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Generalized non-dimensional depth-discharge rating curves tested on Florida streamflow
h [electronic resource] /
by Auristela Mueses-Prez.
[Tampa, Fla] :
b University of South Florida,
ABSTRACT: A generalized non-dimensional mathematical expression has been developed to describe the rating relation of depth and discharge for intermediate and high streamflow of natural and controlled streams. The expressions have been tested against observations from forty-three stations in West-Central Florida. The intermediate-flow region model has also been validated using data from thirty additional stations in the study area. The proposed model for the intermediate flow is a log-linear equation with zero intercept and the proposed model for the high-flow region is a log-linear equation with a variable intercept. The models are normalized by the depth and discharge values at 10 percent exceedance using data published by the U.S. Geological Survey. For un-gauged applications, Q10 and d10 were derived from a relationship shown to be reasonably well correlated to the watershed drainage area with a correlation coefficient of 0.94 for Q10 and 0.86 for d10. The average relative error for this parameter set shows that, for the intermediate-flow range, better than 50% agreement with the USGS rating data can be expected for about 86% of the stations and for the high-flow range, better than 50% for 44% of the stations. Testing the model outside West Central Florida, in some stations at North Florida, and South Alabama and Georgia, show some reasonable relative errors but not as good as the results obtained for West Central Florida. Using a model with a different slope, developed specific for those particular stations improved the results significantly.
Dissertation (Ph.D.)--University of South Florida, 2006.
Includes bibliographical references.
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Adviser: Mark Ross, Ph.D.
Open channel flow.
West Central Florida.
x Civil Engineering
t USF Electronic Theses and Dissertations.
Generalized Non-Dimensional De pth-Discharge Rating Curves Tested on Florida Streamflow by Auristela Mueses-Prez A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Civil Engineering College of Engineering University of South Florida Major Professor: Mark Ross, Ph.D. Ahmed Said, Ph.D. Jeffrey Cunningham, Ph.D. Muhammad Rahman, Ph.D. Jeff Geurink, Ph.D. Mark Stewart, Ph.D. Billy Lewelling, B.S. Date of Approval: November 1, 2006 Keywords: rating curve, runoff, open channe l flow, data management, West Central Florida, non-dimensional streamflow Copyright 2006, Auristela Mueses-Prez
DEDICATION Mary Kay Ash used to say Â“If you think you can, you can. If you think you canÂ’t, youÂ’re rightÂ”. Today I know Â“I canÂ” and I owe this fr uit of my labor: To G od for his strength and support, to Eduardo and my babies, Adriana and Alfonso, for their unconditional love and to my wonderful advisor, Dr. Mark Ross, for his help and encouragement.
ACKNOWLEDGMENTS The author graciously acknowledges the i nvaluable assistance provided by Bill Lewelling, David Fulcher, Dona ld Herndon, Michael Holmes a nd Richard Kane, at U.S. Geological Survey. They provided data, in terpretation, and insi ght concerning rating relationships and analysis th at was greatly appreciated.
i TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES v ABSTRACT vii 1. INTRODUCTION 1 1.1 Description of Simple Rating Curves 1 1.2 Scope of Research 2 2. LITERATURE REVIEW 5 3. DATA 11 4. METHODOLOGY 12 4.1 Identification of Most Significant Variables 12 4.2 Approaches for the Determination of the Non-Dimensional Ratings 14 4.2.1 Approach 1: Normalization by Drainage Area 14 4.2.2 Approach 2: Normalization by Discharge and Depth at Full Channel 17 4.2.3 Approach 3: Normalization of Discharge and Depth at Ten Percent Exceedence Flow 19 5. MODEL DEVELOPMENT 24 5.1 Theoretical Background for Model Development 24 5.1.1 Analysis for Simple Control Sections 26 5.1.2 Analysis for Compound Control Sections 31 5.2 Model for the Intermediate-Flow Region 34 5.3 Model for the High-Flow Region 37 6. MODEL VERIFICATION 40 7. MODEL VALIDATION 41 8. RESULTS AND DISCUSSION 49 9. CONCLUSIONS 59
i iREFERENCES 62 APPENDICES 66 Appendix A: Results from Weir Anal ysis in Channel Controlled by a Horizontal_Uncontracted Section 67 Appendix B: Results from Weir Anal ysis in Channel Controlled by a Horizontal_Contracted Section 68 Appendix C: Results from Weir Anal ysis in Channel Controlled by a V-Notch Section 69 Appendix D: Results from Weir Anal ysis in Channel Controlled by a Broad Crested Weir 70 Appendix E: Results from Weir Anal ysis in Channel Controlled by a Compound Weir consisting of a V-Notch and a Trapezoidal Weir 71 Appendix F: Basin Characteristics Data and Relative Error for Stations in Calibration Data Set for Intermediate-Flow Region 72 Appendix G: Basin Characteristics Data and Relative Error for Stations in Calibration Data Set for High-Flow Region 74 Appendix H: Basin Characteristics Data and Relative Error for Stations in Verification Data Set for Intermed iate-Flow Region 75 Appendix I: Drainage Area and Re lative Error for Stations in Validation Data Set for Intermediate-Flow Region 76 Appendix J: Drainage Area and Re lative Error for Stations in Validation Data Set for High-Flow Region 78 ABOUT THE AUTHOR End Page
i ii LIST OF TABLES Table 1. Ranges of Values for Hydraulic Para meters for Calibration Data 11 Table 2. Matrix Correlations for Discharge for Variables in the Calibration Data 13 Table 3. Results from the Stepwise Procedure Â“Maximum R2 ImprovementÂ” 13 Table 4. Discharge and Depth for th e 10% Discharge Exceedance and the Slope of the Corresponding Rating for Calib ration Data Set 21 Table 5. Statistics of the Regr ession Model for the Intermediate Flows Region 35 Table 6. Ranges of Values for Hydraulic Parame ters for Verification Data 40 Table 7. Discharge and Depth for the 10% Discharge Exceedance and the Slope of the Corresponding Rating for Valid ation Data Set 42 Table 8. Results of T-test for 43 Stations in West-Central Florida 51 Table 9. Relative Percent Error for Some Stations in West-Central Florida for the Intermediate and High-Flow Region 53 Table 10. Results from Weir Analys is in Channel Controlled by a Horizontal_Uncontracted Section 67 Table 11. Results from Weir Analys is in Channel Controlled by a Horizontal_Contracted Section 68 Table 12. Results from Weir Analys is in Channel Controlled by a V-Notch Section 69 Table 13. Results from Weir Analys is in Channel Controlled by a Broad Crested Weir 70 Table 14. Results from Weir Analys is in Channel Controlled by a Compound Weir consisting of a V-Notch and a Trapezoidal Weir 71
iv Table 15. Basin Characteristics Data and Relative Error for Stations in Calibration Data Set for Intermediate -Flow Region 72 Table 16. Basin Characteristics Data and Relative Error for Stations in Calibration Data Set for High-Flow Region 74 Table 17. Basin Characteristics Data and Relative Error for Stations in Verification Data Set for Intermedia te-Flow Region 75 Table 18. Drainage Area and Relative Error for Stations in Validation Data Set for Intermediate-Flow Region 76 Table 19. Drainage Area and Relative Error for Stations in Validation Data Set for High-Flow Region 78
v LIST OF FIGURES Figure 1. Location of USGS Stations in West Central Florida 4 Figure 2. Relative Discharge Versus Relative Stage for Stations in West-Central Florida for Approach 1 17 Figure 3. Relative Discharge Versus Relative Stage for Stations in West-Central Florida for Approach 2 18 Figure 4. Relative Discharge Versus Relative Stage for Stations in West-Central Florida for Approach 3 22 Figure 5. Correlation Between Discharge as Q10 and Drainage Area for Approach 3 23 Figure 6. Correlation Between Depth as d10 and Square Root of the Drainage Area for Approach 3 23 Figure 7. Relative Discharge (Log QÂ’) Versus Relative Depth (Log dÂ’) for the Intermediate-Flow Region 25 Figure 8. Relative Discharge (Log Q Â’) Versus Relative Depth (Log dÂ’) for the High-Flow Region 25 Figure 9. QÂ’ vs. HÂ’ for Horiz ontal_Uncontracted Section for Varying Coefficients C 27 Figure 10. QÂ’ vs. HÂ’ for Horiz ontal_Contracted Section for Varying Coefficients C 28 Figure 11. QÂ’ vs. HÂ’ for V-Notch Section for Varying Coefficients C 28 Figure 12. QÂ’ vs. HÂ’ for Broad Crested Section for Varying Coefficients C 29 Figure 13. QÂ’ vs. HÂ’ All Control Sections fo r Typical Values for C 30 Figure 14. QÂ’ vs. HÂ’ for Uncont racted and V-Notch Weirs and ManningÂ’s Equation 32
v iFigure 15. QÂ’ vs. HÂ’ for Compound Weir Consisting of a V-Notch and a Trapezoidal 33 Figure 16. Generalized Log-Linear Rati ng Behavior 34 Figure 17. Data and Regression Equation for Q10 36 Figure 18. Data and Regression Equation for d10 37 Figure 19. Data and Regression Eq uation for Discharge of Departure (Qdep) 39 Figure 20. Data and Regression Equation for Depth of Departure (ddep) 39 Figure 21. Location of USGS Stations in the Areas of North Florida, Georgia and Alabama 43 Figure 22. Data and Regression Equation for Q10 Including Validation Data Set 44 Figure 23. Data and Regression Equation for d10 Including Validation Data Set 45 Figure 24. Relative Discharge (Log QÂ’) Versus Relative Depth (Log dÂ’) for Stations in North Florida, Alabama and Georgia 45 Figure 25. Relation Between Drainage Area and Channel Slope and Water Depth (d10) for Some Stations in Florida, Georgia and Alabama 46 Figure 26. Some Stations that Pa rtially Matched the USGS Rating Data Using Approach 3 50 Figure 27. Comparison of USGS Ra ting with Model Results, for the IntermediateFlow Region, for Some Stat ions in the Study Area 55 Figure 28. Comparison of USGS Rati ng with Model Results, for the High-Flow Region, for Some Stations in the Study Area 57
v ii GENERALIZED NON-DIMENSIONAL DEPTH-DISCHARGE RATING CURVES TESTED ON FLORIDA STREAMFLOW Auristela Mueses-Prez ABSTRACT A generalized non-dimensional mathematical e xpression has been developed to describe the rating relation of depth and discharge for intermediate and high streamflow of natural and controlled streams. The e xpressions have been tested ag ainst observations from fortythree stations in West-Central Florida. The intermediate-flow region model has also been validated using data from thir ty additional stations in the study area. The proposed model for the intermediate flow is a log-linear equation with zero intercept and the proposed model for the high-flow region is a log-linear equation with a variable intercept. The models are normalized by the depth and disc harge values at 10 percent exceedance using data published by the U.S. Geological Survey. For un-gauged applications, Q10 and d10 were derived from a relationship shown to be reasonably well correla ted to the watershed drainage area with a correl ation coefficient of 0.94 for Q10 and 0.86 for d10. The average relative error for this parameter set shows that, for the intermediate-flow range, better than 50% agreement with the USGS rating da ta can be expected for about 86% of the stations and for the high-flow range, better th an 50% for 44% of the stations. Testing the model outside West Central Fl orida, in some stations at North Florida, and South Alabama and Georgia, show some reasonable relative errors but not as good as the results obtained for West Central Florida. Using a model with a different slope, developed specific for those particular stations improved the results significantly.
1 1. INTRODUCTION A discharge rating is a relationship between st age and discharge at a specific point in a river stream or lake outlet st ructure. The rating relations for a site are created from periodic field measurements of discharge and stage and are the industry standard for continuous flow measurement of streamflow via stream stage. These measurements are used to produce a unique math ematical relation which allows for a particular location and usually for a period of time, continuous stage measurements to be converted into discharge. The resulting rating curves are usef ul for interpolating or extrapolating flow measurements and for modeling. 1.1 Description of Simple Rating Curves The simplest form of a rating curve is a two-parameter stag e-discharge relation. Discharge is calculated from field measuremen ts of velocity and channel cross section. To properly develop rating curves, discharges must be measured at all representative stages, using at least 10 to 12 points coveri ng the range of low to high flows (Gupta, 2001). If there is a direct relation between discharge and gage height, the discharge rating is called simple. A simple rating ma y be only one curve but is more often a compound curve consisting of three segments, one each for the low, medium and high water ranges (Kennedy, 1984).
