JOINT PROBABILITY ANALYSIS OF PRECIPITATION AND STREAMFLOW EXTREMES. Chia-hung Lin. A Thesis Submitted to the Faculty of.

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1 JOINT PROBABILITY ANALYSIS OF PRECIPITATION AND STREAMFLOW EXTREMES by Chia-hung Lin A Thesis Submitted to the Faculty of The College of Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Science Florida Atlantic University Boca Raton, Florida May 213

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3 ACKNOWLEDGEMENTS At the moment when I reached the final section of my research, my head full of great people and their unique personalities for supporting and for guiding me to fulfill my academic goals. In one page I cannot express all my appreciation toward them. To complete this research, I have received tremendous assistance from my advisor, Dr. Ramesh S. V. Teegavarapu. His guidance and help were important to the success of this research; his patience, consistent, and positive attitude encouraged me to challenge myself and never stop before reach my truly power. Also, Dr. Noemi Gonzalez-Ramirez s assistance was critical to the progress of the research. It was an important period of time when I worked with Dr. Noemi and learned from her problem solving methodology and great persistence toward tasks. The South Florida Water Management District (SFWMD) must be commended for providing funding for this research. Moreover, I want to thank Aneesh Goly for his assistance with the preliminary data collection and programming. He is very good study mate and friend. Finally, I would like to express a great appreciation to my family and especially my sister, Tailing for encouraging me to continue my education in the United State of America and for supporting me financially. From basic life knowhow to language assistance, I receive plenty of helps from Tailing and she never complains about all the troubles I bring to her. iii

4 ABSTRACT Author: Chia-hung Lin Title: Institution: Joint Probability Analysis of Precipitation and Streamflow Extremes Florida Atlantic University Thesis Advisor: Dr. Ramesh S. V. Teegavarapu Degree: Master of Science Year: 213 This thesis focuses on evaluation of joint occurrence of extreme precipitation and streamflow events at several hydrologic structures in South Florida. An analysis of twelve years storm events and their corresponding peak streamflow events during wet and dry season including annual peaks considering two seasons was performed first. Dependence analysis using time series data of precipitation and streamflow was carried out next. The analysis included use of storm events with different temporal lags from the time of occurrence of peak streamflow events. Bi-variate joint probability was found to be appropriate to analyze the joint occurrence of events. Evaluation of joint exceedence probabilities under two phases of Atlantic multidecadal oscillation (AMO) influencing south Florida was also evaluated. All methodologies are evaluated for application using observations at several structures in the case study region to provide advances and valuable insights on joint extremes of precipitation and streamflows. iv

5 JOINT PROBABILITY ANALYSIS OF PRECIPITATION AND STREAMFLOW EXTREMES LIST OF FIGURES... ix LIST OF TABLES... xvii LIST OF ACRONYMS... xix 1 INTRODUCTION Background Problem Statement Objectives Thesis Outline LITERATURE REVIEW Different Scales of Meteorological Phenomena Larger-Scale Precipitation Systems Smaller-Scale Precipitation Systems Current Study Area Characteristics Globe Climate Perspective Local Metrological Perspective Atlantic Multidecadal Oscillation (AMO) Extreme events, trends, and Seasonality Analysis Dependence Analysis Applications of Joint Probability Analysis v

6 3 METHODOLOGY Data Processing Streamflow, Rainfall, and Storm Events Geospatial Data Analysis and Tools Hydro-meteorological Extremes and Seasonality Analysis Annual Streamflow and Rainfall Extremes Analysis Seasonal Streamflow Peaks with Threshold and Hurricane Analysis Dependence Analysis Correlation Coefficient and Lag Time Analysis Lag Time and Basin Characteristic Validity of Design Discharge and Rainfall Rainfall Distributions Joint Probability Analysis The Probability of Exceedance Box-Cox Transformation Normality Tests Densities Validity of Joint Distribution Threshold Analysis: Lead Times Partial Duration Series Analysis and Peaks over Thresholds (POT) CASE STUDY Background of Case-study Region Homogeneous Rainfall Areas C-4 Basin and C-6 Basin Description of Canals and Control Structures vi

