweekly,hence 4 3 2 1 0 j ,wefindthatforthesiteFL05,both 4 NH and 3 NO haveafourweektime delaywith 518 0 r and 516 r ,respectively(rank1and3 insignificance);however, 2 4 SO isalsorelativelyhighlycorrelatedwithafourweektimedelaywith 362 0 r Note,thisisthemostcentralofthesevenstationsinCitrusCounty. ForthesiteFL41inSarasota,allt hree, 4 NH 3 NO and 2 4 SO haveathreeweek timedelaywith 516 0 r 457 r and 414 0 r respectively(rank2,5and10 respectively.)ForsiteFL23, only 4 NH isfoundrankedinthetopten,withathreeweek delaytime,howeverboth 3 NO and 2 4 SO arecorrelatedwithrelativehighcorrelation coefficients, 224 0 r and 359 r ,respectively. ThesiteFL99,actuallylocatedontheeastcoast,whichisexposeddirectlytothe openwaterandrankedinthetoptenascorrelatedtothemagnitudeofthebloomshowsa

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167 twotothreeweektimedelay.Thismaysimplybedue tothefactthesenutrientsare immediatelymixedwiththeoceanswaterandsloshedbackandfortharoundtheStateof Floridawhereasthenutrientsmeasuredattheotherlocationmustmovethroughsoils beforereachingtheopenwaters.Tworankingcont ributorswiththreeandtwoweektime delay respectively are 2 4 SO (with 425 0 r ,ranked6 th ) and 3 NO (with 419 0 r ranked7 th ). ForthesitesFL08andFL11,bothrankedaoneweektime delayaseightonthis listwith 417 0 r .ThisforsiteFL11isnotsurprisingsincethissiteisdirectlyonthe eastcoastaswell,butfurthersouthandclosertowherethemajorityofthesampleswere takenin2000.However,forsite FL14thisismoresurprisingsincethislocationisinset northinthepanhandleofFlorida,northofFL23whichshowsathreeweektimedelay. ForsiteFL03,nosignificantdelaytimeswerefound.Thisistobeexpectedsince FL03islocatedinthehe artofFloridaandthereismorelandsurroundingthissitethan anyothersite.Itmaytakemorethanamonthforthesenutrientstomaketheirwaytothe openwatersoftheGulfandAtlantic. Unfortunatelythereisinsufficientdatatoregressthismode lforallsitesandall significanttimedelays,muchlessinteractionbetweenthesenutrients.Furthermore,there isalargetemporalscalewhichdoesnotallowformoreprecisedelaytimestobe estimated.Usingthismodel,atmostsixtermscanbecons idered,butasthe 2 R risesto 0.756,the 2 adj R dropsto0.025.Thismeansnofurtherinterpretationscanbemade regardingthismodelwithoutmoreconcurrentdata. Forthesecondmethod,weneedtosetafixedpoint .Firstconsiderthecenterof allthesamplestakenin2000;thispointliesinthemiddleoftheGulfat27.69 latitude (north)and 83.07 longitude(west).Sincethereisabuoy nearVeniceBeach,locatedat 27.07 latitude(north)and 82.45 longi tude(west)whichhasadditionalinformation suchaswinddirection,windspeed,gust,pressure(hPa),atmospherictemperature,water temperatureanddewpoint,fortheyear2000,wewillmakethisourspecifiedpoint. Thisdataisavailablehourly,howeve rtoalignourtemporalscale,wehaveto compilethisdataweekly.Tofurthercompoundtheproblem,thereareonlyseven monthsofdatarecordedatthisbuoyfortheyear2000.Thislimitsthecountto27 data

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168 points withsomeweeksmissingforthemagn itudeofthebloompresent.Thereis significantbuoydataformorerecentyears;however,thedataregarding KareniaBrevis isnotavailable. Variable Description ) ( t T a Theatmospherictemperatureatthespecifiedpoint ) ( t P Theatmosphericpressure(hPa)atthespecifiedpoint ) ( t T d Thedewpointatthespecifiedpoint ) ( t g Thegustexperiencedatthespecifiedpoint ) ( t Thedirectionofthewindatthesp ecifiedpoint ) ( t w Thespeedofthewinatthespecifiedpoint ) ( t T w Thewatertemperatureatthespecifiedpoint j d The(quasi)distancebetweenthespecifiedpointandthesite wherenutrients aregathered. ) ( t M ThemeanmagnitudeofthebloomintheGulfandAtlantic ) ( t C i Themeasureofconcentrationforthethreenutrientoutlined: Table67 :VariablesofInterest Itisimportanttonotethat themean magnitude ) ( t M canberefinedtoaspecific regionandnotjusttheentiresamplingspace.Hence,extendthisdefinitionto ) ( t M k k d where k isintheoctanewherethesamplingpointisfoundrelati vetothespecified point,and k d isthedistancebetweenthe th k samplingpointandthespecifiedpoint. However,withthelimitedamountofdata,thisextensionwillnotbeexploredatthistime. Considerthera nkinglistedinTable 68 ofthenewcontributingvariables.Where onemightexpectthemeanmagnitudeofabloomtodependonthewatertemperature, thisrankssecondtotheatmospherictemperature.Furthermore,whenconsideredalone, theatmospherictemp eratureexplains16.9%ofthevariationinthemagnitudeofthe bloomandisrelativelysignificantwitha p value0.0577;however,whenconsideredwith thewatertemperature,thereisanextremelyhighcorrelationbetweentheatmospheric temperatureandt hewatertemperature( % 90 2 R and % 6 89 2 adj R )and whenbothare consideredtogether inthestatisticalmodel ,bothreadasinsignificant atthe0.05 significancelevel Therearenotenoughdatatoconsiderinteraction.

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169 Rank V ariable CorrelationCoefficient 1 ) ( t T a 41% 2 ) ( t T w 40% 3 w 29% 4 ) ( t T d 27% 5 ) ( t 19% 6 ) ( t P 19% 7 ) ( t g 7% Table 68 :Rankofadditionalcontributingvariables Whencomparingthecontributionsoftheatmospherictemperatureandthemost significantlycontributingnutrient ) 4 ( 2 t C ,theatmospherictemperatureisstillrelatively significantwith a p valueof0.0566,whereasthecompound 3 NO isfoundtobe insignificantwith p value0.2160. 6.11 UsefulnessoftheStatisticalModel ThedevelopedstatisticalmodelcanbeusedtoestimatethemagnitudeofaRed Tidebloom.This isimportantinmonitoringthemagnitudeofabloomasafunctionof time.Theestimateofthenumberoforganismsasafunctionoftimecanbeusedfor publicsafetyandadvisories.Also,theproposedmodelandanalysiscanbeeasily updatedoncemoreda tabecomeavailable. Furthermore,havingabetterunderstandingofregionaldifferencesenablesusto ranktheregionsanddeterminewhereresearcheffortsshouldbeconcentrated.In addition,understandingthetimedelaysbetweentheseregionsestablish preliminary functionswhichcanbebuiltupon. 6.12 Conclusion ThemaincontributortoRedTideis KareniaBrevis .Evenwiththelimiteddata availablewecandeterminethatthe probability distributionofprobablebloommagnitude isbest characterizatio nofthesubjectresponse(themagnitudeofthebloom)is the Weibull probability distribution.Thereisnosignificantdifferencebetweenthetwoand

