Predictability of sea level and currents of Tampa Bay

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Predictability of sea level and currents of Tampa Bay

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Title:
Predictability of sea level and currents of Tampa Bay
Creator:
Zhang, Mingrui
Place of Publication:
Tampa, Florida
Publisher:
University of South Florida
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Language:
English
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ix, 147 leaves : ill., maps ; 29 cm.

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Subjects / Keywords:
Sea level -- Florida -- Tampa Bay ( lcsh )
Tidal currents -- Florida -- Tampa Bay ( lcsh )
Dissertations, Academic -- Marine Science -- Masters -- USF ( FTS )

Notes

General Note:
Thesis (M.S.)--University of South Florida, 1994. Includes bibliographical references (leaves 121-126).

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University of South Florida
Holding Location:
Universtity of South Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
020133877 ( ALEPH )
32526721 ( OCLC )
F51-00116 ( USFLDC DOI )
f51.116 ( USFLDC Handle )

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Book

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PREDICTABILITY OF SEA LEVEL AND CURRENTS O F TAMPA BAY by MINGRUI ZHANG A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Marine Science Universit y of South Florida August 1994 Major Professor: Robert H Weisberg, Ph. D

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Graduate School University of South Florida Tampa, Florida CERTIFICATE O F APPROVAL MASTER'S THESIS This is to certify that the Master's Thesis of MINGRUI ZHANG with a major in Marine Science has been approved by the Examining Committee on July 8, 1994 as satisfactory for the thesis requirement for the Master of Science degree. Examining Committee: H. Weisberg, Ph.D. Member: Carder, Ph.D. Ph.D. Member: Mark E. Luther, Ph. D.

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ACKNOWLEDGMENTS The first year of my assistantship came from the National Ocean Service, National Oceanic and Atmospheric Administration, in grants to Dr. Robert H. Weisberg. Their support is appreciated. I thank Dr. Robert H. Weisberg for leading me into the area o f physical oceanography and providing me with abundant and generous advice. I thank the members of my committee for their suggestions and comments I w ould like to thank the many peopl e in our department for their help, particularly Zhen Li, Jeff Donovan Lee Chapin, Lin Q iao, and my fello w graduate students in our group. Lastly, I wish to thank my wife, Haihua Liu, and my parents for their boundless support and love throughout my education.

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TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv ABSTRACT vii CHAPTER 1. INTRODUCTION 1 Geographic Description of Tampa Bay 1 Previous Oceanographic Studies 4 Objectives of Study 10 CHAPTER 2 METHODS AND TECHNIQUES FOR PREDICTION 14 Tidal Height Prediction 14 Harmonic Method 14 Predicting Tides 19 Constants and Parameters 20 Greenwich Epoch 25 Prediction of Tidal Currents 34 Response Method 37 Basic Equations 38 Quality of Prediction 41 Some Concepts for Signal Analysis 42 Mean Square 42 Scalar Spectra 45 Single Input System 47 Multiple Input System 48 Complex Demodulation 49 CHAPTER 3 PREDICTABILITY OF SEA LEVEL 52 Sources of Data 52 Astronomical Tides 55 The Earth, Sun and Moon System 55 Tides in Tampa Bay 60 Wind-Forced Sea Level Fluctuation 68 Seasonal Variation 76 Seasonal Cycle 79 Water Temperature and Salinity 82 Atmospheric Pressure 84 Meteorological Tides 86 Prediction Errors Based on Residual Sea Level 87 i

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CHAPTER 4. PREDICTABILITY OF CURRENTS 94 Sources of Data 94 Prediction of Tidal Currents 97 Constituents 01 and M2 97 Prediction by Response Method 100 Wind-Forced Residual Currents 102 Residual Currents and Sea Level Variations 110 Prediction Errors Based on Residual Currents 112 CHAPTER 5 SUMMARY AND DISCUSSIONS 118 LIST OF REFERENCES 121 APPENDICES 127 APPENDIX 1. TIDE CONSTITUENTS AND SEA LEVEL FOR 1987 128 APPENDIX 2 THE SEA LEVEL FOR 1982-1988 142 ii

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LIST OF TABLES Table 1. Tidal Table of St. Petersburg for 1987, Prepared by NOS 17 Table 2. Node Factor f for the Middle of Each Year, 1 970 to 1999 23 Table 3 Equilibrium Argument (V0+u) for the Meridian of Greenwich at the Beginning of Each Calendar Year, 1979 to 1989 2 7 Table 4 Differences to Adapt Table 3 to the Beginning of Each Calendar Month 28 Table 5. Differences to Adapt Table 3 to the Beginning of Each Day of Month 29 Table 6. Differences to Adapt Table 3 to the Beginning of Each Hour of Day of Month 32 Table 7. Summary of S(deg/hour), K' (deg), H(ft.), f and Greenwich Epoch (deg) for Tide Constituents at St. Petersburg for 1987 and 1990 35 Table 8 Major Tidal Potential Constituents 59 Table 9. The Partition o f Fluctuation Kinetic Energy for the Sea Level at the St. Petersburg 87 Table 10. Mean and Root Mean Square Values for the Observed and the Residual Sea Level 90 Table 11. Weights Wij' Amplitude Ri, Phase from Response Analysis for the Sunshine Skyway Bridge (2 7D36'N, 82o39'W) 101 Table 12. The Partition of Fluctuation Kinetic Energy for the Currents a t the Sunshine Skyway Bridge 112 Tabl e 13. Mean and Roo t Mean Square Values for the Observed and Residual Currents 114 iii

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LIST OF FIGURES Figure 1. Base map of Tampa Bay Showing Locations Mentioned in this Presentation 2 Figure 2 Tampa Bay Bathymetry Map 3 Figure 3. Intersections of the Planes of the Earth's Equator, the Ecliptic, and the Moon's Orbit with the Celestial Sphere 21 Figure 4 Cosine Curve for One Tidal Constituent with Amplitude of fH 24 Figure 5. One Input/Output Model 47 Figure 6 Two-Input/One-Output Model 48 Figure 7. Sea level station 53 Figure 8 The Production of Unequal Tides due to the Moon's Declination 57 Figure 9. Major Tidal Constituents at St. Petersburg 61 Figure 10. Major Semidiurnal and Diurnal Tide Constituents over January, 1987 Figure 11. Synthesis of Major Tide Constituents, and the Predicted, Observed and Subtidal Sea Level over the Month of January, 1987 Figure 12. Time Series of Observed, Predicted and Subtidal Sea Level for 1987 at 64 65 St. Petersburg 67 Figure 13. Subtidal Sea Level Fluctuations at St. Petersburg, as well as Wind Velocity Components at TIA for January, 1987 71 iv

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Figure 14. Auto-Spectrum of Subtidal Sea Level Fluctuations and the Windstress Components for 1987 Figure 15. Contours of Coherence Squared as a Function of Windstress Component Orientation and the Frequency Figure 16. Coherence Squared, Phase and Amplitude of Transfer Function between Subtidal Sea Level at St. Petersburg and the 3500T or 0800T Components of Windstress 72 74 at TIA 75 Figure 17. Annual Cycles of Sea Level, Water Temperature, Water Salinity Measured at Tampa Bay 77 Figure 18. Annual Cycles of the Monthly Averaged Sea Level at St. Petersburg 78 Figure 19. The Fluctuation Kinetic Energy Distribution of the Observed, Predicted and Subtidal Sea Level 88 Figure 20. The Amplitudes of Two Major AlongChannel Tidal Current Constituents M2 and 01 at Sunshine Skyway Bridge 96 Figure 21. Along-Channel Component of Water Velocity Measured at 3.0 meter Depth under the Sunshine Skyway Bridge 98 Figure 22. Residual Currents at the Sunshine Skyway Bridge along with Wind Velocity Components at TIA for August, 1990 99 Figure 23. Auto-Spectrum of Residual Water Velocity for Along-Channel Component during August, 1990, to June, 1991 103 Figure 24. Contours of Coherence Squared as a Function of Wind Velocity Component Orientation and the Frequency 105 Figure 25. Coherence Squared, Phase and Amplitude of Transfer Function with 18 Degrees of Freedom between Subtidal AlongChannel Water Velocity Component and the Wind Velocity Components Oriented at 340T and 700T 106 v

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Figure 26. Coherence Squared, Phase and Amplitude of Transfer Function with 18 Degrees of Freedom between Subtidal AlongChannel Water Velocity Component and the Wind Velocity Components Oriented at 3400T 107 Figure 27. Coherence Squared, Phase and Amplitude of Transfer Function between Subtidal Along-Channel Water Velocity Component and the Wind Velocity Oriented at 0600T 108 Figure 28. Coherence Squared, Phase and Amplitude of Transfer Function with 18 Degrees of Freedom between Derivative of Sea Level at St. Petersburg and the Residual Currents at Sunshine Skyway Bridge 111 Figure 29. The Fluctuation Kinetic Energy Distributions of the Observed, Predicted and Subtidal Along-Channel Component of Water Velocity 113 Figure 30. Major Semidiurnal and Diurnal Tide Constituents, and Time Series of the Predicted, Observed and Subtidal Sea Level for Each Month of 1987 at St. Petersburg 129 Figure 31. Time Series of Observed, Predicted and Subtidal Sea Level for 1982-1988 at St. Petersburg vi 143

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PREDICTABILITY OF SEA LEVEL AND CURRENTS OF TAMPA BAY by MINGRUI ZHANG An Abstract Of a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Marine Science University of South Florida August 1994 Major Professor: Robert H Weisberg, Ph.D. vii

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Sea level, water velocity and meteorological records for Tampa Bay, Florida have been examined to assess the errors associated with sea level and current predictions based on tides alone. The analyses focus upon sea level data at St. Petersburg and water velocity data along the main shipping channel under Sunshine Skyway Bridge. The evaluation of sea level prediction shows that 14 major tidal constituents can account for 52% of the RMS sea level fluctuations. The random error of the sea level prediction is em and it is associated with both coastal set-up and direct set-up by the synoptic scale wind. The seasonal cycle of sea level is significant. On climatological average, sea level in January is lower than sea level in September by 20 em, owing to the seasonal variations of water temperature, water salinity, and the atmospheric pressure. This seasonal variation of sea level contributes to the systematic error in the prediction. Tidal currents are predicted by applying the Response Method to the synthesis of tidal heights at St. Petersburg using 1 2 major diurnal and semidiurnal constituents. About 70% of the observed currents fluctuations could be predicted by this method. The random error is .7 cm/s, owing to both coastal set-up and direct set-up by the wind. The systematic viii

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.error which is 5.7 cm/s is attributed to the buoyancy-driven factors. Abstract Major Professor: Robert H Weisberg, P h D Professor, Department of Marine Science Date ________ __

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1 CHAPTER 1. INTRODUCTIO N Estuarine hydrodynamics is an area of increasing scientific concern. The need for practical answers to problems concerned with fisheries, pollution and navigation have driven observational, theoretical and numerical investigations. Our primary i nterest in the present study is the predictability of the sea level and currents within Tampa Bay. Geographic Description of Tampa Bay Tampa Bay is situated along the West Florida continental shelf. The West Florida continental shelf is a broad semienclosed area with the 100 m isobath generally lies some 200 km offshore along its 1000 km path from the Florida keys in the south to Cape San Blas in the north. Just west of Cape San Blas is the Desoto Canyon where the shelf narrows appreciably. Like many east coast estuaries, Tampa Bay is a drowned river valley. As shown in Figure 1 and Figure 2, the bottom topography can be characterized as relatively wide and shallow with the navigational channels dredged to about 15 m.

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2 82.90 1!2.80 82.70 82.60 82.50 82...40 215.05 27.95 27.85 27.7S 27.&5 27.55 Figure 1. Base Map of Tampa Bay Showing Locations Mentioned in thi s Presentation. Solid circles represent tidal stations and solid squares represent the location where the water velocity has been sampled.

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Scale 1:350,000 Projection 'liTM Datum NAD Z1 0 ST. PETERSBURG miles s Depth (meters mlw) : 0 <=1m 0 <=2m 0 <=3m El <= 4m il <= 5m <= 10m >10m 8 Nav Channel Figure 2. Tampa Bay Bathymetry Map. Map is prepared by Coastal Environmental, Inc. Data source: NOAA, digital hydrographic data. 3

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4 The northwestern and northeastern portions of the estuary are k n own as O l d Tampa Bay and Hillsborough Bay. The central part of the bay is referred to as middle Tampa Bay and the lower Tampa Bay is defined as the region adjacent to the West Florida Shelf. The average depth of the Tampa Bay is about 4 m. The Tampa Bay area is characterized by a temperate climate, with maritime tropical air affecting the area for roughly nine months out of the year. Summers in Tampa Bay are long, hot and humid. The prevailing summer winds are easterly interceded upon by a diurnal sea breeze. In winter, the passage of cold fronts from the northwest over the West Florida Shelf are frequent events. Winds are southerly in advance of the front, and then rotate clockwise, becoming strongly northerly upon frontal passage. Previous Oceanographic Studies An NOS (National Ocean Service) survey of tidal currents in Tampa Bay during 1963 was described and summarized by Dinardi (1978) His report included data for 39 current meter sites and was used for revising the existing NOAA Tidal Current Tables and in developing the tidal currents charts for Tampa Bay. Sea level of Tampa Bay has been described by Goodwin ( 1987) as being of mixed semi-diurnal and diurnal types.

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5 Two-dimensional vertically integrated barotropic models have been developed independently by Ross et al. (1984) and Goodwin et al. (1984, 1987, 1989) based on the same assumption that Tampa Bay was a vertically well mixed estuary and hence the density driven circulation is negligible. Both of these models suggested that the non-tidal circulation is dominated by a series recirculating gyres, but the existence of these gyres has not been confirmed by observations. The Office of Ocean and Earth Science (OES) o f the NOAA National Ocean Service conducted the Tampa Bay Oceanography Project (TOP). TOP, which began in June, 1990, consisted of an intensive 15-month survey of currents, water levels, water temperature and salinity, winds, and other meteorological parameters. It was undertaken to provide documented, qualityassured data to the research community and to increase knowledge of the physical oceanography of Tampa Bay to support maritime commerce and environmental management of the estuary. This first data set has been used to describe the flow field and its variations within the bay. Initial findings on the circulation of Tampa Bay were presented by and Williams (1991) They pointed out that the bay is vertically well mixed in salinity and does have jynamically significant horizontal salinity (density) :Jradients, owing to the distribution of inflowing fresh Nater. These horizontal gradients and surface wind forcing JOth imply a fully three-dimensional circulation. Evidence so

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6 far shows three major forcing components of the circulation, driven by tides, buoyancy and winds, respectively. Within the deep shipping channel at the Sunshine Skyway bridge, the tidal currents are observed to be nearly uniform with depth, with peak speeds of about 0.77 m/s. There exists a steady buoyancy-driven mean flow directed into the estuary over the entire portion of the water column sampled with a magnitude of about 10% of the tidal currents. This buoyancy driven convection is accounted for by the baroc1inic pressure gradient associated with the longitudinal salinity gradient between the relatively fresh water at the head of the estuary and the relatively salty waters at the mouth of the estuary. Wind-induced motions were also found to be significant. A three-dimensional time-dependent ocean circulation model was applied to Tampa Bay by Galperin et al (1991). In the absence of wind, the subtidal circulatio n in Tampa Bay model also shows a classical estuarine pattern with surface, fresher currents flowing seaward and saltier bottom currents flowing toward the head o f the Bay. Winds modify this as evidenced in the data. The circulation and sea level within Tampa Bay are also expected to respond to variations imposed by the coastal ocean. A study of relationships between sea level, currents and winds on the West Florida Shelf by Mitchum and Sturges (1982) showed that currents and sea level are coherent with the alongshore wind stress component within a bandwidth

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7 defined by the passage of synoptic weather systems (-0. 05-0.25 cycle per day) and to lag it by approximately 12 hours. In contrast, little coherence is found with the cross-shelf wind stress component. At their inshore mooring (22m depth), the dominant momentum balance in the alongshore direction was found to be between the wind and the bottom stresses. At their offshore mooring (44 m depth), the upper cross-shelf component, which was large relative to that at the inshore mooring, was consistent with Ekman transport while the lower record showed a compensating return flow. This study showed the sea-level fluctuations to be consistent with a geostrophic balance in the cross-shelf momentum equation with an offshore length scale o f about 170 km (approximately equal to the shelf width) This finding of high coherence between c urrents, sea level and alongshore wind stress in the "synoptic" band is widely supported. Cragg et al. ( 1981) analyzed records from Pensacola, Cedar Key, and St. Petersburg for the period 1965-67. They found that sea-level response to winds of varying direction is a maximum for alongshore winds, a typical response being 60 em for a 1 dyn/cm2 alongshore stress with sea level lagging the wind forcing by 102 4 hour. Marmorino (1982) expanded upon the study o f Cragg et al. (1981) by examining coastal tide gage and meteorological records from Pensacola to Key West for the period January-April 1978 for low-frequency fluctuations, the basic approach being similar.

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8 This study showed the dominant 6-days period signals in sea level, alongshore wind stress, and atmospheric pressure were coherent over the entire shelf and propagated southward, consistent with the movement of cold fronts through the area. Sea level response lags the local wind by 18 hour for the northwestern part of the shelf, and as little as 9 hour for the southern part. The response for a unit wind stress of 1 dyn/cm2 varies from 60 em where the shelf is widest (200 km) to 30 em elsewhere. Clarke et al. ( 198 6a, 198 6b) have formulated a model describing the large-scale, low-frequency response of continental shelf waters to synoptic-scale wind stress in terms of a sum of forced waves. This model includes friction and time dependence and provides an efficient method for calculating the response. They showed that to first order, the sea level response to wind stress on the West Florida Shelf consists of a sum of f orced waves traveling southward with the wind stress pattern and free waves propagating northward from the southern boundary at the Florida Keys. Almost all of the wind-induced energy on the West Florida Shelf is accounted for by wind forcing actin' g on the West Florida Shelf itself, but a small energy flux from the eastern Florida Shelf wave guide was also considered. Coastal tides are influenced by several factors and one of the most important of these is the character of the adjacent continental shelf. Clarke and Battisiti (1981)

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9 derived a simple method for estimating tides on smooth continental margins and explained the general trends in terms of semidiurnal and diurnal coastal tidal amplitudes. They suggested, in accordance with observations, that semidiurnal tides should be amplified on wide shelves in mid-and lowlatitudes, unlike the diurnal tides; since continental shelf tidal resonance occurs when the shelf scale ga./(ol-f2 ) (a= shelf bottom slope, W=tidal frequency) is approximately equal to the shelf width. Battisiti and Clarke (1982) applied their theory to describe barotropic tidal currents on "smooth" continental margins off the Atlantic and Pacific coasts. Their first model includes linearized bottom friction on variable topography H(x). The model shows that currents should be proportional to the coastal sea surface height and a linear function of the ratio: (distance offshore) I (water depth) Their results and observations showed that off the Pacific coast, where the continental shelf is narrow (10-30 km), M2 tidal currents are weak ( 0 0 2-0.08 m/s), highly elliptical (E-0 .1) [Here ellipticity is defined as the ratio: (minor ellipse axis)/(major ellipse axis), with positive (negative) values denoting clockwise (anticlockwise) rotation] and the semi-major axis oriented in the alongshore direction Unlike the west-coast M2 tidal currents, the tidal currents along the relatively wide Atlantic continental shelf (100 km) are much stronger (0.10-0.25 m/s). The semi-major axis of the current

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1 0 allipse is oriented perpendicular to the coast, with ellipticity -0.3 to -0.6. The semidiurn a l tides along both the Pacific and Atlantic coasts of United States are largely bar otropic. Using a tidal response analysis 1966; Cartwright et al. 1969), Mayer (Munk and Cartwright (1984) established a linear relationship between sea level and the north component of the depth-averaged tidal velocit y in the straits of Florida. This relationship was used as a one-dimensional model to predict barotropic tidal currents across the straits, and it can account for at least 70% of the variance in the diurnal and semidiurnal tidal band. In summary, variation s in sea level and currents i n Tampa Bay are driven by both local and coastal ocean forcing. The primary time series of t h e sea level or currents appear to be associated with the tides, the synoptic scale weather systems and with the seasons due in part to steric effects. Objectives of Study The entrance to Tampa Bay at the Egmont channel was established as a tidal current reference station in 1950 based on observations of 1948-50. Daily tidal current predictions are given for this reference station and prediction at secondary stations in the Bay are referenced t o these.

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1 1 Since the 1963 survey, the construction o f P ort Manatee, additional channel dredging, deposition of dredge spoil, natural shoaling and the construction of the Sunshine Skyway Bridge have changed the bottom configuration of the bay and hence the circulation. I n response to Mariners observations that NOAA's tide and current predictions did not reflect actual conditions experienced at certain locati ons in the Bay. The Tampa Bay Oceanography Project (TOP) was planned and conducted by the National Ocean Service (NOS) The measurement program o f the TOP included an extensive survey of currents, water levels, temperature, salinity, winds and other meteorological parameters. A portion of these data, collected from May 1990 to September 1991, are used in the present study. Given data on sea level and current variations, a question arises regarding the predictability o f these variations based upon tidal forcing alone. T h e objective of the present study is to determine t h e errors associated w ith sea level and currents predictions by tides alone and to account for the residuals based upon the other important causative agents. The analysis will focus upon data sets having the most extensive data available: sea level at St. Petersburg and currents under the Sunshine Skyway Bridge. The specific objectives as these pertaining to sea level and current are listed below:

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1 2 On sea level prediction 1. Predict the tidal heights for St. Petersburg using 14 tidal constituents by harmonic method, and then evaluate the root mean square error associated with this tidal prediction. 2. Evaluate the contribution of the wind stress to the root mean square error in the tidal sea level prediction. 3. Describe the seasonal variability of the sea level for Tampa Bay and its contribution to the tidal prediction error. O n currents prediction 1. Predict the tidal currents within the main shipping channel beneath the Sunshine Skyway Bridge by the response method, and determine the root mean square error associated with tidal currents prediction. 2 Evaluate the contribution of the wind stress to the root mean square error in the tidal currents prediction. 3. Discuss the root mean square error introduced by mean currents associated with buoyancy driven factors. The remaining four chapters are organized in the following manner. CHAPTER 2 on Methods and Techniques for Predictions introduces the techniques used for sea level and currents predictions. CHAPTER 3 on Predictability of Sea Level discusses the tidal prediction by the harmonic method in Tampa Bay. We first predict the sea level based on the known tidal constituents. Then the residuals are determined

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1 3 by subtracting the prediction from the observation for the same period. Important factors contributing to these residuals, such as winds, atmospheric pressure, seasonal variability etc., are discussed and evaluated. CHAPTER 4 on Predictability of Currents is organized in a similar manner as CHAPTER 3 except that the response method has been applied to predict the tidal currents based on the tidal constituents. The last chapter, CHAPTER 5, summarizes the findings and offers discussions on some of the points raised.