7 4.3.2 Land Use of C-4 and C-6 Basins RESULTS Computer Software Application of Annual Extremes Analysis Application of Annual Streamflow Peak Analysis Application of Annual Rainfall Extreme with Wet and Dry Analysis Application of Streamflow Peaks with Threshold and Hurricane Analysis Applications of Dependence Analysis High Correlations and Rainfall-runoff Domain Basins Low Correlations and Non rainfall-runoff Domain Basins Variation of Accumulated Rainfall and Peak Discharge for Different Lag Times with Hurricanes Comparison of Synthetic Rainfall Distributions and Historical Rainfall Distribution in Different Durations Applications of Joint Probability Analysis in C-4 and C-6 Basins Optimal Box-Cox transformation parameter Estimation Probability Plots and Box-Cox Transformation Joint Cumulative Probability Plots Results for Basins C-4 and C Peaks over Threshold (POT) Analysis Joint Probability Analysis with Cool Phase and Warm Phase Normality Tests and Box-Cox Transformation Cool and Warm Phases Analysis with different Lag Times Peaks over Threshold Analysis Peak Discharge for Different Sets of Discharge Thresholds vii

8 5.7.2 Rainfall and Annual Peak Discharge Correlations with Different Discharge Thresholds CONCLUSION Contributions of the Study Dependence Analysis Rainfall Distribution Curves Based on Historical Data Joint Probability Analysis: Limitation of the Study Recommendations for Future Research APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J APPENDIX K APPENDIX L GLOSSARY REFERENCES viii

9 LIST OF FIGURES Figure 1 Plot for streamflow (Blue), threshold (Red), and storms event (Green) from year 1998 to year Figure 2 A window for streamflow (Upper blue), threshold (Red), storm event (Green), and paired rainfall (Lower blue) in year Figure 3 Graphical representation of streamflow and rainfall annual peaks Figure 4 Selection of streamflow peaks with threshold and hurricane Figure 5 Schematic representation of peak discharge and Accumulated Rainfall Figure 6 Criteria for the evaluation of design conditions for each structure using discharge as the comparison parameter Figure 7 Criteria for the evaluation of design conditions for each structure using rainfall as the comparison parameter Figure 8 Probability of exceedance of discharge and stage knowing the design conditions for each selected outlet structure Figure 9 Schematic representation of occurrence of peak discharge and accumulated rainfall along with definition of lead time Figure 1 Schematic representation of the selection process for POT in the partial duration series analysis Figure 11 Steps for the selection of POT in the partial duration series analysis Figure 12 South Florida map with counties Figure 13 Streamflow stations and rainfall stations used in the study Figure 14 Location of selected rain gauges and streamflow stations in C-4 and C-6 basins ix

10 Figure 15 Annual streamflow peak in wet (Gray) and dry (White) (Continued) Figure 16 Annual streamflow peak in wet (Gray) and dry (White) (Continued) Figure 17 Annual streamflow peak in wet (Gray) and dry (White) (Continued) Figure 18 Annual streamflow peak in wet (Gray) and dry (Gray)... 7 Figure 19 Average analysis of Annual streamflow peak in wet (Black) and dry (Gray) Figure 2 Annual rainfall extreme in wet (Gray) and dry (White) (Continued) Figure 21 Annual rainfall extreme in wet (Gray) and dry (White) (Continued) Figure 22 Annual rainfall extreme in wet (Gray) and dry (White) (Continued) Figure 23 Annual rainfall extreme in wet (Gray) and dry (White) Figure 24 Average analysis of Annual rainfall extreme in wet (Gray) and dry (White) Figure 25 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and nonhurricane related (Orange) (Continued) Figure 26 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and nonhurricane related (Orange) (Continued) Figure 27 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and nonhurricane related (Orange) (Continued) Figure 28 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and nonhurricane related (Orange) Figure 29 Average Correlations for Accumulated Rainfall and Peak Discharge Lag Time from Day Zero to Day Tenth Figure 3 High Accumulated Rainfall-Peak Discharge Correlations for Different Lag Times from Day Zero to Day Tenth Figure 31 Low Accumulated Rainfall-Peak Discharge Correlations for Different Lag Times from Day Zero to Day Tenth x