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170 three parameterWeibull probabilitydistributionfunction .Usingthisinformationwecan estimatethatb loomscanreachahighof139millionorganismspersampleeveryten years,butup2.7timesthateveryhundredyears. Thereareregionaldifferencesmainlynorthernandsoutherndifferences. Thismaybeduetotheproximityoftheriversandstreamsto theopenwater,ortheebb andfloweffecttheopenoceanshaveontheGulfandshorelinessurroundingFlorida. Thereisacorrelationbetweenthenutrientsreleasedintothesoilandsurfacewaterswith somedelayeffects,butwithoutmoredataonamor erefinedtimescale,theseexact correlationsanddelayeffectscannotbeaccuratelymodeled.Foramoredetailed analysis, adatabankof periodicdata(preferablyhourly)ofthe original responsevariable ( organismcount )andthevariouscontributing entities suchassalinity atseveralfixed locations needstobeestablished Furthermore,accordingtorecursiveanalysis,therelativelogisticgrowthrateis estimatedas1.4.Thisisbasedontheassumptionthatthemaximummagnitudewithina sample( capacity)is50. Finally,themostsignificantcontributingvariabletothemagnitudeofbloom appearstobetheatmospherictemperature.Theatmospherictemperatureexplainsonly 16.9%ofthevariationinthemagnitudeofthebloom.Thislackofexplan ationmight easilyberesolvedtemporallywithafinertimescaleandspatiallywithconcurrentdata. Additionalvariablesthatneedtobeincorporatedintotheoutlinedmodelarerainfall, stormconditionsintheGulfandAtlantic(thepresenceofatropi calstormorhurricane), andriverineflow.

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171 CHAPTER7: BIVARIATE PROBABILITY DISTRIBUTIONOFTEP HRA FALLOUT 7.1 Introduction Inthepresentchapterwestudythedispersionofashfallfromavolcaniceventin twoparts.First,weconsidertheempi ricalprobabilityofagivenVolcanicExplosivity Index(VEI);thatis,theassociatedproportionofthevolcaniceruptionsatCerroNegro whichareagivenVEI:1,2or3.(Connor,Hill,Winfrey,Franklin,andLaFemina,2001) Secondly,wewillperformp arametricinferentialanalysisonthemassoftephra measuredat80sitesaroundwhereashfallandwaspresentedbyConnorandHill(1995) [15].Iftherewerenoexternalforcesotherthangravityandallparticleswereperfectin shape(round),wewould expectthedispersiontobebivariateGaussian(normal) probabilitydistributiontocharacterizethekeyvariable,butwiththerotationoftheearth andtheresultingwindshear,thedistributionisskewed(Genton,Editor,2004). Therefore,fourvariati onsofthestandardbivariatenormalarebeingconsideredinthe presentstudy.Thefitoftheseprobabilitydistributionswerecomparedusing 2 and 2 R todeterminethebest fitprobabilitydistributionandperce ntofempiricaldistribution explainedbythestatisticalmodelwhenbestcharacterizesthebehaviorofthesubject phenomenon. Establishingtheprobabilitydistributionofthesubjectvariable(massincubic meters)enablesustoestimatetheamountofm assthatislikelytolandinagiven location.Thisisextremelyimportantinurbandevelopmentaswellasforstrategic planningandriskanalysis.

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172 PhotobyWilliamMelson, Cerr oNegro1968 SmithsonianInstitution Inourpresentstudywewilladdressthefollowingquestions: 1. Identifythevolcanoaddressedandwhy. 2. Wh atistheprobableVEIinavolcanicevent? 3. Whatistheprobab ilitydistribution oftephra ,combinedandbygrainsize ? 4. Whatisthebest fitbivariate probability distribution? ThevolcanoofinterestinthisstudyisCerroNegro,Nicaragua.Locatedat 12.5N and86.7W,thisvolcanohasanelevationof2214feet(675meters)andasummitof2388 feet(728meters).Sinceitsbirthin1850,there havebeenapproximately24eruptions;thelast eruptionwasin1999.At155years,thisisthe youngestofCen tralAmericas volcanoesinthe MaribiosVolcanicrange. Therearemanyuncertaindatafromthe datesoferuptionstothemagnitudeofthe eruptions.SearchingCerroNegro,Nicaragua, therearemanysites,whichofferinformationon volcanoes.TheGlob alVolcanismProgram, maintainedbytheSmithsonianInstitution,has postedinformationonthedurationoferuptions, thevolcanoexplosivityindex(VEI),column height,thetephrafallout,thelavavolumeandthe sourcearea,seeTable69.Additionalinf ormation gleanedfromothersourcessuchasEstimationof VolcanicHazardsfromTephraFallout,(Connor,Hill,Winfrey,FranklinandLaFemina, 2001). Table69givestheeruptionsdataforCerroNegroincludingtheyearthevolcanic eventoccurred,thea pproximatedurationoftheevent,thecumulativevolume(cubic

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173 meters),theapproximatefallvolumeforthegivenevent(cubicmeters),thetephra (cubicmeters),lavavolume(cubicmeters),columnheight(meters)andareaaffect. Year Duration VEI Cum.(T) Volume 3 m Fall Volume 3 m Tephra Volume 3 m Lava Volume 3 m Column Height m Area 1850 10to44 1to2 6.0E+06 4.3E+05 6.5E+05 5.4E+0 6 Formation 1867 16 2 1.0E+07 7.4E+06 8.6E+06 3.0E+03 NE/SW 1899 7to8 1to2 1.1E+07 1.7E+06 1914 6to6 2 1.2E+07 2.8E+06 2.8E+06 1919 10 0to2 1.2E+07 1923 49 2to3 3.9E+07 1.7E+07 3.6E+07 1.0E+07 2.0E+03 Summit NRidge 1929 19 0 to2 3.9E+07 1.0E+05 1947 13to24 3to3 5.1E+07 2.3E+07 3.1E+07 3.8E+06 6.0E+03 Summit NFlank 1948 1 0to2 5.1E+07 1949 1 0to2 5.1E+07 1950 26 2to3 6.8E+07 2.8E+06 3.8E+07 1.0E+05 1.5E+04 1954 1 0to2 6.8E+07 1957 20 2 7. 4E+07 2.8E+06 2.8E+06 4.5E+06 2.0E+03 Summit EastFlank 1960 89 1to3 9.5E+07 1.1E+06 3.4E+07 5.2E+06 1.0E+03 Summit SouthFlank 1961 1 0to2 9.5E+07 NEFlank 1962 2 1 9.6E+07 1963 1 0to1 9.6E+07 1964 0 2 9.6E+07 1968 48 2to3 1.2E+08 9.7E+06 2.7E+07 6.9E+06 2.0E+03 Summit SouthFlank 1969 10 0to1 1.2E+08 1971 10.6to11 3 1.4E+08 3.0E+07 5.8E+07 5.0E+03 Summit EastFlank 1992 3.6to5 3 1.5E+08 2.3E+07 2.6E+07 5.0E+03 1995 79to191 1to2 1.5E+08 5.8E+06 3.7E+0 6 2.3E+03 1995 13to15 2 1.6E+08 2.8E+06 1999 2to3 1to2 1.6E+08 8.4E+05 1.0E+06 6.0E+05 1.0E+03 SouthFlank Table 69 :EruptiondataforCerroNegro 7.2 AnalysisofVolcanicExplosivityIndexofCerroNegro,Nigeria ConsidertheVEI forthevo lcaniceruptionsatCerroNegro,Nigeria.T hereare severalcapriciousdatasources,seeFigure 105andFigure106 Figure115showsVEIof