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1 4 CHAPTER 2. METHODS AND TECHNIQUES FOR PREDICTION Tidal Height Prediction The methods for tide prediction can be classified as harmonic and non-harmonic. By the harmonic method, the elementary constituent tides, which are represented by harmonic constants, are combined into a composite tide. This harmonic prediction is only used for a relatively few "reference stat ions" or "standard ports" where sufficient data are available. The majority of locations are usually presented as "secondary ports," for which predictions are produced by applying time and height differences and/or ratios to the daily harmonic predictions at the appropriate reference station. The technique for calculating these differences and ratios is the non-harmonic method. Parker (1991) gives a general review on the use of this non-harmonic technique. HARMONIC METHOD Under the equilibrium theory of tides, the height of a theoretical tide at any particular place can be expressed

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1 5 mathematically as the sum of a number of harmonic terms involving the relevant astronomical data. The actual tides, however, deviate from the equilibrium tides for several reasons. The depths of the oceans vary from the deep ocean basin to the coastal zones; the presence of land masses or the shape of the ocean basins constrains the tidal flows; the rate of the Earth's rotation is too rapid for the water masses to set up an immediate equilibrium tide; and the water movements are subject to the Coriolis force. Though the actual tides observed do not conform to the equilibrium tide, the observed tides can still be composed of the sum of a number of harmonic constituents having the same periods as those f ound in the tidal-producing force. It i s possible through the analysis of data at any place to obtai n certain constants which can be introduced into the theoretical f ormulas and thus adapt them for the computation of the tide for any des ired time. Schureman ( 1941) published detailed descriptio n of the harmonic method. He suggested, that in formulas expressing the height of the tide at any place, the theoretical coefficient of equilibrium theory may be replaced by a coefficient determined from an analysis of observations for that particular place. The harmonic analysis of tides is based upon an assumption that the rise and fall of t h e tide in any locality can be expressed mathematically by the sum of a series of harmonic terms having certain relations to astronomical

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1 6 conditions. The conceptual equation for the height Z of the tide at any time t may be written: Z = H0 + H1 cos(s1t1 ) + H2 cos(s2t2 ) +etc. ( 2. 1) in which H o is the height of the mean water level. The coefficients H1 H2 etc. are the amplitudes of the constituents. The coefficients of s1 s2 etc., represent the rates of change in the phase and are called the speed of the constituent and are usually expressed in degrees per solar hour. They are derived from astronomical data and are independent of the locality of the tidal station. The symbols and 2 refer to the initial phases of the constituents at the time when t equals zero. Both amplitudes (H1 H2 ) and initial phases ( 2 ) of constituents are called harmonic constants and may be derived from observed tidal data through harmonic analysis. Harmoni c analysis, as applied to tides, is the process by which the observed tidal data at any place are separated into a number of harmonic constituents. Harmonic prediction i s then accomplished by adding or totaling the elementary constituents. To predict the tide, we must first obtain the tidal constants by a harmonic analysis of the data from a tide gauge record. Traditionally, this has been accomplished in two ways: ( 1) by the procedure of solving for one constituent at a time and then removing the effect (sidebands) of constituents at nearby frequencies using "elimination" procedures, and (2) by a least squares solution

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1 7 Table 1. Tidal Table of St. Petersburg for 1987, Prepared by NOS. Tidal constants are obtained using Least Squares Harmonic Analysis Constit. 1-M2 2-S2 3 -N2 4-Kl 5-M4 6-01 7-M6 8 -MK3 9-S4 10-MN4 11Nu2 12-S6 13-Mu2 14-2N2 15-001 16-Lamd 17-Sl 18-Ml 19-Jl 20-Mm 21-Ssa 22-Sa 23-Msf 24Mf 25-Rhol 26-Ql 27-T2 28R2 29-2Ql 30-Pl 312SM2 32-M 3 33-L2 34-2MK3 35-K2 36-M8 37-MS4 H (ft.) 0.54 0.176 0.095 0.529 0.012 0.491 0 .009 0.011 0.003 0 .006 0.027 0.000 0 .031 0.023 0.007 0.019 0 .057 0 .001 0.01 0 0 .104 0.068 0.149 0.048 0.039 0.023 0.095 0.010 0 .007 0 0 1 7 0 .151 0 .008 0 .008 0 .028 0.012 0.080 0 .003 0.005 k' (degree) 36.02 51.05 28.03 328.19 280.13 315.51 105.67 227.53 344.75 264.08 43.34 302.50 225.99 286.9 4 14.72 40.58 60.37 250.22 25.94 332.26 247.03 138.95 67.47 303.86 348.63 303.53 41.31 357.13 283.49 337.47 318.97 114.95 74.65 248.63 49.08 27.22 292.86 K '-k (degree) 20.32 15.24 23.04 7.41 40.64 12.90 60.96 27.73 30.48 43.36 22.68 45.72 25.40 25.76 1.92 17.96 7.62 10.14 4.69 -2.72 -0.41 0 .21 -5.03 -5.49 15.26 15.63 1 5 .45 15.03 18.35 7 .83 10.16 30.48 17.60 33.22 14.83 81.28 35.56 K' (degree) 56. 3 66.3 51.1 335.6 320.8 328.4 166. 6 255. 3 15.2 307. 4 66.0 348. 2 251.4 312.7 16.6 58.5 68. 0 2 60.4 30. 6 329.5 246.6 138.7 62.4 298. 4 3.9 319.2 57. 3 12. 2 301.8 345.3 329.1 145.4 92. 2 281.9 63.9 108.5 328.4 Speed (de g /h) 28.9841 30.0000 28.4397 15.0411 57.9682 1 3 9430 86.9523 44.0252 60.0000 57.4238 28.5126 90.0000 27.9682 27.8954 16.1391 29.4556 15.0000 14.4967 15.5854 0.5444 0.0821 0.0411 1.0159 1. 0980 13.4715 13.4715 29.9589 30.0411 12.8543 14.9589 31.0159 43.4762 29.5285 42.9271 30.0821 115.9364 58.9841

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1 8 of normal equations, solving for all constituents simultaneously. Dennis and Long ( 1971) describe the first method. The second method was developed by Zetler and Cumming (1967, 1982). Since the St. Petersburg is a primary tidal station, the tidal constants s h own in Table 1 for the St. Petersburg have been provided by the NOS using their least squares harmonic analysis. If the tidal constants are not available for direct use for the place under consideration, the two tidal analysis procedures referenced above could be applied. The last column of Table 1 is the speed, which represents the angular speed of any constituent per unit of time This is usually given in degrees per mean solar hour. The time unit in the application of harmonic method is the mean solar hour. The values of the speeds of the different constituents hav e been calculated from astronomical data by formulas derived from the development of the tide-producing force. These speeds, compiled in Table 2 of Schureman' s (Table 1), are essentially constant for all times and places. Take constituent S2 as an example; its speed is 30.00 degrees/per hour. As one complete cycl e for the phase of S2 is 0 -360, so the period of this constituent is simply computed from dividing 360 by 30 which is 12 hours. Since the period of the S2 constituent i s half a day, it is called a semidiurnal tide and denoted by "2." The same notation is also applied to other constituents.

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1 9 PREDICTING TIDES With each constituent represented by a cosine curve with its own period, amplitude, and phase, the expected tidal height at any future time can be determined simply by adding together all the constituents for a given time, with appropriate correction for the long-period variations associated with nodal regression. Thus, the tidal height at any time t may be represented harmonically by the equation (2.2). z = H 0 + L jH cos[st + (V0 + u) -k] ( 2 2) in which z height of tide at any time t. Ho mean height of water level above datum used for prediction. H mean amplitude of any constituent. f factor for reducing mean amplitude H to year of prediction. s the speed of the constituent (degree per solar hour) t time reckoned from some initial epoch such as beginning of the year of prediction. (V o+u) equilibrium argument of the constituent when t=O.

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20 k epoch of constituent or the phase lag of the actual tide from the equilibrium tide. In the above formula, all quantities except z and t may be considered as constants for a particular year and place, and when these constants are known, the values of z, or the predicted height of the tide, may be computed for any value of t, or time. The harmonic method of predicting tides, therefore, consists essentially of the application of the above formula. The exact value of t for the times of high a n d low water may be obtained by setting the first derivative of equation 2.2 to zero, or from dz =-IsfHsin[st+(V0+u)-k]=O dt ( 2. 3) A comparison of equations (2.2) and (2.1) shows that the amplitude in equation (2.2) consists of a product between the factor f and mean amplitude H, and the initial phase of 1 or 2 has been replaced by k(V0+u) Explanations of these factors now follow. CONSTANTS AND PARAMETERS The constant H0 of equation ( 2. 2) is the depression of the adopted datum below the mean level of the water at the place of prediction. For places on the open coast, the mean water level is identical with mean sea level, but in the upper portions of tidal rivers that have an appreciabl e

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21 Figure 3. Intersections of the Planes of the Earth's Equator, the Ecliptic, and the Moon's Orbit with the Celestial Sphere. slope, the mean water level may be somewhat higher than the mean sea level. The datum for the predictions may be more or less arbitrarily chosen, but it is customary to use the low-water plane that has been adopted as the reference for the soundings on the hydrographic charts for the locality. For all places on the Atlantic and Gulf Coasts of the United States, this datum is mean low water. The astronomical tides are primarily forced by the sun and the moon, the orbits of which relative to the plane of the earth's equator are shown in Figure 3. The three great circles: the plane of the earth' s equator, the ecliptic (sun's orbit), and the moon's orbit, intersect. The angle m between the ecliptic and the celestial equator is known as the obliquity of the ecliptic and has a nearly constant value of 23.5. The angle I between the moon's orbit and the celestial equator might be called the obliquity of the moon's

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22 orbit. It varies between 18.5-2 8. 5 with a period of approximately 18.6 years. The 18. 6-year periodic motion of the obliquity of the moon's orbit introduces an important inequality in the tidal movement; it affects the amplitude of tidal constituent. Thus, node factor f is introduced to reduce the mean amplitude of the constituent H to the true amplitude fH The factor f for any single constituent, therefore, passes through a cycle of values. Because the change is slow, it is customary to take the value of the middle of t h e year for which the predictions are being made, and assume this as a constant for the entire year. The error resulting from this assumption is negligible. Each constituent has its own set of values for f, but these values are the same for all localities and have been compiled in Table 14 of Schureman's for convenient use (Table 2) for the middle of each year. The phase (
PAGE 35

Table 2. 23 Node Factor f for the Middle of Each Year, from 1970 to 1999. Prepared by U.S. Coast and Geodetic Survey. Constituent 1970 IWI 1973 IQ74 1976 1976 1m 19711 1 979 -----------------------------------1t. ..... ..... I. ISS K---------r :tos KJ ..... ,,__ ____ 1. 289 Lt----------------0 882 Mt................... 1.11117 O.Q&e Ma .......... _________ 0 950 M , MN............. O.Q34 1.'1 32 1.232 0 668 3.176 0.973 0.960 O.Q48 1.097 1 063 1. 150 1.118 I. el)3 0.983 o.ws 0.1167 1.051 1.029 1.055 1.270 1 012 0 .9Q5 o.m o QQ1 O.QQ5 O.QQI 0 957 1.014 1. 63.5 1.008 1.012 1.0111 0. 9.16 0 961 0.871 0.808 1.m 1.020 1.029 l..,, ,.,, .., 1. 030 Ma .... _____________ 1.046 M , MN -------1 .061 1.021 1.03 1 1.042 1.009 1.013 1.018 O.QQ7 0.994 O.QQ3 0.984 0.977 0 .11611 0.974 0.11112 0.949 0.1167 0. 961 0.936 0. 1164 0. 96-4 0 946 o. 9 4 7 o.m o.930 0.1169 0. 954 0. 939 M a-----------------1.092 1.063 1.027 0.11811 0 954 0.924 0.904 0.894 0.8116 0.910 M------------------1.125 1.036 0.986 0 939 0 .901 0 87 4 0.862 0.86-4 0.881 Ot, Q, 2Q .PI---------0. 858 0. 916 0. 979 I. 04 1 I 0911 1 140 1.168 1.182 I .I!Kl I. 161 oo __________________ o.SII6 o.m o.921 1.137 1.361 1 .600 1.706 1.778 t.ioo 1.668 MK .................. 0.941 0 .967 0.9116 1.022 1 .043 1.068 1.008 1.072 1.071 1.065 2MK--------------0.11611 0.987 1.005 1.019 1.027 1 .031 1.032 1.032 1.032 1.032 M L .-------------0. 716 0. 820 0. 949 I. 088 I. 221 I. 333 1. 412 1. 4 50 I. 443 I. 392 Mm----------------1.103 1.070 1.029 0.986 0.94 4 0.909 0.884 0.872 0.874 0.891 _ c_o_ns_t_ltu_en t ____ _1993 ],,,____ ___ ___________ I. 1 20 Kt .. -----"-------1.079 Ka------------------I. 203 Lt-----------------I 216 Mt-------------------I. 334 M, ,N, ,2N,X tt"'"' o.m M s ....... ----------0. 966 M,, MN ... 0.966 M a....... ............ 0. 9 32 Ma .... ............... 0. 911 O t, Q,, 2Q, P I --------I. 128 00.--------------I. 605 MK.................. 1.054 2MK.------------1 .030 ML................. 1.303 Mm ....... ___ ________ 0 9 1 8 I O!Kl 1.051 1.116 1.248 1.1611 0. 988 0 982 0. 976 0.1164 0. 1.081 1.296 1.038 1.025 1.184 0.9611 1.030 1.016 1.016 0.898 1.778 1.000 1.000 1.000 1.000 !.COO 1.02 4 1.072 1.016 1.015 1.048 0.998 0. 972 0.976 0.922 0. 801 1.829 1.013 1.019 1.025 1.038 1.051 0 960 0 8(;3 0 988 1.000 0 .910 1.042 0 914 0.937 0 .842 1.077 I. 282 1.02 4 1.0311 1.048 1.072 I.C98 0.897 0 688 0 969 0.982 0 :1!6 1.081 Factor 1 of MS, 28M, and MSC are each equal to factor I o r M1. Factor 1 o f P , Rt, s., s,, s,, s,, T , Sa, and Ssa are each unity. 0.8114 0.905 0. iSS 1.208 0 800 1.032 1.048 1.065 I .OQQ 1.134 0 .844 0. 934 0. 96-4 0.691 1.11 0 0.833 0.886 0.754 1.107 1.083 1.037 1.056 1.075 1.115 I.IM 0.812 0. 498 0.918 0. 952 0 .1136 1.128 o. 829 0 852 0 883 0. 897 0. 750 o. 772 0.921 0.893 1.487 1.038 1.034 1.057 1.051 1 076 1.069 1.117 1.106 J. I. 143 0.808 O .ll32 0.489 0 .538 0. 916 o. 928 0. 951 0. 959 0. 629 0 669 1.130 I. 117 0.896 0 926 0 8 2 1 1.096 I. 214 l.OZ7 !.OW !.OM 1.08 2 I. Ill 0 879 o.rJ43 0. 950 0 9 7 6 0 7S2 1.091

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z 0 Figure 4. k Water t Cosine Curve for One Tidal Constituent with Amplitude of fH. 24 water phase for that constituent is known as the phase lag or epoch of the constituent and is represented by the symbol k in equation (2.2). Bottom friction a n d geomorphology of the basin and adjacent land masses generally account for this. The other part of the initial phase is the equilibrium argument of constituent at t=O. In Figure 4, the vertical line through T may be taken to indicate any instant of time under consideration. If t=O represents the time of the high water for the equilibrium tidal constituent, the phase of the constituent argument at time T is reckoned from t=O and is expressed by the symbol (V0+u) The V0 or uniformly varying portion of the argument, refers to the initial epoch, while the u, or slowly varying portion of the argument, is due to changes in the longitude of the moon' s node.

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25 The amplitude H and the epoch k for each constituent tide to be included in the predictions are the harmonic constants determined by harmonic analysis or least squares harmonic analysis of the observed tides. Each place will have its own set of harmonic constants, and when once determined, these will be available for all times, except they are slightly modified by a more accurate determination from a better series of observations or by changes in the physical conditions at the locality, such as may occur from dredging, by the depositing of sediment, or by other causes. GREENWICH EPOCH The quantity (V0+u) are the values of the equilibrium argument of constituents at the initial instant from which the value of t is reckoned. These values, mentioned as local values of (V0+u), are different for different constituents, tide stations and the initial epoch in tide analysis or prediction. Since it is impossible to tabulate for each tide station around the world, the values of (V0+u) have been modified to be easily adapted t o other initial epochs and the locations. The main ideas behind this modification are that the values of (V0+u) for the Greenwich epoch can be easily calculated from the astronomical data. The difference between the true local values of (V0+u) and their values o f Greenwich epoch may be added to the k of equation (2 2) resulting in

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26 k'. This k' can be easily obtained through tidal analysis using the computer program provided by NOAA. After the modification of k, the practical expression for the angles of equations (2.2) may now be written st + (V0 + u)-k = st + Greenwich(V0 + u)-k' ( 2. 4) where t is the local time pertaining to the tide station which the harmonic method has been applied. The values of Greenwich (V0+u) have been tabulated in Tables 15-17 of Schureman ( 1941) Thus, by applying the corrections indicated in (2.4) to the k's for any station, a modified epoch is obtained. These will remain the same year after year and permit the direct use of the tabular Greenwich(Vo+u) 's in determining the actual constituent phases at the beginning of each calendar year. The equation (2.2) is now: z = H0 + LfH cos[st + Greenwich(V0 + u)-k'] ( 2 5) for the height of the tide at any time. As an example, let us find out the tidal constants for semidiurnal constituent M2 The beginning o f our prediction is 0000 on January 1, 1 987. Looking through parameters provided by NOS (Table 1), the speed s is 28. 9841 degree/hr, with a mean amplitude H of 0.54 ft and a k' of 56.3 degrees. The mean sea level H0 is also provided by NOS along with these parameters; it is 4.572 ft.

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Table 3 --C o n stituent 1070 1 071 1072 . -270. 0 1.8 03. I 1--. ------------1 3 4 15.5 17. 1 t --------------207. 2 211.6 214. 8 ------------262 8 60.3 245.2 1 -------------155. 2 4 8.6 307. 8 1 -------------------165. 4 266 8 8 0 J ---------------68 1 40 2 11.0 ----------330. 0 173.6 1 5.0 -------------------1 36.3 80 4 23.0 -------------301. 7 347. I 31.8 1 -------------------270. I 282.7 295 2 !. ...... ... ...... . ...... 14. 8 298. 7 222.4 :>:: ::::::::::::::::::::::: 150. 7 249.0 347. 8 68.7 326. 4 232.6 ------349 8 350. 0 350 2 ------------------2.55. 4 205. 0 276 0 1---------------....... 0 1 281.0 202.2 1 ---------------------1 n 8 tn.6 tn. 3 ---------------------180 0 180.0 1 80. 0 4 .'----------------0 0 0 0 0 0 ---------------2 2 2.4 2.7 ----------------285. 6 107.4 100.1 -----------330 0 172. 1 14. 0 ------1 ----225. 3 156. 1 86 8 -------210. 6 136. 4 66.6 K ....................... 178. 0 282.3 25.1 .fK ....................... 3 1 7 4 158. 1 3 58 8 N . ..................... 7 6 6 189.5 303. 2 s ........................ 166. 4 266.8 8.0 2S M ... -----........... 104.6 9 3 2 352.0 (_ _______ ____ _ ______ ____ 44. 0 308.7 212.4 sr.. ------------------194. 6 93. 2 3 52.0 m .......... 255.3 344. 0 72 8 -------------280. 2 280 0 270. 8 a.-------------------200. 6 200. 0 100. 5 ---Equilibrium Argument (Vo+u) for the Meridian of G reenwic h a t t h e Beginning o f Each Ca lendar Year, 1979 to 1989. Prepared b y U S Coast and Geodetic Survey. 1 98 1 11982 1983 1 1984 1 1985 1986 1987 1088 1975 1 9 7 6 t9n 1078 1970 1 980 ---------. . . . . . . 0 1 07. 4 286 6 1 4 3 100. 2 198. 0 281.8 4 0 89 1 188. 0 276. 5 5 6 95 8 201.0 292. 7 24 7 8 10. 0 10.2 1 8 2 16. 2 14. I tO. A 6 8 3 7 2 5 1.5 1.5 2.4 4 9 6 0 0.2 11. 0 218. 4 2 1 8 2 2 1 5 8 2 1 1.5 207. 6 200.9 194.2 1 88 2 185. 6 1 83. 1 1 82 6 1 84.2 1 89. 3 1 0 3 5 198. 3 203. 3 81. 8 282.4 00. 0 276. 4 00. 7 207. 0 1 3 0 9 310 2 116. 6 313. 4 159. 2 350.3 135 8 :1.14 5 188. 0 25. 5 234. 0 1 58. 7 61.8 302. 5 1 98.6 135. 7 41.8 291.5 170.8 8 7 9 30.0 291.6 1 71.7 82 0 31.2 200 1 84.5 1 85 3 285.8 26. 0 1 01.8 201. 8 3 01. 9 4 2 0 117. 0 2 1 8.4 3 1 0 1 60 0 1 36 8 238. 1 330.0 81.0 306 8 m 9 248. 6 219. 0 152.7 122. 7 02.8 63 0 356 8 327. 6 298. 6 270 0 205 2 tn. 2 149 3 121.6 169. 1 1 0 5 211. 5 52 1 203. 6 4 3 7 243.8 84.0 235. 8 76.7 278. 1 120. 0 273. 6 116.2 3 1 0 1 162. 1 253. 6 8 1 3 7 2 78.1 305. 3 245 6 185.7 126. 0 3 53.7 295. 1 237. 2 180. 0 50.3 354. 3 298.7 243. 1 338. 2 21.0 63. 0 104. I 47. 1 87.3 127.6 168. 0 111.6 153.4 196.2 240 0 1 87 I 232. 4 278.2 324. 1 270 0 282 0 293. 8 305. 3 279 3 200 6 301.0 313. 3 287. 4 200. 2 3 11.2 323 4 298 4 3 11.0 323. 7 336. 4 4 1 8 7 3 01.8 224 6 06. 8 10. 4 302.0 22 4 7 97 0 20 0 303. 2 226 7 1 00 0 23. 8 307. R 231.9 61.7 162.0 263. 6 6 6 85 7 1 01.3 206 0 41. 6 119.4 221. 1 3 21.4 60.8 1 3 4 2 232. 6 330 7 68. 8 163.9 64. 6 3 21.1 2 1 2 6 126. 6 0 8 253.0 130.2 57. 6 3 1 3 6 2 1 3 9 117 6 61.0 318. 5 227.0 135 6 349. 5 3 49.7 350. 0 350. 2 349. 5 3 4 9 7 3 49.9 350 2 340. 4 3 4 9.7 3 4 0 9 350. 2 3 4 9 4 3 4 9 6 3 4 0 0 360 1 24 7 2 258. 7 2 71.6 285. 8 263. 2 280.1 297 0 312 0 289 0 3 01.9 3 13.5 32 4.2 295 8 305. 4 3 14.8 32 4 2 72 6 4 270. 5 205. 1 80 7 8 8 297 0 224 3 08. 5 22 7 305. 6 2:?1. 5 07. 4 1 8 3 200. 0 2 1 0 6 178. 0 171.8 177. 5 1 77.3 178. 0 177. 7 tn.5 177. 2 178 0 ITT. 7 177. 4 tn. 2 177.9 11i 7 tn.4 177. 2 1 80 0 1 80 0 1 80 0 180 0 180. 0 180. 0 1 80.0 180.0 1 80 0 1 80 0 1 80 0 1 80.0 180.0 180.0 1 80 0 1 80 0 0 0 0 0 0 0 0 0 0.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o 0 0 2 0 2.2 2 6 2 7 2 0 2.3 2 5 2 8 2 0 2.3 2.6 2 8 2.1 2 3 2.6 2 8 7 6 278. 8 189. 8 1 00 6 358. 2 268 8 179 4 90 0 347. 8 258.8 170.0 81.4 3 4 0 0 0 163. 9 7 6. 0 1 67 0 8. 4 209. 7 50 7 202. 8 4 3 6 244 4 85 3 237.6 78 8 280. 2 1 21.9 275. I 117 2 3 1 9.4 1 61.6 3 41.5 271. 7 201.7 1 31.4 25.3 314. 0 244.4 174. 0 68.0 368. 0 288.2 2 1 8 6 113. 5 44. 3 335.2 266. 2 3 1 8 7 2 48. 4 170. 4 Ill. 9 0 2 304. 3 239.4 173. 6 89.6 0 7 200. 5 2 1 9 4 110. 0 38.8 326.4 254. 0 103. 6 204. 4 304. 0 42.2 115 9 2 1 2.3 306. 7 4 5 7 120. 4 210. 8 320.6 0 2.4 141.7 24 5 0 3 l 8 8 02.6 150. 0 351.4 103. 3 35 9 180.4 33 2 237.0 8 0 4 233. 3 76 3 276.6 117.6 288. 6 100. 3 300 9 1 60 5 354.5 107.2 2 1 9 5 331.3 21.1 1 32 4 243.8 355 3 45.3 1 5 7 6 242.2 23. 4 7 6 I 1 89 1 303. 2 8 4.5 1 8 5 3 285 8 26. 0 1 01.8 201.8 301.9 4 2 0 117 9 218 4 319 1 60 0 136. 8 238 I 339. 6 81.0 275.5 1 7 4 7 7 4 2 33 4 0 258. 2 158. 2 58. 1 3 1 8 0 242. 1 141.6 4 0 0 300. 0 223. 2 121.9 20 4 270. 0 141.1 41.3 298. 8 1 93 0 110. 4 350 3 2 48.0 138.8 50. 0 310. 2 2 1 6 2 118 4 4 8 4 3 1 3 0 218. 1 123. 4 275. 5 174. 7 7 4 2 33 4 0 258. 2 1511. 2 68. 1 3 1 8 0 242.1 141.6 4 0 9 300 0 223. 2 121.0 20. 4 279. 0 1 7 4 6 263. 3 352. 0 80.7 182. 5 271.2 0.0 8!1. 7 190. 5 279.2 7 9 06. 6 1 08. 4 287 1 1 5 9 104. 6 280 5 280.3 280. 0 279. 8 280. 6 280 3 280.1 270. 8 280. 6 21l0 3 280.1 270. 8 280 6 280. 4 280.1 270. 9 201.0 200 5 200 0 100. 6 201. J 200. 6 200.1 100. 6 201 1 I 200. 6 200 2 1 90.7 201.2 200. 7 200 2 1 90 8 1980 0 222.8 14. 8 210 0 16ft. 0 170. 8 158. 1 5 7 1 316. 2 114. 2 272. 3 311.7 105. 3 141.6 71.2 3 4 9 4 295 3 811. 0 1n.9 1 80 0 0 0 2 1 33 4 8 315. 0 1 61. 4 145 0 1 72 H 301.4 1 00 8 168. I 201.0 54. 8 201.9 206. 4 280 e 201.3 N