11 Figure 32 Variation of Accumulated Rainfall and Peak Discharge for Different Lag Times for Station: S97_S Figure 33 Map of Selected Rain Gauges for Rainfall Distribution Analysis Figure 34 Comparison of Synthetic and Historical Rainfall Distributions in 24 Hours Duration Figure 35 Comparison of Synthetic and Historical Rainfall Distributions in 72 Hours Duration Figure 36 Optimal Box-Cox Transformation Parameter for Peak Discharge for C4.CORAL and MIAMI.FS_R Figure 37 Optimal Box-Cox Transformation Parameter for Peak Discharge for C4.CORAL and MIAMI.FS_R. (Lag Time = 3) Figure 38 Probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R before and after Box-Cox transformations Figure 39 Joint cumulative probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 4 Joint exceedance probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 41 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 42 Joint return period plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 43 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 24 hour rainfall duration for structure C6.NW Figure 44 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 72 hour rainfall duration for structure C6.NW xi

12 Figure 45 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 24 hour rainfall duration for structure C4.Coral Figure 46 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 72 hour rainfall duration for structure C4.Coral Figure 47 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C4.CORAL and rainfall station Miami.AP.R Figure 48 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days structure C4.CORAL and rainfall station Miami.FS.R Figure Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C4.CORAL and rainfall station S335_R Figure 5 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for C4.Coral. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R Figure 51 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C6.NW36 and rainfall station Miami.AP.R Figure 52 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C6.NW36 and rainfall station Miami.FS.R Figure 53 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C6.NW36 and rainfall station S335_R Figure 54 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for C6.NW36. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R Figure 55 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure G93 and rainfall station Miami.AP.R xii

13 Figure 56 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure G93 and rainfall station Miami.FS.R Figure 57 Variation of accumulated rainfall and peak discharge for 24 and 72 hour for structure G93 and rainfall station Miami.FS.R Figure 58 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure G93 and rainfall station S335_R Figure 59 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for G93. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R Figure 6 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure S_25B and rainfall station Miami.AP.R Figure 61 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure S_25B and rainfall station Miami.FS.R Figure 62 Variation of accumulated rainfall and peak discharge for 24 and 72 hour for structure S_25B and rainfall station Miami.FS.R Figure 63 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure S_25B and rainfall station S335_R Figure 64 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for S25_B. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R Figure 65 Optimal Box-Cox transformation parameter for peak discharge for C4.CORAL- MIAMI.FS_R Figure 66 Optimal Box-Cox transformation parameter for accumulated rainfall (lag=3) for C4.CORAL-MIAMI.FS_R xiii

14 Figure 67 Joint cumulative probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 68 Joint exceedence probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 69 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 7 Joint return plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Figure 71 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2313 in cool phase Figure 72 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2313 in warm phase Figure 73 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 233 in cool phase Figure 74 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 233 in warm phase Figure 75 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2329 in cool phase Figure 76 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2329 in warm phase Figure 77 Probability plots for peak discharge and accumulated rainfall (lag =) for station 2329 before and after Box-Cox transformations Figure 78 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for structure C4. CORAL at lag time = xiv

15 Figure 79 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for structure ID 2329 at lag time = Figure 8 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for station C4.CORAL Figure 81 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for structure ID Figure 82 Joint return plots for peak discharge and accumulated rainfall (lag =5) for station C4.CORAL Figure 83 Joint return plots for peak discharge and accumulated rainfall (lag =5) for structure ID Figure 84 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution (lag =5) for structure ID 2313 in cool phase Figure 85 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution (lag =5) for structure ID 2313 in warm phase Figure 86 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in cool phase Figure 87 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in warm phase Figure 88 Joint return plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in cool phase Figure 89 Joint return plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in warm phase Figure 9 Number of peaks for different % of max peak discharge thresholds Figure 91 Location of POT events for 7% and 75% of max peak discharge threshold for C4.Coral xv

16 Figure 92 Location of POT events for 8% and 85% of max peak discharge threshold for C4.Coral Figure 93 Location of POT events for 9% and 95% of max peak discharge threshold for C4.Coral Figure 94 Correlations between rainfall and annual peak discharges for different discharge thresholds. C4.Coral - Miami.F.S Figure 95 Variation of individual rainfall for different lag times from to 1 days and peak discharge for C4.Coral Figure 96 Correlations between rainfall and annual peak discharges for different discharge thresholds. C6.NW36 - Miami.F.S Figure 97 Variation of individual rainfall for different lag times from to 1 days and peak discharge for C6.NW xvi