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174 0,1,or2(Smithsonian)whereastheFigure106showsVEIof0,1,2,2.5,3(Connorand Hill).One source has aVEI ofzerowhenthereislittleappreciabletephrafallout,no columnheightandnoorlittlelavaflow, while othersourceshaveVEI of1 ,forthesame eruption.Suchast heeruptionin1995,onesourcestates therewasaneruptionwhich lasted79days,bu tshowsaVEIof0,andforthatsameyearaneruptionwhichlasted only13daysexpelledasignificantamountofvolumewithaVEIof1.Whereas, accordingtotheSmithsonianandhistoricaldatatheeruptionthatoccurredin1995show aVEIof2. While thetwosources,(Smithsonian)and(Connor,Hill,Winfrey,Franklinand LaFemina,2001),aredifferent;however,theyarehighlycorrelatedwitha nestimateof the correlationcoefficient % 8 71 2 R .ComparetheVEIfromthetwosources, Ta ble 70 .Thesecondsourcehasseveralzeroswhereasthefirstsourceonlyhasrecordofone orhigher.Thatis,thefirstsourcestatesalleruptionshaveindexatleastone.Ifwe considertheproportionsassociatedwiththevariouslevelsofVEI,wes eethatthesecond sourceindicatesthat35%ofalleruptionsareinsignificantwithaVEIof0,andaVEIof 1or2isequallylikelyat26%,butthataVEIof3islikelytooccur13%ofthetime. Morerealistically,thefirstsourceindicatesthataVE Ioftwoismostlikelyat61%ofthe eruptions.WhereasaVEIofthreeisthesecondmostlikelymagnitudeoferuptionat 30%andaVEIofoneoccursthe remaining9%ofthetime.Note, bothsources indicate that therehasneverbeenaneruptionswithV EIoffourorfive;thispowerfuleruptionhas notoccurredatCerroNegro,yet.

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175 Figure 105 :LinegraphofVEI overtheyears(firstsource) Figure 106 :LinegraphofVEI overtheyears(secondsource) VEI Frequency(1) Frequency(2) Pr obability(1) Probability(2) 0 0 8 0% 35% 1 2 6 9% 26% 2 14 6 61% 26% 3 7 3 30% 13% 4 0 0 0% 0% 5 0 0 0% 0% Table 70 :Frequenciesandproportionsforthetwodatasources Undertheassumption,thefirstsourceismore realisticwewillproceedwit hthe analysis of theremainingvariables;namely,thedirectionofthedeposit,andthe thicknessofthedeposit. 7.3 TrendAnalysis Considerthecumulativevolumesincubic meters ofvolcanicfallout. Throughout time,volcaniceruptionsofmagnitude3 a recommonly followedbyeru ptionsof magnitudetwoorone. Let ) ( ) ( t VEI n t x i i bethecumulativefrequenciesforeach of thethreemainmagnitudes, 3 2 1 i ,showninFigure 108 aretheoverallcumulative frequencies defined by 3 1 ) ( ) ( i i t x t n .Thentheprobabilityofaneruptionofagiven magnitudeis V E I ( 1 ) V E I ( 2 ) Year Year

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176 ) ( ) ( ) ( t n t x t p i i for 3 2 1 i asillustratedinFigure 109 Figure109showtheconvergenceofthepercentageofgivenVEIovertime;that is,appr oximate9%ofvolcaniceruptionsatCerroNegrohaveVEIof1,61%ofvolcanic eruptionsatCerroNegrohaveVEIof2,andapproximately30%ofvolcaniceruptionsat CerroNegrohaveVEIof3. Figure 107 :Linegraphofcumulativevolume 3 m sincethebirthofCerroNegro VEI C u m u l a t i v e V o l u m e ( m 3 ) Year

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177 Figure 108 :Linegraphofcumulative frequencies Figure 109 :Linegraphofpercentages overtime(probabilities) Weseethattherearetwolargegapsinthelinegraphs givenby Figure s 108 a nd Figure 109 ,thefirstgap appears between1867and1899(32years)andbetween1971 and1992(21years);buton the average,thereisaneruptionevery6.2years.Inaddition, thefirstfeweruptionsafterthistwenty plusyearlullwhereoneofmagnit ude3andthen twoeruptionsofmagnitude2overasevenyearperiod.Infact,overhalf of the differencesindicateaneruptionevery3years.However,therewasatime that this volcanolaydormantforthreedecades.This study willnotaddressthepr obabilitythat thereisaneventin a givenyear,buttheconditionalhazardthatrelatestotheprobabilities thatagiveneventhasaspecifiedmagnitudeandwhere,relativetothemainventofthe volcano,theprobabilitythatthetephrawillfallina specifieddirection. 7.4 DirectionofDeposition:ConicSections DatacollectedatCerroNegrobytheUniversityofSouthFloridasGeology department,themassoftephrabygrainsizethatcanbeusedtoanalyzethe probability distributionoftephra fallout.Withthemainventofthevolcanosetastheorigin, considertheeightsections(45 each)enumeratedcounter clockwiseofftrueeast.These i x Year i p Year V E I

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178 eight conical regionsdefinedfindingtheangleoffdueeastandthendividedintoequal sectors 1 45 int s Themajorityofthetephrafalloutisinthesixthsector asshowninFigure110.As giveninTable71, 81%ofthemassfallstothesouth southwestofthevolcano. Also, all massfallssouthofthemainventandverylittle(11%)fal lssoutheast.Thedifferences betweenthelocationswherethedataiscollectedandthewheretephrafallsshouldbe minimal;thatis,thedataisassumedtobegatheredatrandom(systematically)selectedto representalllocationswheretephrafalls. Fi gure110illustratesthattheprobability distributionofthegivendataisnotbestcharacterizedbythesymmetricbivariatenormal probabilitydistribution. Figure 110 :ScatterplotofTephraFalloutbyconicalsection Sectors N o r t h e r n Eastern