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Table 4 Difference t o Ad apt Tab l e 3 t o the B eginning o f Eac h Calendar M onth. Prepared by u .s. Co a s t and Ge odetic Survey. Month or yaar C:oustltuent I Apr. May 1une 1uly Auc. SepL Oct. ---------------------. 0 0 0 0 0 0 0 00 67 108. gg 184. 68 248.08 321.65 23. 17 116. 74 174.31 Zlll 82 311 .311 1:1.111 0 .00 30.68 58 15 88 71 118. 28 148. 83 178.40 208.118 239 .61 :!ell 08 21111. 54 329.21 0.00 81. 11 116 31 177. 42 236. 66 297.66 368.80 67.111 1111.02 178. 18 2311. 27 2118. u 0 .00 11.19 52.33 81.54 82.02 111.21 111. 71 120 .110 130 W 161).611 1611. 78 180 29 0 .00 346 54 7. 3 2 362 .86 3.50. 48 338 .02 333 54 3111. 18 304 .72 302.34 287 88 286 60 0 .00 312.W 0 73 34 2 83 337 11 3111. 20 313 47 295.68 277 .65 271 .113 264 01 248.29 0 .00 324. 17 I. 411 66 314 22 278 .311 266.116 231.12 183 86 148.02 136.68 0 .00 306.211 2. 24 291.33 237 .511 ;;!20.42 1118.88 112.114 116. 78 42 04 24.87 0 .00 288.35 2 08 291.33 268. 44 1116. 79 173.911 102.24 30 .611 7 70 206.06 273.16 0 .00 252.62 4 48 257 00 222.66 115 1 R 80. 86 333. 37 226.811 1111.66 84.111 411.74 0 00 2111.69 6 .117 222.66 17ft. 88 33 .611 347 .80 204 .411 61.18 16.40 232 10 186 32 0 .00 2711.18 310 66 229 .82 188 42 1 06 68 62.18 341.34 2611.60 217 .11 136 27 112.87 0 .00 234 14 2flll 82 133.117 58 .82 292.77 217 .42 Ill. 57 326 .71 260 36 124. 61 411.18 0 .00 2113. 112 303 34 236 .118 195 .114 129 68 8S. M 22. 10 316. 78 274 77 208 30 167 37 0 .00 127 .411 172 97 300.46 40 .111 Jfi8.10 268.211 35 76 163.24 263 311 30.811 131.04 0 .00 329.44 301.86 271.29 241. 72 211. 17 JRI.60 151.04 120 .49 IIO.P2 60. 36 30 .79 0. 00 248. 60 252 60 141. 11 fJII. 14 316 75 243 78 132.311 20.119 308.03 1116. 63 123.67 0 .00 203 69 201.67 4 6.28 300 34 143.113 311. 02 242.61 86.20 341 28 184 87 711.116 0 .00 30.66 68 .16 88 7 1 llll 28 148.83 178 40 208 .118 2311. 51 2fili .08 m M 3211. 2 1 0 .00 0 00 0 00 0 00 0 00 0.00 0 .00 0 .00 0 .00 0 00 0 .00 0 00 0 .00 3211.44 301.85 271.29 2U.72 211 17 181.60 161.04 120 .411 110.112 60 .36 30.711 0 00 314.118 JW. l 6 2114. 15 232.20 187. 19 155 .24 110 22 65 .21 33. 28 348 .24 316. 211 0 .00 288. 2.118 2111. 3.1 268. (4 1116.711 1 7 3 90 102. 24 30.611 7 70 296.06 273 18 0 .00 333.36 53 82 27.18 36.24 9 60 18. 66 352.02 326 .311 334 .44 307 .81 3 1 8 .87 0 .00 302. 81 366 66 2116.47 277 .116 220 77 200.211 143 .111 85 .87 65 36 8. 17 347 .66 0 .00 354 7 3 69. 64 54 37 72. 50 87 .23 86.36 80.08 74 81 112.113 87 66 106 .711 0 .00 267 .711 304 83 202.62 150 16 47.118 366 60 253 .211 161.08 116.62 366 .41 303 .116 0 .00 2 4 3 33 312. 1 5 196.48 140 54 23.117 3211. 1 3 212.47 116.80 40.118 284. 211 229 .46 0 00 324 17 I 49 I 326 .66 314 .22 278.311 288 .115 231.12 1116.30 183.86 148. 02 136 .68 0 00 36 .83 J58 6J I 34 34 4 6. 78 81.61 113. 06 128. 88 164. 70 176. 1 6 2 11.116 223 4 2 0 00 116.114 114. S2 211. 76 282.34 Ill 27 89.86 186. 711 283.73 364 .31 Ill 26 181.83 0 00 35.83 368. 51 34 34 46.78 81 81 113.06 128.88 154 70 176. 16 211.116 223 4 2 0 00 46 .02 60.84 116. 86 127.80 172.111 204.78 2411.711 2114. 711 328 7 4 II. 76 4 3 71 0 00 30.66 68 16 88. 71 118. 28 148 83 178.40 208.118 2311. 61 2811.09 21111. 54 3211.21 0.00 81. 11 116.31 177 42 236.68 2117. 66 368.80 67 .111 1111. 02 178. 18 2311. r. 2118.41 I 'Tbls table was dMicned lor direct we lor common year.o For a leap tear tbe values ror the months or to December, Inclusive, apply to the last d a y or the premonth, but may be used directly, provided an allowance I s made In t e dAy or month as Indicated In tbo lollowlnS t a b l e 1 he llrst line lor constituent M,,lves the dlllerence as based upon the rormula In table 2; the second line e lves tbe ltl'erences as derived lrom tbe half speed o! oorutltuent Ma. The dlllerenOIW lor constituents Sa, So, s., etc., are eacb uro lor every month. N 00

PAGE 41

Table 5. Differences to Adapt Table 3 to the Beginning of Each Day o f Month. Prepared by u s Coast and Geodetic Survey. D a y of month Constituent I I 2 a 4 6 6 7 8 9 10 11 ---------------------------------0 0 0 0 0 0 0 0 0 0.00 14.05 28 1 0 4 2 15 56 20 70 25 84. 3C 98. 35 112.41 126. 4 6 140.51 0.00 0 99 1.97 2 96 3 0 4 4 03 6.01 6 00 7.88 8 87 9 86 0 00 I. 97 3.94 6 .91 7 88 9 86 11.83 1 3 80 15.77 17 .74 19 7 1 0.00 3 4 8 88 337.37 326.05 314 .73 303. 4 2 292 10 280.78 2G9.47 258 .15 Zl6 84 0.00 347.92 335.84 323.76 311.68 200.60 287.52 275.44 263.37 2.Sl. 29 239. 2 1 0 00 347.81 335.62 323. 4 3 311.24 200.05 286.86 274. 66 2ti2. 4 7 250 28 238.00 0 00 335.62 311. 2 4 286. 86 262. 4 7 238.00 213 7 1 1 80.33 1 64.95 140 .57 116.1 8 0.00 323.43 286.86 250.28 213. 7 1 177.14 140 57 103.99 67.42 30.85 354.28 0.00 311.24 262.47 213.71 164. 05 116. 1 8 67. 42 18 66 329. 00 281. 1 3 232.37 0.00 286 86 213.71 140.67 67.4 2 35 4 28 281.13 'Jm.OO 134.84 61.70 348.56 0 00 262. 47 1 64.95 67.4 2 329 00 232.37 134.84 37.32 2'.10. 79 202.27 104.74 0.00 322.55 285.11 247.66 2 1 0.21 1 72.77 135.32 07.88 60.4 3 22.98 345. 54 0.00 300.49 258.98 20!!. 47 1 67.95 1 07 .44 56. 93 6. 42 315.0 1 265.4 0 214. 8 0 0.00 33 4 .63 300.27 283.00 258.63 233. 1 6 'JJJ1. 80 1 82 43 157.00 1 31.70 1 00.33 0.00 27 .34 64.68 82.02 100. 35 1 36 69 164.03 1 91.37 218 .71 246. 05 273. 38 0.00 -0.99 -1.97 -2.96 -3. 94 4 93 -6.01 -6.00 -7. 88 -8. 87 -9.86 0.00 321.6 7 283.14 244. 70 206.27 167.84 129.41 OO.Il8 52. 54 14. II 335.68 0 00 257.0 1 205.5 1 154.01 102.5 1 61. 02 359.52 3011.02 256.63 205.03 0 00 0 .99 1.97 2.00 3.94 4 .113 5 .01 6.00 7 88 8 87 9.86 0 00 0 00 0 00 0 00 0.00 0.00 0.00 0.00 0.00 0 00 0 00 0.00 0 .99 1.97 -2.96 -3. 94 4.93 -5.91 -6.00 -7.88 8 87 -9. 86 0 00 346 93 333.87 320.80 3(11 7 4 294.68 281.61 268.64 255. 4 8 242. 4 2 229 35 0 00 311.24 262.4 7 2 13.71 164.95 116. 18 67. 4 2 18.66 329.00 28 1 .13 232. 37 0 00 324 30 288.60 252 9 1 2 1 7.21 181.5 1 145 8 1 IIO 11 7 4 4 2 38.72 3.02 ::::: :::::::::::::::::::::: : : 0 00 323 .32 286. 63 249. 95 2 1 3 27 1 76.68 1 39 00 103.21 66.53 211.85 363 1 6 0.00 336.60 3 13.21 289.81 266 4 2 243.02 2 19.62 100.23 1 72.83 140 .44 1 26.04 2MK-------------------------------------------------------0 00 3 1 0.25 260.60 2 1 0.76 1 61.00 111.26 51.61 11. 76 322.0 1 272.26 222. 5 1 MN. ---------------------__ ----------------_ ----__ ----______ 0.00 298.17 236.34 1 7 4 .62 112.6 9 60.86 3 4 9 03 287. 20 225.38 1 03 56 101.72 MB-------------------------------------------------------0 00 335.62 311. 24 286.86 262. 47 238. 00 2 13.71 189.33 164. 95 140.57 116 1 8 28M---------------------------------------------------0.00 24.38 48.76 73.14 97.63 121.91 146.29 170. 67 1 95.05 219.43 243. 82 M l ---------------------------------------------------------0 00 26.36 62.71 79.00 1 06.41 131.76 1 68. 1 2 184.47 210. 82 237. 1 8 263.53 M B f----------------------------------------------------------0 00 24.38 4 8 76 73.14 97.63 121.91 146 .29 170.67 195. 05 219.43 243.82 Mm--------------------------------------------------------0.00 13 07 26.13 39 20 62.26 65.32 78 39 91. 4 6 104. 62 117.58 130 65 B --------------------------: ______ _____________________ _______ 0 00 0 99 1.97 2.00 3 .94 4 93 5.91 6 00 7.88 8 .87 9 86 Sea ________________________ __ __ __ __ ___________________ _________ 0 00 1.97 3 94 6.91 7 88 9.86 11.83 13.80 15.77 17.74 19 .71 Tbe table Ill adapted directly tor use wltb common years, but It tbe required date f alls between Mar. I and Dec. 31, Inclusive, In a Jeep year the day o r month should be lncr86Sed by one belor e entarlng tbe tab l e tTbe first line l o r constituent M, gives the ditferen oes as based upon the lormulal n table 2 the second line gives the dilferenoes as derived from the hall speed of constituent M t Tbe dltrerenoes for constituents 8,, s , s,, S., etc., are each e r o t o r tbe beginning o f every dar. N \0

PAGE 42

Table 5. Constituent (continued) -====== ================================== = 12 1M 58 10.84 21.68 52 '01.13 226 .110 91.80 317. 70 183. 61 276 .41 7 21 308 09 1114. 37 80.116 300.72 10 84 297.26 163 63 10. 84 0 .00 -10.114 218 28 183.81 327.32 48 102. 86 1 72.76 :w. 89 91.80 268 .20 2119. 88 268.20 143. 7?. 10 84 21.68 13 1 88 81 11.63 23.fl6 224. 20 215 .06 213 71 6 7 42 :!81. 13 134. 84 202 .27 2110.611 270 .114 113 .86 66 5 9 328 06 -11. 83 268.81 102. 03 11.83 0 .00 -11.113 203.22 134. 84 291.62 2711.80 711. 1 23 02 338 .06 67 .42 292. 68 31.23 292.68 l r.tl.78 11.83 23. M H 182 M 12.81 2 5 63 212.88 202 97 201.52 43 04 244. 58 86.08 129 .12 172 .18 233 .20 113. 35 30 23 366 .40 -12.81 120 .38 60. 64 12.81 0 .00 -12. 81 1110. 16 811. 08 243 II 65.86 73 27 276. 24 43.04 318 .116 342.511 316.11'1 169 84 12 .81 26 63 1 6 1116.71 13.80 27 .60 201.57 1110. 89 189. 33 t 8 M 207.99 32 66.98 74.64 1 95 75 4 86 22.74 -13. 80 181.95 35 9 04 1 3 .80 0 00 -13.80 177. 09 37 32 220 .23 2011. 43 32.411 23 52 214.41 18.66 8.94 341.34 182.91 13.80 27.60 Oay of month 1 6 210 76 14. 78 29 57 1110. 26 178.81 1 7 7 .14 JM. 28 171.42 348 58 342 83 337 II 168 .30 322 .3.1 339 49 60.08 -14. 78 lt3. 52 307 M 14. 78 0 .00 -14.78 184. 02 348 00 184 63 189 9 06 333. n 152 68 JM. 28 5 i 2 35.29 5. 72 195 .116 14. 78 29.57 17 224 81 15.77 31.M 178. 94 166 73 184.95 329.110 134. 84 299 .711 289.611 239. 58 120 .86 271.82 314.13 n.42 -ts. n 106 .09 2.'16. 06 1s. n 0 .00 -1s n 160 .116 299 .711 148.83 t:J-1.06 345.67 284 .02 110.75 329 .110 30 .10 61.84 30. 10 2011. 04 ts. n St. 84 18 238.811 16.76 33 .51 167. 62 164.66 76 305.52 116. 27 261.03 1116. 64 142.06 83 .41 221.30 268 76 104.75 -16.78 66.66 204.65 16.76 0.00 -16.76 137.110 261.03 113 13 116. 38 322 27 234. 27 28.92 306.62 64 48 88 00 M .48 222.10 16 76 33 .51 19 262.91 1 7 74 3&.48 158 30 142. 57 140. 57 281.13 81. 70 202. 27 123.40 44.63 45.116 170.711 263.311 132.09 -17.74 28.22 06 17. 7 4 0 00 17 74 124. 83 202.27 77 .44 69.1!11 2116. 88 184 62 327 10 281 .13 78.87 114. 35 78.87 235. 17 17.74 35 .48 20 2116.116 18.73 37 46 144.99 130 49 1211.38 256.76 26. 13 163. 60 60.28 307 .01 8 62 120 28 238. 02 18.73 349.79 101.66 18.73 0 .00 -18.73 Ill. 76 163. 00 41.74 23. 01 276 48 134 78 286 27 266.75 103. 140 .70 103.26 248. 24 lA. 73 37 .46 21 281.01 19.71 43 133. 67 118 41 116 19 232. 37 348 58 104.74 337 II 2011. 48 3.11. 07 oo. n 212 M 186 77 -19. 71 60 .06 Ill 71 0 .00 -19.71 116.70 104. 74 6 04 346.33 262.08 86.03 203 .44 232 37 127. 63 167. 06 127. 63 281.30 19.71 311.43 22 296 011 20.70 41.40 122.36 106 33 103.99 207.99 311 .116 66 .116 263.97 Ill. 95 293 .62 19. :JII 187.29 214 .11 -20. 70 272 .112 358.58 20.70 0 .00 -20. 70 86 84 66.98 330. 34 309 84 2211. 611 35 28 141. 61 207.99 1 52 01 1113. 41 1 52. 01 274 3G 20.70 41.40 "The table b adapted directly ror use with common yean, but If the required date ralb between Mar. 1 and Dec. 31, Inclusive, In a leap year tbe day of m on th be In creaaed by ooe bef ore the table !The ftnt line f11r comtituentM, gives the di1Terenoe1..., ba.o;ed upon the formula In tabl e 2 ; the oo d line rives the di1Terenoa1 L'l derived fr om tbe h alf o r constituent Mr. IT be diJJerenoea for comtltuents s,, Br, s , s,, etc., are each &ero for tbe bellnnlnr ol every day. VJ 0

PAGE 43

Table 5. (continued) Day of montb CollliUluant I 23 24 24 211 71 28 211 ao 31 32 -----------------------------0 0 0 . 0 0 30Q. II 323 .18 337 .22 361.71 6 32 Hl. 37 33 t2 4 7 47 61.52 75 5 7 21.811 22.87 23 .68 24. 84 24. 83 211.81 71.110 28.68 29.57 30. 58 43 37 46 34 4 7 31 49.28 61. 25 63. 22 65.20 67 17 59. It 81. II lll.()f 119.72 88.40 n.ou 65 .77 54. 4 6 43.14 31.82 20 50 II 19 94.24 82.18 70.10 68.02 46.94 33.86 21.78 II. 70 357 62 345 .64 n80 79 .81 87 42 65.23 43. ()f 30.85 18 .86 6 47 3M 28 342 00 183 .61 159 23 13t 84 110 46 86 .08 61.70 37 .32 12 .94 348 58 324 17 276 .41 238.84 202 .27 1115.611 1211.12 92.65 65.118 Ill tO 312 .1!3 306 .26 7 .21 318 4 6 269. G9 220 92 172 16 123. 4 0 74.84 25. 87 3:17. II 2!18. 35 190.82 117.811 4 4 .63 331.39 268. 24 1 85. 1 0 Ill. 96 38.H1 325.66 262 52 14.43 278.90 1 79 .38 81.85 344 32 246.80 1411.27 61. 7 5 31t. 2'2 216. till 256.18 218 73 181.28 1 43 84 108.39 811.94 31.60 354 05 3 1 6 .60 2711. 16 328. 76 718. 24 227. 72 tn. 21 1211.70 78.111 25.811 33 5.17 284. 66 23t 15 181.112 138. 68 Ill. Ill 85 .82 eo. tS 36.0U 9 72 344 .35 318 .00 2113.62 241.4 6 2118.78 2118.12 323 .48 350 .80 1 8. It ts. 411 72 .81 100 .15 127. til -21.811 -22.87 -23.86 -24.84 -25.83 -211.61 -27.110 -28.511 -211 .57 -30. 58 234.411 1116.08 167 83 IJ9. 20 80.78 42.33 3 .90 324 .47 287. ()f 24ij. 60 am.08 265.67 204 07 152. 57 101.07 411.68 368 .0S 306 .68 255 .0Q 203 .w 21.811 22.87 23. M 24. 84 25. 83 211.81 27. 60 28 .68 29. 5 7 30.58 0.00 0 .00 0.00 0.00 0 .00 0 .00 0 .00 0 00 0 00 0 .00 -21.811 -22.87 -23. oe -24. 84 -24. 83 -211.81 -27. eo -28. 68 -29.57 30.56 72.67 611.60 48.44 33 38 20 .31 7. 24 364. 18 3 4 1 12 328 .05 314.118 7 .21 318. 46 269./ill 220 .112 172. 18 123.4 0 74.84 25 M7 337 II 288. 35 294.84 248.96 223 .24 1 87.65 161.85 JIG 16 80.46 44. 76 9 .011 333 36 712.116 23a. 28 1 00 .69 1211.22 89.64 62.86 16.17 339 49 302 .81 :106.29 181.90 168.60 136.10 Ill. 71 88.31 64.92 41.5 2 18. 12 364 .73 346.63 2118. 78 248.03 1116.28 146.64 96.711 47. ()f 357 .29 307 .64 257 79 111.78 17 .116 3111. 13 264.30 192.4 7 130.84 88.82 e oo 305 16 2t3. 33 18J. 6 1 1611.23 134 84 110 48 86.08 81.70 37 .32 12 .94 3411.68 32t. 17 176 39 200.77 226.18 249 .64 273.112 2118.30 322.811 347.08 11.44 3 5.1!3 2111. 76 248.11 272 .47 298. 82 325.17 351.62 17. 88 44.23 70 .58 96.64 178 39 200 .77 226 18 249 .64 273 G2 2118.30 32'l. 811 347 06 II. 4t 35 .83 'JB7. 43 aoo. 60 IIS.68 3211.62 339 811 352 78 5 82 18. 88 31.115 4 5 02 21.68 22.87 23 M 24 .84 25.63 211.81 27.110 28. 58 29 67 30.56 43.17 46.14 4 7.31 411.28 61.24 63.22 65. 20 67.17 59. u 81. II I The table Is adapted dl.reeUy lor WICiwltb common y-. bullf lberequlred date fall bet wilD Mar. 1 and Dec 31. Inclusive, In a l1>11p year the day of month sboulll ho i u rreased by one before enterl111 tbe teble. tTho ftrst line lor OODIUluent M o r,voa the ditfeceoooa a.a b...OO upon tbe f ormula In Tab le 2; the second II no the ditfero nces as derived !row tbe ol co n sti tuuut Mo. !Tbe d l lfereuceeloc oonatltuen&l , a , a,, a,, etc., are c.b -o lor tbe beclnn lnK of every day. VJ -