17 LIST OF TABLES Table 1 Perspectives of hydro-climatology and hydro-meteorology... 6 Table 2 Different Scales of Meteorological Phenomena... 7 Table 3 Meteorological events in year Table 4 Details of the Spatial Coordinate System Table 5 24-hour Duration Rainfall Curve (Source: FDOT, 24) Table 6 72-hour Duration Rainfall Curve (Source: FDOT, 24) Table 7 List of streamflow stations and rainfall stations Table 8 Basin information and associated structures for C-4 and C-6 basins Table 9 Classification of Land use for C-4 basin Table 1 Classification of Land use for C-6 basin Table 11 Summaries of Peak Discharge and Accumulated Rainfall Correlations and the Corresponding Lag Times Table 12 Best Correlation and the Lag Time Occurrence in Days Table 13 Classification of Streamflow Stations Based on Rainfall and Discharge Best Correlations Table 14 Selected Rain Gauges for Rainfall Distribution Analysis Table 15 Design Conditions and Control Levels for Basins C-4 and C Table 16 List of Structures and Three Paired Rain Gauge Stations paired with Available Data Length Table 17 Probability of occurrence and exceedance for structures C6.NW36 and C4.Coral xvii

18 Table 18 Comparison between design conditions and historical data for structure G93 (Miami.FS.R) Table 19 Comparison between design conditions and historical data for structure S_25B (Miami.FS.R) Table 2 Best Correlation and the corresponding lag time for the evaluated structures in C- 4 and C-6 basins Table 21 Data length for selected paired stations in C-4 and C-6 basins Table 22 Duration of cool phase and warm phase for stations in C-4 and C-6 basins Table 23 Data length for new selected paired stations Table 24 Duration of cool phase and warm phase for new stations Table 25 Summary of Normality tests after Box-Cox transformation at lag time day three Table 26 Correlations for a set of discharge thresholds in C-4 basin Table 27 Correlations for a set of discharge thresholds in C6 basin xviii

19 LIST OF ACRONYMS AMO Atlantic Multidecadal Oscillation DBHYDRO SFWMD s Water Quality and Hydrology Database DMC Deep Moist Convection DSA Desk Study Approach ENSO El Niño/Southern Oscillation FEMA Federal Emergency Management Agency FGDL Florida Geographic Data Library GIS Geographic Information System HARN High Accuracy Reference Network IDW Inverse Distance Weighting JPA Joint Probability Analysis JPM Joint Probability Method MCCs Meso-Scale Convective Complexes NAD83 North American Datum of 1983 xix

20 NCDC National Climatic Data Center NOAA National Oceanic and Atmospheric Administration NRCS Natural Resources Conservation Services PDO Pacific Decadal Oscillation PDS Partial Duration Series POT Peaks over Threshold QA/QC Quality Assurance and Quality Control SCS Soil Conservation Service SFWMD South Florida Water Management District USGS United States Geological Survey USHCN The United States Historical Climatology Network xx

21 1 INTRODUCTION 1.1 Background The demands of water usage rise in high population areas; nevertheless, water resource lost quickly due to urbanization. Moreover, extreme meteorological events from global climate changes cause water resources management more complex and decision making based on flooding and water scarcity. The spatial and temporal variability of precipitation events and their link to peak flooding events are important aspects to civil engineers; particularly hydrologists, water resource managers, urban and regional planners, and climate scientists. All related field engineers are required to emphasize the probability of two dependent events occurs simultaneously for better understanding and analyzing current environmental situation. 1.2 Problem Statement The idea of traditional hydrologic structure design is deterministic and empirical. The design and decision is made based on the collection and evaluation of maximum historical records or empirically based on the return period. Indeed, the failure of structure caused by flooding is mainly a combination of rainfall and streamflow extreme events. If these two events are considered as independent variables, the probability of failure can be calculated easily. However, this incorrect assumption would lead to an under or over design. These two variables should be considered as dependent; therefore, the proper method for multivariate statistics should be developed to estimate the probability of failure. 1