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179 Sector Freq Probability 1 0 0% 2 0 0% 3 0 0% 4 0 0% 5 6 8% 6 64 81% 7 9 11% 8 0 0% Table7 1 :Databydirection(eightsectors) Furtherrefinethisdirectionalpartitionintotwenty fourconicalsectorsshown by Figure 111 wheretheconesdefinedfindingtheangleoff dueeastandthendividedinto equalsectors 1 15 int s Thisrefinedpartitionshowsthatwhileon3.8%ofthe samplesaretakennearthemainventofthevolcano,20.7%ofthemassfallinthis direction. Themorerefinedthesectors,them orenormalthedistributionappearsas showninTable 72 Figure 111 :ScatterplotofTephraFalloutbyconicalsection Sectors N o r t h e r n Eastern

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180 Sector Count ProbabilityCone MeanMass ProbabilityMassinCone 13 1 1.3% 282.418 8.7% 14 2 2.5% 220.426 6.8% 15 3 3.8% 380.958 11.7% 16 9 11.4% 358.461 11.0% 17 29 36.7% 457.632 14.0% 18 26 32.9% 333.708 10.2% 19 5 6.3% 436.064 13.4% 20 3 3.8% 674.304 20.7% 21 1 1.3% 116.635 3.6% Table 72 :Databydirection(twenty foursectors) Thisanalysisshowsthatthedata isnotGaussian;thedispersionofthetephrais notsymmetricallywithrespecttothecenterofthemainvent.Thisanalysisalsoshows thatevenifthestandard(non correlated)bivariatenormalisassumed,thedatamust eitherberotatedtoaprimarya ndsecondaryaxisorusedthegeneral(correlated)bivariate normal. 7.5 RadialAnalysis Letthelocationof the mainventbethecenterofourvolcaniceruption. Assumingtheconvertedlatitudeandlongitudedenotedinmeters(northingandeasting, Univ ersalinaTransverseMercatorcoordinate),andthenwecancomputethedistance fromthiscentermarkeraswellastheangle. Thatis,the distance 2 2 c i c i i y y x x d between i i y x isthelocationofthe th i sampleand c c y x isthelocationofthemain ventorcenterandtheangle c i c i i x x y y 1 tan is offdueeast.Thenwecanbest analyzethedistanceandangleindependently.

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181 Considerthehistogramofthedistancessampled as shown by Figure 112along withthebasicstatisticsthatdescribethedata .Moresamplesweretakenclosertothe mainvent,andfewerweretakenmorethan10,000metersfromthemainvent,butallin allthenumberofsamplesareuniform.Percentagesfordistance (withthirtycontours) are giveninTable 73 ,butmoreinteresting,themassmeasuredatthesevariousdistances illustrated byFigure112 Alsointhetablethatfollows,inadditiontothedistance contour,itshowsthecount,presentbydistance,sampl emeanofthemasspresentandthe percentofmassateachdistance.Thebasicdescriptivestatisticsarealsogivenbythe accompanyingtable. Figure 112 :Histogramoftheestimateddistancefromthecenter(mainvent) includingdescriptivestatist ics Statistic Estimatio n Count 80 Mean 4962.338 Median 4660.535 StdDev 2952.392 Variance 8716618 Range 10936.3 Min 0 Max 10936.3 IQR 5458.585 25th% 2131.45 75th% 7590.035 Distance m

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182 Distance Contour Count Percentby Distance MeanMass Percentby MassatDistance 1 1 1.3% 850.883 7.7% 3 2 2.5% 294.861 2.7% 4 4 5.1% 904.855 8.2% 5 4 5.1% 481.218 4.3% 6 2 2.5% 1111.812 10.0% 7 4 5.1% 348.172 3.1% 8 2 2.5% 1132.425 10.2% 9 2 2.5% 404.726 3.7% 10 4 5.1% 633.359 5.7% 11 2 2.5% 639.388 5.8% 12 4 5.1% 508.198 4.6% 13 2 2.5% 385.726 3.5% 14 5 6.3% 390.416 3.5% 15 3 3.8% 384.823 3.5% 16 3 3.8% 326.406 2.9% 17 3 3.8% 271.232 2.4% 18 3 3.8% 277.526 2.5% 19 3 3.8% 274.951 2 .5% 20 4 5.1% 198.025 1.8% 21 2 2.5% 244.189 2.2% 22 4 5.1% 177.359 1.6% 23 3 3.8% 198.457 1.8% 24 3 3.8% 153.463 1.4% 25 2 2.5% 92.025 0.8% 26 4 5.1% 212.001 1.9% 27 3 3.8% 123.487 1.1% 30 1 1.3% 62.848 0.6% Table 73 :Massbydistance

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183 Figure 113 :Scatterplotofmass 2 / m kg anddistancefromthecenter(mainvent) Furthermore,considerthedistributionoftheangle i ,samples appeartobe normallydistributionasillustrated byFigure114 with ameanof252.3 offdueeastwith astandarddeviationof47.1 ;however,thisissimplythesamplingdistr ibution. However,asFigure115 illustratesthedistributionofmassisalsocenteredaboutthis angleaswell. Thebasicdescriptivestatisticsar ealsogivenbytheaccompanyingtable. Figure 114 :Histogramoftheestimatedanglesoffthehorizonincludingdescriptive statistics Statistic Estimation Count 79 M ean 252.324 Median 252.363 StdDev 19.536 Variance 381.648 Range 112.276 Min 189.036 Max 301.312 IQR 21.265 25th% 241.697 75th% 262.961 Angle k g / m 2 Distance Sectors

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184 Figure 115 :Scatterplotofmass 2 / m kg andangleoffdueeast Further analysis ofthemass byangle indicatesthatthedistributionoftheangleof falloutisnotnormallydistributed.Thenormalplotandboxplotgiven by Figure 116 and Figure 117 ,respectively,indicatethatthedataismoreuniformlydistributednearthe centralangledetermi nedbytherotationoftheearthandthedirectionofthewindnearthe mainvent.Allotherdirectionsareoutliers asillustratedintheboxplotgivenbyFigure 116,whereanoutlierisanypointwhichfallsfurtherthanthreesamplestandarddeviation fromthemean. T heseanglesoftrajectorywouldbeuniformlydistributedinoverall 360;thisisduetothefactthatwithouttheexternalforces(andassumingperfectly sphericalanduniformparticlesize)thedispersionoftheashfallwouldbebivari ate normal(Gaussian). M a s s ( k g / m 2 ) Angle Figure116 :Normalprobabilityplotfor directionoffallout Figure117 :Boxplotfordirectionof tephrafallout z score F r e q u e n c y A n g l e Angle