2A simple stage-discharge relation has a power form given by the following equation: na h A Q ) ( (1) where, Q = discharge h = gage height a = gage height at zero flow A, n = constants When plotting this equation in log-log paper, the rating is transformed to a straight line. A straight line is preferred b ecause (1) it can be extended or extrapolated, and (2) it can be described by a simple mathematical equation (Gupta, 2001). The resulting stage-discharge curve repres ents Q as a function of stage, datum correction, channel slope and ManningÂ’s Coeffi cient (n). The proce dure is costly and time consuming, and dangerous or imprac tical during high floods Thus typically, streamflow rating only exists for limited station locations and with limited data at high flow conditions. 1.2 Scope of Research As it will make clear from the literature re view, most of the conventional methods to determine stage-discharge ratings are base d on open channel flow, uniform flow and ManningÂ’s equation and channel properti es. These methods require a roughness coefficient, estimation for the channel cr oss sectional area and slope, field survey measurements, or some other data specific to the channel. The difficulty in estimating an
3adequate ManningÂ’s roughness co efficient or discharge coef ficient incorporates some uncertainty to these empirical models. Als o, channel cross secti onal flow data is expensive to acquire and often is not avai lable, particularly for un-gauged streams, limiting the widespread use of these methods. Although the existing methodologies cited in the references explain general concepts and traditional approaches to determine rating curves such as the variables involved in the ratings, scaling factors and non-dimensional parameters, and discharge as a function of area, none of the mentioned st udies investigated non-dimensional discharge rating curves related to watershed characteristics. This study proposes a generalized mathem atical expression to describe nondimensional depth-discharge relations from the perspective of the watershed characteristics, particular flow indices and the general behavi or of stage-discharge curves. Forty-three stations in West -Central Florida have been analyzed. Locations of USGS stations in West Central Fl orida are shown in Figure 1. Da ta used in this study were obtained from USGS known rating data reported between 1987 and 1998 and correspond to the best fit rating curves used by USGS to translate stage to discharge. The accuracy of the prediction capability of the proposed model is evaluated using statistical errors of the estimated values compared to USGS rating data.
4 Figure 1. Location of USGS Stations in the West Central Florida
5 2. LITERATURE REVIEW Several methods have been studied to explai n the empirical relation among the variables involved in the rating relations Most of these methods have been developed by the U.S. Geological Survey and are summarized in the work by Kennedy (1984) and Rantz et al. (1982), among others. Many of the current empirical methodol ogies that explai n the relationship between the variables involved in the rating curves define the relation between the stage and discharge by assuming the discharge as uniform open channel flow with constant slope, and incorporating th e ManningÂ’s equation and roughness coefficient (Kennedy, 1984; Rantz et al., 1982; Atabay and Knight, 1999; Dawdy et al., 2000). Others have treated the discharge as flow in a constricte d weir section (Kennedy (1984), Rantz et al (1982)). Although the rating curves are commonly used to convert stage into discharge, the literature recognized that there are errors associated with the development of ratings curves. Dymon and Christian (1982) identifie d three types of errors influencing the random error of a single discharge measuremen t. These were rating curve error, water level measurement errors and an error cause d by ignoring the physical parameters that affect discharge. They reviewed the methods for evaluating the three types of errors. Similarly, Tillery et al. (2001) derived potenti al errors for individual measurements and rating relations for 17 stations in Arizona In the study, they explored the errors
6associated with developing rating curves base d on direct measuremen ts of discharge and stage, indirect measurements of discharge a nd theoretical weir and culvert computations. They concluded that errors are greate r when based on indirect measurements. In more recent attempts at defining a theoretical relation based on the channel properties Dingman et al. (1997) proposed a multiple-regression equation that is independent of the roughness coefficient. The di scharge is described in terms of the cross sectional area, hydraulic radius and wate r surface slope. Dingman (1999) extended his work to incorporate acquisition of remotely sensed channel data. The method can be applied to un-gauged streams in the New Hamp shire area, with values of width and slope similar to the ones considered in developing the model. The idea of non-dimensional stage-discharg e rating curves has not been widely studied. Some investigations i nvolving scaling factors for di scharge were made by Ervine et al. (1993), Savage et al. ( 2001), and Sivapalan et al. (2001). Ervine et al. (1993) used a non-dimensional discharge coefficient to quan tify the effect of the main parameters affecting conveyance in meandering compound ri ver channels. Bruce et al. (2001) used non-dimensional discharge curves to compare results from their study on flow parameters calculation over a standard ogee-crested spillw ay using a physical model, a numerical model and existing literature. Sivapalan et al (2002) used a scali ng relationship based on area to explain the non-linear response of the watershed due to the dependence of a catchment hydrological property on area. A recent investigati on by Shesha et al. (2003) developed a generalized linear head-discharge relationship for flow over sharp-crested inclined inverted V-notch weirs. In those studies, the scaling fact or was introduced to present and compare their results and not as part of the methodology.
7Another investigation describing a theo retical expression for non-dimensional rating relations was made by Kamula (2000). He established a new procedure for fish ways design and suggested a different equa tion for each of the main existing fish way types. In his procedure, he created a general scaling equation for dimensionless discharges, based on the depth of flow over the weir and the eff ect of pool length (Kamula, 2000). The determination of discharge from theore tical ratings considering the effect of unsteady, non-uniform flow in the stage-di scharge relation was examined by Schmidt (2002). In his work, Schmidt addressed the ra ting hysteresis problem and the effects of different downstream backwater conditions. The rating relations are calculated from Hydraulic Performance Graphs (HPG) curves and require the channel cross section and the stage data points at each end of a stream reach. Westphal et al. (1999) also explored the hysteresis effect by recognizing that stagedischarge relations fo r the rating curve are not single-valued, in measurements in the Midd le Mississippi River. He also attributed to replication errors as the possible cau se for the stage-discharge pattern. Some investigations present methodologies to determine discharge and stage in ungauged stations. Moramarco and Singh (2001) proposed a simple method for relating local stage and remote discharge. The method reconstructs the disc harge hydrograph at a river section, based on water level recorded at the cross section and a discharge value recorded at another upstream cross section. Re sults show the method as reliable when the hydraulic conditions at the site are unknown. Si milarly, Franchini et al. (1999) proposed a methodology for synthesizing the rating curve pr ovided with stage data, when a reliable rating curve and stage data is available in an upstream cross section. The proposed
8methodology uses the Muskingum-Cunge model for routing the flood wave to the site of interest. Szilagyi et al. (2003) proposed a similar approach but using a reservoir-cascadetype formulation. The method allows for stag e predictions using physically based flow routing of known discharges from an upstream station, in river sect ions where no values of discharge are available. Some studies explored the adequacy of the rating curves in converting water levels to flow rates for high flows. Fent on and Keller (2001) found that rating curves errors impose severe limitations in developing methods to estimate floods due to the bias introduced at site and regional flood fr equency estimates. They proposed some improvements in the existing practices to be able to obtain a more reliable estimate of high flows. On the same topic, Lewis (1998) proposed some enhancements of river flood forecasts by using a dynamic hydraulic flow routing technique to properly extend the rating curve to account for hydraulic condi tions that may cause overbank flow. Jothityangkoon and Sivapalan (2003) inves tigated changes in the rating curve due to the effects of overbank flows during the transition from normal to high flows. The authors attributed the changes in the rating to the effects in the interactions between the main channel and floodplain sections and proposed a distributed flood routing model based on non-linear storage-discharge re lationship to account for the effects. Discharge has also been calculated base d on channel character istics of streams. This is particularly useful for ungauged streams where a bankfull channel can be identified by use of field indi cators and related to discharge. Cinotto (2003) used channel dimensions to develop regional curves from the regression analys es of the relations between drainage area and the cross-sectional area, mean depth, width, and streamflow of
9the bankfull channel (Cinott o, 2003) at 14 stations in Pe nnsylvania and Maryland. The analysis showed bankfull cross sectional ar ea and bankfull discharg e with the strongest correlation to drainage area. Bankfull wi dth and depth showed a moderate linear correlation to area. Leopold and Maddock back in 1953 alre ady recognized the physiographic implications of the hydraulic geometry of st ream channels. In their study, they related discharge and some hydraulic characteristic s of a given cross section Â– depth, width, velocity and suspended load Â– with a power function. They found that in a hydrologically homogeneous region there is a pattern when plotting thes e hydraulic characteristics against discharge. Interesti ngly, they also found that among the stations, the respective lines are generally parallel. The station curves differed much in ordinate but were similar in slopes. The rest of the study focused on exploring the relations of depth, with and velocity as a function of the lo ad transported in the channel. Some studies involving ba sin characteristics were developed using the U.S. Geological Survey (USGS) fl ood-regionalization procedures (in Florida, Hammet and DelCharco, 2005) that relate flood characteristics to watershed and climatic characteristics through the use of regressi on analysis. These equa tions use hydrologic characteristics like drainage area, channel slope percent wetlands and rainfall, to estimate the magnitude of flows for specific recurrence in terval, such as the 2-, 10-, 25-, 50-, 100-, and 500-year. These procedures are particularly useful for predicting or estimating flood flows at stream sites where little or no st reamflow information is available. These equations do not represent a stage-discharge rating relation and are limited to specified recurrence intervals.
10Mazvimavi et al. (2006) also explored the influence of basin characteristics on the spatial variation of river fl ows. The study investigated the correlation of three basin characteristics and river flows. The study id entified mean annual precipitation, mean annual potential evaporation and median slope as the most important basin characteristics affecting the variation of river flows. More complex analyses have been done using artificial neural network (ANN) and fuzzy neural network (FNN). ANN has been widely used in water-related research with reasonable good results. Bhattacharya and Solomatine (2000) applied ANN to develop a stage-discharge relationship for rive r Bhagirathi in India. The model provided a solution to the random fluctuations often e xhibited by the ratings by using more data generated from the ANN model. Deka and Chandramouli (2003) presented a FNN model to derive a stage-discharge relationship. Th e study shows the adva ntage of using a FNN approach by simplifying the amount of inform ation required in terms of the number of inputs. The study also shows that the FNN model has a better prediction abil ity that the single ANN and conventional methods. However, despite the high efficiency of ANNÂ’s and FNNÂ’s in modeling non-linear relationships, creating and tr aining the network is still complex and difficult to interpret.