PAGE 44

Table 6 Differences to Adapt Table 3 to the Beginning of Each Hour of Day of Mon th. Prepared by u.s. Coast and Geodetic Survey. Hour or day Constituen t I 0 I 2 3 6 6 7 _8_1 _9_1_1_ o _1_1 1 0 0 0 0 0 0 0 0 0 0 00 1 5 59 31.17 46.76 62.34 77. 93 93 .51 109 10 1 2 4.68 140.27 155.85 1 71.44 0 .00 15.04 30. 08 4 5.1 2 60.16 76.21 90.25 105.29 120 .33 135.37 150.41 165 .45 0 .00 30.08 60.16 90 25 120.33 1 50 .41 180.49 210.57 240.66 270.74 300.82 330.90 0. 00 29.53 59.06 88.59 118. 11 147 64 177. 17 206 .70 236. 23 265. 76 295.28 324.81 0 .00 14.50 28.99 43. 49 67. gg 72. 48 86.98 101.4 S 115.97 1 30 47 144.97 1 59.46 0 .00 14. 49 28 98 43.48 67.97 72. 46 86 95 101.44 115.94 130.43 144. 92 1 59.41 0 .00 28.98 57.97 86 .95 115.94 144 .92 173.90 202 89 231.87 260.86 289.8 4 318.83 0 .00 43. 48 86. 95 130 43 173.00 217 .38 260.86 304.33 347.81 31.29 74.76 118. 24 0 .00 57.97 115 .94 173.00 231.87 289.84 347.8 1 45.78 103 75 161.71 219.68 277.65 0.00 86. 95 173. 90 260.86 347.81 74. 76 161.71 248 .67 335 62 62.57 149.52 236 48 0 .00 115. 94 231.87 347.81 103 7 5 219 .68 335.62 91.55 207. 49 323.43 79.36 195.30 0.00 28 44 56.88 85.32 113. 76 142 20 170.64 199.08 227.52 255.96 284. 40 312 8 4 0 00 27. 90 55.79 83 69 111.58 139 48 167. 37 195.27 223 16 251.00 278 95 300.85 0. 00 13.64 27.89 41.83 65.77 69.72 83.66 97.60 111.54 125.49 139. 15.1. 37 0 00 16.14 32. 28 48. 42 64.56 80.70 96.83 112.97 129. II 145. 25 161.39 177. 53 0 .00 14.96 29.92 44.88 69.84 74. 79 89.75 104. 71 119.67 134. fJ3 149. 5U 164. 5S 0.00 1 3 40 26 80 40 20 63.59 66 .99 80.39 93.79 107 1 9 120.59 1 :!3.99 147 39 0 .00 12.85 25 .71 38.68 61.42 64. 27 77. 13 89.98 102 .83 115.69 128 .54 141.40 0 .00 30 04 60.08 00.12 120 1 6 150 .21 180 .25 210 29 240 .33 270.37 300 .41 330. 4S 0 .00 15.00 30. 00 4 5. 00 60 00 75.00 90 00 1 05.00 120 .00 135. 00 150 .00 165 .00 0 .00 30 00 60 00 00 00 120.00 150 .00 180 00 210.00 240 .00 270.00 300 .00 330 .00 0 .00 60 .00 120 00 180 .00 240.00 300.00 0 00 60.00 120 :00 180.00 210.00 300.00 0 .00 90.00 180 00 270 00 0.00 00.00 180 00 270.00 0.00 90 00 180 .00 270 .00 0.00 29.96 69. 92 89.88 119.84 149.79 179.75 209.71 230.67 2119.113 209.59 329.55 0.00 29. 46 58.91 88. 37 1 17.82 147. 28 176. 73 206 1 9 235.65 265.10 294 5fl 324.01 0 .00 27.97 65. 94 83.90 111.87 139.84 167.81 1 95.78 223. 75 251.7 1 279 .68 307 65 0 00 28.51 57 .03 85 .64 114 05 142 56 171.08 1 99.59 228. 10 2.'i6. 6 1 285 .13 313 .64 0 .00 13. 47 26.64 40 .41 53 89 67.36 80.83 94. 30 107.77 1 21.24 134.72 148. 1 9 0 00 44.03 88.05 132. 08 176 10 220.13 264.15 308.18 352 20 JR. 23 80.25 124 28 0 .00 42.93 85.85 128.78 171.71 214.64 257.56 300 4 9 343 42 26 .34 69.27 112 20 0 .00 57. 4 2 114.85 172 27 229 70 287.12 344 .64 41.97 99 39 1 56.81 214 .24 271.00 0 .00 58.98 117 97 176 95 235 .94 294.92 353.90 6 2.89 Ill. 87 170 .86 229.84 288 83 0 .00 31.02 62.03 93.05 124.00 1 55.08 186 .10 217 .11 248 .13 279 14 310.16 341. 17 0.00 I. 10 2. 20 3.29 4 39 6. 4 9 6 59 7. 69 8 78 9 88 1 0 .98 1 2 .08 0.00 1.02 2 03 3.05 4.06 6.08 6 .10 7 II 8. 13 9. 1 4 1 0 16 11. 17 0.00 0 64 1 09 I 63 2.1 8 2.72 3 27 3 .81 4.35 4. 00 6.44 6.99 0 .00 0.04 0 08 0 12 0 1 6 0.21 0 .25 0 29 0 .33 0 .37 0. 41 0. 4 5 0 .00 0 08 0.16 0 .25 0 33 0. 4\ 0.49 0 .57 0.66 0. 74 0.82 0 .90 / tTbe fir s t line (or constituent Mt gives tbo dUTerenoos as based upon tbe rormuJaln table 2; tbe second line gives tbe dlaerenoos as dorlved from the baH speed or N

PAGE 45

Table 6 (continued) Con.sUluenl H our olday 12 13 It 16 18 17 18 18 20 2 1 22 23 ----------------------1---------------------------------------tfir:::::: ::::::::::::::::::::::::::::::::: ::::::::: 2MK ........... ...... MN ... .................. ............ MS .. ................................ 28M ... ... ...... ......... . .... ML ................................ . . MSL ................. ........ ... .......... Mm ....... . ................................. 8 ---------------------------... Sa . ____ _____ .......... -... ---.------..... --187.03 180 411 0 Gil 364 3 4 173 .116 173.90 347 .81 161. 71 335 62 3 23 43 311. 24 3 41. 28 334 74 167.32 193 87 178 6 1 160 78 164.U 0.411 180 .00 0 .00 0 00 0.00 3511. 6 1 363 47 336 .82 342. 1 6 181.M 188 .30 166. 1 3 328 .011 147 8 1 12. 18 1 3 18 12. Ill 8 63 0 48 0 .811 202. 61 195.53 31.07 23 87 188 4 6 188 .40 16 .711 205 1 9 33 .59 50 .38 67 17 II. 72 2.84 181.28 2011.81 lk47 174. 18 167 .11 3D.63 1g5 .oo 30.00 ao.oo 110.00 28 4 7 22.92 3 .6G IO. M 176.13 212.33 1 118.06 28 61 44.18 43.21 14. 27 13 21 7 08 0.63 1 .U7 2 1 8. 20 210.67 61.1 6 63. 4 0 202 95 202. 8 9 45.78 248.67 91.65 137.33 183.11 38. 1 6 30.63 1116 20 22li.G5 2011. 4 3 187. 68 178 .116 60. 67 210 00 80.00 120 .00 180.00 611.43 62.38 31.65 38.18 188 .60 2M.36 240.U8 83.113 1()5. 78 74.22 16.37 14 .22 7 .113 0. 67 1 .16 233.78 225.62 Ill. 23 82.113 217 4 6 2 17.38 74.76 282 14 1411. 62 224 .28 l!W .05 M .60 58 4 3 2011. 1 6 242.011 224.38 200 U8 1112.81 90.62 225 .00 90.00 180 .00 270 .00 811. 38 81 83 6G.62 67 CMI 202. U7 300 .38 283.111 141.36 164 7 6 106. 24 Ul.47 16.24 8. 1 7 0.113 1.23 240 3 7 240 00 121.31 112.46 231.115 231.87 1 03 7 6 335.62 207 .411 311.24 64.118 115. Of 84.33 223. Oil 258 23 238 34 214 .38 205.67 120.66 24D. 00 120 .00 l40. 00 0,00 1111.34 111.28 87.48 G6.20 216. 64 3. 40 3211.83 1118.78 223.76 136. 26 17 6 7 10 .25 8 7 1 0 66 1 31 264 0 5 255 .70 161.40 141.U8 24 6. 44 24 6 .36 132 TJ HI.OO 2115. 4 6 38. 111 170 .112 123 4 8 114 22 237 03 274 .36 254 .30 227 78 2 18.62 150 .70 266.00 160.00 300 00 90. 00 1411.30 140 76 116 46 124 .71 228 .02 28 4 3 II. 7 6 266 .21 282 73 1 67 27 18.67 17 27 11. 26 0 7 0 1.40 280.64 270 74 181.48 171. 61 260 .114 260 .84 161. 7 1 62.67 323. 4 3 1 25 14 286 .84 161.112 142. 1 2 250 .117 290 50 21141. 28 241.18 231.38 180. 74 270 .00 180 .00 0 .00 180.00 178 28 170 .20 143 43 163 23 242.411 72 4 6 62.68 313 63 341.71 JU8. 28 19 .76 18.28 8 .80 0 74 I 48 2116. 12 286 78 211.68 201.04 275 44 27 5 3 5 190 7 0 106.05 381. 4 0 212.011 4 2 711 180 .35 170.01 264 .112 306 .84 284 22 254 67 244.23 210.78 285 .00 210.00 80.00 270 .00 2011. 22 1811. M 171.40 181.74 255. Ga 116 48 116.62 11. 06 40.70 228 .30 20.84 111.30 10 .34 0. 78 1 .60 311. 7 1 3 00 82 241 .64 230 57 288 G3 2119.84 218 68 1411. 62 '711.36 m .05 1 58. 73 208. 711 19 7 9 1 278 .84 322. 7 8 21111. 18 287 .117 U7.0ll 240 .82 300 00 240 00 120. 00 0 00 2:111. 18 2211. II 1811.36 2 1 0 25 2611. 4 3 180 60 138 64 88. 4 8 811. 88 21.88 20 .32 1 0 .811 0. 82 I.M 3 27 28 315.86 271.72 260 10 304 4 3 304 .33 24 8 67 1113. 00 137 33 28. 00 27 4 00 237 23 225.80 28 2 80 338 .112 314 14 281.37 21141.114 270 .84 315 .00 270 00 180.00 90 .00 269 .14 2511.6 7 227 .33 238 76 282 .90 204. 53 181.47 1 26 .110 158. 67 281.33 23 .06 21.33 I I. 4 3 0 .86 I. 72 3 42.88 33 0 .110 3 01.81 2811. 63 318 .113 3 1 8 .83 m M 236 4 8 1 95 .30 112. 9 5 30 .60 26 5 67 253 .70 306 7 5 365.06 328 1 0 2114. 77 282.711 300 .110 330 00 300 00 240 00 1 80 00 21MJ. I O 288 02 255 30 267 28 ZGG. 37 24 8. 65 224 4 0 183. 3 2 217.05 322.35 24 1 6 22.35 11.118 0 .110 1 .81 3.'>8. 47 345. 94 331.811 3 1 9 1 6 333. 4 2 333 .32 3 06 63 27 9 95 253 27 1911.110 146 64 284 II 281. 6 9 320 .611 11. 20 3 44 06 308 17 28 5 05 330 94 34 5 .00 330 00 300 00 270 .00 3211. 06 317 48 2113.27 2115 .711 3011. 8 4 28 2 68 26 7 3 2 240 75 27 6.63 353 37 26.25 23 3 7 1 2 .52 0 .114 1.80 tTbe llnllloe lor conallluenl M, 1 lvee Ule dllle r e uCOlS aa blii6d upon lbe formula ID lable 2; lbe eeooud UD&IIVM tbe dUJerauces u derived fr o m lbe o f coostilueul M,. w w

PAGE 46

34 The Node factor f is found in Table 14 of Schureman (1941), ( see Table 2). For 1987, it is 0. 964 for the M2 constituent. The Greenwich(V0+u) is the sum of values (V0+u) of Greenwich epoch at the beginning of each calendar year, month, day and hour. From Table 3 through Table 6, these are 339.6, 0.00, 0.00, 0.00, degrees respectively, as the total Greenwich(V0+u) is 339.6 degrees for the M2 constituent here. Following these procedures, the tidal parameters of s, H, f, Greenwich(V0+u), k' for tidal constituents at St. Petersburg (27'N, 82'W) are listed in Table 7. Prediction of Tidal Currents Pioneered by Munk and Cartwright (1966), an analysis procedure termed the Response Method was first applied to pelagic tide prediction and analysis. This method is the first successful major departure from the traditional solutions where the tidal oscillations are described by the amplitudes and phase lags for a finite set of predetermined frequencies. The method reveals, in addition to the astronomical constituents, a spectrum of non-gravitational ones as well as a noise c ontinuum (which arises from nonperiodic effects such as meteorological fluctuations) The response method is therefore useful for storm-surge studies since it allows for non-periodic effects.

PAGE 47

Table 7 Summary of S (deg/hr), k (deg), H (ft.), f s k H (1987) f (1987) f (1988) f (1990) f (1991) Greenwich(Vo+u) for 1987 January lth. 0 am (year) (month) (day) (hour) k'-Greenwich(Vo+u) Greenwich(VO+u) for 1990 August 22th. 20pm (year) (month) (day) (hour) k'-Greenwich(VO+u) M 2 28.98 56.3 0.56 0.964 0.964 0.977 0 .988 339.6 0.00 0 .00 0.00 -283.4 259.4 231.12 207.99 219.68 -141.89 and Greenwich Epoch (deg) for Tide Constituents at St. Petersburg for 1987 and 1990. Mm, S5a, Sa, M5t1 are not listed out since they are not used in currents prediction. s2 30.0 66.3 0.176 1. 00 1. 00 1. 00 1. 00 0.0 0.00 0.00 0.00 66.3 0 .00 0.00 0.00 240.0 -173. 7 N2 28.44 51.1 0.0985 0.964 0.964 0.977 0 .988 323. 7 0 .00 0 .00 0 .00 -272.6 324.3 341.34 293.62 208.79 -36.95 v2 28.51 66.0 0.028 0 .964 0.964 0 .977 0.988 335. 2 0 .00 0 .00 0 .00 -269. 2 92.2 352.02 330.34 210.25 -198.81 ll 2 27.97 251.4 0.032 0.964 0.964 0 .977 0.988 319.4 0 .00 0.00 0 .00 -68.0 157.1 102.24 55.98 199.36 96.72 2N2 27.90 312.7 0 .0239 0.964 0.964 0.977 0.988 307. 8 0.00 0.00 0 .00 4.9 29. 2 91.57 19.26 197.91 -25.24 L 2 29.53 92. 3 0.0225 1. 244 0.749 1.216 1. 248 188.0 0.00 0.00 0 .00 -95.8 2.2 120. 9 122.35 230.57 -23.72 (j.) VI

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Table 7. (continued) K1 01 S1 Q1 P1 Mm Ssa Sa Msf s I 15.041 13.94 15.00 13.40 14.96 0.544 0.082 0.041 1.016 k' 335.6 328.4 68.0 319.2 345.3 329. 5 246.6 138.7 62. 4 H (1987) 0 .476 0.415 0.057 0.0804 0.151 0.119 0 .068 0.149 0.0498 f (1987) 1.112 1.182 1. 00 1.182 1. 00 0. 872 1. 00 1. 00 0.964 f (1988) 1.111 1.180 1. 00 1.180 1. 00 0 .874 1. 00 1. 00 0.964 f (1990) 1.079 1.128 1. 00 1.128 1. 00 f (1991) 1. 051 1. 081 1. 00 1 .081 1. 00 Greenwich(Vo+u) for 1987 (year) 9.2 330.7 180.0 314.8 349.9 15. 9 200.2 280.1 20.4 January (month) 0.00 0.00 0.00 0 .00 0.00 0.00 0 .00 0 .00 0 .00 lth. (day) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 0 0 0 am (hour) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 k'-Greenwich(Vo+u) 326.4 -2.3 -112.0 4.4 -4.6 313. 6 46.4 -141.4 42.0 Greenwich(VO+u) fOI 1990 (year) 16.7 240.1 180.0 305.0 349.6 August (month) 208.96 22.16 0.00 132.39 151.04 22th. (day) 20.70 187.29 0.00 272.92 -20.70 VJ 20pm (hour) 300.82 278.86 300.00 267.97 299.18 0'1 k'-Greenwich(VO+u -211.58 -40.01 -52.0 60.92 -73.82

PAGE 49

37 RESPONSE METHOD For any linear system, an input function CCt) and an output function Uct(t) can be related according to ud = 't)w('t)d't + noise(t) ( 2. 6) where w('t) i s the impulse response of the system. The Fourier transform of w('t) Z (f) = fo-w( i2nf-r = R(f)ei9U> (2. 7) is defined as the system admittance at frequency f. In practice, the integrals are replaced by summations and w and Z are generally complex. The discrete set of w values are termed response weights. In general, R (f) represents the relative magnification of the amplitude, and (/) represents the phase lead at frequency f. The basic motive underlying the response method for analyzing and predicting tidal height or currents is to evaluate the station transfer function (w or Z) between a suitable input series and the observed tidal height or currents. T o predict the tidal currents at the Skyway Bridge by the response method, (since St. Petersburg is close to the Skyway Bridge), the tidal currents at the Skyway Bridge (27'N, 82'W) are assumed to be closely related to the tides at the St. Petersburg (2 746'N, 82'W) tidal station. The sequence of the analysis and the prediction of tidal currents are enumerated as follows:

PAGE 50

38 1) Prepare the time series of the observed currents at Skyway Bridge. 2) Synthesize the tidal constituents of St. Petersburg tidal station for the same period as the Skyway Bridge's observations. 3) Compute response weights for optimum response prediction of the observed currents at Skyway Bridge based on the synthesis of tidal constituents at St. Petersburg. 4) Use the response weights and the synthesized tides to predict the currents at Skyway Bridge. 5) Compute the residuals for the entire series. 6) Perform spectral and coherence analyses on the residuals to determine their origin. BASIC EQUATIONS In practice, equation (2.6) is usually replaced by the summation. ud(t) = L w( -rJ,(t--r) + noise(t) ( 2 8) s Following one of the practical procedures o f Response Analysis derived by Mayer (personal communication), w e take ud(t) to be the current component at the Skyway Bridge and '(t) as the predicted tidal height at St. Petersburg. The basic assumption of this analysis is that the prediction of current component ud(t) can be the weighted sum of the present and past

PAGE 51

39 values of the predicted tidal heights. The lagged values of the predicted tidal height are expressed by '(t--r). According to the harmonic prediction of tidal height, the tidal height and its lagged counterpart may be expressed as '(t) = Lfil cos[st + Greenwich(V0 + u)-k'] ( 2 9) and '(t-'t's) = Lfil cos[s(t-r) + Greenwich(V0 + u)-k'], (2.10) respectively. Equation (2.8) can be further expressed as uit) = w11' 1 (t) + w12 1 (t't'1 ) + w21, 2(t) + w22, 2(t-r-2 ) + u/t) (2.11) where we have separated '(t) into two groups; '1(t) is the synthesis of the diurnal tidal constituents, and '2(t) is the synthesis of the semidiurnal tidal constituents, with '1(t--r-1 ) and '2(t--r-2 ) as their lagged counterparts. Munk and Cartwright (1966) have experimented and demonstrated that it is not necessary to take more than 4 arithmetic lags for Mayer (1984) used the lag of approximately 90 degrees for real constituents. Following that approach, -r-1 for diurnal constituents, chosen to be 6 hr, and for semidiurnal constituents is 3 hr. The constants wii are called weights and are computed from auto-and cross-covariance of the current and the synthetic tide records. From the definition of the covariance function at arbitrary fixed v a 1 u e s of t1 = t and t2 = t + 't', we solve equation(2.11)

PAGE 52

40 for the residual current u/t), square the residual current and sum over the time series. C"(O) = Cdd(O) + (0) + w 1 2 2 C11 (0)-2wuC1d(O)-2w12C 1i -t"t) 2w11w12C11( 't'1 ) + wi1 C22(0) + 2w21Cu(O) -2WzzCu( 't'z) + 2wztw22Czz( 't'z) where: 1 N C"(O) = N 1 N Cij(O) = N 1 k=N-'t1/.6.t Cii('t) = N I L\t) 1 N cid co) = -I (k)ud (k) N k=t 1 N-'t1/.6.t cid ( 'ti) = L (k)ud (k'ti I L\t) N k=t (i=r,d) (i,j=1,2) (i=1,2) ( i=1,2) (i=1, 2) (2.12) (2 .13) Here N is the total number of points in the current record. The cross covariance between the diurnal and semidiurnal tides are all zero. Since C,(O) is the function of the weights, we may minimize the acrT = o awil we have C"(O) and determine the weights by acrr = 0 (i=1,2) awi2 wuC11(0)+w12C11('t 1 ) = C 1 d(0) w"C"('tt)+wtzCu(O) = Ctd('tt) WtzC22(0) + w22C22C 't'z) = Cu(O) wztCzz ( 't z ) + w22Czz(O) = Czd ('tz) Solving for the weights, we get (2.14) (2. 15)

PAGE 53

w = C;d (O)C;; (0)-C;d ( 't; )C;; ( 't;) u (O)-< 't;) W;z = (i=1, 2) C;; (0)-C;; ( 't;) 41 (2.16) From the weights and the reference series (synthetic tides), we can get the weighted currents (the predicted tidal currents) as: (2.17) QUALITY OF PREDICTION Zetler and Munk(1975) have studied the optimal number of weights needed to minimize the residual variance of a record segment. They found that the optimum number of weights in response analysis depends directly on the length of record, and inversely, on noise level in a tidal band; more weights degrade the prediction and generate an artificial wiggliness in the admittance. In this paper, a single pair of weights has been used for the diurnal band and another pair for the semidiurnal band. Weights can be used to generate a prediction of tidal currents. The quality of this prediction is then measured by computing a variance reduction. It is: VR=i-(;:J (2.18) where ar=standard deviation of residual currents ad=standard deviation of predicted tidal currents