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185 7.6 Particle SizeProbability Distribution Theseeruptionsproducedanash richcolumnextending2kilometers.Herewe seethesmallestparticlesofashfallingawayfromthemushroomingcolumnaswellas vibratordustw hereasthemajorityoftheparticlesformamoreliquidousbuoyantstate. Ingeneral,strombolianeruptionsarecharacterizedbythesporadicexplosionor spewingforthbasalticlavafromasingleventorcrater.Eacheventiscausedbythe releaseofvo lcanicgases,andtheytypicallyoccurperiodically sometimeswithan appearancepatternsandothersmorerandomly.Thelavafragmentsgenerallyconsistof partiallymoltenvolcanicbombsthat becomeroundedastheyflythroughtheair. Theseparticlesw eregatheredandsiftedintosixteendifferentparticlesizes phi d 2 log where d istheparticlediametermeasuredinmillimeter Therangesin diametersareaslisted inTable7 4 Diameter mm d Size d 2 log Mass 2 km kg 16.00 4 325.40 11.31 3.5 264.48 8.00 3 572.21 5.66 2.5 1079.65 4.00 2 1662.49 2.83 1.5 2955.97 2.00 1 3921.00 1.41 0.5 4905.60 1.00 0 4792.37 0.71 0.5 4370.21 0.50 1 2621.48 0. 35 1.5 1463.36 0.25 2 777.45 0.18 2.5 428.41 0.13 3 276.76 <0.09 >3 872.66 Table 74 :Massbydiameterandsize Considerwhenthemassisplottedfirstversusdiameter asshownby Figure 118 andthenversusphi shownby Figure 119 .Furthermore,consid erthe probability distributionofthediameterandphiusingmassasthefrequency.

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186 Figure 118 :Scatterplotofmass 2 / m kg bydiameter d Figure 119 :Scatterplotofmass 2 / m kg bysize d 2 log Phidoesdemonstrateamorenormal probability distribution,butdoesnot compensateforthedistributionsinthetail.Thebest fit probability distributionisthe Log Normal probability distribution,seeTable 75below ;however ,manyvolcanologists usethenormal probability distribution whichwillgivemisleadingresults Test:Diameter(Phi) Normal Lognormal Exponential Weibull Kolmogorov Smirnov <0.010 <0.001 <0.001 <0.010(<0.001) CramerVonMiser <0.005 <0.001 <0.001 <0 .005(<0.001) AndersonDarling <0.005 <0.001 <0.001 <0.005(<0.001) Table7 5 :Testforbest fitdistribution Heretheempiricaldistributioniscomputedbythemassofthevariousparticle size;thatis, i i M j M j P ) ( ) ( ) ( where isthesetofparticlesizesdefinedby d 2 log with d being thediameter sizeoftheparticleinmillimeters 0 d .Note:thereislossofmasswhenconvertingthe M a s s M a s s Diameter Size

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187 percentageparticlesizea teachgivenlocationbacktocomparablemassunit.The manipulateddataisaccurateupto % 2 oftheactualrecordedpercentmass. Thenormalprobabilitydistributionofthesizephiisgiven by 2 2 2 exp 2 1 ) ( f 69 where istheexpectedvalue( truemean)ofthesize and istheassociated standarddeviation ,where therecordedmass is thefrequency.Itmaybenecessaryto inclu deaseparateweighingsystemtobreakthemassintofrequencyorcountofnumber ofparticlesofagivensizeinagivenmass.Toconsiderthisinterpretation giventhe numberofparticles,theprobabilitythatagivenparticleisofagivensize wou ldrequire estimationsonthemassofparticlesofagivendiametersize. Herewehavethe cumulativeprobabilitydistributionof givenby 2 2 2 exp 1 ) ( F andthatof d givenby 2 ln exp 1 ) (ln ln ln ln ) ( 2 2 d d F d P d D P d D P d F D Wecansimplifythiscumulativeprobabilitydistributionif d isgivenby 2 2 2 2 ln exp 1 ) ( d d d F D 71 Furthermore,notethatwecanwrite

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188 d d f d f D ) (ln ) ( whichistheprobabilitydistri butionfunctionofthemassbyparticlesize(diameterof tephra ) 7.7 Statistical Modeling of TephraFallout Considerthethreevariables, mass distance and angle associatewithtephra fallout .Letthemassofthetephraatagivenlocationbedenote dby m ,thenwecan considerthelinear statisticalmodelgivenby 2 1 0 d m 72 wherethe s i aretheweightsthatdrivetheestimateofthesubjectresponseand is therandomerror. Statistically ,wefindthatonlythedistanceawayfromthemainventissignificant ascontributingvariables with p value<0.0001explaining36.5%ofthevariationinthe amountofmassrec ordedatagivenlocatio n. Thedirectioninwhichthemassoftephra foundisdependentonthewind,butdoesnotsignificantlycontributetothedispersionof themass. Thus,anacceptableestimateofthestatisticalmodelgivenbyequation 72 is, d m 0613542 0 368 810 73 Thatis,theresponsevariableonlydependsondistancefrommainvent.Also,t urning theserolesaround,considerthedistancesofthemassbyparticlesize i m isgivenby 16 1 0 i i i m d 74

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189 wherethe s i aretheweightsthatdrivetheestimateofthesubjectresponseand is therandomerror. Thedevelopedstatistical modelexplains52.7%ofthevariat ioninthedistance, butnooneparticlesizewasfoundtobesignificant. Thismodelisimportantwhen consideringthatmostadvectionequationassumethatthelocationwheretephrais expectedtofalldependsonparticlesize.However,thepresentstud yshowsthatthe location,atleastintermsofdistance,doesnotdependonparticlesize. 7.8 BivariateDistribution Considerthe mass m overthenorthingdistance y andeastingdistance x shown by Figure 120 .Weseethatthemajorityofthemassfallsnearthemainventand dependingonthedirectionthewindisblownfromthevolcano.Thiswinddirectioncan bemeasured,thenthemajorandminoraxiscanberotatedoffofthenorthande astatthe requiredangle,afterwhichasimplenon correlatedbivariateNormaldistributioncanbe used;however,therearemanycontributingfactorsandthewindisno ttheonly determiningfactor.

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190 Figure 120 :Scatterplotofmass 2 / m kg Considerthestandardcorrelatedbivariate normalprobability distributiongiven by 2 2 2 2 ) 1 ( 2 1 exp ) , , | ( y y x xy x xy xy y y x x z z z z K y x f w here x x x x z y y y x z 1 1 xy and 2 1 2 1 xy y x K ,for y x Noticeth efunctionestimates givenbytheaboveprobabilitydistribution areana differentscale,thisissimplyduetothefactthatthetrueempiricalprobabilityis ) ( ) ( ) ( y x m y x m y x P East North 2 / m kg

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191 where isthepercentofthemass collected;thatis,ifthetotalfalloutmassexpelledby thevolcanois m and m y x m ) ( isthepercentofthetotalmassmeasuredinthe sample However,evenwithdifferentscales,weseeasshown by Figure 121 (a),t he distributionofthedatacollectedisskewedtowardthevolcanosmainvent;whereas,as shown by Figure 121 (b)and 121 (c),thenon correlatedandthecorrelatedbivariate Gaussianare symmetrical 121 (a) Empirical 121 (b) 0 0 xy 121 (c) 504 0 xy (dependent) Figure 121 : Empiricalprobabilitydistributionforthegivennorthernandeastern coordinates Whencomparingtheempiricalprobabilitydistributionwiththatofthegeneral non correlatedbivariateGau ssian probabilitydistribution andthecorrelatedbivariate Gaussian probability distributions,neitheraccountfortheskewnessofthedatas distributiontowardthevolcanosmainvent.Comparethecontourplotsforthenon correlatedbivariateGaussian probabilitydistribution andthecorrelatedbivariate Gaussian probabilitydistribution asshownby Figure 122 (a)and1 22 (b) respectively.