22 1.3 Objectives The main objectives of this research start from reviewing spatial and temporal precipitation events and their link to peak flooding events at outlet structures in South Florida and end with providing multivariate statistics results. This research will provide insightful observations and meaningful conclusions relevant to floods and their links to storm magnitudes to South Florida flood modeling projects currently. The obtained results from this research may be used by state and federal agencies such as Federal Emergency Management Agency (FEMA) and the South Florida Water Management District (SFWMD) Hydrologic System Modeling and the Regulation Department. The objectives of this thesis are as follows: 1. Statistical modeling of spatial and temporal precipitation events with link to peak flooding events at critical water conveyance structures in South Florida. 2. Different temporal storm events and streamflow trends falling under meteorologically homogeneous precipitation areas. 3. Dependence analysis of streamflow nad rainfall in different time lag and the relationship of correlation and basin characters. 4. Comparison of synthetic and historical rainfall distribution. 5. Bivariate Normal distribution and conditional probability analysis with different basins in cool and warm phases. 1.4 Thesis Outline The chapters of this thesis are organized as follows: Chapter One: Introduces the difficulties of water management and the importance of the Joint Probability as well as the objectives of this study. 2

23 Chapter Two: Provides a review of literature. Chapter Three: Methodology 1. Collection of data (12 years storm related data and 5 years for long term analysis) 2. Developing and analysis relationships among peak flooding events and storms. (Seasonal characteristics, variability, trends, shifts) 3. Extreme events at meteorologically homogeneous precipitation area. 4. Dependence Analysis a. Basin response time to flooding ( Regional lag times by correlation) b. Dependence analysis of peak streamflow, rainfall, and accumulated rainfall. 5. Rainfall discharge curve a. CASE A: Rainfall used for design conditions and the discharge versus accumulated rainfall from historical data are known. b. CASE B: Rainfall for design conditions and discharge versus accumulated rainfall from historical data are given. 6. Rainfall distribution 7. Joint Probability Analysis a. Box-Cox Transformation b. Normality Tests c. Bivariate Normal Distribution d. Conditional Probability Densities e. Validity of Joint Distribution f. Threshold Analysis: Lead Times g. Partial Duration Series Analysis and Peaks over Thresholds (POT) h. AMO analysis 3

24 Chapter Four: Case Study Chapter Five: Presents all results from different analysis and methodologies. Chapter Six: Presents conclusions, contributions, limitations of this work, and recommendations for further research. 4

25 2 LITERATURE REVIEW An extensive literature review relevant to flooding issues in South Florida is the fundamental processes of the research. To explore the causes of floods, the interactions around meteorological, climatic, hydrologic, and drainage basin characters respect to the long-term climatic and short-term meteorological phenomena are the essential factors. Joint probability analysis has been used in different study areas and is selected in this study. The advantage of joint probability analysis is addressed with applied applications. This chapter presents a literature review including different scales of meteorological phenomena, property characteristics of current study area, dependence analysis, and applications of joint probability analysis. 2.1 Different Scales of Meteorological Phenomena Ward and Robinson broadly define modern hydrology is the science that studies the occurrence and movement of water on and under the Earth s surface, water s chemical and physical properties, water s relationship to biotic and abiotic environmental components, and human effects on water. Peixoto states climatology is an applied science that examines the fluxes of energy, mass, and momentum among the land and ocean surfaces and the atmosphere. These fluxes are integral parts of the climate system modulated by both external and internal factors. The word hydro-climatology is defined by American hydrologist Walter Langbein as the study of the influence of climate upon the waters of the land (qtd. in Shelton, January 12, 29). Moreover, modern hydro-climatology requires a more holistic view that emphasizes a process orientation and a role in a variety of environmental systems ranging from water quantity and quality to stream habitats (Shelton, 29). 5

26 The theory of flood hydro-climatology is typically applied to atmospheric causes of flooding and is an approach to analyze floods through the temporal context of their history of development and variation and the spatial context of the local, regional and global atmospheric processes and circulation patterns from which the flood develops. The antecedent conditions, regional relationships, large-scale anomaly patterns, global-scale controls, and long-term trends can also affect the flood variability. This approach is based on the meteorological and hydrological processes which directly cause flooding. Therefore, a meteorological scale perspective is addressed (Hirschboeck, 1988, 1999, and 2). Table 1 summarizes the perspectives of hydro-climatology and hydro-meteorology. Table 1 Perspectives of hydro-climatology and hydro-meteorology Hydro-Climatology Weather, short time scales Local, regional spatial scales Forecasts, real-time warnings Hydro-Meteorology Seasonal, long-term perspective Site-specific and regional synthesis of floodcausing weather scenarios (Regional linkages and differences identified) Entire flood history context; benchmarks for future events The spatial and temporal rainfall distributions and its corresponding watershed characteristics are main influencing factors for occurrence of floods. The phrase hydrometeorological flood refers to a process or phenomenon of atmospheric, hydrological or oceanographic natures that could cause losing lives, injuries, health impacts, and property damages, such as loss of livelihoods or receiving social or economic damages. Hydrometeorological floods occur when precipitation cannot be accommodated in by the drainage network and the storage structures in a watershed. The flood characteristics depend on the varying precipitations delivered by different storm systems. The different scales of the meteorological phenomena are related with the length and duration of the flooding event. 6