PAGE 54

42 VR=1 represents the perfect prediction. With a record length of 2 years, the weights could be calculated from the first half of the data and then applied to the second half t o evaluate the prediction procedure. Some Concepts for Signal Analysis Since certain concepts of statistics and digital signal processing continually arise in applications of time series analysis, the purpose of this section is to define key parameters such as scalar spectra, coherence, the mean and variance. This section follows Bendat and Piersol (1986) and Weisberg (1982) MEAN SQUARE Data are generally the result of several different physical processes. In the present case, the prediction of sea level is generally affected by the influences of winds or meteorological processes or other factors superimposed upon the astronomical tides. These extraneous effects lead to inaccuracies when we attempt to predict time series based upon the tides alone. Suppose that a large number of identical noise generators have been turned on at some time in the past and left to run and the probability density function f(x,t) is

PAGE 55

43 associated with the output of all these generators. The expected value of any function of g(x), denoted by E[g(x)], is defined as E{g[x(t0)]} = r:g[x(t0)}t(x,t0)dx (2. 19) In particular, the Jl(to) = r: X(t0)f(x,t0)dx mean and variance are given by (2.20) f]x(to)-Jl{to)t /(X, t0)dx (2.21) and the standard deviation is the positive square root of the variance. If the random process is stationary, Jl{t0 ) and are independent of time origin; under the ergodic hypothesis, ensemble averages may be replaced by time averages. Because a random variable can only be sampled over a finite record, even under the conditions of stationary and ergodicity, the sample mean and sample variance will differ from the true mean and variance defined in (2.20) and (2.21). For discretely sampled, finite, record-length data, the sample mean and the unbiased sample variance are given by 1 N-1 m = x = L x(i) ( 2 2 2 ) and N i = O 1 N-1 2 s2 =--I[x(i)-x] respectively N -1 i = O (2.23) [If the divisor of the expression for s2 were 1/N rather than 1 / (N-1), then the expected values of s2 could be [(N -1) I N]
PAGE 56

44 Some attributes of statistical estimates, such as bias, variance and mean square error may be stated as follows: bias: (2.24) variance of estimate: mean square error: ljf; = + = ]2 ; ; ; 1\ (2.25) (2.26) where cf> is the true parameter; cf> is an estimate of true parameter cf>; E[] denotes the expected value or time averaged value. In order to explain how these definitions are applied to evaluate the prediction, let us take sea level prediction as an example. The same process is applied to currents prediction. If the actual sea level is represented by then the predicted sea level can be represented by Because the expected value of cf> remain s same, that is E[] = cf>, so we can write equation (2.24) in the form 1\ = E[ c{>-cf>] ; (2.27) 1\ The c{>-cf> is defined as the difference between the observation and prediction of the sea level, which is the residual sea 1\ level. The value of E[c{>-], the expected value, is carried out by averaging the residual sea level. The bias error tells u s how much our prediction deviates from the actual sea 1\ level under the maximum probability. The value of E[c{>-]2 the mean square error of prediction, is carried out by squaring and averaging the residual sea level. Given the mean square 1\ and the bias errors of cf>, the variance of the prediction may

PAGE 57

45 be obtained from equation (2.26). This variance provides a quantitative measure of the prediction based upon the astronomical tides. Factors such as winds, atmospheric pressure, seasonal variations of temperature, and salinity provide sources for the prediction error. Whether the various sources of error result in bias or random contributions depends upon the length of sea level record being evaluated. Preliminary knowledge of the seasonal sea level variations shows sea level to be lowest in February and highest in September. For record lengths of several years duration these seasonal variations should be random, but when we analyze shorter records, e.g. the months of August and September, the seasonal variations will result in a bias error. For shortterm analysis the seasonal variation of sea level becomes the systematic error in the prediction. For the prediction of currents, wind-driven flows contribute to random errors of the prediction and buoyancydriven flows contribute to the systematic errors of prediction. Careful evaluation of these factors will help us improve prediction accuracy. SCALAR SPE CTRA Consider a pair of associated sample records xk(t) and yk(t) from stationary random processes {xk(t)} and {yk(t)} For a

PAGE 58

46 finite time interval 0 T, the cross-spectral density function between {xk(t)} and is defined as where 1 Sxy(f,T,k) = -Xt T Xk(f,T) = J: Yk(f,T) =I: Yt(t)e-i2trftdt (2.28) (2.29) The x;(f,T) is the complex conjugate of For the special case of x(t) = y(t), the auto spectral density functions are (2.30) The auto spectral density functio n gives the distribution of variance (mean square value) of a time series as a function of frequenc y Auto spectral density functions are symmetric about f=O, allowing for the definition of one sided spectra over as Gn(/) = 2Sn(/) G:t:t (f) = 2S" (/) The cross spectral density is defined between a pair of time histories, one is usually thought of as the input to a system and the other an output. Its amplitude gives the amount of variance that the two time series share in common at a given frequency, while its phase gives the offset in radians between oscillations occurring i n each of the two time series at a given frequency. The real part or co-spectrum is

PAGE 59

47. symmetri c and the imaginary part, or quadrative-spectrum, is antisymmetric about f=O. SINGLE INPUT SYSTEM Figure 5. One Input/Output Model. Consider a linear system with a single input random variable x(t) and single output random variable y(t). If x(t) is a stationary random variable with a zero mean, the y(t) will have similar properties. The transfer function between x(t) and y(t ) is defined as the Fourier t ransform of the impulse response function h(t): H(f) = J: h(t)e-j2trf'dt (2.31) and H(f) obtaine d via linear mean square estimation as given by (2.32) Therefore, with knowledge of the input auto spectral density and the cross spectral den sity, the frequency response function for a linear system i s completely determined, including both gain and phase.

PAGE 60

48. The coherence function r!Cf) is a real-valued quantity defined as 2 ls"'(/)12 r "'(/)-Su(f)S11(/) (2.33) It gives the fraction of variance for y(t) that may be accounted for by linear operation on x(0. MULTIPLE INPUT SYSTEM Figure 6. Two-Input/One-Output Model. Consider a linear system having two input random variables x1(t), x2(t) and a single output random variable y(t). The Fourier Transform Y(f) for the total outpu t is Y(f) = H1 (/)X1 (/) + H2 (/)X2 (/) + N(f) ( 2 3 4) in which N(f) is the Fourier Transform of uncorrelated noise input. Using the linear mean square estimation, the transfer function may be estimated from the auto-and cross-spectra by S1y = H1S11 + H2S1 2 ( 2 .35)

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49 The multiple coherence function provides a measure of the output variance that may be accounted for by linear operations on the input variables. Thus, the multiple coherence ry: z(f) is given by r:,. = = 1-[ (2.36) where s_ is the spectrum of the uncorrelated input noise. For the case of uncorrelated inputs, s .. = [ + riy ]s" (2.37) the multiple coherence reduces to the sum of the ordinary coherence, or 2 2 2 ry:;a, = r1y + r2y (2.38) COMPLEX DEMODULATION The time series under our consideration consist of signal and noise. The signal has a discrete spectrum and the noise a continuous spectrum. The complex demodulation is applied to detect the presence of such s ignals, and to estimate their amplitudes and phases. A typical model for such a process is, k x(t) = I,a/t)eilil,t + z(t) ( 2 3 9) r=l where, z(t) is the noise term, a,(t), the amplitude of the signal which is a slowly changing function of t and OJ,, which is the frequency of the signal.

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50 The basic ideas may be explained as follows. If the OJ,.. is known, we can study the time-dependent behavior of the rth term in the equation (2.39) by shifting its frequency down to zero, and this is easily accomplished by multiplying x(t) by exp(-ico,..t), leading to x' (t) = e-iru,'x(t) = a,..(t) + I,as(t)ei(ru,-ru,)t + e-iru,'z(t) (2.40) Then we pass x' (t) through a low-pass filter which will remove all the terms other than a,..(t) The time-dependent amplitude of the rth term is given by lx. (t)l, and similarly, the timedependent phase is arg{x' (t)} To explain this technique in more detail, let g(-r) be a symmetric filter (g( -r) = g( --r)), OJ0 the shifting frequency, and write u(t) = I,g( -r)x(t--r)cos OJ0 (t-r) 't'=--v(t) = I,g( -r)x(t--r)sin OJ0(t--r) (2.41) 1'=-so that w(t) = u(t)-iv(t) = I,g( -r)x(t--r)e -iruo(t--r) (2.42) 't'= Now consider the case where there is only one signal and noise is absent (z(t) = 0), so that x(t) may be .writ ten in real terms as x(t) = lZt (t) cos( OJ1t +
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51 = lZt (t)r( m0 m1 ) exp( -i {( m0 m1 )t1}) + (t)r(m0 + W1)exp(-i{(m0 + m1)t+ 1}) (2.44) .. where r(ro) = :Lg2ro0 then the second term in (2.43) is effectively zero, compared with (2.41) we have 1 u(t)--lZt (t)r(m0 -m1)cos[(m0 -W1)t-4>.1 2 v(t) -..!_ lZt (t)r( m0 W1 ) sin[( m0 W1 )t1 ] 2 Here we have and -{v(t)} ) A. 1jl(t) =tan -(m0 -W1 t'l't u(t) (2 .45) (2.46) (2.47) (2.48) Phase 1jl(t) is a linear function of t. Since W0 i s known, we may easily get 4>1 and m1 from equation ( 2 4 7) and estimate lZt(t) by equation (2.46).

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52 CHAPTER 3. :PREDICTABILITY OF SEA LEVEL Among the many tidal prediction methods, the harmonic method is the one that is most widely used. This method of tidal analysis was developed by Lord Kelvin and Sir George Darwin during the late 19th century. In this section, the harmonic method will be applied to hindcast sea level at St. Petersburg, Florida, to evaluate the errors associated with the hindcast and, hence, with tidal prediction. The two primary factors resulting in tidal prediction errors are: 1) Meteorologically forced sea level variations due to the synoptic weather systems and 2) seasonal variations of sea level Sources of Data Sea level data for Tampa Bay have been recorded since 1929. The primary stations at the St. Petersburg and at Clearwater Beach are shown in Figure 1 and are part o f the National Water Level Observation Network (NWLON) operated by the NOAA National Ocean Service. A schematic of a typical tide gauge station is shown in Figure 7.

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T1DE STAFF (IC ... MCUre4 tO 2x4 lretld board) EXISTING PIER Figure 7. Sea Level Station. ADII Tllll QAGI! I" OR GALVANIZED PIPE SfCURED TO PIER WITll GAL.V....nD a;==*=>-STU!. IRACKETS 53 The tide station at St. Petersburg is defined as "reference station," for which independent daily predictions are given in the Tide Tables for the East Coast of North and South America, published by NOS. The majority of locations are presented as "subordinate stations," for which the corresponding predictions are obtained by means of differences and ratios. For example, in the Tide Tables, Table 1 consists of daily predictions for 4 9 harmonically analyzed reference stations, while Table 2 consists of time and height differences and/or ratios for more than 2000 subordinate stations. Predictions at a subordinate station in Table 2 are produced by applying these differences and/or ratios to the daily harmonic predictions for the appropriate reference station in Table 1.

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54 The tide gauges used in this study consisted of a stilling well, to dampen the effects of short-period wind and boat waves, and a water-level recorder housed in an instrument shelter attached to the top of the stilling well. Digital, battery-powered, water-level recorders were used to record tidal stages at either 5-or 15-minute intervals with sampling intervals accurate to seconds per month. Water stage data accurate to the nearest one-hundredth of a foot were punched on paper tape at each gauge. The reference level for these tidal-stage data is a bench mark, which is set at the St. Petersburg. Presently, the Next Generation Water Level Measurement equipped with part of the Systems (NGWLMS) station at St. Petersburg is specialized data transmission equipment as Physical Oceanographic Real-time System (PORTS) for Tampa Bay. PORTS, as one of three major components of Tampa Bay Oceanography Project (TOP) is the nation's first fully-integrated system, which incorporates information on currents, water levels, and winds at locations where these data are critical for safe navigation. Ancillary data for use with the sea level analyses consisted of various water property parameters and surface wind data. The water properties were collected by the Environmental Protection Commission (EPC) of Hillsborough County, the Southwest Florida Water Management District

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55 (SFWTWMD), and the city of Tampa Bay Study Group (COT), spanning 17-year period 1974 to 1990. The wind velocity data, recorded at the Tampa International Airport (TIA) were obtained from the National Climatic Data Center (NCDC) in Asheville, N.c., for the period of January 1, 1970, to December 30, 1990. Astronomica1 Tide s The astronomical part of the sea level variations consists of diurnal and semidiurnal tides along with weaker tidal constituents at both lower or higher frequencies. This part of the sea level record is investigated by tidal synthesis using harmonically determined constituent parameters. Theses are introduced according to their origin as follows. THE EARTH, SUN AND MOON SYSTEM From the earliest times it has been realized that there is some relation between the tides and t h e sun and the moon. Because the relative motions of the Earth, sun and moon are complicated, it follows that the tidal variations are equally complicated. In order to describe the astronomical tides of Tampa Bay, we must first define certain terminology describing the Earth, moon and sun system.

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56 The Earth and the moon mutually revolve around their common center of mass for a period of 27.3 days. This generates a centrifugal force on the Earth which, when combined with the gravitational force exerted by the moon on the Earth, forms the tide-producing force. The tractive component (parallel to the earth's surface) of the tideproducing force is what causes the water to move, resulting in the lunar tides. The same arguments, when applied to the Earth and Sun system, accounted for the solar tides. Because the Moon revolves about the Earth-Moon center of mass once every 27.3 days, in the same direction as the Earth rotates upon its own axis within a perio d of 24 hours, the period of the Earth's rotation with respect to the Moon is 2 4 hours and 50 minutes. This explains why successive tidal oscillations occur about an hour later on each successive day. The moon's orbit is not in the plane of the Earth's Equator, but is inclined at an angle of I, as shown in Figure 3. The obliquity of the moon's orbit I varies between 18.5 -28.50 within the period of about 18. 6 years. For one complete revolution of the moon, the declinatio n of the moon (line joining the center of the Earth to the moon) ranges up to about 28o on either side of the equatorial plane, over a cycle of 27.2 days (different from the 27. 3-day period of the Earth-moon system's rotation) To an observer on Earth, successive paths of the moon across the sky appear to rise

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tide bulge (exaggerated) Earth's axis 57 large 'high tide' ,., toMoon declination of Moon Figure 8. The Production of Unequal Tides Due to the Moon's Declination. and fall over a 27.2 day cycle, in a similar way to the variation of the sun's path over a year cycle. When the moon is at a large angle of declination, the plane of the two tidal bulges will be offset with respect to the equator, and hence the heights reached by the semidiurnal tides will show diurnal inequalities (Figure 8) An observer at Y would experience a higher high tide than would an observer at X. Twelve hours and 25 minutes later, their would be reversed, each observer would notice a diurnal inequality. The revolving orbit of the Earth-Moon system is not circular but elliptical. The consequent variation in the distance from the Earth to the moon results in corresponding variations in the tide-producing forces. When the moon is

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58 closest to the Earth, it is said to be in perigee, when the moon is farthest from the Earth, it is said to be in apogee. The sun also plays its part as a tide-producing agent. The revolving orbit of the Earth-Sun system is shown in Figure 3. The magnitude of the sun's tide-producing force is about 0.46 that of the moon, since the sun is s ome 360 times farther from the Earth than the moon. The solar tide has a semidiurnal period of 12 hours. Just as the relative heights of the two semidiurnal lunar tides are influenced by the moon's declination, there are diurnal inequalities in the solar-induced components o f the tides because of the sun's declination. The sun's declination varies over a 1 year cycle (compared with the 27.2-days cycle for the moon), and ranges between 23 on either side of the equatorial plane. The orbit of the Earth around the sun is also elliptical, with a consequent minimum Earth-Sun distance, when the Earth is said to be at perihelion, and a maximum distance at aphelion. Consider all the frequencies of the interest in the moon-sun-earth system; every time-varying term in the tideproducing forces must have a frequency which i s the certain combination of all these frequencies. Each term has been grouped according t o their frequency and their origin, and assigned a symbol. Table 8 shows some of the major tidal potential constituents f ollowing Schureman's Table 2 (1941).

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59 Table 8 Major Tidal Potential Constituents. s ol Period (hour) LUNAR SEMIDIURNAL M2 12.42 N2 12.66 2N2 V2 J.l2 LUNAR DIURNAL 12.91 12.63 12.87 01 25.82 Ql 26.87 LUNAR LONG-PERIOD Mm 661.31 354.37 COMBINATION DIURNAL K1 23.93 s ol Period (hour) SOLAR SEMIDIURNAL s2 12.00 SOLAR DIURNAL p l 24.07 s 1 24 .00 SOLAR-LONG PERIOD Ss a 4382.92 S a 8765.82 COMBINATION SEMIDIURNAL L2 12.19 Based on the interaction of the solar and lunar tides, when the tide-producing forces of the sun and moon are acting in the same directions, the tidal range produced is large, i .e. the high tide is higher, and low tide is lower than the average. Such tides are known as spring tides. When the sun and moon act at right angles to each other, and the solar and lunar tides are out of phase, the tidal range is correspondingly smaller than average. These tides are known as neap tides. The following terms are defined: New Moon: Sun and Moon in conjunction; Spring tide. First quarter: Sun and Moon in quadrature; Neap tide.

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Full Moon: Last quarter: Sun and Moon in opposition; Spring tide. Sun and Moon in quadrature; Neap tide. TIDES IN TAMPA BAY 60 Major tidal constituents for Tampa Bay as determined by the NOS at the St. Petersburg location are shown in Figure 9 for May 198 7. Among these rna jor constituents, the three largest ones are the lunar semidiurnal M2 the combination diurnal K1 and lunar diurnal 01 Each of these has an amplitude larger than 0.15m. semidiurnal Figure 10 shows how s2) and the principal the two principal two diurnal (K1, 0 1 ) species are added together for the month of January 1987. The character of the tidal variations depends upon how these individual constituents constructively or destructively add to each other. For the semidiurnal species (M2+S2 ) in Tampa Bay, there are two high and two low tides each day. The high tides occur approximately every 12 hours. During the particular month of January, 1987, the range of the tide decreases for the first 7 days and then increases until about January 15. It decreases again until about January 22 and then builds up t o another great range at about January 29. Compared with moon's phase, the times of the highest tides follow the f ull moon or the new moon by a couple of hours. These are called the spring tides of semidiurnal species and occur near the times

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C\1 Cl.l C\1 z 0 .... 01 .... Cl.l 8 :::s 6 1 0.1 1 00 or -Q.1 UVUVUVYVU09JVUVU9J00h9NUVV091919J9JOUOVVI/VUUVVOVVUVVVVO VVVO 0.1 1 a a a a a a a a a a a 0 n n n n a 0 n n 0 n n a a u u u u u u u u u u u u u u u u u u u u u u a u u u u u a 0.1 0.0 -f====-----=======----==== -0.1 0.1 0.0 -t ____ """""""' _______ ......,.. ___ -0.1 I I I I I I I I I I I I I I 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 M 1987 Figure 9. Major Tidal Constituents at St. Petersburg. The time in days for May 1987 is given o n the bottom of the figure.

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62 when the moon and sun are either in opposition or conjunction. The low tidal ranges, neap tides of semidiurnal species, occur after the first (January 6) and last (January 22) quarters of the moon when the moon is in quadrature. These spring and neap tides can be explained by the beat effect of the lunar semidiurnal M 2 and solar semidiurnal s2 tides. The spring tides are the results of the constructive combination of M 2 and S2, while the neap tides are the results of the destructive combination of them. This beat effect was also found between the combination diurnal K 1 and lunar diurnal 0 1 tides. The difference is that the beat effect of the diurnal species does not follow the moon's phase as the semidiurnals do. These four constituents comprise the greater part of sea level variations of Tampa Bay. Since the tidal ranges of diurnal species and semidiurnal species are comparable, and the diurnal species are comparatively larger, the interference patterns that results is quite complicated, as shown as Figure 11. In this figure, the second time series is the sea level prediction when considering the total of 14 constituents; the third is the actually observed sea level variations and the subtidal sea level is the result of subtracting the observed sea level by prediction. For the particular month of January, the amplitude of synthesized semidiurnal and diurnal species (M2 +S 2+K1 +OI) first decreases from high water at January 1 to the lowest

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63 water, then increases to second high water at January 2, which is called lower high water. The lower high water is usually followed by a higher low water. The lower high water is far lower than the preceding high water. This phenomena, known as diurnal inequality, is attributed to the declination of the moon. The neap (i.e. January 7) and spring (i.e. January 15) tides in Tampa Bay do not follow the phase of the moon but depend upon the constructive or destructive patterns of semidiurnal and diurnal species. It might be more clear when we compare the following two cases. At September 16, the tidal range of the semidiurnal species (M2+ S2 ) is minimum while the range of diurnal species (K1 +01 ) i s maximum. The spring tide of combination (M2+S2+K1+01 ) is at September 16, and the difference between the spring tides and neap tides is very small, 0.2m, due to the destructive combination of these two species and their comparative amplitudes. At December, both species have the maximum ranges at 2 1 and the minimum ranges at 14. Thus, the spring tides are most amplified and the range difference between the spring tides and neap tides is up to 0.6m, due to the constructive combination of the two species. "Form Ratio, F= (K1 +01 ) I (M2+S2 ) is the ratio o f the sum of the amplitudes of the two main diurnal constituents of the actual tide to that of two main semidiurnal amplitudes. This ratio gives the relative importance between semidiurnal and

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N U) Q) + ...., N Q) )1 s 64 o.o A A A A A A A A A A A A A Afi A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 0.2 -0.2 l!Vv v v v v wv v v v v v vvv v vv vv v v v vv v ij v v m v v v v1 vrv vrrv v vv v v1wv v 0.2 1 0 0 AOOAOOOOI\OOAI\It./\AAI\Afti\I\1\AOAAit./\11 VV o 011 o v vo-v .. vlrvvovlJVll 0 0 0 011 0 0 0 vvvv'IJ'oOA 'A o a A A " a 1\ 1\ a A A " A 1\ o A ,.. A v v v VV WVllif VVV\f VVVWVOVWliV -0.2 0.4 0.2 0.0 -0.2 -0.4 oo JAAAAAAAAOAAAAAAAOAAAAAAAAAAAf 0.2 j -0:2 v v v v v vrv=v v=v vrv v v v v vrv v v vrv=v v v 00(\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\(\(\(\(\(\(\I 0.2 f 0 : 2 v=v vrv10 v v=v v v10 vlJlUIO v=v v v v v=vrv-vrv 0.4 0.2 0 Q) i; t 0.0 s -0.2 -0.4 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 J 1987 0 () first quarter full Moon last quarter new Moon Figure 10. Major Semidiurnal and Diurnal Tide Constituents over January, 1987. The time in days is given on the bottom of the figure.