11 3. DATA Forty-three stations located in West-Central Florida were used to develop the model as the calibration-data set. The verification data-s et consisted of thirty additional stations, also within the West Central Florida study area. Following standard practice, the stations in the verification data set were not used to develop the model. Stat ions in the calibration and verification data set were selected randomly, based on th e stations where rating and basin characteristics da ta were available. Table 1 gives the range of values for hydr aulic parameters for the calibration data set. Locations of all USGS stations in th e study area are shown in Figure 1. The rating data for the calibration set corresponds to USGS known data reported between 1987 and 1998. Data for the verification se t corresponds to the rating da ta actually in use by the USGS for 2005-2006. Table 1. Ranges of Values for Hydrau lic Parameters for Calibration Data Parameter Units Mean Std DevMinimum Maximum Discharge Q CMS 31 64 0 790 Depth d m 1.99 1.44 0.03 6.65 Drainage Area A Sq. mi138.32 214.31 4.3 1373 Channel Slope ChS% 3.41 1.62 1.23 7.01 Watershed Slope S m/m 0.007430.004260.001 0.025 %Wetland W % 18.58 8.22 5.1 36.82 DTWT WT m 1.19 0.24 0.92 2.07
12 4. METHODOLOGY 4.1 Identification of Most Significant Variables To identify all variables that could be considered as most significant predictors for the model, a statistical analysis was performed using the SAS statistical software package. The six watershed characteristics considered as potential predictors of discharge were: water depth, drainage area, percent wetlands, depth to water table, watershed slope and channel slope. A correlation matrix was constructed using values of Q10 for discharge and d10 for depth, because these were measured values at each station. The average 10% exceedence discharge values, Q10, and corresponding depths, d10, are readily available (i.e. normally reported) for measured flow stations from annual data summaries and published rating tables, respectively. The Q10 value was taken from the station record, from the Water Resources Data of Fl orida, USGS (2003). The average depth corresponding to that discha rge was calculated from the publ ished USGS rating for that station. Table 2 summarizes the correlations among the variables in the calibration data; results showed water depth and drainage area with the highest correlation to discharge. However, the correlation was very low to other expected parameters including basin slope and percent wetlands. A least square regression analysis was us ed to develop the discharge model. The stepwise procedure Â“maximum R2 improvementÂ” in SAS was used to identify which of the potential predictor variables besides water depth, would be include d in the regression
13model. The stepwise procedure combines the variables to develop the best two-variable model, three-variable model and so forth, that produces the highest correlation coefficient R2. Other statistical measurements as Cp and Fvalue were also used to define the best model. The stepwise procedure was performed using the rating data from the 43 stations in the study area. Results from this analysis (Table 3) showed that, again, the most significant variables are water depth and drainage area; any additional variable included did not improve the R2 significantly. Thus, the model development focused on a twovariable model including only wa ter depth and drainage area. Table 2. Matrix Correlations for Discharg e for Variables in the Calibration Data Q d A ChS SlopeW WT Q 1.00 0.77 0.97 -0.420.07 -0.070.02 d 0.77 1.00 0.70 -0.26-0.13 -0.140.06 A 0.97 0.70 1.00 -0.450.21 -0.050.08 ChS -0.42 -0.26-0.451.00 -0.06 -0.450.09 S 0.07 -0.130.21 -0.061.00 -0.200.57 W -0.07 -0.14-0.05-0.45-0.20 1.00 -0.07 WT 0.02 0.06 0.08 0.09 0.57 -0.071.00 Table 3. Results from the Stepwise Procedure Â“Maximum R2 ImprovementÂ” Variables R2 Cp F value Q d 0.46203.2802.4 d, A 0.5512.3 576.5 d, A, S 0.558.7 388.0 d, A, S, ChS 0.557.8 292.4 d, A, S, ChS, W 0.555.3 235.6 d, A, S, ChS, W, WT0.557.0 196.3
144.2 Approaches for the Determinatio n of the Non-Dimensional Ratings Three different approaches were examined for the determination of a non-dimensional relationship between discharge and depth. The approaches are explained below. 4.2.1 Approach 1: Normalization by Drainage Area It was first hypothesized that discharge and depth could be normalized as a function of basin drainage area and total annual rainfall. The logic for this was that discharge would increase by increasing drainage area and/or increased long term annual rainfall if all other factors were equal. Hence, the stream cro ss section characteristics and resulting rating would be function of these variables. The general non-dimensional equations for discharge and depth for th is hypothesis are given by: RA Q Q (2) 5 0' A d d (3) where Q is dimensionless (relative) discharge, Q is measured discharge (L3T-1), R is annualized rainfall intensity (LT-1), A is watershed area (L2), d is dimensionless (relative) water depth and d is stream water depth (L). For the analysis, annualized rainfall intensity was assumed constant, c onsidering that the l ong-term annual rainfall intensity in West-Central Florida was fixed and readily determined from gage record. A fundamental departure from USGS rating curves was that the proposed relation in Equation (2) uses depth of wa ter instead of stage elevation. This is desirable because: The stream stage is measured from an ar bitrary datum that is not always defined to a standard datum, e.g., National Ge odetic Vertical Datum of 1929 (NGVD-29).
15This makes it difficult to correlate to a reference when using a generalized relation in an un-gauged stream (i.e., the streambed elevation are not known). Equations that define the hydraulic behavior of streams are based on depth, thus this would be more realistic in terms of the physical representation of the system. Live bed environments typical of natura l streams are always changing, greatly affecting low flows. However, it was hypothesized that the streambed inverts change while, on a long-term basis in na tural channels, similar slope and friction conditions prevail. Thus, absolute depths should be similar in long-term record to maintain a fixed Q (d) relationship. The use of water depth instead of stage is inconvenient since a change in depth (as hydraulic depth) does not necessarily produce the corresponding change in discharge, as opposed to measured stage. For example, th e average hydraulic depth may decrease as the flow transitions out-of-banks in increasing flood flow conditions. This will result in a misrepresentation of the rating relation. To overcome this inconvenience, an effective depth (d) related to stage was de fined using the following equation: 0 0h h d d (4) where, d = effective depth (L) d0= minimum hydraulic depth (L) h = stage (L) h0 = stage at zero flow (L)
16The minimum hydraulic depth was estimated from USGS field data, where depths are the average values calculated from cro ss sectional area and channel width. The stage is the data point from the USGS rating curves and the relative stage at zero flow was determined from the USGS rating data. The relative discharge and depth were used to derive a log-linear relation of the form: b d m Q log log (5) In this general relationship, the values of the parameters m and b must accurately represent the overall be havior of the stage-discharge ratings for a great number of stations. Figure 2 shows the rela tive discharge versus relative depth for all stations used in this study, using the USGS rating data. Note worthy is that most of the stations have similar slopes. This gives credibility to the idea that a unique relation can be determined to represent the depth-discharge characteristic s of many stations. An average value of the slopes and intercepts was used as the m and b parameters for the generalized nondimensional relation. The final equation is defined as: 79 6 log 72 1 log d Q (6) Results from this model were compared to the USGS rating data. Some stations provided a Â“good fitÂ” for only a portion of the data, other stations differed completely. Discharge values estimated w ith the model were subtract ed from the USGS known rating data and divided by the USGS data to obtain the relative error. The average relative error
17compared to the USGS data varied from 18% to 50% for 46% of the stations, 51% to 100% in 30% of the stations and more than 100% in the remaining 24%. -5 -4 -3 -2 -1 0 1 2 -6.4-5.9-5.4-4.9-4.4-3.9-3.4-2.9-2.4 Log Relative Depth d'Log Relative Discharge Q' Figure 2. Relative Discharge Ve rsus Relative Depth for Stati ons in West-Central Florida for Approach 1 4.2.2 Approach 2: Normalization by Discharge and Depth at Full Channel Discharge and depth were normalized as a func tion of discharge and depth at full channel flow within banks. These two parameters were approximated from the rating data, as the point where a change in the rating slope o ccurs (rating break point). The general nondimensional equations for disc harge and depth are given by: fullQ Q Q (7) fulld d d (8)
18where Q is dimensionless (relative) discharge, Q is measured discharge (L3T-1), Qfull is discharge at full channel bank flow (L3T-1), d is dimensionless (rela tive) water depth, d is stream water depth (L) and dfull is water depth at full channel bank flow (L). As in the first approach, equation (7) is described in terms of depth of wate r instead of stage. The generalized relation base d on relative discharge and depth was defined with a log-linear equation as in A pproach 1. For this model, the non-dimensional dischargedepth relation was defined with zero intercep t. Figure 3 shows the relative discharge versus relative depth for all stations unde r study, using the USGS rating data. A general value for slope was defined and the final equation for this approach is: log 9 1 log d Q (9) -4 -3 -2 -1 0 1 2 3 -2-1.5-1-0.500.511.5 Log Relative Depth d'Log Relative Discharge Q' Figure 3. Relative Discharge Ve rsus Relative Depth for Stati ons in West-Central Florida for Approach 2
19A value of discharge and depth at full cha nnel banks is required to generate the rating in un-gauged stations. Known values of discharge at full channel banks were related to drainage area and known values of de pth were related to the square root of the drainage area. The resulting trend line showed a small correlation of the two parameters, 0.45 for discharge and 0.20 for depth. Results from this model were compared to the USGS rating data. As in the first approach, some stations matched a portion of the data, while other stations completely differed. The average relative error compared to the USGS data varied from 8 to 50% for 33% of the stations, 51 to 100% in 26% of the stations and more than 100% in the remaining 41% (poorer performance than Approach 1). 4.2.3 Approach 3: Normalization of Discharge and Depth at Ten Percent Exceedence Flow Discharge and depth were normalized as a func tion of the value of the discharge reported to be exceeded ten percent of the time a nd the corresponding depth. The rationale for selecting the ten percent exceedance value as the scaling factor, is that most rivers in the West Central Florida area have well defined incised channels and most discharges at or below the 90th percentile (10% exceedance) are contained within banks (Lewelling, 2004). Because bankfull discharg e typically increases with drainage area, high flow values as Q10 in this case, were considered to repres ent a better relation to drainage area. For streams with rela tively high friction of groundwater flow, Q10 is also indicative of a runoff dominated flow period.
20The general non-dimensional equations fo r discharge and depth are given by: 10' Q Q Q (10) 10' d d d (11) where Q is dimensionless (relative) discharge, Q is measured discharge (L3T-1), Q10 is discharge exceeded ten percent of time (L3T-1), d is dimensionless (relative) water depth (L), d is stream water depth and d10 is water depth correspon ding to the 10% discharge exceedence (L). As in the previous approaches the relative depth is described in terms of depth of water instead of stage. The generalized relation ba sed on relative discharge and depth was also defined with a log-linear equation with zero intercept. Table 4 shows data of discharge and depth for the 10% discharge exceedance and the sl ope of the corresponding relative discharge versus relative depth rating, for the thirty-fiv e stations where data were available. A very high correlation coefficient of 0.996 was obtained from the fitted USGS data for this approach using specific slopes (Slope10 in Table 4). Thus, these slopes represent the rating of each particular stati on quite well. Plots of these ra tings are shown in Figure 4. The slope was defined and the fina l equation for this approach is: log 77 1 log d Q (12) A value of discharge and depth exceeded 10% of the time was required to generate the rating in un-gauged stati ons. Known values of discharge as Q10 were related to drainage area and values of depth as d10 were related to the squa re root of the drainage
21area. The resulting trend line s howed a better correlation of the two parameters, 0.94 for discharge and 0.62 for depth. Figures 5 and 6 show data and trend line for discharge and depth, respectively. Table 4. Discharge and Depth for the 10% Discharge Exceedance and the Slope of the Corresponding Rating for Calibration Data Set ID USGS Station Name USGS Gage # Slope10 Q10 (CMS) d10 (m) 5 Payne Creek Nr Bowling Green, Fl 2295420 1.34 7.5 1.3 6 Peace River At Fort Meade, Fl 2294898 1.73 15.5 1.8 7 Peace River At Arcadia Fl 2296750 1.74 76.5 2.8 8 Little Manatee River Nr Wimauma, Fl 2300500 1.72 11.0 1.9 9 Little Manatee River Nr Ft Lonesome Fl 2300100 1.72 2.2 1.3 10 North Prong Alafia River At Keysville Fl 2301000 2.1 8.5 1.8 11 Alafia River At Lithia Fl 2301500 1.67 20.5 1.9 12 Peace River At Bartow Fl 2294650 2.57 17.1 1.6 13 Horse Creek Nr Myakka Head Fl 2297155 1.6 2.1 0.8 15 Charlie Creek Nr Gardener, Fl 2296500 1.98 21.2 2.4 16 South Prong Alafia River Nr Lithia Fl 2301300 1.9 6.3 1.5 17 Horse Creek Nr Arcadia, Fl 2297310 1.9 15.0 2.4 18 Peace River At Zolfo Spring, FL 2295637 1.61 41.1 2.6 19 Trout Creek Near Sulphur Springs Fl 2303350 2.34 1.6 0.7 20 Cypress Creek At Worthingt on Gardens FL2303420 2.37 3.6 1.3 21 Peace Canal Nr. Wahneta 2293987 2.48 9.0 1.3 22 Bowlegs Creek Nr. Ft. Meade 2295013 2.27 2.2 0.9 24 Joshua Creek @ Nocatee 2297100 2.19 7.8 1.7 28 Myakka River Nr Sarasota 2298830 2.71 19.4 2.0 29 Deer Prairie @Power Line 2299120 2.6 3.8 1.3 30 Big Slough Canal Nr Myakka 2299410 2.29 2.7 1.0 31 Walker Creek Nr Sarasota 2299861 1.92 0.4 0.4 35 Braden River Nr Lorranie 2300032 1.91 2.4 1.2 38 Bullfrog Creek Near Wimauwa 2300700 1.95 2.3 1.0 41 Blackwater Ck Nr Knights 2302500 1.84 5.3 0.9 42 Hills R. Nr Zephyrhills 2303000 1.96 14.9 1.6 43 Baker Creek At Mcintosh Ro 2303205 2.22 1.2 0.5 48 Delaney Creek Nr Tampa 2301750 2.74 0.7 0.4 50 Tiger Creek Nr Babson Park 2268390 2.53 1.9 0.9 51 Livingston Ck Nr Frostproof 2269520 1.91 3.7 1.1 52 Carter Creek Nr Sebring 2270000 2.64 1.5 1.0 53 Josephine Ck Nr Desoto 2271500 2.3 4.8 1.1 57 Brooke Creek @ Van Dyke 2307200 2.85 0.3 0.4 62 Anclote River Near Elfers 2310000 2.1 4.9 1.7 63 Pitha R. Nr Fivay Junction 2310280 2.7 0.5 0.4
22 -4 -3 -2 -1 0 1 2 3 -1.5-1-0.500.511.5 Log Relative Stage d'Log Relative Discharge Q' Figure 4. Relative Discharge Ve rsus Relative Depth for Stati ons in West-Central Florida for Approach 3 Results from this model were compared to the USGS rating data. As in the previous approaches, some sta tions provided a good fit while others completely differed. However, in this approach, the general m odel matched a larger portion of the data compared to the other approaches. This suppor ts the idea that this model could better represent a larger range of the stream de pth-discharge behavior The average relative error compared to the USGS data varied from 20 to 50% for 42% of the stations, 51 to 100% in 48% of the stations and mo re than 100% in the remaining 10%.