27 According to Hirschboeck (2), there are 4 meteorological phenomena scales that can be enumerated as it is shown in Table 2: Table 2 Different Scales of Meteorological Phenomena System Scale Typical Length Scale Typical Time Scale Larger-Scale Precipitation Systems Smaller-Scale Precipitation Systems Macro-scale More than 1, km Approximately 1 month Synoptic-Scale Meso-Scale Storm-Scale Between 1, km to 2, km Between 2, km to 5 km Between 5 km to 5 km. Less than 1 month to few days Few days to couple hours Approximately 1 hour Larger-Scale Precipitation Systems The large-scale precipitation systems are often associated with anomalous and persistent middle-level atmospheric wave patterns, slow moving or stagnant features, such as blocking anticyclones and cutoff lows, or nearly stationary synoptic frontal zones. When such features occur, the larger-scale processes produce heavy rains over large regions and affect extensive geographic areas. These systems are: 1. Macro-Scale Systems: The interactions between the atmosphere and the earth s surface can affect globe regions on seasonal or long time period. The Monsoon rainfall (India), El Niño / Southern Oscillation (ENSO) are examples. The wave patterns in this system can support the development, persistence, or sequential recurrence of smaller-scale precipitation systems which lead to floods. 2. Synoptic-Scale Systems: The heavy rains in this system can occur along and ahead of cold fronts or along and to the cool side of warm and stationary fronts. This type of situation can also indirectly contribute to heavy rains by setting up an environment very 7

28 conductive to the development of smaller-scale precipitation systems. Extra-tropical cyclones are examples. The floods associated with synoptic weather features are usually widespread, so flooding occurs over large river systems during an extended period Smaller-Scale Precipitation Systems Meso-scale and storm-scale features produce heavy rains are generally convective in nature. Convective storms, form within an atmosphere, are conditionally unstable. Intense convective rains can bring among of precipitation on a small area in short time. These systems include: 1. Meso-Scale Systems: Convective clouds do not typically evolve as isolated entities; rather they tend to organize into narrow lines or bands or into clusters or complexes of individual storms, particularly thunderstorms. Wildly speaking, meso-scales systems in the mid-latitudes can be classified as: a) Precipitation Bands occur most frequently over the oceans and adjacent coastal region result during the cool season when the circulations within the parent cyclone produce an unstable thermal stratification that will support convection. b) Squall Lines are most frequent over the continents and usually occur in the warm, moist air ahead of cold fronts. These are basically long narrow lines of intense thunderstorm cell. c) Meso-Scale Convective Complexes (MCCs) are large, organized convective cloud systems that lack the distinct linear structure of squall lines (Maddoxx et al., 1986). The precipitation structures embedded within MCCs can be very complicated, consisting of short squall lines, rainbands, cluster of thunderstorms, and individual thunderstorms as well as widespread areas of steady and light-to-moderate rainfalls. 8

29 d) Tropical Systems are basically short-wave troughs in the easterly wind flow that manifest as convective cloud clusters. Large floorings could happen if the storm moves slowly and has interactions with environmental surroundings. 2. Storm-Scale Systems: Isolated thunderstorms can cause localized rainfall rates and amounts sufficient to produce local flash flooding. Especially, when the intense rainfall produced by isolated thunderstorms is concentrated within a small drainage basin or if the storm moves slowly across the basin. 2.2 Current Study Area Characteristics In subtropical areas, or in regions with extensive regions of warm oceans, the land-sea breeze circulations can be important in developing Deep Moist Convection (DMC). This role for land-sea breeze circulations can be enhanced for islands and peninsulas (Pielke, 1974). For example, subtropical peninsular Florida is noted for its frequency of warm-season thunderstorms even though the synoptic flows in the region remain generally weak (Doswell and Bosart, 2). The convective features over South Florida present a large variation in both space and time. Consequently, rainfall amounts and areal coverage can both vary by orders of magnitude and the location of convention may differ considerable from one day to the next (Blanchard and Lopez, 1985). Several factors can be responsible for the observed large day to day fluctuations in convective patterns: 1. Variations in the large synoptic regimes should develop distinct spatial and temporal patterns of convection. 2. Surface features in the large water elements like Lake Okeechobee, water conservation areas, and the coastal configuration should influence in the generation, maintenance, and decay of circulations on the peninsular and local scales. 3. Synoptic and regional scale flow. 9