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0 6 0.4 0 2 0 0 0 2 -0.4 -0. 6 0 6 0 4 0 2 0 0 0 2 -0. 4 -0.6 1.2 1.0 0.8 0 6 0.4 0.2 0 0 -0.2 -0.4 0 6 -0.8 0.8 0 6 0.4 0.2 0 0 -0.2 0 .4 0 6 1 J 1987 first q uarter 65. 21 23 25 27 29 31 0 '() full Moon last quarter new Moon .Figure 11. Synthesis of Major Tide Co n stituents, and the Predicte d Observed and Subtidal Sea Level over the Mon t h of January, 1987. T h e sea level is predicted using 14 tide con stituents (M2r S 2 N 2 Y2 J.l.2 2N21 L2, K1, 01r S1 Q l P1, Mm, Mst).

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66 diurnal tidal species. Tidal classification based on Form Ratio is as follows: F=O to 0.25 semidiurnal tides. F=0.25 to 1.5 mixed, mainly semidiurnal tides. F=1.5 to 3.0 mixed, mainly diurnal tides. F>3.0 diurnal tides. For the Tampa Bay St. Petersburg station, the amplitudes of the tidal constituents K1 01 M2 S2 are 0. 476, 0. 415, 0. 56, 0.176 respectively, giving F=l.2. Thus, Tampa Bay is classified as having mixed, mainly semidiurnal tides dominating the Bay, but with large inequalities in range and time between the highs and lows each day. Local effects can modify the amplitudes and phases of tidal constituen t s from year to year. Harbor modifications, shifting sand bars, and changing mean sea level can cause variations in the boundary conditions and thus in the response to the tidal forcing function. The quarter-diurnal component M4 (twice the frequency of M2 ) and the one-sixthdiurnal component M6 (three times the frequency of M2 ) are generated in addition to the semi-diurnal component due to the local non-linear interactions among the principal components of the M2 harmonics. In Tampa Bay, the amplitudes of M4 and M6 are only around 0.003m, far less than that of the principal constituents.

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67 1.0 0.8 0.6 "0 0.4 ........... Q) 0.2 Q) .... 0 0 Q) Q) til e -0.2 ,:; 0 --0. 4 -0.6 -0.8 -1.0 0.6 "0 0.4 Ql -.... 0.2 C) Ql ... +' 0.0 "0 Ql Ql s -0.2 -Q., -0.4 -0.6 0.8 0 6 ....... -0.4 a:s "0 Q) 0.2 ... +' .... l=l Q) 0.0 0 s -0. 2 z --0.4 -0.6 J M A M J J A s 0 N D 1987 Figure 12. Time Series of Observed, Predicted and Subtidal Sea Level for 1987 at St. Petersburg. Subtidal sea level is obtained by subtracting predicted sea level, which is the synthesis of 14 tide constituents, from the observed sea level.

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68 In some bays, the tidal range is very large compared with the range in the ocean near the mouth of the bay. This phenomena is often attributed to resonance-the water in the bay having a natural period of oscillation close to that of the astronomical tide and therefore accumulating energy from it. As for Tampa Bay, with an average depth of 4m and length of 60 km, the fundamental resonant period is T=2.66 hr, so resonance is not expected. Comparing sea level variations at the coastal Clearwater (27'N, 82'W) station with that of St. Petersburg shows that the tidal ranges in these two locations are similar. Thus, resonance at the diurnal or semidiurnal frequencies is not an important factor for tides in Tampa Bay. Wind-Forced Sea Level Fluctuation From Figure 11, a large residual sea level variation could be observed. This residual exists throughout the year in varying measures as shown in Figure 12. Much of this residual can be related to the wind forcing which will be examined. Wind forced sea level fluctuations are due to both coastal ocean set-up a n d the local effect of the wind stress acting on the estuaries' surface. From bulk formulae, e.g. Large and Pond (1981), the wind stress is empirically related to the wind velocity by

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69 ( 3. 1) where kg/m3 w is the wind velocity (vector) and lwl is the wind speed (scalar), and a drag coefficient C o = 1.5*1 0-3 i s used following Marmorino (1983). The components of the wind stress are thus: -r .. = 1. 5 1 o-3 p ( u2 + v2 i'2 u (3 2) u and v are North and East components of the wind velocity. Wind velocity t ime series for one year period of 1987, as recorded at TIA, were obtained from the National Weather Service and transformed to wind stress. Figure 13 shows the TIA wind velocity in the north-east Cartesian coordinate system along with the residual sea level. The sea level data was provided by NOS. Throughout 1987, the sea level in St. Petersburg exhibited significant subtidal fluctuations with a maximum range 1 2 meter (Figure 12). The subtidal variation, as shown in Figure 13, is dominated by a sequence of events occurring at intervals of 3-6 days. When the wind has a component blowin g from the south to the north at January 4 1987, through the Ekman transport, the water flows into the Bay and the sea level has been raised. When the wind has a component blowing from the north to the south in the following Januar y 5, the water flows out of the Bay and hence the sea level has been lowered. Spectral analysis of the sea level records (Figure 14) shows that the variance is

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70 concentrated in two frequency bands centered at 0.01 cph and 0.004 cph. These bands also appear in the spectrum of wind. As a first analysis of the relationship between the residual sea level and the winds, coherence was calculated between the wind stress and the residual sea level for different wind stress vector orientations. The results are shown in Figure 15 as contours o f coherence squared as a function of wind stress component orientations and the frequency. The coordinates in this analysis have been rotated anticlockwise from 00T, the orientation of 100 in figure 15 means 10 degrees westward relative to the north, which is 350T; the orientation of 150 represents the component oriented at 210T which is parallel t o the orientation of 30T. The analysis shows that the residual sea level is coherent with the wind stress at TIA. This coherence is highest over the frequency band (period of between approximated 2-10 days) that includes the synoptic scale wind fluctuations. Within this synoptic band, one wind stress component that is most coherent with the residual sea level is the component that is oriented at approximately 350T. This orientation is approximately aligned with the local isobath of the West Florida Shelf. Thus, the residual sea level fluctuations are mostly driven by an alongshore wind stress component. There is a secondary peak in coherence squared with the preferred orientation parallel to 30T, which coincides with the orientation of Tampa Bay. The two f o r cing

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...c:l ...-.. (/) +' +' r. 0 0 z .=-r.0 (/) +' +' u 0 Q) >.=16 12 8 4 0 -4 -8 -12 1 J 1987 0.8 RESIDUAL SEA LEVEL 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 1 J 1987 2 3 4 5 7 1 6 7 8 9 1 0 Figure 13. Subtidal Sea Level Fluctuations at St. Petersburg, as well as Wind Velocity Components at TIA for January, 1987.

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72 NONTIDAL SEA LEVEL (87) FREQUENCY (CYCLE/PER HOUR) Figure 14. Auto-spectrum of Subtidal Sea Level Fluctuations and the Windstress Components for 1987. Spectrum is frequency averaged for approximately 18 degrees of freedom.

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73 NORTH COMP. OF WINDSTRESS (87) 10' EAST COMP. OF WINDSTRESS FREQUENCY (CYCLE/PER HOUR) Figure 14. (continued)

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180 160 ,-..140 Q) Q) J-4 120 Q) s:l 0 ..... 80 +) as +) 60 s:l Q) ..... ... : : :. !91 :..-. 180 :.r.: @/ 160 140 r 120 100 [ili 80 f ; fP] v 60 0 I I i \,!\ 0 .00 I I I I I ;. I I I I t\ I ,1 ,._ I o / I I II I J-4 40 0 20 40 20 0 0.003 0.006 0.009 0.012 0 015 0 018 0.021 Frequency Figure 15. Contours of Coherence Squared as a Function of Windstress Component Orientation And the Frequency. The coordinates have been rotated anticlockwise from the north (00T) 0 .024 -...) .p.

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75 250.000 125 .000 83.33 3 ,.c:l 1.00 0 0.80 C) cv 0 .60 0 .40 .... 0 .20 :;:j a 0.00 3 1400 E-< cv 1.5700 l7l 0 0 .0000 ID M P<-1.5700 -3.1400 1.00 cv 0 .80 't:l E-< :;:j 0 .60 0 .... ID 0 .40 "" a 0.20 CIS 0.00 3 1400 cv 1.5700 E-< l7l 0 0 .0000 co P<-1.5700 -3. 1400 1.00 cv 0 .80 't:l :;:j 0.60 E-< 0 .... co 0.40 s 0.20 CIS 0.00 0 .000 0 .004 0.008 0 .012 0.0 1 6 0 0 2 0 0.024 Frequency (cycle/per hr) Figure 16. Coherence Squared, Phase and Amplitude o f Tran s fer Functio n b etween Subtidal Sea Level at St. P etersburg and the 350T or 080T Components o f Windstress at TIA.

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76 mechanisms may be identified as the coastal set-up and the direct set-up of sea level by the wind. In order to address how much of the residual sea level can be accounted for in terms of the winds, multiple coherence calculation is applied. Based upon the results of Figure 15, the coordinate system along the West Florida Shelf of approximately 350T has been taken into our following analysis. The results of the linear mean square estimation analysis are given in Figure 16. Multiple squared coherence is the portion of the residual sea level variance that is accounted for by linear operation upon both orthogonal wind components. Figure 16 shows that over the synoptic weather band more than 80% of the residual sea level fluctuations are accounted for by the winds. The cross-spectra analysis further shows, within this range, responding to 1 dyne/ cm2 wind stress component oriented at 350T, sea level fluctuations reach up to 50 em. This response lags behind the northward blowing wind by several hours. The same results have also been revealed by same analysis applied to the coordinate system aligned with the along-axis direction of approximately 30 T of Tampa Bay. Seasonal Variation Given the hourly sampled sea level data in St. Petersburg for 12 years (1977 to 1988), we performed

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77 32. 0 tz:l 28.0 v :;:l "0 E-< --30.0 E-< ........ z ..... 28.0 p.. p.. -< 26. 0 en 24.0 c:: 22.0 < > 20.0 u:i -;:;-en "' 1.4 tz:l ..... X < a 1.3 .._.. en 1.2 X X 1.1 J F M A M J J A 0 Figure 17. An nual Cycles of Sea Level, Water Temperature, Water Salinity Measured at Tampa Bay. Atmospheric pressure anomaly measured a t TIA while sea level measured a t St. Petersburg. The cross x represents the maximum or minimum monthly averaged sea level for 1 2 years (1977-1988)

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78 1.5 -1-t 1.4 Cll co .., 0) 1.3 .... 1.2 1.5 ---.... 1.4 M Ill co ...., 0) 1.3 .... 1.2 1.5 -1-t 1.4 co ...., 0) Ill 1.3 .... a -1.2 1.6 1.5 1-t I() Ill co ...., 1.4 0) Ill .... 1.3 1.2 1.6 1.5 1-t (0 co ..., 1.4 0) .... a -1.3 1.2 1.5 l .... r-co ...., 1.4 0) .... 8 1.3 J 0 Figure 18. Ann ual Cycles of the Monthly Averaged Sea Level a t St. Petersburg. Seasonal pattern is recognized for 1982 through 1987.

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79 statistical calculations including monthly averaged means, climatological means, and deviation of the sea level. Parts of the results are shown in Figure is 17 and Figure 18. The the average over every monthly averaged mean sea level month for each of the 12 years. The climatological mean is the average of the monthly averaged mean sea level over the 12 years of record; thus, the climatological January is the mean over 12 Januarys. The deviation is the difference of the maximum and minimum monthly averaged means around the climatological mean. The same statistical calculations were applied to the meteorological data, water temperature and water salinity. SEASONAL CYCLE The most obvious o f the longer term sea-level variations at nearly all coastal sites is the seasonal cycle. Pattullo ( 1966) found that the seasonal variations range from only a few centimeters in the tropics to amounts on the order of 20 em or more at high latitudes. Over most of the world the lowest sea level during the annual cycle occurs in spring and the highest in fall. This is true of both hemispheres, the northern and southern hemispheres, which oscillate in opposite directions according to the seasons. Higher frequency sea level fluctuations are also modulated by the annual cycles. Figure 12 shows a plot of the

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80 observed sea level, the sea level prediction by tides, and the subtidal sea level which is the difference between the observation and the prediction. The average range of residual sea level fluctuations is about 0.2m. However, this average range only applies at certain times of year. The range may be as large as 0. Sm in winter and as small as 0. 15m in summer. The larger fluctuations of sea level in winter are due to a series of cold fronts from the continental United States and moving southeast cross Florida. The subtidal sea level fluctuations during the winter are dominated by a series of events with periods of ten days or shorter, and by one-day events within the summer. This behavior may be seen from 1982 to 1988 (see Appendix I I) showing the seasonal modulation of the subtidal sea level fluctuation in Tampa Bay. Li (1993) described an annual cycle of wind velocity for the Tampa Bay region with winds blowing toward the southwest in fall and winter, and toward the northeast in spring and summer. However, compared with the higher frequency wind velocity fluctuation due to frontal passage, the monthly mean wind velocity is relatively small. These synoptic scale weather induced fluctuations are larger than the means for all months with the largest fluctuations occurring in the winter months (with the exception of the occasional tropical cyclones in summer) Thus, it is the fluctuations of the winds, rather than the mean winds, that are most important

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8 1 for driving the wind-induced portion of the circulation in Tampa Bay. These higher frequency wind fluctuations are concentrated over two frequency bands, the synoptic weather band with time scales of roughly 2-6 days and the diurnal sea breeze. Variance within the synoptic band varies seasonally with a maximum variance occurring in winter and a minimum in summer. The sea breeze, in comparison, peaks in spring and summer, thereby coinciding in time with the minimum variance in synoptic scale energy. Thus, the total wind variance remains fairly uniform throughout the year, but with fluctuations occurring at different time scales and for distinctly different reasons. The seasonal variation of sea level is consistent with the wind at TIA. At the synoptic scale, the response of the sea level is maximum in winter and minimum in summer. The climatological mean of sea level reveals another important distinctive seasonal variability ( see Figure 17 ) At Tampa Bay, the climatological mean of sea level is a minimum in January to February, with values as low as 1.25 meters. In contrast, it is generally at maximum value in September; its value can reach up to 1.45 meters. Examples of the monthly averaged mean of sea level are shown in Figure 18 for 1982 to 1987. The trends of monthly averaged means, as expected, follow the climatological mean very closely. Interpretation or prediction of this seasonal cycle of sea

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82 level are difficult because most of the driving mechanisms are seasonal and highly coherent, that is, exactly in phase with one another. In the following sections, we will try to study the contributions of these factors to this seasonal cycle. WATER TEMPERATURE AND SALINITY The forces affecting the ocean in the state of rest are the forces of gravitation, and external and internal pressures. The field of internal pressure in a stratified ocean is partly determined by the field of mass. The total field of pressure, including atmospheric pressure and internal pressure, is considered as the driving mechanisms for seasonal sea level variation in this study. The present section will focus on internal pressure; external pressure will be discussed in the following section. Preliminary information on the climatological mean value of a,, temperature, and salinity in Tampa Bay is shown in Figure 17. Large seasonal salinity variability was observed with 6 ppt seasonal change as well as an annual pattern of minimum salinity in September. Historical data of water temperature shows summer maximum of 28 C to 30 C from June to August, and winter minimums of 16 C to 18 C from December to February. This cycle may be related, in part, to thermal effects. Variations in the water temperature bring about the

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83 change in density of water; salinity has only a secondary effect. Let us consider a simple example. The static pressure, p, at any depth z below the sea surface is given by the weight of the water column of unit cross section between the surface and the depth z. If p is the mean density of the water column, then p = pgz and the pressure change due to the change of water density is llp = !l.pgz. This pressure does not include the atmospheric pressure added at the sea surface; p is called the sea pressure. Since the sea surface is the uppermost isobaric surface in the sea with sea pressure zero, the increase of the water density will induce the depression of the water. The pressure of 1 dbar ( 1 bar=10 dbar) corresponds approximately to the sea pressure exerted by a water column of 1 meter. Based on the definition of sea pressure, as an example, assuming the temperature and salinity of the water in Tampa Bay are relatively constant through the depth of 10 rn, the climatological sea pressure is expected as in Figure 17 with its lowest point reaching 1.015b at September. Sea pressure from December to January reaches 1. 021b. The difference between these seasonal changes is about 6 millibars, corresponding to 6 ern sea level variation. Since the depth of water at Tampa Bay is not uniform, the estimation of these seasonal changes of sea level varies from place to place.

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84 ATMOSPHERIC PRESSURE A significant portion of the annual change in sea level can be ascribed to variations in atmospheric pressure. The variations of atmospheric pressure over the sea cause additional deviations of the sea surface from a level surface. This effect is best studied in the case of a motionless sea and homogeneous water. Because the isobaric surface within a sea at rest must be level, it follows that in areas of low atmospheric pressure the sea surface must be higher, and in areas of high atmospheric pressure it will be lower than a level surface. This difference depends on the atmospheric pressure difference in the different time period. At any level of depth h below the surface (in the state of rest) the pressure, Pn must be constant, it follows that Pn = Pa + pg(h+ z) =constant (3.3) where Pa denotes the atmospheric pressure, g the gravity acceleration, p the constant density of the water, and z the elevation or depression of the sea surface. As Pn=constant, and h=constant, the elevation of the sea surface could be written as: /::,.z = -/::,.pajpg (3. 4) The factor ljpg for sea water is about 10-3 and if all quantities are expressed in CGS units, ru = -10-3 /::,.pa If /::,.pa is

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85 expressed in millibars and in em, it follows that for all practical purposes = -tJ.pa(mbar) (3. 5) Hourly sea level pressure data for Tampa International Airport was obtained from the National Climatic Center of NOAA for the period of Jan. 1970 to Dec. 1990. Averaging these sea level pressures month by month, and subtracting 1000 mbar from the monthly averaged sea level pressure, the climatologically averaged atmospheric pressure variation is shown in Figure 17. Monthly climatology of atmospheric pressure reveals seasonal variability. Common at all locations in Tampa Bay, it is a minimum in September, with values as low as 16.0 mbar. Maximum atmospheric pressure is expected in December-February with 21. 0 mbar. The difference of this pressure in September and January is 5.0 mbar, which can yield a 5 em variation of sea level. Tampa Bay may be characterized as having pronounced seasonal variability for its temperature and salinity and the monthly mean sea level. The analyses on the climatological means show that the annual variation in sea level may be accounted for the seasonal variation of temperature, salinity, and atmospheric pressure.

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86 METEOROLOGICAL TIDES In addition to the elementary constituents obtained from the tide-producing forces of the moon and the sun, there are a number of harmonic terms that have their origin in meteorological changes. Two of them are the Sa and Ss a constituents. The period o f Sa is 365 days, the period of Ss a is 182.5 days. The theoretica l magnitude of the solar tide p r oducing force (Sa, S5a) is less than 2*10-5 part of the total tide-producing force o f the moon and sun ; i t is usu ally disregarded altogethe r On the contrary, h a r mon i c analysis s h o ws the actual magnitudes of these solar constituents ( Sa S5a) are more than one-hundredth of the tides a t Tampa Bay. Further study shows that t h e amplitudes of Sa and Ssa are 0.049, 0.023 meters at 1987, one half o f that at 1986, which are 0.108 and 0.062 meter respectively. It differs from the conclusio n drawn from t h e theor etical derivation that these amplitudes should be constan t through the years. Thus the constituents S and S are not astronom'cal constituents in a sa Tampa Bay. Variations in temperature, barometric pressure, and in the dir ecti o n and force of wind may be expected to cause fluctuat i ons in water level. Although in general s uch fluctuations are very irregular, there are some season a l

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87 variations wh ich occur with a rough periodicity that admit of being e xpressed by harmonic terms. The meteorological constituents usuall y taken into account in the tidal analysis a r e Sa a n d Ssa with periods corresponding respectively to the tropical year and the half tropical year. They are more likely driven directly by the water temperature, salinity, a n d atmospheri c pressure as we have discussed in the last two sections. Prediction Errors Based on Residual Sea Level Sea level has been monitored almost continuously at the St. Petersburg since 1929. The data set along with the results of the Least Squares Harmonic analysis were supplied by NOS. The comparison between observations and predictions was done in the following way. Sea level was predicted by Table 9. The Partition of Fluctuation Kinetic Energy for the Sea Level at the St. Petersburg. Constituent Variance (percent) High-frequency fluctuations 0 Semidiurnal tides 30 Diurnal tides 48 Other random forcing 22

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88 104 10 1 02 10' 10 1.0 z 0 0.8 i= 0 (.) li.. a::: w >Ul 0 .4 (.!) CD 0::: 0 w z 0.2 w 0 0 1.0 I. I. .. 1, z 0 0.8 -i= 0 (.) -w 0.4 -(.!) 0::: 0::: 0.. w z 0.2 w 0.0 'I 'I 'I I 1.0 z 0 0.8 i= (.) _J -z 0.4 0 (.!) z 0::: w z 0.2 w 0.0 1010-' 10-: 10-' 10" FREQUENCY (C.P.H. ) Figure 19. The Fluctuation Kinetic Energy Distributions of the Observed, Predicted and Subtidal Sea Level.

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89 synthesizing the 14 tidal constituents, and the residual sea level or subtidal sea level is the result of subtraction of the prediction from the observation. Both mean and root mean square (RMS) values were achieved by applying auto spectra analysis. One way of describing the sea level fluctuations is in terms of their kinetic energy distribution functions (EDF), shown in Figure 19. These integrated and normalized spectral density functions give the partition of the fluctuation kinetic energy as the function of frequency. Table 9 summarizes the partition of the sea level variance. Roughl y 30% of the fluctuation kinetic energy resides at the semidiurnal tides, another 48 % is due to the d iurnal tides. A small amount of 22% is associated with the other low frequency random forcing, such as winds. The quantity which is of foremost practical importance is the RMS error between the sea level actually observed and the predicted sea level. This would be concerned by a mariner trying t o steer his ship by the tide tables The results are given in Table 10 for the period of 1 987 The RMS value of the observed sea level is about 0.25m, and the residual is 0.12m. The 14 major tidal constituents under this study only account for 51% of the sea level fluctuations at Tampa Bay.

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90 Table 10. Mean and Root Mean Square Values for the Observed and the Residual Sea Level. Observed sea level Residual sea level Mean (meter) 0.000140 0.000094 RMS (meter) 0.25 0 .12 According to our previous discussions, the RMS value is the sum of two parts; the first part is a variance term describing the random portion of the error and the second part is the square of a bias term describing the systematic portion of error. The second part of this error, the average discrepancy (bias term) between the two time series, is approximately 0 (0. 9*10-4 m); indicates that the systematic error of this prediction is negligible. The RMS value for the first part, however, amounts on the average to 0.12 m much larger than the bias term. On average, The RMS value which consists of only a random portion of the error in this case, is around 0.12 m. Sea level prediction based on harmonic method can be 12 em higher or lower than the actual observation. The high RMS error r eflects the variability of the sea leve l due to other random processes that cannot be improved by harmonic analysis, no matter how refined it may be. By this study, the sources of error and uncertainty exist for the sea level prediction at Tampa Bay include 1) wind stress, 2) seasonal variability, 3) non-linear effect of shall ow water.