23 y = 0.0522x R2 = 0.9453 0.0 20.0 40.0 60.0 80.0 100.0 050010001500 A (sq mi)Q10 (CMS) Figure 5. Correlation Between Discharge as Q10 and Drainage Area for Approach 3 y = 0.0763x + 0.4815 R2 = 0.6539 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 010203040 A0.5 (mi)d10 (m) Figure 6. Correlation Between Depth as d10 and Square Root of the Drainage Area for Approach 3
24 5. MODEL DEVELOPMENT From the analysis of the three different a pproaches, it is recognized that even though the slopes in the intermediate-flow region have a si milar behavior, they vary within a specific range (Figure 7). Also, the high-flow regi ons exhibited log-linear behavior with considerably different slopes than the in termediate-flow regions (Figure 8). Based on this, two different mathematical relationships are proposed for each of the flows regions. 5.1 Theoretical Background for Model Development Certain physical characteristic s at the gauging section or in the channel bed, known as station controls, stabilize th e stage and discharg e relation (Gupta, 2004) It is common practice to locate the streamfl ow gauging station in a sectio n control with the purpose of making the rating more stable. A section control can be a natural or man-made obstruction, such as a riffle, rock, weir or sp illway. For engineered structures as in the case of weirs and spillways, by creating an obstr uction, critical depth is forced to occur and a unique relationship between stage and discharge results. This simple relation between stage and discharge also exists for natural obstructions. The control characteristics of the station will determine the particular features of the gauging site rating curve, includi ng the rating slope.
25 -4 -3 -2 -1 0 1 2 3 -1.5-1-0.500.511.5 Log d'Log Q' Figure 7. Relative Discharge (Log QÂ’) Ve rsus Relative Depth (Log dÂ’) for the Intermediate-Flow Region -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -0.200.20.40.60.811.2 log d'log Q' Figure 8. Relative Discharge (Log QÂ’) Versus Relative Depth (Log dÂ’) for the High-Flow Region
26Hence, the difference in streamflow beha vior ( ie. rating slop e) between the two regions and within the same fl ow region can be examined by presuming simple control at stream sections. 5.1.1 Analysis for Simple Control Sections Consider that streamflow is downstream cont rolled at a section by one of the following cases: sharp crested uncontracted; sharp cres ted contracted; V-notch ; Broad crested weir. All weir relations are represented by nH K f Q , with K being all the constant parameters (discharge coefficient, weir le ngth, contraction) and n=3/2 for uncontracted, moderately contracted recta ngular and trapezoidal weirs a nd 5/2 for Vnotches. Using some hypothetical data for stage and di scharge and normalization parameters, nondimensional rating curves (QÂ’ vs. HÂ’) were developed for the assumed control sections. The dimensional form of C was used for the weir equations. Assuming a horizontal_uncontracted section, the weir equation is: 2 3CLH Q (13) with C values between 3.2 and 4.4 and a typical value of 3.3. Figure 9 shows the nondimensional discharge and depth, plotted on loglog scale for this case. Data used in this figure is shown in Appendix A.
27 Horizontal Uncontracted 0.01 0.10 1.00 10.00 0.010.101.0010.00 H'Q' C = 3.3 C = 3.5 C = 4.0 C = 4.4 Figure 9. QÂ’ vs. HÂ’ for Horizontal_Uncontra cted Section for Varying Coefficients C For a horizontal_contracted section, the weir equation is: 2 3) 2 0 ( H H L C Q (14) with C values between 3.2 and 4.4 and a t ypical value of 3.3. Figure 10 shows the nondimensional discharge and depth, plotted on loglog scale for this case. Data used in this figure is shown in Appendix B. For a V-Notch Section section, the weir equation is: 2 3CH Q (15) with C values between 2 and 4.3 with a typi cal value of 2.5 for 90 V-Notch. Figure 11 shows the non-dimensional discharge and dept h, plotted on log-log scale for this case. Data used in this figure is shown in Appendix C.
28 Horizontal Contracted 0.01 0.10 1.00 10.00 0.010.101.0010.00 H'Q' C = 3.3 C = 3.5 C = 4.0 C = 4.4 Figure 10. QÂ’ vs. HÂ’ for Horizontal_Contr acted Section for Varying Coefficients C V-Notch 0.00 0.00 0.01 0.10 1.00 10.00 0.010.101.0010.00 H'Q' C = 2.0 C = 2.5 C = 3.0 C = 4.3 Figure 11. QÂ’ vs. HÂ’ for V-Notch S ection for Varying Coefficients C
29For a Broad Crested Weir section, the weir equation is: 2 3CLH Q (16) with C values between 2.34 and 3.3 and a t ypical value of 2.63. Figure 12 shows the nondimensional discharge and depth, plotted on loglog scale for this case. Data used in this figure is shown in Appendix D. Results for the typical C value for each of the evaluated control sections are shown in Figure 13. Broad Crested 0.01 0.10 1.00 10.00 0.010.101.0010.00 H'Q' C = 2.3 C = 2.63 C = 3.0 C = 3.3 Figure 12. QÂ’ vs. HÂ’ for Br oad Crested Weir Section fo r Varying Coefficients C From Figures 9 to 12, the effect of vary ing the discharge coefficients for each control section shape is basically a shift in the resulting rating. The discharge coefficient lumps together the effects of the approach ve locity head with the contraction and head losses. Even though the analysis did not consider the variat ion of C with head, both the velocity head and discharge coefficient increas e with stage. Thus, for the same head and crest length, a larger discharge coefficien t will result in a larger discharge.
30 All Control Sections 0.00 0.00 0.01 0.10 1.00 10.00 0.010.101.0010.00 H'Q' Hor_Uncontracted Hor_Contracted V-Notch Broad Crested Figure 13. QÂ’ vs. HÂ’ for All Control Sections for Typical Values for C Figure 13 presents the results for all control section shapes Similarly to the effect of varying C, the effect of varying the sect ion shape is a shift in the resulting rating except for the V notch. The discharge capacity of a rectangular versus a broad crested weir is different, therefore the shift in th e ratings, however the three equations use the same exponent. The exponent in the V-No tch section is different/larger than the rectangular and broad crested sect ions which is reflected in th e different and in this case steeper slope. It is worth to notice that the slope of the non-dimensional relations does not change with the varying C for the same contro l section shape. This is a clear indicative that the relation slope is a di rect function of the exponent an d is not affected by all the constant parameters included in K.
31By observing the slopes in the non-dim ensional rating curves for each of the stations being studied, most of the values fa ll between 1.5 and 2.5. Therefore, the degree of contraction could easily e xplain the variation in slope within the intermediate-flow region. Thus, from the weir equati on for the control section: nH K Q log log log or H n K Q log log log (17) Where n=3/2 or 5/2 is the slope of the non-dimensional relations. It can also be shown that uncontracted (e.g., uniform) wide flows approach n values of 5/3 as an upper lim it for turbulent flows. To e xplore this, a plot with the uncontracted and V-notch sections and non-dimensional curves using ManningÂ’s equation was created (Figure 14) for a rectangul ar and trapezoidal section. The behavior of the ManningÂ’s relation is similar to the we irsÂ’ behavior but w ith a different slope. 5.1.2 Analysis for Compound Control Sections In compound control sections the low/normal flows are managed by, for example, a Vnotch, and larger flows would require, for ex ample, a trapezoidal weir. For these cases, the two profiles would form a compound weir The simplest way to calculate the discharge over a compound weir is applyi ng the discharge equa tion corresponding to each segment of the weir and up to the head of that segment. Assuming a compound weir consisting of a V-Notch or Horizontal and a Trapezoidal (Cipottelli) Weir (standard equations follow) the non-dimensional relation was created. Figure 15 show results using typical values for C. Data used in this figure is shown in Appendix E.
32 0.00 0.00 0.01 0.10 1.00 10.00 100.00 0.010.101.0010.00 H'Q' Hor_Uncontracted V-Notch Manning's_Rect Manning's_Trap Figure 14. QÂ’ vs. HÂ’ for Uncontracted and VNotch Weirs and ManningÂ’s Equation The equation for a trapezoidal section is: 2 3CLH Q (18) Figure 15 could explain the different/steeper slopes observed in the high-flow regions. In compound high-flow control sect ions (channel and out-of banks flood plain sections) there is a different degree of cont raction. The low/normal flows are managed by cross sections with a relative degree of contraction (similar to a V-notch or uncontracted weir), and larger flows would be managed by cross sections with a higher degree of
33contraction (similar to a trapezoidal weir). For these cases, when plotting Q vs. H, the two profiles would form a compound weir as shown in Figure 15. This will result in a break in the resulting rating at the point where the flow cha nges from channel flow (low and intermediate) to floodplain flow (high); there is a break in the rating curve, with the high-flow range having a steeper slope. 0.00 0.01 0.10 1.00 10.00 100.00 0.101.0010.00 H'Q' Intermediate_2.5 High Transition Intermediate_1.5 Figure 15. QÂ’ vs. HÂ’ for Co mpound Weir Consisting of a VNotch and a Trapezoidal Rating data also show a third flow re gion corresponding to extreme low flows. Low flows are dominated by contribution of groundwater baseflow. Hydraulically they may not be fully turbulent, exhibiting parabolic (non log-linear) relationship for discharge-depth rating. Most ra tings are also very unstable for low flows as channel debris and shifting live bed conditions highly influence the rating behavior. For example,
34most low-water discharge measurements have different gage stage at zero flow (GZF) and consequently, do not define the same rating or adapt to logarithmic plotting (Kennedy, 1984). This condition renders it harder to represent in a general model, thus this study did not consider the mathematical re presentation of low flows. At present, the author believes that it is doubtful if any generalized mathematical model can be developed for the very lowest flow conditions. 5.2 Model for the Interm ediate-Flow Region The model for the intermediate-flow region wa s developed using a least square regression analysis. Only discharge and water depth va lues corresponding to the intermediate-flow region were included in the analysis. The uppe r limit of the intermediate-flow region was approximated from the rating data, as the point where a change in th e rating slope occurs (i.e., rating break poi nt) (See Figure 16). -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -0.8-0.6-0.4-0.200.20.40.6 Log Q'Log d' Figure 16. Generalized LogLinear Rating Behavior Intermediate Flows (within-banks) Very Low Flows High Flows Break point
35The generalized relation base d on relative discharge and depth was defined with a log-linear equation with zero in tercept. For the forty-three stations in the study, the best fit slope was found to be 2.0 for the West-Cen tral Florida study ar ea. The mathematical model was described as: log 2 log d Q (19) Table 5 shows the statistics of the model. Table 5. Statistics of the Regressi on Model for the Intermediate Flows Region R2 Adj R2 RMSE Coeff VarF Value Pr > F 0.95 0.95 0.1862 291.3 21319.3 <.0001 Variable CoefficientSE t Value Pr > |t| VIF d 2.0 0.01445146.01 <.0001 1 Results from this model were compared to the USGS rating da ta. The accuracy of the prediction was measured in terms of the relative error. Predicted values were first calculated using values of Q10 and d10 specific for each station. The average relative error was less than 20% for 58% of the stations; 21% to 30% for 16% of the stations; 31% to 50% for 12% of the stations; and larger than 51% for 14% of the stations. Basin characteristics data and relative error for the specific model is shown in Appendix F. A value of discharge and depth exceeded 10% of the time is required to generate the ratings in un-gauged stations. As in the Approach 3, known values of discharge as Q10 were related to drainage area, but a bette r correlation was found by relating values of depth (d10) to Q10. The correlation coefficient for d10 was improved from 0.62 to 0.86.