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192 122 (a) 0 0 xy 122 (b) 53 0 xy Figure 122 : Non correlatedandcorrelated bivariateGaussiandistribution Inthenon correlatedbivariate,thewindsheareffectisnotpresent;thatis, assumingthatthedirectionsareindependent.Acommonpracticetocompensateforthis istousetheinitiallydefinedadvectiondiffusioneq uationsundertheassumptionthat eachlayerintheatmospheremovescollectivelyandfallinanormallydistributionpiles whenconsideredbygrainsize;thatis,theassumptionisthatthefalloutissymmetricalto acenterpointandnotskewedtowardth emainvent.Thiskernel likeapproachneednot directlyaccountforthewindshear,butmoreoverdoesnotaccountfortheskewnessof thefallouttowardthemainventofthevolcano. Furthermore,whilethecorrelatedbivariateGaussiandoesaddressthe issueof orientationwithoutanyintermediatetransformations,itdoesnotaddresstheskewnessof thedatatowardthemainventasillustrate by Figure 123 (a).Therefore,eitherweneedto sift(profile)thedataaccordingtoparticlesizeinanattempt tojustifythegeneral bivariateGaussian probabilitydistributionfunction ortestthegoodness of fitofa skewedprobabilitydistributionsuchassomeformcontinuedwithinthegeneralized extremevaluedistribution(GEVD); ( theWeibull,theGumbel,the FrechetorthePareto) ortheskewedNormaldistribution. Furthermore,thecategorizationofthetephrafalloutbyparticlesizedefined by ) ln( d ,where d isthediameterofthetephra ,isnormalizedby thescaleand homogenize d thevariance.Forthemajorityoftheparticlesizes,thedistributionis skewedtowardthemainventofthevolcano,butastheparticle'ssizebecomessmaller, indicatedbythelargevaluesofphi,thedistributionlossesitscenterofconce ntrationand becomesmorevariegated.Therefore,itispossibleforthelargerparticlesizestobe

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193 combinedandcharacterizedbythesameparametricdistribution,whereasnon parametric techniquesmayneedtobeforthesmallerparticlesize. 123(a) phi 4 123 (b)phi 3.5 123 (c)phi 3 123 (d)ph i 2.5 123 (e)phi 2.0 123 (f)phi 1.5 123 (g)phi 1.0 123 (h)phi 0.5 123 (i)phi0.0 123(g)phi0.5 123 (h)phi1.0 123 (i)phi1.5 123(j)phi2.0 123(k)phi2.5 123 (l)phi3.0 F igure 123 : Tephrafalloutbyparticlesize

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194 Thegeneralempiricaldistributionsforthemajorityofthelargerparticlesizesare verysimilarinskewnessanddispersiontothedistribution ofthecombinedinformation. Thesmallerparticlesize,themoreu niformthedistribution,yetthedispersedisinthe samegeneralregionasthelargeparticlesizes.Hence,theoverallcorrelatedbivariate probability distributioncanbeappliedtothetotalmassandnotbyparticlesizeto approximatethebestprobab ilisticbehaviorofthesubjectphenomenon(locationof fallout)usingoneofthethreeformsofthebivariatenormaldistribution:therotatednon correlatednormal probability distribution,thenon rotatedcorrelatednormal probability distributionandt herotated(independent)skewnormal probability distribution. ForthedatagatheredatCerroNegro,thecorrelationcoefficientbetweenthe northingandeastingdistancesis 508 0 xy usingdatalocationinlistedformand the estimateofth ecorrelationcoefficient 5378 0 xy usingtheassociatedmassas weights. The sample meannorthcoordinateis 525944 x ( 527329 x )and sample meaneastcoordinateis 1380440 y ( 1380896 y ),with sample standarddeviations of 3222 x ( 3 2751 x )and 1720 y ( 5 1416 y ).Comparethenorth south andtheeast westcross sectionoftheempiricaldataversusthecorrelatedbivariate normaldistributionasshown by Figures 123 thru 126 ;thesefiguresillustratethatthere thestandardbivariatenormalisnotthebest fitdistribution.Thereisskewnessinthedata isnotsimulatedbyasymmetricnormalbivariatedistribution;thisap pearsas aleaninthe dataasillustrateby Figure 123 However,theempiricalprobabilitydistributionshownin Figure125isbettercharacterizedbythenormalprobabilitydistributionshowninFigure 126.

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195 Furthermore, considerthethree dimensionalplotoftheempiricalprobability distributionandthecorrelatedbivariatenormaldistributionwithrespecttothemapof distancenorthbydistanceeastshown by Figure1 28 andFigure1 39 ,respectively.This furtherillus tratesthatthe contourplotsshown by Figure 130 andFigure 121 ,respectively. Inthesecontourplots,thelinesrepresent x ( true horizontal mean )and y ( true vertical mean )andthemainventistheindicated bythedashedline. E m p i r i c a l North B e s t f i t N o r m a l North E m p i r i c a l East B e s t f i t N o r m a l East Figure124 :Scatterplotfortheempirical probabilitydistributionanddistan cenorth Figure125 :Scatterplotforthe correlatedbivariatenormaldistributionand distancenorth Figure126 :Scatterplotfortheempirical probabilitydistributionanddistanceeast Figure127 :Scatter plotforthecorrelated bivariatenormaldistributionanddistanceeast

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196 Figure 128 :Three dimensionalscatterplotoftheempiricalprobabilitydistributionover theunderlyingdistancenorthanddistanceeast. Figure 129 :Three dimensionalscatterplotofthecorrelatedbivariatenormaldistr ibution overtheunderlyingdistancenorthanddistanceeast. East North Empirical East North xy y x y x y x f , , |

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197 Figure 130 :Three dimensionalscatterplotoftheempiricalprobabilitydistributionover theunderlyingdistancenorthanddistanceeast. Figure 131 :Three dimensionalscatterplo tofthecorrelatedbivariatenormaldistribution overtheunderlyingdistancenorthanddistanceeast. 7.9 Comparisonof F our F ormsoftheBivariateNormal ProbabilityDistribution Recall that wedefinedthedistance 2 2 c i c i i y y x x d between i i y x isthelocationofthe th i sampleand c c y x isthelocationofthemainventorcenterand