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9 1 Going back to the section Prediction of wind-forced sea level fluctuation," 1 dyne/cm2 wind stress fluctuation can drive the sea level fluctuation up to 50 em at the time scales of 3-6 days. The RMS error with a period centered between 3-6 days is most likely induced by the wind stress fluctuation in the same period. The subtidal sea level fluctuation does show some variation with seasons. To further our evaluation of sea level prediction, the RMS value have been calculated independently for January through April, and May September. During the summer of 1987, for example, through the RMS value was 7 em, while it was 16 em in winter or spring. The RMS error of sea level prediction for the summer was significantly smaller than that during the winter or spring due primarily to the overall reduction in the variance of the wind stress. Through analysis of the climatological mean of sea level, a fairly regular annual cycle is observed with the lowest sea level from January to March, and the highest sea level from August t o October. The difference (0.20m) can be primarily attributed to the annual cycle of atmospheric pressure, water temperature, water salinity and steric effects. Although harmonic analysis reveals this annual cycle by Sa and Ssa constituents, these constituents are not astronomical tides and their amplitudes change from year to year. For the estuary with prominent seasonal cycles like Tampa Bay, the results of constituents S a and S sa from

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92 harmonic analysis are not suitable for the prediction in future. Tampa Bay is such a wide, shallow estuary as we described previously, that it cannot be free from the shallow water effects. It is well known that the nonlinear effects, particularly those associated with shallow water, can lead to significant distortions of the tidal profile. Although nonlinear interactions are accommodated in harmonic prediction by allowing constituents at sums and differences of the frequencies of the main linear constituents, the number of additional constituents required increases rapidly with increasing nonlinearity. Both of the factors stated here are the erro r sources existing for the sea level prediction at Tampa Bay, and since they are out of prediction by the harmonic method, they become the largest part of the RMS error of the harmonic prediction. Since the days of George Darwin, tide records have been analyzed for the amplitude and phase at those particular periods (and their harmonics), which have been previously observed in the orbital motion of the moon and sun. The assumption was implicit that tide records could be accounted for at any desired degree of precision if only a sufficient number of such periods were included. But for practical procedure, tidal records or predictions are associated largely with irregular oscillation due to wind,

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93 atmospheric pressure, nonlinear effects and large storm events such as Hurricanes, et al.

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94 CHAPTER 4. PREDICTABILITY OF CURRENTS Sources of Data In June, 1990, the Estuarine and Ocean Physics Branch of NOAA's National Ocean Service (NOS) launched a 15-month survey of Tampa Bay's currents, water levels, hydrographic properties and meteorological parameters for the purpose of improving the annually published sea level and current tables. The NOS and Tampa Oceanography Project (TOP) provide the first extensive set of physical oceanographic measurements for Tampa Bay. The primary instrument used to measure currents was an acoustic Doppler current profiler (ADCP) manufactured by RD Instruments San Diego, CA. The ADCP uses the Doppler shift from range gated acoustic signals to measure the speed and direction of the currents through the water column (RD Instruments, 1988 and 1989) These units were deployed in an upward-looking mode on the bottom. The Sunshine Skyway Bridge Station (27o38'N, 82o4o'w), located at the intersection of the Sunshine Skyway Bridge and the main shipping channel of Tampa Bay, provides the longest duration record.

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95 Presently, water velocity data at this station continues to be collected by a real-time velocity measuring system. The present study utilizes water velocity sampled hourly at 3 meter depth for a nine month period August 22, 1990 t o June 11, 1991 along with simultaneous wind data from the Tampa International Airport (TIA) Initial findings on the circulation of Tampa Bay derived from the Sunshine Skyway Bridge data have been presented by Weisberg and Williams (1991). They found that while Tampa Bay may be vertically well mixed in salinity, the dynamically significant horizontal salinity (density) gradients give rise to a slowly varying mean circulation. Thus, the fully threedimensional circulation of Tampa Bay is driven by tides, winds and buoyancy, as is the case for most estuaries. Both semidiurnal and diurnal tidal currents, and the wind-driven flows distribute nearly uniformly with depth over the portion of the water column sampled. Because of this, the following analyses on the prediction of currents and the relationship between the wind and the residual currents are representative of the water column sampled within the main shipping channel, not just for the 3 meter depth.

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96 30.00 -20.00 en ............. 0 )! 10.00 0.00 60.00 50.00 40.00 .-.. en C\1'-... )! )! 30.00 u -20.00 10.00 0.00 2200 2200 2200 2200 2200 2200 2200 2200 2200 2200 22 22 22 22 22 22 22 22 22 22 A S 0 N 0 J F' M A M 1990 1991 Figure 20. The Amplitudes of Two Major Along-Channel Tidal Current Constituents M2 and 01 at the Sunshine Skyway Bridge. The amplitudes are obtained by using complex demodulation.

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97 Prediction of Tidal Currents CONSTITUENTS 01 AND M2 In order to see if the tidal constituents are constant or dependent on the time interval, particularly for the diurnal and semidiurnal constituents, we applied complex demodulation to the observed currents. The problem of detecting the presence of tidal currents constituents from the observed time series, and estimating their amplitudes and phases, have been studied by complex demodulation. The basic ideas and the corresponding mathematics of this technique have been discussed in Chapter 2. Auto-spectrum analysis was first applied to the nine-month water velocity time series, followed by the complex demodulation. The amplitudes of the two important constituents 01 and M2 shown in Figure 20. Their amplitudes vary significantly with the amplitude of M2 ranging from 40 cm/s to 50 cm/s. Theoretically, the amplitudes and the initial phases of the tidal constituents are considered as constants by NOAA in their t ides or tidal currents predictions, these significant variation of the amplitudes will introduce prediction error.

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120.0 80.0 40.0 0.0 -40. 0 -80.0 -120.0 120 0 80.0 40. 0 0.0 -40. 0 -80. 0 -120. 0 -160.0 120.0 80.0 40.0 0.0 -40. 0 -80.0 -120. 0 2200 22 A 1990 2200 2200 2200 21 20 20 0 N D 2200 19 J 1991 98 2200 2200 2200 2200 18 20 19 19 F M A M Figure 2 1 Along-Channel Component of Water Velocity Measured at 3 meter Depth under the Sunshine Skyway Bridge. Current is predicted by applying response method. Twelve diurnal and semidiurnal tide constituents (K1, 0 1 S1, Q l P1, M 2 S 2 N 2 V2, ll2r 2 N 2 L 2 ) of St. Petersburg have been used. Subtraction of predicted currents from the observed currents result in the residual currents.

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99 I I I I I I I I I I I 22 23 24 25 26 27 28 29 30 31 1 A S 1991 40. 0 Residual Currents 30.0 20.0 10.0 0.0 -10.0 -20.0 -30.0 22 23 24 25 26 27 28 29 30 31 A 1990 Figure 22. Residual Currents at the Sunshine Skyway Bridge along with Wind Velocity Components at TIA for August, 1990. The time in days is given on the bottom of the figure.

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100 PREDICTION BY RESPONSE METHOD Our application of the response method produces the prediction for the along and across-channel water velocity component at the Sunshine Skyway Bridge for August, 1990, to July, 1991, as shown in Figure 21 and Figure 22. Because St. Petersburg is in the vicinity of the Skyway Bridge, the tidal currents at the Skyway Bridge are assumed to be closely related to the tides at St. Petersburg. Twelve major tidal and divided into diurnal and semidiurnal groups. Within each group, the tidal c onstituents are synthesized with appropriate phase lags. The weights for the different groups and phase lags have been calculated from the currents and the synthesized tides following the procedure discussed in Chapter 2 Finally, the currents' predict ion is made by summing up these weighted groups following equation (2.17). Results of response analysis are s h own in Table 11. The amplitudes of weights and the phase lead of the predicted currents with respect to the maximum height at St. Petersburg are computed by Ri = + (i,j=l,2) (i=1, 2 ) ( 4 1) For example, the amplitude of weight R1 and the phase lead <1>1 for the along-channel component o f the diurnal species are 0.51 s 1 and 55.47o. Responding to 1 meter sea level

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101 fluctuation at St. Petersburg the diurnal tidal currents at the Sunshine Skyway Bridge may reach 51 cm/s with phase lead of 55. 47 Together, both R i $i (i=1,2) are the band-averaged frequency response functions within each o f the diurnal and semidiurnal tidal bands. Table 11. Weights wii' Amplitude Ri, Phase $ i from Response Analysis for the Sunshine Skyway Bridge (27o36' N, 8239'W). wll W12 w21 w22 R1 R2 4>1 4>2 (S-1) (S-1) (degree) Along-channel 0.289 0 420 0.956 -0.256 0 .51 0 .99 55.47 15.00 Across-channel -0.0084 0 .0046 0 .012 0 .016 0 .0096 0.02 28.70 36.87 Revealed by response analysis, the we ights of the across-channel component of water velocity are much smaller than that of the along-channel. This result coincides with the results from the rotary spectra analysis (Li, 1993) that across-channel is the direction in which the variance of tidal currents has been minimized. The phase lead of the predicted current at the Skyway Bridge with respect to the maximum height of tides at St. Petersburg is 55 47 (4 hr) ir. diurnal and 15.00 (1 hr) in semidiurnal frequency bands for along-channel of currents. To evaluate the prediction of tid a l currents, thE response method was applied t o the first half part of thE data to derive the weights, and then the weights were used tc

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102 predict the tidal currents for the second half period of the current data. The variances and the mean values were calculated for both the observed currents and the residual currents for the along-channel component. The quality of the prediction was then measured by computing a variance reduction VR. Here VR=0.864. Since a perfect prediction has VR=l, we could see the prediction is still good. Wind-forced Residual Currents The circulation of Tampa Bay is not only driven by tides, but also by winds. Along with the tidal currents, there exists appreciable non-tidal fluctuations occurring at the synoptic weather bands. Throughout 1990-19 91, residual currents at the Skyway Bridge exhibited significant non-tidal fluctuations with a range larger than 50 cm/s, as shown in Figure 21 and Figure 22. The spectrum and coherence analyses have been performed on the along-channel and across-channel components separately. Figure 23 shows the auto-spectrum of the alongchannel current component at 3 meter depth at the Sunshine Skyway Bridge. Inspection of these non-tidal, along-channel current component fluctuations show a broad, lower frequency peak centered at 0.01cph, indicating synoptic scale variations in residual currents. In addition to this energy

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103 FREQUENCY (CYCLE/PER HOUR) Figure 23. Auto-Spectrum of Residual Water Velocity for Along-Channel Component during August, 1990 to June, 1991. Spectrum is frequency averaged for approximately 18 degrees of freedom.

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104 peak, a narrow high energy peak occurs at the semidiurnal frequency. To study the relationship between the residual currents and winds, coherence was calculated between the residual water velocity and the wind velocity vector orientations. The result, shown in Figure 24, shows that the residual currents are highly coherent with the wind velocity at TIA. This coherence is highest over the synopti c weather band. Within this synoptic band, one wind velocity component that is most coherent with the residual currents is the component that is oriented at approximately 3 4 0T, which i s approximately aligned with the local bathymetry and coastline of the West Florida Shelf. Northward winds resul t in a coastal set-up, as water will be transported onshore through the Ekman transport, and conversel y for southward winds. Another high coherence occurs at 240T, which is parallel with the alignment ( 060T) of the shipping channel under the Sunshine Skyway Bridge. Two forcing mechanisms can be identified as the coastal set-up and the direct set-up by the winds. Based upon the results of Figure 24, the coordinate system a long the West Florida Shelf of 34 0 T has been considered in the following analysis. The results of the multiple coherence analysis are given in Figure 25. The results show that over the synoptic weather band more than 80% of the residual currents are accounted for by the winds. Cross spectrum analyses (Figure 26) show that within this

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180 i @/\ ,. \ f 1 AI \ 1 @] ll .. N 1 180 160 160 -...140 Q) Q) M 1 2 0 bG Q) 0 ..... 80 60 Q) ..... M 40 0 20 0 0 .00 : ... .:: : \ 0.003 0 .006 0 .009 0.012 Frequency r.," 0 r @ 0 015 @] I \,'' .. @ V 0.018 : l '\ll !' '"'' iol! t-:!_1 ;; 0.02 1 Figur e 24. Contours of Coherence Squared as a Function of Wind Velocity Component Orientation and t h e Frequency. The coordinates have been rotated anticlockwise from the north (00 T) 1 4 0 120 100 80 60 40 20 ....... 0 U1

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1.00 .!!l 0.80 0.60 '3 0.40 s 0.20 0.00 3 1400 E-< 1.5700 0 .0000 M -3. 1400 2.00 1.50 1.00 0.50 0.00 3.1400 Cl) 1.5700 E-< Ul 0.0000 -3.1400 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.000 106 250.00 125.00 83.33 0.004 0.008 0 .012 0.016 0.020 FREQUENCY (cycle/per hr) Figure 25. Coherence Squared, Phase and Amplitude of Transfer Function with 18 Degrees of Freedom between Subtidal Along-Channel Water Velocity Component and the Wind Velocity Components Oriented at 340 T and 070T.

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107 Residual currents and 340T TIA wind 250.0 125.0 50.0 (l) 1.00 0 0 7 5 = (l) 0.50 (l) ..l:l 0.25 0 0 0.00 4.00 3.50 (l) 3.00 "0 2.50 :::::1 +.J ..... 2 .00 .... t:l4 a 1.50 as 1.00 0.50 0.00 3.14 1.57 (l) D'l as 0.00 ..l:l t:l4 -1.57 -3.14 0.0000 0.0040 0.0080 0.0120 0.0160 0.0200 Frequency ( cycle/per hr) Figure 26. Coherence Squared, Phase and Amplitude of Transfer Function with 18 Degrees of Freedom between Subtidal Along-Channel Water Velocity Co mponent and the Wind Velocity Component at 340T.

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Residual currents vs. residual TIA wind Q) (.) Q Q) ... Q) ..c= 0 (.) Q) 1'1.1 1.00 0.75 0 .50 0.25 0 .00 4 .00 3 .50 3 .00 2.50 2.00 1.50 1.00 0.50 0 .00 3.14 1.57 ;l 0.00 1=1. -1.57 3.14 0.0000 0 .0040 0.0080 0.0120 0.0160 Frequency (cycle/per hr) 108 50. 0 0.0200 Figure 27. Coherence Squared, P hase a n d Amplitude of Tran s f e r Function betw een S u b tidal A l o n g -Channel Water Velocity Compo nent and the Wind Velocity Compon ent a t 0 60o T The fluctuati o n s induced by 3 40T wind componen t have been subtracted.

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109 range, responding to 1 m/s wind velocity component oriented at 340T, residual currents fluctuations may reach 1.0 cm/s. The currents' response leads the northward blowing wind (at TIA) by approximately 10 hours. The fluctuations of residual currents within the bay are forced primarily by two mechanisms. The first one is cooscillation with coastal set-up at the mouth of the bay, as we have discussed above. The second one is local set-up induced by wind forcing. In order to address how the local effect of wind drives the currents, we need separate the coastal set-up from the local effect of wind. The crossspectrum has been computed between the along-channel currents and the wind velocity component oriented at 340oT. This wind component multiplied by the transfer function, one of the outputs of cross spectrum analysis, represents the portion of currents which is driven by wind component at 340T. This alongshore wind driven portion of residual currents was subtracted from the residual currents, the resulting residuals are currents' fluctuations whic h might be only due to the local effects of wind. The alongshore wind driven portion of the 060T wind component was subtracted from the 0 60oT wind component by applying same procedure. The cross spectrum of these two residuals is then calculated, as shown in Figure 27. The along-channel residual currents are coherent with the local effect of wind over the frequency band centered at 0.008 cph, which is corresponding to 5-day

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1 1 0 period. Wind with velocities of 1 m/s can drive 0.50 cm/s currents, and the currents' fluctuations are out of phase with these wind fluctuations. As the wind blows into the bay, the water will flow out of the bay. Residual. Currents and Sea Level. Variations Sea level variations are closely related to the currents; the water flows into the bay, the sea level wil l be elevated. Water flows obey the mass conservation law, the relationship between the time derivative of sea level elevation and the water velocity is governed b y equation ( 4 2) S d7J =Au dt ( 4 2) Here u is the averaged water velocity flowing into or out of the bay past some cross section area A, is the sea level elevation and S is the surface area. Equation (4.2) can be further simplified into dry -=Cu dt ( 4 3) where c is a constant, defined by the dimensions of Tampa Bay. To test this relationship, a cross spectrum analysis was performed between the residual currents at the Skyway Bridge and the time derivative of sea level elevation at St. Petersburg, as shown in Figure 28. It can be seen that coherence is high over vast frequency bands and the amplitude

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1 1 1 250.0 125 0 83.3 Q) 1.00 C) = 0 .75 Q) 0.50 .... Q) ..c:l 0 .25 0 C) 0.00 0 .0020 Q) 0.0016 = ..., 0 0012 1"'4 'a.o.OOOB s < 0 .0004 0.0000 3.14 1.57 Q) Ol < 0.00 ..c:l 1.57 -3.14 0.0000 0 .0040 0 .0080 0.0120 0.0160 0.0200 0.0240 FREQUENCY (cycle/per hr) Figure 28. Coherence Squared, Phase and Amplitude of Transfer Function with 18 Degrees of Freedom between Derivative of Sea Level at St. Petersburg and the Residual Currents at Sunshine Skyway Bridge.

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1 1 2 of transfer function is about 0.001. Within synoptic weather band, for one hour period, with 1 cm/s residual currents flowing into the bay, sea level at St. Petersburg will be raised by 3.6 em, and this sea level elevation is in phase with the residual currents. Prediction Errors Based on Residual Currents Currents were sampled within the main shipping channel beneath the Skyway Bridge for the period from August 22, 1990 to June 11, 1991. The comparison between observations and predictions for the along-channel component has been done in the following way. St. Petersburg is chosen as the reference site to predict tidal currents at the Sunshine Skyway Bridge. Twelve semidiurnal and diurnal tidal constituents, provided by NOS least squares harmonic analysis, are utilized to synthesize the semidiurnal and diurnal tidal heights at St. Table 12. The Partition of Fluctuation Kinetic Energy for the Currents at the Sunshine Skyway Bridge. Constituent High-frequency fluctuations Semidiurnal tides Diurnal tides Other random forcing Variance (percent) 2 68 28 2

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1 1 3 10. 10J 10: 10' 10" 1.0 z Q 0.8 1-0 u <( w 0.6 a::: > lL. a::: w >-(/) 0.4 "' (Il a::: 0 w z 0.2 w 0.0 1.0 z 0 0.8 i= 0 u <( w 0.6 a::: 1-lL. u i5 >w 0.4 "' a::: a::: a.. w z 0.2 w 0.0 1.0 z 0 0.8 i= u ...J <( <( 0.6 a::: :::l lL. 0 >-Vi 0.4 w "' a::: a::: w z 0.2 w 0.0 1o- 10-J 10: 10' 10 FREQUENCY (C.P.H.) Figure 29. The Fluctuatio n Kinetic Energy Distributions of the Observed, Predicted and Subtidal Along-Channel Component of Water Velocity.

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114 Petersburg. The tidal currents are thus predicted by applying the response method to these synthesized heights. The residual currents is the result of subtracting the prediction from the observed currents. Mean and root mean square (RMS) values were achieved by applying auto spectra analysis. The kinetic energy distribution functions (EDF) have been employed to describe the current fluctuations, shown in Figure 29. Table 12 summarizes the partition of the currents variance. Roughly 68 % of the fluctuation kinetic energy resides at the semidiurnal tides, another 28% is due to the diurnal tides. The very small amount of 4 % is associated with the high frequency fluctuations and other random forcing, such as winds. Table 13. Mean and Root Mean Square Values for the Observed and the Residual Currents Observed currents Residual currents Mean (cm/s) 5.70 5.70 RMS (em / s ) 46.86 15.47 The RMS error between the currents actually observed and those predicted is the most important practical parameter used to evaluate the prediction. The results are given in Table 13. The RMS value of the observed currents is 4 6. 86 cm/s, and of t h e residual is 15.47 c m / s The 12 major diurnal and semidiurnal constituents account for to 67 % o f the current fluctuations within the shipping c hannel. Similar to the RMS error discussion on sea level, the RMS error is the

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115 sum of two parts; the first part is a variance term describing the random portion of the error, and the second part is the square of a bias term describing the systematic portio n of error. The mean value of the residual currents is about 5.7 em/sec, indicating that t here is a systemati c error in our prediction with an average magnitude up to 5.7 em /sec. This result i s consistent with analysis done by Weisberg and Williams ( 1991) They suggested that there is a buoyancydriven flow withi n the shipping channel with magn i tudes abou t 10% the magnitudes of the tidal currents. The mean currents provides a persistent flux of water, which at t his location, is directed i n t o the estuary over the entire portion of the water column sampled. This systematic error induced by the buoyancy-driven f low can be easily reduced if we add these mean currents to our tidal currents prediction. The RMS value of the residual currents, which amounts on the average to 15.47 cm/s, is much larger than the bias term. The currents' prediction based on the response method within the shipping channel can be 1 5 4 em/sec higher o r lower than the actual observation. T h e high RMS error reflects the variability of the currents due to other random processes that cannot be improved by currently used prediction method. The sources of error and uncertainty may exist for the current s prediction at Tampa Bay include 1) wind, 2) nonlinear effect of shallow water.

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1 1 6 Studies of the wind-driven residual currents show that the major portion of non-tidal current fluctuations are driven by the alongshore wind oriented at 340oT within synoptic weather band. Each 1 m/s of wind fluctuatio n drives, on average, a current fluctuation of 1. 0 cm/s, and these along-channel residual current fluctuations lead the wind fluctuations (as measured at TIA) by about 10 hours. Since the TIA winds lag the winds at the farther offshore, this lead is physically reasonable. Another portion is driven by the local effect of wind; it is estimated that with 1 m / s wind blowing into the bay, currents of 0.5 cm/s flow out. Thus, the RMS error with a period centered at 3-6 days is most likely induced by both the alongshore wind and the local effect of wind within the same time intervals. Tampa Bay is such a wide shallow estuary, as we have discussed previously, that it cannot be free from the shallow water effects. It i s well known that nonlinear effects, particularly those associated with shallow water, can lead to significant distortions of the tidal currents. Compared with the time series at sea level with that of the currents, we can see that though sea leve 1 variation is dominated by diurnal tides most of the time, the variation of currents is dominated by semidiurnal constituents. The semidiurnal constituents contribute more to currents than they do to the sea level variation. More detailed discussions of non-linear effects are beyond the purpose of this study.

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117 Based on the above discussion, we can conclude that the RMS error of the currents' prediction comes from systematic error and random error. Buoyancy driven mean flow is systematic error, and the wind driven flow and the non-linear shallow water effects are important portions of random errors. In the future, these factors should be considered to improve the currents' prediction at Tampa Bay.