36The corresponding regre ssion equations for Q10 and d10 are: A Q 0522 010 (20) 4053 0 10 106505 0 Q d (21) Where Q10 is in CMS, drainage area is in square miles and d10 is in meters. Figures 17 and 18 show data and regression equations for discharge and depth, respectively. Discharge was then predicted using the model with Q10 and d10 estimated from the regression equations. The average relative error compared to the USGS data was less than 20% for 23% of the stations; 21% to 30% fo r 27% of the stations; 31% to 50% for 24% of the stations; and larger than 51% for the remaining 26%. These results are comparable to the ones obtained with the specific known values. y = 0.0522x R2 = 0.9453 0.0 20.0 40.0 60.0 80.0 100.0 050010001500 Drainage Area (sq mi)Q10 (CMS) Figure 17. Data and Regression Equation for Q10
37 y = 0.6505x0.4053R2 = 0.8636 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 0.020.040.060.080.0100.0 Q10 (CMS)d10 (m) Figure 18. Data and Regression Equation for d10 5.3 Model for the High-Flow Region The slopes in the high-flow region, although diffe rent from the slopes in the intermediate region, exhibit similar behaviors among them but in this case, w ith a very different intercept. The model for this region was develope d by error minimization using the least mean square procedure. Not all stations in the study area have data in the high-flow region, thus the analysis was performed for those stat ions where high flows were available, for a total of twenty-seven st ations. Using the specific intercept for each station, the best fit slope was found to be 5. 0 for the West-Central Florida study area. The mathematical model was described as: stationb d Q log 5 log (22) Where bstation is the specific inte rcept for each station.
38The model was tested using the specific intercept for each station and results were compared to the USGS rating data. The averag e relative error was less than 20% for 33% of the stations; 21% to 30% for 7% of the st ations; 31% to 50% for 4% of the stations; and larger than 51% for 56% of the stations. Basin characteristics data and relative error for the specific model is shown in Appendix G. A value for the intercept is required to generate the rating in un-gauged stations. The intercept is related to the discharge where the high-flow region departs from the intermediate-flow region, believed to be the out-ofbank condition above which floodplain flows become hydraulically significan tly (shown as break point in Figure 16). Therefore, those values of discharge of departure (transition to out-of bank flow) (Qdep) were identified from the rating data and were related to drainage area. The corresponding depth of departure (ddep) was related to Qdep. The resulting trend line showed a correlation, R2 of 0.90 for Qdep and 0.86 for ddep The corresponding regressi on equations for discharge of departure (Qdep) and depth of departure (ddep) are: A Qdep16 0 (23) 428 0479 0dep depQ d (24) Where Qdep is in CMS and drainage area is in sq uare miles. Figures 19 and 20 shows data and regression equation for Qdep and ddep, respectively.
39Data from the high-flow re gion were tested using a linear model with a fixed slope of 5.0 and an intercept calculated from Qdep and corresponding ddep. The results were very poor with most of the stations exhibiting relative errors exceeding, in some cases, 100%. Therefore, it is apparent that more work needs to be done on the general departure threshold. y = 0.1592x R2 = 0.8996 0.0 50.0 100.0 150.0 200.0 250.0 050010001500 A (sq mi)Qdep (CMS) Figure 19. Data and Regression Equa tion for Discharge of Departure (Qdep) y = 0.4788x0.4277R2 = 0.863 0.00 5.00 10.00 15.00 20.00 25.00 010002000300040005000600070008000 Qdep (CMS)ddep (m) Figure 20. Data and Regression E quation for Depth of Departure (ddep)
40 6. MODEL VERIFICATION To measure the success in pr edicting rating discharge for the intermediate region, the model was verified using data from thirty sta tions that had not been previously used in development. The stations in the verification data set were randomly selected within the study area. The range of discharg e and drainage areas was sim ilar to the stations in the calibration set. Table 6 gives the ranges of va lues for discharge, de pth and drainage area for the verification data set. Table 6. Ranges of Values for Hydrau lic Parameters for Verification Data Parameter Units Mean Std Dev Minimum Maximum Discharge Q CMS 31.5 54.7 0.0 393.9 Depth d m 2.00 1.47 0.00 5.89 Drainage Area A Sq. mi214.39 507.49 1.46 1825.00 The model was tested using the Q10 and d10 calculated from the regression equations. Results from the model were compared to the USGS rating data and the accuracy was measured as the relative error. The average relative erro r when compared to the USGS data for the intermediate flow wa s less than 30% for 43% of the stations, 31% to 50% for 27% of the stations and larger than 51% for the remaining 30%. Thus, the error was comparable to the results obtain ed with the calibration data set. Basin characteristics data and relative error for the specific model is shown in Appendix H. Because of the lack of accurate results in the high-flow region using the intercept values from the regression equations, the high-flow model was not validated using the verification data set.
41 7. MODEL VALIDATION To evaluate the applicability of the proposed model in an area outside West Central Florida, the model was tested at forty stations located in th e areas of North Florida, and South Georgia and Alabama (See Table 7 and Figure 21). These sta tions comprise the validation data set and were selected randomly with drainage areas varying from 17 to 13600 square miles, with an average of 1830 square miles. The drainage area and Q10 values were taken from the station record, from the Water Resources Data of Florida, Georgia and Alabama, USGS (2004). The rating data corresponds to data actually in use by the USGS for 2005-2006. Following the same approach as before, this validation was performed using two separate models for data in the intermedia te and high flow regions. The limit between the intermediate and high-flow region was also approximated from the rating data, as the point where a change in the rating slope occurs. As with the stations in West Central Flor ida, the model was tested using both, the specific values of Q10 and d10, and general values obtained from a regression equation. For this purpose, a new relation was devel oped for the non-dimensional parameters. The regression equations for Q10 and d10 including the new stations in the validation set are: A Q 0522 010 (25) 4053 0 10 106505 0 Q d (26)
42Table 7. Discharge and Depth for the 10% Discharge Exceedance and the Slope of the Corresponding Rating for Validation Data Set Gage # USGS Station Name Slope10 Q10 d10 CMS m 2369800 Blackwater River Near Bradley Al 1.12 106.55 4.67 2361500 Choctawhatchee River Near Bellwood Al 1.65 89.55 2.28 2378300 Magnolia River At Us 98 Near Foley, Alabama 2.80 1.13 0.49 2376500 Perdido River At Barrineau Park, Fl 1.28 40.52 1.52 2374250 Conecuh River At State Hwy 41 Near Brewton, Al 1.80 203.75 3.65 2373000 Sepulga River Near Mckenzie Al 2.02 47.89 1.25 2361000 Choctawhatchee River Near Newton, Al 1.24 54.98 1.80 2363000 Pea River Near Ariton Al 1.12 39.11 1.28 2372422 Conecuh River Bel Pt A Dam Nr River Falls, Al 0.84 99.19 1.22 2364500 Pea River Near Samson Al 1.63 106.55 3.68 2377570 Styx River Near Elsanor, Al 1.43 22.98 1.26 2374950 Big Escambia Cr At Sardine Br Nr Stanley Crossroad 1.06 16.97 0.62 2362240 Little Double Bridges Creek Nr Enterprise, Al 2.04 1.70 0.82 2323000 Suwannee River Near Bell, Florida 1.13 427.91 4.01 2319800 Suwannee River At Dowling Park, Florida 1.21 306.06 4.13 2319000 Withlacoochee River Near Pinetta, Fla 1.24 129.51 2.36 2322500 Santa Fe River Near Fort White, Fla 1.03 72.26 0.87 2321500 Santa Fe River At Worthington Springs, Fla 1.66 31.17 2.54 2330150 Ochlockonee River Nr Smith Creek, Fla 1.1 121.86 2.65 2330100 Telogia Creek Nr Bristol, Fla 1.64 12.24 1.74 2359000 Chipola River Nr Altha, Fla 0.94 78.21 2.03 2365500 Choctawhatchee River At Caryville, Fla 1.84 320.23 4.63 2368500 Shoal River Nr Mossy Head, Fla 1.42 12.36 0.90 2370500 Big Coldwater Creek Nr Milton, Fla 1.41 25.16 1.44 2376500 Perdido River At Barrineau Park, Fl 1.23 40.52 1.53 2322700 Ichetucknee R @ Hwy27 Nr Hildreth, Fl 1.11 10.03 0.47 2226500 Satilla River Near Waycross, Ga 1.34 81.90 3.23 2228000 Satilla River At Atki nson, Ga 1.33 170.03 3.41 2314500 Suwannee River At Us 441, At Fargo, Ga 1.52 74.25 3.44 2226000 Altamaha River At Doctortown, Ga 1.2 889.83 3.59 23177483 Withlacoochee River At Mcmillan Rd, Ga 2.1 34.57 2.87 2327500 Ochlockonee River Near T homasville, Ga 2.1 35.99 2.84 2329342 Little Attapulgus Creek At Attapulgus, Ga 1.32 0.68 0.38 2353000 Flint River At Newton, Ga 1.42 368.40 3.33 2353265 Ichawaynochaway Creek At Ga 37, Near Morgan, Ga 1.39 12.92 1.34 2354410 Chickasawhatchee Creek Near Leary, Ga 3.2 8.08 0.66 2354800 Ichawaynochaway Creek Near Elmodel, Ga 1.56 55.83 1.65 2355662 Flint River At Riverview Planta tion Nr Hopeful, Ga 1.35 402.41 3.92 2357000 Spring Creek Near Iron City, Ga 1.93 33.44 2.63 2316000 Alapaha River Near Alapaha, Ga 2.47 42.51 3.19
43 Figure 21. Location of USGS Stations in the Areas of North Florida, Georgia and Alabama Where Q10 is in CMS, drainage area is in square miles and d10 is in meters. Figures 22 and 23 show data and regressi on equations for discharge a nd depth, respectively. As shown, when adding the stations in the valid ation data set, the correlation coefficient improved slightly for Q10 compared to the regression with just the West Central Florida stations. The regression for d10 shows a poorer correla tion but still moderate. Data for the intermediate flow region wa s initially tested using the same model developed for West Central Florida stations (Equation 19). The average relative error using values of Q10 and d10 specific for each station was less than 30% for 33% of the
44stations and less than 50% for 68% of the stations. The average relative error using values of Q10 and d10 estimated from the regression equations was less than 30% for 23% of the stations and less than 50% for 63% of the stations. Drainage area data and relative error for the specific model is shown in Appendix I. In an attempt to improve these results, a model for the intermediate-flow region specific to the data in the validation set wa s developed using a l east square regression analysis. The generalized rela tion based on relative discharge and depth was defined with a log-linear equation with zero intercept. For th e forty stations in the validation data set, the best fit slope was found to be 1.35. The mathematical model was described as: log 35 1 log d Q (27) This difference in the rating slope can be observed in the plot of relative discharge versus relative depth for the stations in the validation data se t (Figure 24). When compared to the same plot for the stations in West Central Florida (Figure 4), most of the ratings in the validation set show a lower slope. y = 0.0589x R2 = 0.9555 0.0 200.0 400.0 600.0 800.0 1000.0 050001000015000 Drainage Area (sq mi)Q10 (CMS) Figure 22. Data and Regression Equation for Q10 Including Validation Data Set
45 y = 0.6126x0.3171R2 = 0.7514 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0.0200.0400.0600.0800.01000.0 Q10 (CMS)d10 (m) Figure 23. Data and Regression Equation for d10 Including Validation Data Set -6 -5 -4 -3 -2 -1 0 1 2 3 -3.5-3-2.5-2-1.5-1-0.500.511.5 Log d'Log Q' Figure 24. Relative Discharge (Log Q Â’) Versus Relative Depth (Log dÂ’) for Stations in North Florid a, Alabama and Georgia
46The lower rating slope can be expl ained by looking at the watershed characteristics. The stations in the validation data set have a much larger drainage area than the stations used in the calibration and ve rification data set. As indicated, the mean drainage area in the data validation set is 1830 square miles, that compared to 138 sq. miles in the calibration set and 214 sq. miles in the verification set, in average is much larger. Large drainage areas are associated with flatter average channel slope and consequently larger water depths for th e mean discharge than the smaller sized watersheds. As a result, the ratings at the larger watersheds will exhibit higher values of depth for the mean discharge, resulting in flatter rating slopes. Figure 25 shows the relation of drainage area and ch annel slope and water depth, for some stations in Florida, Georgia and Alabama. In this fi gure, water depth is shown as d10 as a representative value for the streams. 0.000 2.000 4.000 6.000 8.000 10.000 12.000 0.00500.001000.001500.00 Drainage Area (mi2)ChSlope(m/mi), d10(m) Channel Slope d10 Figure 25. Relation Between Drainage Ar ea and Channel Slope and Water Depth (d10) for Some Stations in Florida, Georgia and Alabama
47The bankfull channel dimensions could also be a factor in the flatter slope. A study for regional channel characteristic s in Florida stre ams (Metcalf, 2004) demonstrated that streams in Northwest Fl orida have much larger bankfull areas and discharge than other areas of Florida. The st udy attributes the differe nce to precipitation and geologic runoff factors in the Northern Fl orida area. Also, some reviewed studies on development of regional curves (Cino tto, 2003; Keaton, 2005; Moody, 2003; Kuck, 2000) show drainage area as the main factor for the variability in bankfull characteristics (ie. bankfull area). This supports the rati onalization presented pr eviously of larger drainage areas in the validation data set as the possible reason for the flatter slope. When testing Equation 27, the averag e relative error us ing values of Q10 and d10 specific for each station was less than 30% for 58% of the stations and less than 50% for 78% of the stations. These results are much better than the ones obtained with Equation 19 and comparable with the results obtained for West Central Florida stations. The average relative error using values of Q10 and d10 estimated from the regression equations was less than 30% for 25% of the stations and less than 50% for 63% of the stations. These results are comparable to th e ones obtained with Equation 19. A similar approach was used for the high flow region. A model specific to the high flow data in the validation set was developed by the least square regression procedure. Using the specific intercept for each station, the best fit slope was found to be 2.9 for the stations in the validation set. The mathematical model was described as: stationb d Q log 9 2 log (28) Where bstation is the specific inte rcept for each station.