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198 theangle c i c i i x x y y 1 tan offdueeast.Wecanthereforetransformthenorthernand easterndirections intoacentralizedCartesianplanebyconsiderthetransformeddata c i i x x x : and c i i y y y : thenin cis notation,wehave i i i i d y x r 2 2 which correspondingradiansgivenby i Torotatethisdatatoaprimaryandsecondaryax is wherethereisminimumorno correlation,wecanrotatethedatabyagivenangle by defining ) cos( i i i r x and ) sin( i i i r y .Wecanestimatethenecessary value of by consider ing theregressedslopebetween x and y setequaltozero.Accurate tothefirstdecimalwehave 8 18 whichshowsanassociatedslopeof 000357 0 m and simple correlationof 000801 0 r .Scatterplotsofthesetwocoordinate systemsare asshownby Figure 132 andFigure 133 Figure 132 :Scatterplotof original data Figure 133 :Scatterplotoftherotateddata However, evenwiththisrotationthemassisskewedandthemajorityofthe tephrafallmoreinonedirectionthaninanyother,namelythedirectioninwhichthe windisblowing. Tocomparethese statistical modelstotheempiricaldistribution,considerthe uni tized probabilitydistribution. Recall thesamplesetsconsistis

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199 80 ,..., 2 1 : ) ( i y x i i andtheempiricalprobabilityatgivenlocationisdefinedin termsofthemassatthatlocation ) ( ) ( ) ( ) ( y x i i i i i y x M y x M y x P p .Then wecancompute thebest fitcorrelate dbivariatenormal probability distributionforthetransformeddata ) ( i i y x f atthevariouslocationsforwhichwehavedataandthendefinetheassociated probabilityas ) ( ) ( ) ( ) ( y x i i i i i i y x f y x f y x f .Similarly,wecancomputethebest fitnon c orrelatedbivariatenormaldistributionforthetransformedandrotateddata ) ( i i y x g as defined by ) ( ) ( ) ( ) ( y x i i i i i i y x g y x g y x g .Thenwecandeterminewhichdistributionyields thebest fit. Since x and y areindependent,then ) ( ) ( ) ( y f x f y x f Y X ,where y x y x | , X x x X | and Y y y Y | .Inthisrotated coordinatesystems,thedistributionofthemin oraxisisnormallydistributed ;thatis, 2 ~ y y N y .However,thedistributionofthemajoraxisisskewed;thatisthe variable x isbetterdescribed bytheskeweddistributiongivenbye quation 7 7 where ) ( x g isthestandardnormalp robabilitydistributionfunctionwithparameters x and 2 x .Theskewedprobabilitydistributionfunctionisgivenby x G x g x f X 2 75 where G ist hecumulativenormalprobabilitydistributionfunction, istheskewing factorand x istherotatedcoordinatedefinedabove. Thus, wehavefour probability distributionstoconsider:thetransformednon correl atedbivariatenormal(TNCN),thetransformedcorrelatedbivariatenormal(TCN), thetransformedrotatednon correlatedbivariatedistribution(TRNCN)andfinally,the (independent)skewedtransformedrotatednon correlatedbivariatenormal(STRNCN).

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200 To determinewhichofthefourprobabilitydistributionbest fitsthedata,we proceedtodeterminewhichofthefourprobabilitydistributionm inimizingthemeasure of i i i i i p p y x f 2 2 ) ( and i i i i i adj p p y x f 2 2 ) ( ,wecanimprovethefitof thedistri butionbyletting 23 5 ,thisskewnessyieldsasignificantdecreaseinthe 2 Accordingtothisstatistic,thetransformedcorrelatedbivariatenormalisthebest fit.Let 2 0 representtheminim ummeasured 2 ,thenthelargersuchstatisticsaregivenasa multipleofthisminimumstatistic,seeTable 76 .Furthermore,withtheadjusted statistics, 2 adj wehavethatthefirsttwo probabilitydistributions ( statistical models ) ,the TNCNandtheTCNareverysimilar,buttheSTRNCNshowsavastimprovementover alltheremaining statistical models.Therefore,wecanalsoconsiderthecorrelation between i p and i f for thegivenprobabilitydistributionfunction,respectively. We c omparethesepro babilityestimatesgivenby Figures 134 thru 137 ThisagainindicatesthattheSTRNCNexplainsmorevariationintheempirical probabilitydistributionthantheother statist ical modelswith % 3 39 2 R ;howeverwhen weconsiderthe simple correlationbetween i p and i f forthegivenprobability distributionfunction,respectively theTRNCNisthemostexplanatory.Note: alldata considerinitstranspos edformwithgraphgivenby Figure 134thru137 inthecoordinate system y x InFigures134thru137,areondifferentscales;wheretheempiricalprobabilityin Figure149sumtoone,theestimatedpr obabilitiesinFigures135thru137arebasedon thebivariatedistributionwhichisonlyasmallportionofthetotalprobabilityasstated previously.

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201 Statistic y x TNCN TCN 2964.3 637.82 2964.29 637.82 2 2751.46 1416.3 2751.46 1416.27 0 0.5378 2 R 28.90%(52.9%) 35.20%(54.6%) 2 1.3 2 0 2 0 2 adj 0.413 0. 422 Statistic y x TRNCN STRNCN 3196.3 396.23 3196.34 396.23 2 2760.22 1137.5 2760.22 1137.48 0.000 0.000 2 R 17%(58.9%) 39.30%(45.1%) 2 11.9 2 0 16.2 2 0 2 adj 0.607 0.265 Table 7 6 :Descriptivestatisticsandregressedslopeandcorrelationcoefficient 0.0567 0.0508 0.0449 0.0389 0.0330 0.0271 0.0212 0.0152 0.0093 0.0034 EMPIRICAL -4000 -2000 0 2000 y -10000 -5000 0 x Figure 134 :Contourplotofempiricalprobabilitydis tribution

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202 1.2849708E-08 1.1671353E-08 1.0492997E-08 9.3146422E-09 8.1362869E-09 6.9579316E-09 5.7795763E-09 4.6012211E-09 3.4228658E-09 2.2445105E-09 TNCN -4000 -2000 0 2000 y -10000 -5000 0 x Figure1 3 5: Contourplotfortheestimatednon correlatedbivariatenormaldistribution 1.5395378E-08 1.4291473E-08 1.3187567E-08 1.2083662E-08 1.0979757E-08 9.8758518E-09 8.7719466E-09 7.6680413E-09 6.5641361E-09 5.4602309E-09 TCN -4000 -2000 0 2000 y -10000 -5000 0 x Figure 136 :Contourplotfortheestimatedcorrelatedbivariatenormaldistribution

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203 1.2025014E-09 1.1712527E-09 1.1400039E-09 1.1087552E-09 1.0775064E-09 1.0462577E-09 1.0150089E-09 9.8376019E-10 9.5251144E-10 9.2126268E-10 TRNCN -4000 -2000 0 2000 y -10000 -5000 0 x Figure 137 :Contourplotfortheestimatedrotatednon correla tedbivariatenormal distribution. 1.2933270E-09 1.1708505E-09 1.0483740E-09 9.2589751E-10 8.0342100E-10 6.8094450E-10 5.5846799E-10 4.3599149E-10 3.1351498E-10 1.9103848E-10 STRNCN__I_ -4000 -2000 0 2000 y -10000 -5000 0 x Figure 138 :Contourplotfortheestimatedrotatednon correlatedskewedbivariate normaldistribution(assumingindependence).