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11 8 CHAPTER 5. SUMMARY AND DISCUSSIONS The National Ocean Service (NOS) conducted the Tampa Bay Oceanography Project (TOP) for the purpose of revising the existing NOAA Tidal Current tables for Tampa Bay that were based on data from the 1963 survey. Combined with l ong term sea level observation at the St. Petersburg tidal station, these data provide a comprehensive data set for evaluating sea level and current predictions for Tampa Bay. The present study evaluates sea level and current predictions for the St. Petersburg and the Sunshine Skyway Bridge locations, respectively. The evaluation of the sea level prediction shows that 51% of the sea level fluctuations can be predicted by the harmonic method using 14 major tide constituents. The root mean square error of this prediction is 2 e m. The systematic error comes from the seasonal variation of sea l evel. The RMS error seems to be due to such random processes as the wind, and perhaps nonlinear effects. Within the bandwidth of the synoptic weather fluctuations, sea level fluctuations driven by 1 dynes/cm2 wind stress with orientation at 350T or 030T can reach 50 em and lag behind the wind stress by several hours. Inspection of long-term sea level records shows t here is appreciable seasonal cycl e at

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1 1 9 Tampa Bay. The sea level is high (1.45 m) during the period from August to October, and the lowest sea level of 1 2 5 meter is occurring from January to March. According to this study, this seasonal difference at sea level can be attributed to the annual cycle of water temperature and salinity, and the seasonal variation of the atmospheric pressure over Tampa Bay. The evaluation of current predictions shows that the response method (using 12 major semidiurn a l and diurnal constituents) can predict 67% of the c urrent fluctuations within the main shipping channel. Buoyancy-driven flow with the average value of 5.7 em/sec exists as a systematic error in currents prediction. Wind-driven flow is the main source of random error; both the alongshore (340T) wind and the local effects of wind ( 0 60 T) contribute to it. Wi thin the time scales of 3-6 days, 1 m/s wind oriented a t 340 T drives 1. 0 cm/s currents. The currents' fluctuations are out of phase with the wind oriented at 060T. Responding to 1 m/s 060oT wind, currents' fluctuations are 0.5 cm /s. On average, the RMS error for current predictions is 15.47 cm /s, current predictions within the main shipping channel can be 15. 4 7 cm/s higher or lower than the actual observations. One point should be mentioned here; since the c urrent predictions are evaluated only within the main shipping channel, it is unwise to conclude that 67% of currents fluctuations can be predicted for the entire Tampa Bay, or

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120 currents are more predictable than the sea level at Tampa Bay. Reasons suggested by Godin (1983) includes: 1) Tidal elevations are coherent and change smoothly over hundreds of kilometers, while currents correlate poorly over a few hundred meters in the horizontal. 2) The measurement of water level is simple and direct, while the measurement o f currents is difficult. 3) We have years of records on the water levels in our tidal stations, while current measurements cover a few months. 4) The tide has only one degree of freedom, the vertical, while the currents have three degrees of freedom. According to these reasons, we can conclude that our evaluati o n of sea level prediction is applicable to most of Tampa Bay, but it might not be the case for current predictions. This presentation is trying to respond to the mariners' concerns about the accuracy of sea level and currents prediction at Tamp a Bay. From our present knowledge, predictions can be improved by taking wind, buoyancy, and seasonal variability into account. Numerical models could be developed to integrate these findings for the sea level and current predictions.

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1 2 1 LIST OF REFERENCES Battisti, D S., and A. J., Clark. Estimation of nearshore tidal currents on nonsmooth continental shelves, J. Geophys. Res., 87(C10) : 7873-7878; 1982. Bendat, J. S., and A G. Piersol. Random data: analysis and measurement procedures, Wiley-Interscience, New York, 2nd edition; 1986. Boler, R (editor), Hillsborough County water quality, 1984-1985, Hillsborough C ounty of Environmental Protection Community, Tampa;1986. Brooks, D. A. Subtidal sea level fluctuations and their relation to atmospheric forcing along the North Carolina coast, J. Phys. Oceanogr., 8: 481-493; 1978. Brooks, I. H Fluctuations in the transport of the Florida current at periods between tidal and two weeks, J. Phys. Oceanogr., 9: 1048-1053; 1979. Cartwright, D. E., W Munk, and B. Zetler. Pelagic tidal measurement-a suggested procedure for the analysis, EOS Trans. Amer. Geophys. Union., 50: 472-477; 1969. Clarke, A. J., and D S., Battisti. The effect of continental shelves o n tides, Deep-Sea Research, 28A: 665-682; 1981. Clarke, A. J., and K. H Brink. The response of stratified, frictional flow of shelf and slope waters to fluctuating large scale, low frequency wind forcing, J Phys. Oceanogr., 15: 439-453; 1985. C larke, A. J., and S. Van Gorder. A method for estimating wind-driven frictional, time dependent, stratified shelf and slope water flow, J Phys. Oceanogr., 16: 1013-1028; 1986. Dennis, R. E., and E. E Long. A user' s guide to a computer program for harmonic analysis of data at tidal frequencies, NOAA Technical Report NOS 41, U.S. Department of Commerce, NOAA; 1971.

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122 Dinardi, D.A. Tampa Bay circulatory surve y National Ocean Survey Oceanographic Circulatory Survey Report No.2, NOAA; 1978. Doodson, A. T. The analysis of tidal observations, Phil. Trans. Roy. Soc. London, A227: 223-279; 1928. Doodson, A T. The analysis and prediction of tides in shallow water, Int. Hydrogr. Rev., 34(1): 85-126; 1957. Douglas, B. C. Global sea level rise, J. Geophys. Res., 96(C4): 6981-6992; 1991. Fernandez-Partagas, J., and C. N. K. Mooers. A subsynoptic study of winter cold fronts in Florida, Mon Wea. Rev., 103: 742-744; 1975. Galperin, B., A. F. Blumberg, and R. H Weisberg. A time dependent, three-dimensional model of circulation in Tampa Bay, Proceedings: Tampa Bay area scientific information symposium 2 Treat, S. F. and P A Clark, eds. TEXT, Tampa, FL, 77-97; 1991. Godin, G. The analysis of tides, University Toronto Press; 1972. Godin, G. On the predictability of currents, International Hydrographic Review, 60(1) :119-126; 1983. Goodwin, C. R., and D. M., Michaelis. Tides in Tampa Bay, Florida: June 1971 to December 1973, US Geological Survey Open File Report FL -74004; 1976. Goodwin, C R Changes in tidal flow, circulation and flushing caused by dredge and fill in Tampa Bay, Florida, Geological Survey Open File Report 84-447, Tallahassee, Florida; 1984. Goodwin, C R. Circulation of Tampa and Sarasota Bays, NOAA Estuary-of-the Month, No.ll, Proceedings of a seminar, Washington, D .C., 49-64; 1989. Hendershott, M. C., and A. Speranza. Co -oscillating tides in long, narrow bays, the Taylor problem revisited, Deepsea Research, 18 (10): 959-980; 1971. Horn, w. Some recent approaches to tidal problems, Int. Hydrogr. Rev., 37(2): 65-85; 1960. Hsueh, Y., G. 0 Marmorino, and Linda L. Vansant. Numerical model studies of the winter-storm response of the West Florida shelf, J. Phys. Oceanogr., 12: 1037-1050; 1982.

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123 Ianniello, J. P. Tidally-induced residual currents in Long Island and Block Island Sounds, Estuarine Coastal and Shelf Science, 12: 177-191; 1981. Large, W. G., and S. Pond. Open ocean momentum flux measurements in moderate to strong winds, J. Phys. Oceanogr., 11: 324-336; 1981. Lisitzin, E. Sea level changes, Elsevier, New York; 1974. Marmorino, G. 0. Variability of currents, temperature, and bottom pressure across the West Florida continental shelf, J. Geophys. Res., 88: 4439-4457; 1983. Mayer, D. A., K. D. Leaman, and T. N Lee. Tidal motions in the Florida current, J. Geophys. Res., 14(10): 1551-1559; 1984. Mayer, D A., and J C. Larsen. Tidal transport in the Florida current and its relationship to tidal heights and cable voltages, J. Phys. Oceanogr., 16(12): 21992202; 1986. Mitchum, G. T., and W. Sturges. Wind-driven currents on the West Florida shelf, J Phys. Oceanogr., 12: 1 3 1 0 -1317; 1982. Mitchum, G T., and A J Clarke. The frictional nearshore response to forcing by synoptic scale winds, J. Phys. Oceanogr., 16: 1029-1037; 1986a. Mitchum, G. T., and A. J. Clarke. Evaluation of the frivtional, wind-forced long-wave theory on the West Florida Shelf, J. Phys. Oceanogr., 16: 1029-1037; 1986b. Munk, W. H., and D E. Cartwright. Tidal spectroscopy and prediction, Phil. Trans. Roy. Soc. London, A259: 533-581; 1966. Mysak, L. A., and B. V. Hamon. Low-frequency sea level behavior and continental shelf waves o f f North Carolina, J. Geophys. res., 74: 1397-1405; 1969. Neumann, G., and W. J Pierson. Principles o f physical oceanography, Prentice-Hall, Englewood Cliffs, NJ; 1966. Palumbo, A., and A. Mazzarella. Mean sea level variations and their practical applications, J. Geophys. Res. 87(C6): 4249-4256; 1982.

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124 Parker, B.B. (editor), Tidal hydrodynamics, u.s. Department of Commerce, NOAA; 1991. Parker, B. B. Sea level as an indicator of climate and global change, J Mar. Technol. Soc., 25(4): 13-25; 1992. Pattulo, J W. Munk, R. Revelle, and E. Strong. The seasonal oscillations in sea level, J. Mar. Res., 14: 88-156; 1955. Pickard, G. L., and W. J Emery. Descriptive physical oceanography: an introduction, Pergamon Press, 5th Edition;1990. Review and synthesis of historical Tampa Bay water quality data, Technical Report #7-92 (Final Report), Tampa Bay National Estuary Program, Nov.; 1992. Robinson, A. R. Continental shelf waves and the response of sea level to weahter systems, J. Geophys. Res., 69: 367-368; 1964. Ross, B.E. The hydrology and flushing of the bays, estuaries, and nearshore areas of the eastern Gulf of Mexico. In: A summary of knowledge of the esatern Gulf of Mexico, St. Petersburg, Fla., Florida Institute of Oceanography, IID1-IID45; 1973. Ross, B.E., Ross, M.A., and P.D., Jenkins. Waste load allocation study, Volume !-Hydraulic model, Volume IITransport model, Volume II-Nutrient box model, Prepared by Civil Engineering and Mechanics Department, University of South Florida, Tampa, Florida. Prepared for Florida Department of Environment Regulation, Tallahassee, Florida; 1984. Schureman, P. Manual of harmonic analysis and prediction of tides, U.S. Coast and Geodetic Survey, Washington D. C.; 1941. Smith, J. A., B. D. Zetler, and S. Brodia. Tidal modulation of the Florida current surface flow, J. Mar. Technol. Soc., 3: 41-46; 1969. Supplement to manual of harmonic analysis and preidction of tides (special publication No. 98), U.S. Department of Commerce, NOAA;1982. Tide Tables 1987, High and low water predictions, Coast of North and South America, U .S. Department of Commerce, NOAA.

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125 Walters, R. A., and J. W. Gartner. Subtidal sea level and current variations in the northern reach of San Francisco Bay, Estuarine Coastal Shelf Sci., 21: 17-32; 1985. Wang, D. P., and A. J. Elliott. Non-tidal variability in the Chesapeake Bay and the Potomac River, evidence for nonlocal forcing, J. Phys. Oceanogr., 8: 225-232; 1978. Wang, D. P. Subtidal sea level variations in the Chespeake Bay and relations to atmospheric forcing, J. Phys. Oceanogr., 9 : 413-421; 1979a. Weisberg, R. H., and W. Sturges. Velocity observations in the West Passage of Narragansett Bay: A partially mixed estuary, J. Phys. Oceanogr., 6 : 345-354; 1976a. Weisberg, R. H. The non-tida l flow in the Providence River of Narragansett Bay: A stochastic approach to estuarine circulation, J. Phys. Oceanogr., 6: 721-724; 1976b. Weisberg, R. H., and L. J.Pietrafesa. Surface wind field analysis in the South Atlantic Bight, Technical Report No. 82-5, Department of Marine, Earth and Atmospheric Sciences, North Carolina State University;1982. Weisberg, R. H., and L. J. Pietrafesa. Kinematics and correlation of the surface wind field in the South Atlantic Bight, J. Geophys. Res., 88(C8): 4593-4610; 1983. Weisberg, R. H., and R. G Williams Initial findings on the circulation of Tampa Bay. Proceedings: Tampa Bay area scientific informatio n symposium 2, Treat, S F. and P. A. Clark, eds. TEXT, Tampa FL 49-66; 1991. Williams, R. G., M. C. Connolly, and H R. Frey. Data collected during the NOS circulation survey of Tampa Bay Proceedings: Tampa Bay area scientific information symposium 2, Treat, S F. and P. A. Clark., eds. TEXT, Tampa, FL, 35-47; 1991. Wong, K. C., and J. H. Trowbridge. Some observational evidence on the effect o f atmospheric forcing on tidal variability in the upper Delaware Bay, J. Geophys. Res., 93 (C9): 16229-16241; 1990. W K C and R. E. Wilson. Observations of low-frequency ong, . variability in Great South Bay and relations to atmospheric forcing, J. Phys. Oceanogr., 14: 1893-1900; 1984.

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126 Zervas, C.E. Tampa Bay oceanography project: physical oceanographic synthesis, NOAA Technical Report NOS O ES 002, U.S. Department of Commerce, NOAA;1993. Zetler, B. D., and D V. Hansen. Tides in the Gulf of MexicoA review and proposed program, Bull. Mar. Sci., 20: 57-69; 1970.

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1 27' APPENDI CES

PAGE 140

128 APPENDIX 1. T IDE CONSTITUENTS AND SEA LEVEL FOR 1987

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Figure 30. Major Semidiurnal and Diurnal Tide Constituents, and Time Series of the Predicted Observed and Subtidal Sea Level for Each Month o f 1987 at the St. Petersburg. The Moon's phase is also shown. The predicted sea level is made by using 14 tidal constituents (M2 S2 N2 V2 J..L2 2N2 L2 K1 01 S1, Ql, P11 Mm, Mst.)

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130 APPENDIX Cll 1:' 0 2 B oo -o:2vv v Wv rv v vo 0 0 rrro v v lf\fO vv M VYV v v v vvYv v owo o rv v vv v v vv Vv M1 ..... 0 + ..... "' Ill + .. N Q) C/) s +Cil ::s ,_ Q) "' > Q) "' .. Q) Q) ] s o--(IS "' "0 Q) ..... .., .., Q) 8 C/) --0.4 0.4 0 2 0 0 -0.2 -0.4 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0 6 0.4 0.2 \[VV vv 0.0 -0.2 -0.4 -0. 6 1.2 1.0 0.8 0.6 0.4 0.2 0 0 -0.2 -0.4 -0.6 -0.8 0.8 0 6 0.4 0.2 0.0 -0.2 -0.4 -0.6 1 J 1987 3 5 7 9 11 13 t) first quarter 15 17 19 21 23 25 27 29 31 0 () lul l Moon lut quarter new Moon

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1 3 1 APPENDIX 1 ( continued) .... '"' 0 Cll + +> .... Cll ::.:: e -.... 0 + .... '"' Cll + .oJ C\l Cll U) 8 +C\l 0 4 -o:2VVv v11v v 0 0 0 0 vlJ v v v rv v lflfv v v v v VVv1V1111 v v v vlJ tf\fv ovm v v v v v -0.4 a 0 (\ {\ A A A A A {\ 1\ 0 0 o D 1\ A A A A A A {\ f\ r 0 4 -0:2 v 0 v v vlV v-vrv=v=v v=vv 0 v v v v-vrv=vrv vrv=v -0. 4 A 0 0 A. A A A A A A oA An 0 0 A. A A A A J oA 0 0.4 vv v V'V v v ij 'V vvv v V' v n V' r -0.6 0.6 0.4 0.2 0.0 -0.2 -0. 4 -0.6 0 6 0.4 0.2 0.0 -0.2 -0. 4 -0.6 -0.8 0.4 0.2 0.0 -0.2 -0. 4 -0.6 1 3 5 7 9 11 13 15 17 19 21 23 25 27 F 1987 () 0 () firs t quarter f ull M oon last quarter new Moon Figure 30. (continued)

PAGE 144

APPENDIX 1 (continued) ......... ... 0 Q) + .... Q) 8 -... 0 +_..-... ... :.:: Q) + .... C'll Q) Ul El +C'Il 'd_..-... Q) I>' Q) ""' .... Q) Q) ] s o132 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 M 1987 () 0 () first q u arter (continued) ful l M oon last quarter new Moon Figure 30.

PAGE 145

A P PENDI X 1 ( continued) .... '"' 0 Ill + ..., .... Ill ::0:: s 0 + .... '"' ::0:: Ill + ..., Cll Ill {/) e + Cil :::l!l 133 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 A 1987 0 () first quarter full Mooo l ast quarter new Moon Figure 30. (continued)

PAGE 146

APPENDIX 1 (con t inued) 134 -0 +...-.. -"" ::.::: Q) + ... N Q> tl} 8 +'-' N ::s 0.4 -0:2\fV VWvvv OlJlJV lflf\fQ lfV V V VlJ VV V VV \fV V lfV OYvv o 011 rv 0 V VVlJ \f\fVlJV V VV V VlJ1 V\ -0.4 0.4 A A A A A 1\ o o 1\ /\ A A A A A A f\ {\ a 1\ A A A A A A -0:2 ;mv ttv ++v *.f.v ++v ++v -0.4 0 6 0 4 0 2 0.0 -0.2 -0.4 -0.6 0 6 0.4 0.2 0. 0 -0.2 -0.4 -0. 6 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 0.2 :t J:! 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 M 1987 0 () first quart e r t ui l Moon last quarter new Moon Figure 30. (continued)

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135 APPENDIX 1 (continued) -C\1 U':l Q) + ., C\1 Q) s --... 0 Q) + +' ... Q) ::lid a -... 0 +.-... ... ::lid Q) + +' C\l Q) U':l a + C\l 0.4 fl fl ft ft r. G A fl 0 A 0 A A AhA A 6 6 0 A A 0 0 0 fl Aft G Gr. ft ft 0 A 0 0 A A 6 A A A A 0 0 A 0 -o:2 v rowwv vro rv rv v v v v v v v v v v rv rvv rm o vwo ovv rvv v wv WVvlJVvlJ \ -0.4 0.4 o.2 A A -0. 4 0.6 0.4 0 2 0.0 -0.2 -0.4 -0.6 0.6 0 4 0.2 0.0 -0.2 -0.4 -0.6 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0.4 0.2 0.0 -0.2 () f irst quart e r 0 () f ul l M oon last quarter ne w M oon F igure 30. (continued)

PAGE 148

136 APPENDIX 1 (continued) ,-... C\1 '"' (I) Q.) + ..., C\1 Q.) s -.-.. ...... "" 0 Q.) + ..., ...... Q.) ::0:: s -...... 0 + ......... ...... "" ::0:: Q.) + ..., C\1 Q.) (I) s + C\1 --ld '"' "0 Q.) .... ..., ..., Q.) s (I)'-" o" o" o" "o o o o A A 0 A A A A 0 h o o A a o o "" o o" " o o A o A A 0 A 0 A 0 0 o o o o o 0.4 -0:2 V lflfv ifv vVv V VV V Vlfffi V V V lfv VVV 0 vVO v lflflfV11 (V V V VlJ V Ql{QVf01flfl -0. 4 0.4 -0.2 -0.4 0.6 0 4 0.2 0.0 -0.2 -0.4 -0. 6 0 6 0 4 0 2 o.o -0.2 -0.4 -0.6 0 8 0.6 0.4 0 2 0.0 -0. 2 -0.4 -0. 6 0.4 0.2 0.0 -0. 2 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 J 1987 0 () first qu arter t ull Moon l a s t q uart e r new Moon Figure 30. (continued)

PAGE 149

APPENDIX 1 (continued) ... 0 +..-.. ... ,.. v + ..... N a.> C/) s +.__, N f) first quarter Figure 30. (continued) 1 3 7 0 () lull Moon last quarter ne w Moon

PAGE 150

APPENDI X 1 (continued) ..--. -< J.. 0 Q) + ..., .... Q) 8 -...... 0 +..--. .... ... Q) + ..., C\l Q) (/) 8 +C\l ,; "0,-.. Q) ... I> Q) ... ..., Q) Q) $ 8 o() 0 first quarter lull M oon Figure 30. (continued) 138. () () last quarter new M oon first quarter

PAGE 151

139 APPENDIX 1 (continued) ...... 0 11) + ..., ...... 11) ::..:: 8 -...... 0 + ...... 11) + ..., N 11> rtl 8 +N ::::s 0.6 0.4 0.2 oAAAAAAAAA 0 vrv vrv v vrv v v=v 0.0 -0.2 -0. 4 -0.6 0.6 0.4 0.2 0 0 -0. 2 -0.4 -0.6 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 J:i 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 0 1987 0 () full Moon Figure 30. (continued) last quarter new Moon first quarter

PAGE 152

APPENDIX 1 (continued) -< 1-o 0 Q) + .., .... Q) E .... 0 +-.... 1-o Q) + .., N Q> [/) s + N ::a 0.4 0.2 0.0 0 2 -0.4 0.4 0 2 0 0 -0.2 -0.4 0 6 0.4 0.2 0.0 -0.2 -0.4 -0. 6 0.6 0.4 0.2 140 0.0 -0.2 -0.4 0.6 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 0.4 0 2 0.0 -0.2 -0.4 0 tun Moon Figure 30. (continued) () last quarter new Moon first quarter

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1 4 1 APPENDIX 1 (continued) ..... 0 +.... ,.. ::..:: + ..... C\l (/) e +C\l ::s "0..-. ,.. > .. ..... E e o -...... "0 t ......... ..... e (/) -0.4 0.2 0 0 -0.2 -0.4 0.4 0.2 0.0 -0.2 -0.4 0.6 0.4 0.2 0.0 -0.2 -0. 4 -0. 6 0.6 0 4 0 2 0 0 -0.2 -0.4 -0.6 0.6 0.4 0.2 0 0 -0. 2 -0.4 -0.6 -0. 8 -1.0 0.4 0.2 0 VIJl) 0.0 -0.2 -0.4 -0.6 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 D 1987 0 () full M oo n l ast q uarter new Moon first quart e r Figure 30. (continued)

PAGE 154

142 APPENDIX 2. THE SEA LEVEL FOR 1982-1988

PAGE 155

Figure 31. Time Series of Observed, Predicted a n d Subtidal Sea Level for 1982-1988 at St. Petersburg. The subtidal sea level is obtained by subtracting the predicted sea level, which is the synthesis of 14 tidal constituents ( M 2 S2 N2 v2 2N2 L 2 K1 011 S1 Q1 P1 M m M5u ) from the observed sea level

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APPENDIX ( .rc;qam) paA.rasqo (.ra'.j.am) (.ra'.j.atn) repnuoN 144

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PAGE 158

APPENDIX 2 ( c ontinu e d) paA.zasqo (.xalaw) papwa.xd (.xalaw) ["BPHUON <0 146 '0 (j) ;:::l c:; -rl .j...J c:; 0 () rl (') (j) H ;:::l tTl -rl

PAGE 159

APPENDIX 2 (continued) I I I I I I I I I I (J:alaw) papJpaJ:d (J:alaw) rapnuoN z 0 (/) < ., ., cc -,cc 147 'd Q) ::J c; rl .j...J c; 0 u ...-l (Y') Q) H ::J ty\ rl


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