48The model was tested using the specific in tercept for each station and results were compared to the USGS rating data. The av erage relative error using values of Q10 and d10 specific for each station was less than 30% for 36% of the stations and less than 50% for 48% of the stations. Due to the poor result s obtained using a general equation for the intercept for the high flow region in the calibration data set, no general model was developed for the validation data set. Drainage area data and relative error for the specific model is shown in Appendix J.
49 8. RESULTS AND DISCUSSION The main purpose of this research was to de velop a generalized equation for rating stream depth-discharge when there is an absence of a rating measurement data set. The equation will describe the depth-discharge rating by recognizing and individually describing the intermediate and high-flow regions. A sec ondary interest was to provide a simple equation which could be used in model calibration. From the three explored approaches, the general model based on the depthdischarge values using the 10% exceedence flow rate and depth, provided the best overall results. It is important to mention that even though all three explor ed approaches require some normalization parameters, in the case of Approach 2 and 3, the estimation of the scaling factors, may introduce some potential errors. Nevertheless, even factoring in the error into the analysis, Approach 3 yielded the best results. A qualitative assessment of results from Approach 3 is shown in Figure 26. The qualitative comparison shows that those sta tions that provide an apparent Â“good fitÂ” match only partially the USGS rating data. The matching region corresponds to low and intermediate flows. It is clear from these results that this model does not adequately represent the high flow region. For some st ations that upper limit of applicability is around the Q10 and for others is near the rating break point. However, some stations showed good performance for discharges much larger than Q10 and Qfull flows.
50In order to quantitatively evaluate the accuracy of the proposed method, values of discharge were obtained using Approach 3 a nd compared to discharge values from the USGS rating curves. The StudentÂ’s paired T-test was used as the statistical analysis to measure the differences between the discha rge from the USGS rating curves and the discharge estimated by the model. The T-test is commonly used to compare the mean differences between two data sets based on th e paired differences of the data. The test statistic assumes that the sampling distribu tion of the differences is approximately normally distributed and independent of th e magnitude of the discharge (Johnson, 1995). The test was performed for each of the stati ons in the study area. Results are shown in Table 8. 2300032 0 500 1000 1500 2000 2500 3000 3500 0.005.0010.0015.0020.00 d (m)Q (CMS) USGS Model 2294898 0 20 40 60 80 0.001.002.003.00 d (m)Q (CMS) USGS Model 2300500 0 20 40 60 80 100 0.001.002.003.004.005.00 d (m)Q (CMS) USGS Model 2295637 0 20 40 60 80 100 120 0.002.004.006.00 d (m)Q (CMS) USGS Model Figure 26. Some Stations that Partially Ma tched the USGS Rating Data Using Approach 3
51Table 8. Results of T-test for 43 Stations in West-Central Florida Station T testStationT testStationT test 5 -0.77 22 -1.68 44 -3.21 6 -0.66 23 -1.39 48 -6.37 7 -2.85 24 -1.21 50 -0.52 8 -0.19 28 -3.01 51 8.06 9 -1.41 29 -2.63 52 11.42 10 -0.38 30 -2.39 53 -2.26 11 -1.98 31 -5.97 54 -3.10 12 -1.37 32 -8.55 57 -2.16 13 -4.64 34 -2.58 58 -1.71 15 -0.49 35 -3.81 63 -1.60 16 -1.49 37 -5.95 64 -1.77 17 3.06 38 -3.44 44 -3.21 18 -2.07 39 -2.73 48 -6.37 19 -1.98 41 -3.75 50 -0.52 20 0.47 42 -5.06 51 8.06 21 -1.07 43 -8.64 Results from the T-test were provided using a 95% level of confidence. Good model estimation for this level should result in a va lue within the range of 1.96 for the highest sampling size and 2.23 for the lowest. Appr oximately 50% of the stations fall within this range, suggesting that the proposed m odel might provide a good estimation of the rating curve for better than half of the stations. To further analyze these results, the relative percent error in discharge was examined. As mentioned earlier, the aver age relative error of the proposed model compared to the USGS data varies from 20% to 50% for 42% of the stations and 51% to 100% in 48% of the stations. Fo r the flow range between the Q90 and the Q10 discharge, the average relative error is below 30% for 47% of the stat ions, and below 50% for 65% of the stations. Although good pe rformance was seen for many stations for much larger
52than Q10 flows, the higher flows errors indicat ed that a better definition of model parameters and range of a pplicability is warranted. From the analysis of the th ree approaches it is recogni zed that the slopes in the intermediate-flow region vary within a specific range and that the high-flow regions exhibited log-linear behavior wi th considerably different slopes than the intermediateflow regions. Thus, two different mathematical relationships are proposed for each of the flow regions. The model for the intermedia te-flow region using specific values of Q10 and d10 provided good overall results with 74% of the stations exhibiti ng errors below 30%. When testing the model using values of Q10 and d10 as determined from the regression equations, the results are still reasonably good, with 60% of the stations with errors below 30%. Therefore, the regression equations for Q10 and d10 provide a fairly good estimation of these highly sensitive model parameters, with an average relativ e error of 26% for Q10 and 20% for d10. Table 9 shows the comparison be tween the USGS data and model results for some of the stations in the study. A qualitative assessment of results for the intermediate-flow region is shown in Figures 27A and B. These figures show a co mparison using both the specific and general values of Q10 and d10, for some of the stations in the study area. As indicated from the relative errors, when using the proposed model with Q10 and d10 values specific for each station, the matching region (i.e., the region exhibiting excellent performance) with the USGS data is much larger than when th e values are estimated from the regression equations. This would be very useful if using the relationship for calibration purposes, where specific values of Q10 and d10 are known for a particular stream; in those cases, a much better performance of the model could be expected. This would also represent a
53good relationship for the stage-discharge characte ristics of the particul ar stream (perhaps even in un-gauged regions). Table 9. Relative Percent Error for Some Stations in West-Central Florida for the Intermediate and High-Flow Region Station 2300500 Intermediate-Flow Region High-Flow Region Q-USGS1 Q-Mod_Specific2 RE3 Q-Mod_General4 RE Q-USGS Q-Model RE CMS CMS % CMS % CMS CMS % 6.7 5.9 12.36.1 8.0 47.8 48.2 0.9 10.6 10.6 0.2 11.1 5.2 66.0 70.0 6.1 23.0 27.2 18.628.6 24.5170.9 180.4 5.5 40.7 47.6 17.049.9 22.8393.1 414.0 5.3 Station 2301300 Intermediate-Flow Region High-Flow Region Q-USGS Q-Mod_Specific RE Q-Mod_General RE Q-USGS Q-Model RE CMS CMS % CMS % CMS CMS % 0.6 0.4 26.50.4 23.411.6 9.3 19.8 1.6 1.3 19.11.4 15.721.9 19.5 10.7 3.2 2.8 10.73.0 7.0 41.3 35.7 13.4 5.3 5.2 2.3 5.4 1.8 54.0 46.1 14.5 6.0 6.0 0.3 6.2 4.0 Station 2297310 Intermediate-Flow Region High-Flow Region Q-USGS Q-Mod_Specific RE Q-Mod_General RE Q-USGS Q-Model RE CMS CMS % CMS % CMS CMS % 0.3 0.3 24.20.4 9.6 37.2 30.3 18.5 1.9 1.6 16.82.3 20.489.1 66.6 25.3 5.3 4.8 8.9 6.9 31.8121.7 88.3 27.5 12.1 12.1 0.2 17.5 44.3270.0 181.7 32.7 23.2 24.8 7.0 35.9 54.8 1 Q-USGS = USGS Published rating relationship data (Water Resources Data of Florida, USGS, 19871998) 2 Q-Mod_Specific = model results using Q10 and d10 values specific for the station 3 RE = relative error 4 Q-Mod_General = model results using Q10 and d10 generalized parameters for the study region The model for the high-flow region using specific values of Q10 and d10 provided reasonably good overall results with 40% of the stations with errors below 30%. A
54qualitative assessment of results for the highflow region is shown in Figures 28A and B. Table 6 shows the results for some of the stations in the study. As mentioned, when testing the model using values of Qdep as determined from the regression equations, the errors were larg er than 100% in most of the stations. The regression equation for Qdep provides a poor estimation of th e parameter, with an average relative error of 57%. The lack of accuracy in the predic tion of the high-flow region model with a general intercept is mostly due to the lack of an equation that better describes the Qdep parameter. From testing the model with the verification data set, the average relative error when compared to the USGS data for the inte rmediate flow was less than 50% for 70% of the stations. Thus, the error was comparable to the results obtained with the calibration data set. Testing the model in some stations at North Florida, and South Alabama and Georgia, show some reasonable relative erro rs but not as good as the results obtained for West Central Florida. Using a model with a di fferent slope, developed for those particular stations improved the results significantly. Th e larger watershed size in this area could explain the different, lower slope of the ratings, considering that large sized watersheds will exhibit higher values of depth for the mean discharge, resulting in flatter rating slopes. It is important to mention that when comparing a fixed USGS rating relationship to actual historical USGS physical measur ements, the average re lative error among the stations was found to be around 90%. Thus, re sults from the proposed models may still