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204 7. 10 Usefulness Thistypeofanalysisisextremelyimportantonbothaglobalscaleaswe llasona localscale.Globalimpactsarenotjustthepossibilityofashinthearea,butalsothe placementofsuchthingsasnuclearpowerplantsandbiohazardfacilities.Onalocal scale,thereareurbanplanningforeconomicgrowth,andmoreimpor tantlyevacuation planningincaseofavolcaniceruption.Unfortunately,thisisnotanexactscience.In 2001,volcanologistsstatedwith95%confidence,thatCerroNegrowoulderuptagain before2005,butthishasnotyetcomeintofruitionanditis overayearpastthisforecast (Connor,Hill,Winfrey,FranklinandLaFemina,2001)..Doesthismeanweareover due,andhowdoesthisadditionaltimeaffecttheprobablemagnitude(VEI)ofthenext volcaniceruption?Considerthetrendovertime. 7.1 1 Conclusion ThevolcanodataanalyzedinthisstudywasobtainedfromCerroNegro,Nigeria. Wehave80samplesitesfromwhichthetephrawasmeasuredintermsofmassand sieveddownintomassbyparticlesize.ThemostprobableVEIforCerroNegrois2. Intermsofgrainsize,theprobabilitydistributionsarethesameforlargerparticle size;theskewedbivariatenormal.Whereasasthesmallertheparticlesizethemore uniformthedistribution.Themajorityoftheparticles,however,arelargeenoug hto considerthesemassescombined. Todeterminethebivariateprobabilitydistributionwhichbestcharacterizesthe subjectresponse(locationofashfall),fourprobabilitydistributionsaretestedfor goodness of fit: thetransformednon correlatedbi variatenormal probabilitydistribution (TNCN),thetransformedcorrelatedbivariatenormal probability (TCN),thetransformed rotatednon correlatedbivariate normalprobability distribution(TRNCN)andfinally,the (independent)skewedtransformedrotate dnon correlatedbivariatenorma lprobability distribution (STRNCN). Whileboththerotatednon correlatedbivariatenormal probabilitydistribution andthecorrelatedbivariate normalprobabilitydistribution thefit isextremelytight,whenmeasuringthe goodness of fitusingthechi squarestatistics, 2 indicate s thatthecorrelatedbivariatenormal probabilitydistribution bestcharacterizes thedistributionoftephra.However, 2 R indicatesthatthedistributi onfittothedataby

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205 firstrotatingthedatatoaprimaryaxisandthenon correlatedbivariatenormal probabilitydistribution used. Thus ,neitheroftheseprobabilitydistributionsaccountfor theskewnessinthedata.Accordingthethirdwaytodeterm ine thebest fit, 2 adj indicatesthattheskewedtransposed(transformedbycentertothemainventofthe volcano)rotated(basedonremovingthecorrelationbetweennorthernandeastern direction)non corrected(madetobeindependent) bivariatenormal probability distribution bestcharacterizesthebehaviorofthesubjectphenomenon.TheTCN explainsanestimated35.2%to54.6%ofthevariationintheempiricaldistribution;these valuescorrespondtothecorrelationcoefficient 2 R first between i f and i p ,andsecond between i f and i p .TheTRNCNexplainsanestimated17%to58.9%ofthevariationin theempiricaldistributionandt heSTRNCNexplainsanestimated39.3%to45.1%ofthe variationintheempiricaldistribution. Knowingthebivariateprobabilitydistributionwhichbestcharacterizethe behaviorofthesubjectphenomenon(locationofashfall)isextremelyimportanttou rban planning,strategicplanningandriskanalysis.

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ABOUTTHEAUTHOR RebeccaDyanneWootenwasbornonJuly31,1971inTampa,Florida.Shegraduated withbothaB.A.andaM.A.inMathematicsfromtheUniversityo fSouthFloridainMay 1996.AfterstudyingbrieflyattheFloridaStateUniversity,in1997sheenteredthePh.D. programattheUniversityofSouthFloridawhereshecontinuedherstudiesinGraph Theory,Combinatorics,ProbabilityandStatistics.Rebec caWootengraduatedwitha Ph.D.inStatisticsfromtheDepartmentofMathematicsandStatisticsfromtheUniversity ofSouthFloridain2006.


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2006.
3 520
ABSTRACT: This study consists of developing descriptive, parametric, linear and non-linear statistical models for such natural phenomena as hurricanes, lightning, flooding, red tide and volcanic fallout. In the present study, the focus of research is determining the stochastic nature of phenomena in the environment. These statistical models are necessary to address the variability of nature and the misgivings of the deterministic models, particularly when considering the necessity for man to estimate the occurrence and prepare for the aftermath.The relationship between statistics and physics looking at the correlation between wind speed and pressure versus wind speed and temperature play a significant role in hurricane prediction. Contrary to previous studies, this study indicates that a drop in pressure is a result of the storm and less a cause. It shows that temperature is a key indicator that a storm will form in conjunction with a drop in pressure.^ This study demonstrates a model that estimates the wind speed within a storm with a high degree of accuracy. With the verified model, we can perform surface response analysis to estimate the conditions under which the wind speed is maximized or minimized. Additional studies introduce a model that estimates the number of lightning strikes dependent on significantly contributing factors such as precipitable water, the temperatures within a column of air and the temperature range. Using extreme value distribution and historical data we can best fit flood stages, and obtain profiling estimate return periods. The natural logarithmic count of Karenia Brevis was used to homogenize the variance and create the base for an index of the magnitude of an outbreak of Red Tide. We have introduced a logistic growth model that addresses the subject behavior as a function of time and characterizes the growth rate of Red Tide.^ This information can be used to develop strategic plans with respect to the health of citizens and to minimize the economic impact. Studying the bivariate nature of tephra fallout from volcanoes, we analyze the correlation between the northern and eastern directions of a topological map to find the best possible probabilistic characterization of the subject data.
502
Dissertation (Ph.D.)--University of South Florida, 2006.
504
Includes bibliographical references.
516
Text (Electronic dissertation) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 214 pages.
Includes vita.
590
Adviser: Christopher P. Tsokos
653
Parametric analysis.
Extreme value distribution.
Linear and non-linear modeling.
Prediction and forecasting.
690
Dissertations, Academic
z USF
x Mathematics and Statistics
Doctoral or Masters or Specialist.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.1824