55be comparable to the accuracy of a well established rating when compared to a particular flow measurement. Station 2300500 0.0 10.0 20.0 30.0 40.0 50.0 60.0 0.001.002.003.004.00 d (m)Q (CMS) USGS Model_Specific Model_General Station 2301300 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.000.501.001.502.00 d (m)Q (CMS) USGS Model_Specific Model_General Figure 27. Comparison of USGS Rating with Model Results, for the Intermediate-Flow Region, for Some Stations in the Study Area
56 Station 2297310 0.0 10.0 20.0 30.0 40.0 50.0 0.001.002.003.004.00 d (m)Q (CMS) USGS Model_Specific Model_General Station 2298830 0.0 5.0 10.0 15.0 20.0 25.0 0.000.501.001.502.00 d (m)Q (CMS) USGS Model_Specific Model_General Figure 27. (Continued)
57 Station 2300500 0.0 100.0 200.0 300.0 400.0 500.0 0.02.04.06.08.0 d (m)Q (CMS) USGS Model Station 2301300 0.0 20.0 40.0 60.0 80.0 0.01.02.03.0 d (m)Q (CMS) USGS Model Figure 28. Comparison of USGS Rating with Model Results, for the High-Flow Region, for Some Stations in the Study Area
58 Station 2297310 0.0 50.0 100.0 150.0 200.0 250.0 300.0 0.02.04.06.0 d (m)Q (CMS) USGS Model Station 2298830 0.0 100.0 200.0 300.0 400.0 0.01.02.03.04.0 d (m)Q (CMS) USGS Model Figure 28. (Continued)
59 9. CONCLUSIONS This study provides a generalized mathematical expression that desc ribes depth-discharge relationships for stream gauging stations in West-Central Florida, for the intermediate and high-flow regions. The study also provides the parameters required to estimate rating curves when there is limited or no measured data available. Despite using stage-discharge by the c onventional methods, this study used the channel depth instead of stage elevation to define the rating curve. This approach provided the advantage of being easier to corr elate to a reference in un-gauged streams and it was contended to be more realistic in te rms of the physical processes of the system. For the intermediate-flow region, rating cu rves developed in mild-sloped streams of West-Central Florida, provi ded by USGS data plotted in log-log scale, exhibit linear behavior with similar slopes and approximate zero intercept. Also, the high-flow regions exhibited log-linear behavior wi th considerably different slopes than the intermediateflow regions, and different intercepts. Thus two different mathemati cal relationships can be defined for each of the flow regions. In this study, the difference in streamfl ow behavior between the two regions and within the same flow region was examined by presuming simple control at stream sections (e.g., behaving as weirs). The varia tion of the rating slopes within the range of 1.5 to 2.5 in the intermediate-flow region coul d be due to streamflow being controlled at a section that is through a relative degree of contracti on (e.g., behaving either as
60rectangular, trapezoidal or Vnotch section with discharge coefficients between 3/2 and 5/2). For the high-flow region, steeper rati ng slopes are undoubtedly indicating a higher degree of contraction and non-uniform flow. Th e different frictional conditions and cross section characteristics of floodplain flows (e.g., with or without houses ) also plays a role in the behavior. A log-linear equation with zer o intercept and a slope of 2.0 is proposed for the intermediate-flow region for the West-C entral Florida study area as a general mathematical model. The model is define d using the 10% exceedence flow rate (Q10) and corresponding depth (d10). The Q10 was shown to be reasonably well correlated to the watershed drainage area and the d10 showed a reasonable good correlation to the Q10, thus a regression equation was defi ned to obtain general values for those two parameters. A log-linear equation with a slope of 5.0 a nd an intercept specific for each station is proposed for the high-flow region for the West-Central Florida study area. The model uses the same Q10 and d10 parameters defined for the intermediate-flow region. The intercept is related to the discharge where the high-flow region departures from the intermediate-flow region (Qdep). General values for inter cept were obtained by defining a regression equation of Qdep as a function of drainage area. The accuracy of the proposed two-regi on model was examined statistically. Results show that for the intermediate-flow region, the average relative error is below 50% for 86% of the stations when using values of Q10 and d10 specific for each station; and below 50% for 74% of the stat ions when using values of Q10 and d10 from the regression equations. For the high-flow region, the average relative error is below 50% for 44% of the stations when using values of the intercept specific fo r each station. When
61using values of the intercept from the regr ession equations the errors are higher than 100%. Thus, this is not a good predictive appr oach. The accuracy in the prediction of the high-flow region model with a general intercept could be im proved with an equation that better describes the Qdep parameter. Other expressions ma y exist for this parameter but no insight yet exists as to how it could be bett er estimated in the absence of site specific data. Results from verifying the intermediate flow region model in thirty stations that had not been previously used in developing th e model, showed to be comparable to the ones obtained with the ca libration data set. The model was validated by testing in some stations at North Florida, and South Alabama and Georgia. Using the model w ith a slope developed specific for those particular stations showed the best results. A log-linear equation with zero intercept and a slope of 1.35 is proposed for the intermediate -flow region for those areas. The different, lower slope of the ratings in those areas may be due to the size of the watersheds. It is interesting to note that the accu racy of the proposed models may be comparable to (or better than ) the discrepancies between a fixed USGS rating relationship and periodic physical flow measurement. Th erefore, the use of similar general rating relationships in West-Central Florida or else where should prove to be a useful method to populate model data sets (subject to calibra tion through simple parameterization) when limited field data exist for model stream sections.
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65Tillery, A., J. Phillips and J. Capesius, 2001. Potential Errors Associated with StageDischarge Relations for Selected Stream flow-Gaging Stations, Maricopa County, Arizona. U.S. Geological Survey Water Re sources Investigation Report 00-4224, 54 p. U.S. Geological Survey. Water Resources Data Florida, Water Year, 2004. Water Data Report FL-04-4. Westphal, J., D. Thompson, G. Stevens and C. Strauser, 1999. Stage-Discharge Relations on the Middle Mississippi River. Jour nal of Water Resources Planning and Management. 125 (1), pp. 48-53.
67Appendix A: Results from Weir Analysis in Ch annel Controlled by a Horizontal_Uncontracted Section Table 10. Results from Weir Analys is in Channel Controlled by a Horizontal_Uncontracted Section C= 3.3 C= 3.5 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.08 0.06 0.03 0.06 0.09 0.06 0.03 0.15 0.33 0.15 0.11 0.15 0.35 0.15 0.11 0.30 0.94 0.30 0.30 0.30 0.99 0.30 0.32 0.46 1.72 0.45 0.55 0.46 1.82 0.45 0.58 0.61 2.65 0.61 0.85 0.61 2.81 0.61 0.90 0.91 4.86 0.91 1.56 0.91 5.15 0.91 1.65 1.22 7.48 1.21 2.40 1.22 7.93 1.21 2.55 1.52 10.46 1.52 3.35 1.52 11.09 1.52 3.56 C= 4 C= 4.4 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.10 0.06 0.03 0.06 0.11 0.06 0.04 0.15 0.40 0.15 0.13 0.15 0.44 0.15 0.14 0.30 1.13 0.30 0.36 0.30 1.25 0.30 0.40 0.46 2.08 0.45 0.67 0.46 2.29 0.45 0.73 0.61 3.21 0.61 1.03 0.61 3.53 0.61 1.13 0.91 5.89 0.91 1.89 0.91 6.48 0.91 2.08 1.22 9.07 1.21 2.91 1.22 9.98 1.21 3.20 1.52 12.67 1.52 4.07 1.52 13.94 1.52 4.47
68Appendix B: Results from Weir Anal ysis in Channel Controlled by a Horizontal_Contracted Section Table 11. Results from Weir Analysis in Channel Controlled by a Horizontal_Contracted Section C= 3.3 C=3.5 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.03 0.06 0.01 0.06 0.00 0.06 0.01 0.15 0.13 0.15 0.04 0.15 0.00 0.15 0.04 0.30 0.36 0.30 0.11 0.30 0.01 0.30 0.12 0.46 0.64 0.45 0.20 0.46 0.01 0.45 0.22 0.61 0.95 0.61 0.31 0.61 0.02 0.61 0.32 0.91 1.65 0.91 0.53 0.91 0.03 0.91 0.56 1.22 2.39 1.21 0.77 1.22 0.03 1.21 0.81 1.52 3.14 1.52 1.01 1.52 0.04 1.52 1.07 C= 4 C=4.4 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.04 0.06 0.01 0.06 0.00 0.06 0.01 0.15 0.16 0.15 0.05 0.15 0.00 0.15 0.06 0.30 0.43 0.30 0.14 0.30 0.01 0.30 0.15 0.46 0.77 0.45 0.25 0.46 0.01 0.45 0.27 0.61 1.15 0.61 0.37 0.61 0.02 0.61 0.41 0.91 2.00 0.91 0.64 0.91 0.03 0.91 0.71 1.22 2.90 1.21 0.93 1.22 0.03 1.21 1.02 1.52 3.80 1.52 1.22 1.52 0.04 1.52 1.34
69Appendix C: Results from We ir Analysis in Channel Controlled by a V-Notch Section Table 12. Results from Weir Analysis in Channel Controlled by a V-Notch Section C= 2 C= 2.5 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.00 0.06 0.00 0.06 0.00 0.06 0.00 0.15 0.01 0.15 0.01 0.15 0.01 0.15 0.01 0.30 0.06 0.30 0.02 0.30 0.07 0.30 0.02 0.46 0.16 0.45 0.03 0.46 0.20 0.45 0.04 0.61 0.32 0.61 0.05 0.61 0.40 0.61 0.06 0.91 0.88 0.91 0.09 0.91 1.10 0.91 0.12 1.22 1.81 1.21 0.15 1.22 2.27 1.21 0.18 1.52 3.17 1.52 0.20 1.52 3.96 1.52 0.25 C= 3 C= 4.3 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.00 0.06 0.00 0.06 0.00 0.06 0.00 0.15 0.02 0.15 0.01 0.15 0.02 0.15 0.01 0.30 0.09 0.30 0.03 0.30 0.12 0.30 0.04 0.46 0.23 0.45 0.05 0.46 0.34 0.45 0.07 0.61 0.48 0.61 0.08 0.61 0.69 0.61 0.11 0.91 1.33 0.91 0.14 0.91 1.90 0.91 0.20 1.22 2.72 1.21 0.22 1.22 3.90 1.21 0.31 1.52 4.75 1.52 0.30 1.52 6.81 1.52 0.44
70Appendix D: Results from Weir Analysis in Ch annel Controlled by a Broad Crested Weir Table 13. Results from Weir Analys is in Channel Controlled by a Broad Crested Weir C= 2.3 C=2.63 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.06 0.06 0.02 0.06 0.07 0.06 0.02 0.15 0.23 0.15 0.07 0.15 0.26 0.15 0.08 0.30 0.64 0.30 0.21 0.30 0.73 0.30 0.24 0.46 1.16 0.45 0.38 0.46 1.33 0.45 0.44 0.61 1.77 0.61 0.59 0.61 2.02 0.61 0.68 0.91 3.18 0.91 1.09 0.91 3.64 0.91 1.24 1.22 4.80 1.21 1.67 1.22 5.49 1.21 1.91 1.52 6.56 1.52 2.34 1.52 7.50 1.52 2.67 C= 3 C=3.3 H Q H' Q' H Q H' Q' m CMS m CMS 0.06 0.08 0.06 0.02 0.06 0.08 0.06 0.03 0.15 0.30 0.15 0.10 0.15 0.33 0.15 0.11 0.30 0.83 0.30 0.27 0.30 0.92 0.30 0.30 0.46 1.51 0.45 0.50 0.46 1.67 0.45 0.55 0.61 2.31 0.61 0.77 0.61 2.54 0.61 0.85 0.91 4.15 0.91 1.42 0.91 4.57 0.91 1.56 1.22 6.26 1.21 2.18 1.22 6.88 1.21 2.40 1.52 8.55 1.52 3.05 1.52 9.41 1.52 3.35
71Appendix E: Results from Weir Analysis in Channel Controlled by a Compound Weir consisting of a V-Notch and a Trapezoidal Weir Table 14. Results from Weir Analysis in Channel Controlled by a Compound Weir consisting of a V-Notch and a Trapezoidal Weir H Qv Qt Qcomb Htotal H' Q' m CMS CMS CMS m 0.15 0.013 0.331 0.013 0.15 0.01 0.30 0.071 0.935 0.071 0.30 0.02 0.46 0.195 1.718 0.195 0.45 0.04 0.61 0.401 2.645 0.401 0.61 0.06 0.15 16.882 17.283 0.76 0.17 0.30 47.751 48.151 0.91 0.37 0.46 87.723 88.124 1.06 0.63 0.61 135.059135.460 1.21 0.93
72Appendix F: Basin Characteristics Data and Relative Error for Stations in Calibration Data Set for Intermediate-Flow Region Table 15. Basin Characteristics Data and Rela tive Error for Stations in Calibration Data Set for Intermediate-Flow Region Gage # S10 Q10 d10Ch Slope Slope Area % Wetland DTWTRE CMS m % m/m mi2 % m % 2295420 1.34 7.5 1.3 2.14 0.010876 125.2 10.25 4.34 52 2294898 1.73 15.5 1.8 0.0108763479.6 23.89 5.83 16 2296750 1.74 76.5 2.8 1.3 0.0088 1372.919.88 3.55 25 2300500 1.72 11.0 1.9 0.00738 151.7 12.58 3.714 15 2300100 1.72 2.2 1.3 0.0060 30.9 12.59 3.88 12 2301000 2.1 8.5 1.8 4.96 0.01 136 5.1 4.765 2 2301500 1.67 20.5 1.9 3.45 0.0088 339.4 12.87 4.77 23 2294650 1.6 6.2 1.6 1.25 0.0120 404.7 26.03 5.20 10 2297155 1.6 2.1 0.8 5.5 0.0050 40.9 13.01 3.50 53 2296500 1.98 21.2 2.4 1.68 0.0061 326.5 19.63 3.51 4 2301300 1.9 6.3 1.5 4.17 0.0070 112.2 11.52 4.53 11 2297310 1.9 15.0 2.4 2.79 0.004779 217.3 17.71 3.50 8 2295637 1.61 41.1 2.6 1.38 0.00994 839.1 19.55 4.15 15 2303350 2.34 1.6 0.7 2.44 0.005 17.3 27.52 3.46 10 2303420 2.37 3.6 1.3 2.65 0.010444 128.7 30.54 3.68 20 2293987 2.45 9.0 1.3 0.014 170.7 31.14 3.68 21 2295013 2.36 2.2 0.9 4.96 0.007 46.3 25.21 4.135 17 2297220 2.25 1.9 1.1 0.004 47.9 19.27 3.24 29 2297100 2.19 7.8 1.7 4.06 0.004 120.9 10.96 3.37 2 2298830 2.71 19.4 2.0 2.14 0.00455 225.5 22.18 3.28 22 2299120 2.6 3.8 1.3 0.001 26.1 36.82 3.02 20 2299410 2.29 2.7 1.0 1.23 0.002 35.8 21.99 3.20 2 2299861 1.94 0.4 0.4 0.005 6 7.38 3.35 24 2299780 1.7 3.1 0.8 0.005 31.1 8.92 3.32 82 2299684 1.48 0.1 0.2 0.002 4.3 19.88 3.21 98 2300032 1.91 2.4 1.2 0.004 25.2 13.16 3.40 17 2300700 1.98 2.3 1.0 7.01 0.007 28.6 12.10 3.67 14 2301900 4.16 0.5 0.7 6.1 0.009 9.3 12.24 4.75 55 2302500 1.84 5.3 0.9 3.52 0.009011 98.6 17.79 3.23 17 2303000 1.89 14.9 1.6 3.87 0.005 227.2 16.82 4.09 29 2303205 2.22 1.2 0.5 0.009011 21.6 13.4 4.869 3 2303330 8.08 17.2 3.2 0.01206 387.9 18.05 3.87 34 2303800 5.1 6.6 1.3 2.1 0.010081 167.8 32.60 5.00 56
73Appendix F: (Continued) Table 15. (Continued) Gage # S10 Q10 d10 Ch Slope Slope Area % Wetland DTWTRE CMS m % m/m mi2 % m % 2301750 2.74 0.7 0.4 5.2 0.011 14.2 8.35 3.80 41 2268390 2.7 1.9 0.9 0.025 53.1 18.95 6.79 4 2269520 1.91 3.7 1.1 0.013 118.3 32.33 4.13 14 2270000 2.54 1.5 1.0 0.014 39 18.90 6.17 4 2271500 2.34 4.8 1.1 3.81 0.01275 113.2 26.96 5.37 16 2307200 2.74 0.3 0.4 1.62 0.003 5.2 33.14 3.51 38 2307359 2.5 1.4 1.0 2.81 0.00545 33 34.68 3.31 32 2310000 2.1 4.9 1.7 3.54 0.00282169.6 29.61 3.43 4 2310280 2.69 0.5 0.4 2.3 0.015 148.8 13.32 3.82 35 2310300 3.66 2.0 1.0 2.7 0.01342 181.4 17.53 3.82 48
74Appendix G: Basin Characteristics Data and Relative Error for Stations in Calibration Data Set for High-Flow Region Table 16. Basin Characteristics Data and Rela tive Error for Stations in Calibration Data Set for High-Flow Region Gage # S10 b Q10 d10 Ch Slope Slope Area Wetland DTWT RE % CMS m % m/m mi2 % m % 2295420 4.42 -1.54 263 4.13 2.14 0.0108 125.2 10.25 4.34 145 2294898 3.03 0.02 546 6 0.0108 479.6 23.89 5.83 2 2296750 3.95 -0.29 2700 9.24 1.3 0.0088 1372.9 19.88 3.55 8 2300500 4.95 -1.03 389 6.1 0.0073 151.7 12.58 3.71 5 2300100 4.77 -0.34 77 4.3 0.0060 30.9 12.59 3.88 0.5 2301000 5.14 -0.67 299 6 4.96 0.01 136 5.1 4.76 6.2 2297155 5.73 -2.83 74 2.66 5.5 0.0050 40.9 13.01 3.50 70 2296500 4.54 -0.66 747 7.76 1.68 0.0061 326.5 19.63 3.51 47 2301300 5.37 -0.42 224 4.84 4.17 0.0070 112.2 11.52 4.53 13 2297310 5.56 -0.62 530 7.9 2.79 0.0047 217.3 17.71 3.50 27 2295637 3.83 -0.36 1450 8.45 1.38 0.0099 839.1 19.55 4.15 129 2303350 3.83 -0.35 55 2.45 2.44 0.005 17.3 27.52 3.46 223 2297100 7.40 -1.91 275 5.54 4.06 0.004 120.9 10.96 3.37 98 2298830 4.88 0.00 686 6.7 2.14 0.0045 225.5 22.18 3.28 14 2299410 7.50 -1.37 95 3.13 1.23 0.002 35.8 21.99 3.20 87 2299684 4.75 -2.44 4.2 0.55 0.002 4.3 19.88 3.21 14 2300032 4.65 -1.43 85 4 0.004 25.2 13.16 3.40 71 2300700 5.83 -1.87 82 3.16 7.01 0.007 28.6 12.10 3.67 64 2302500 4.01 -0.81 187 2.89 3.52 0.0090 98.6 17.79 3.23 218 2303000 5.52 -1.46 525 5.2 3.87 0.005 227.2 16.82 4.09 4 2303330 4.80 0.08 606 10.53 0.01206 387.9 18.05 3.87 21 2271500 3.80 -0.06 171 3.55 3.81 0.0127 113.2 26.96 5.37 56 2307359 6.95 -0.36 50 3.17 2.81 0.0054 33 34.68 3.31 76
75Appendix H: Basin Characteristics Data and Relative Error for Stations in Verification Data Set for Intermediate-Flow Region Table 17. Basin Characteristics Data and Rela tive Error for Stations in Verification Data Set for Intermediate-Flow Region ID station A RE mi2 % 2 229995067 32 3 2294491145 44 4 229421759.5 12 37 23005307.3 26 44 230330057.7 24 49 226700046 57 54 2270500388.5 68 58 2307323229 108 65 223635041.8 40 66 2310800107.6 45 67 2310947352.8 47 68 2311500419.6 48 69 2312000572.6 24 70 231218088 85 72 2312500805.9 32 73 226290084 36 74 226630084.6 57 75 229876020 15 76 230003225.8 19 77 2300500149 24 78 230677417.8 15 79 23130001825 79 80 230001850.6 25 81 23017382.6 113 83 230030038.4 32 84 23058512.59 29 85 23017406.5 24 86 23017452 78 87 23017931.46 58
76Appendix I: Drainage Area and Relative E rror for Stations in Validation Data Set for Intermediate-Flow Region Table 18. Drainage Area and Relative Error for Stations in Validation Data Set for Intermediate-Flow Region Station Area RE mi2 % 2369800 1182 24 2361500 1280 15 2378300 16.6 40 2376500 394 3 2374250 2661 17 2373000 470 9 2361000 686 7 2363000 498 23 2372422 1273 102 2364500 1182 11 2377570 192 4 2374950 193 21 2362240 21.4 45 2323000 9390 11 2319800 7190 6 2319000 2120 8 2322500 1017 15 2321500 575 57 2330150 2080 54 2330100 126 49 2359000 781 26 2365500 3499 41 2368500 123 4 2370500 237 51 2376500 394 5 2322700 213 344 2226500 1200 55 2228000 2790 49 2314500 1260 41 2226000 13600 47 23177483 502 14 2327500 550 33 2329342 16.9 3
77Appendix I: (Continued) Table 18. (ContinueD) Station Area RE mi2 % 2353000 5740 4 2353265 303 6 2354410 157 106 2354800 1000 17 2355662 7080 3 2357000 485 31 2316000 663 270
78Appendix J: Drainage Area and Relative E rror for Stations in Validation Data Set for High-Flow Region Table 19. Drainage Area and Relative Error for Stations in Validation Data Set for High-Flow Region Station Area RE mi2 % 2361500 1280 9 2378300 16.6 85 2376500 394 2 2374250 2661 73 2373000 470 205 2361000 686 29 2363000 498 82 2377570 192 8 2374950 193 411 2362240 21.4 54 2323000 9390 42 2319800 7190 6 2319000 2120 88 2322500 1017 451 2330100 126 26 2359000 781 41 2365500 3499 3 2368500 123 30 2370500 237 47 2376500 394 3 23177483 502 65 2327500 550 75 2354410 157 68 2357000 485 70 2316000 663 76
ABOUT THE AUTHOR Auristela Mueses-Prez was born on Oct ober 10, 1967 in Dominican Republic. She attended San Judas High School, finishi ng as the best GPA in 1984. In 1987 she graduated Magna Cum Laude with her Bachelor in Civil Engineering from Instituto Tecnolgico Santo Domingo. In 1989 she moved to Puerto Rico to pursue her Master of Science in Water Resources Engineering, at University of Puerto Rico, Mayaguez. She completed her degree in 1991. In 1992 she joins Po lytechnic University of Puerto Rico as an Assistant Professor, and was promoted to Associate Professor in 1999. In 2003, she began Ph.D. coursework in the Mayaguez Ca mpus, and after a year she decides to continue her Ph.D. in Water Resources at University of South Florida. She completed her work in the field of non-dimensional rating curves in the summer of 2006. She already took a position as a faculty member at Polytech University in Orlando.