30 The starting and ending of the summer could be determined by the beginning of the rainfall, a signal that highlights the beginning of the summer season and put an end to the late winter dryness. South Florida is characterized by two predominant seasons. Wet season is identified from May to October and dry season from November to April. The seasonality is determined by precipitation instead of temperature. Summer season is characterized by warm, humid conditions with frequent showers and thunderstorms. Winter season has cooler temperatures, lower humidity, and less frequent precipitation. Autumn and spring are included in the winter season because these two transition periods are drier and cooler than the summer season. The convection with short duration events, such as showers and thunderstorms, has a daily occurrence during summer season, particularly in the late afternoon and early evening. In an easterly wind regime, the precipitation occurs as late night and morning showers or thunderstorms over the coastal areas, and afternoon thunderstorms over the inlands. In a westerly wind regime, afternoon thunderstorms affect inlands and coastal areas identically. In light wind conditions, thunderstorms develop over southeast Florida and affect most areas in the afternoon. Overall, the best indicators of summer season in southeast Florida are dew-point temperatures and minimum temperatures remaining in the 7s, and frequent daily rainfall (NOAA, 21). This finding has been verified recently in a comprehensive investigation of summertime precipitation over the United States (Higgins et al., 1997) Globe Climate Perspective The potential effects of El Niño / Southern Oscillation (ENSO) on the inter-annual variations in regional precipitation and streamflow have been widely studied (Ropelewski and Halpert, 1986 and 1987; Redmond and Kock, 1991). In some areas of the world the ENSOprecipitation-flood relationship is straightforward. But in other areas, a direct relationship leads to 1

31 floods is not evident. Most regions are both affected by hydro-climatic and hydro-metrological mechanisms for generating floods. Anthropogenic caused climatic changes and natural shifts in climate conditions affect hydrologic systems. Researchers report that an analysis of variations in the position, intensity, or frequency of the large scale atmospheric circulation patterns can improve the understanding of the hydro-climatic causes for the spatially distributed flooding. Floods in a single drainage basin can be generated from different types of precipitation events. The type of event that occurs depends in part of the season but also may depend on factors related to the large-scale circulation environment. It is possible to forecast or monitor a developing flood situation over meteorological and short climatic time scale. Hydrologists have been using time series of gauge flood-peak data to evaluate the probability of occurrence of a flood for a given magnitude. Precipitation systems has different intensities and durations and the flood peaks in a given stream may reflect this, resulting in hydro-climatically defined mixed distributions in the overall probability density function of flood peaks (Hirschboeck, 1987a and 1987b). For example, two rivers in the Arizona portion of the Lower Colorado River basin that experience flooding from different precipitation systems have been grouped into three categories: storm-scale precipitation systems, tropical storm-related precipitation systems and synoptic-scale extra-tropical cyclone-frontal precipitation systems (House and Hirschboeck, 1997). This approach can supplement a purely statistical analysis of the flood peaks by revealing the kinds of weather systems that produce different magnitudes of floods. Also, it can be used to evaluate the causes of flood variations over time by different types of storms in several streams throughout a region or by evaluating a long flood record of a single stream in detail (Hirschboeck, 1987b; Webb and Betancourt, 1992). However, Hirschboeck (2), on the later study, concluded that metrological processes directly cause flooding. 11

32 2.2.2 Local Metrological Perspective Convective storms or, more commonly, thunderstorms are produced by a phenomena knows as convective lifting, where air rises as by virtue of being warmer and less dense than the surrounding air, resulting in the precipitation events. Convective storms had been extensively investigated in the past (Byers and Rodebush, 1948; Day, 1953, Gentry and Moore, 1954; Frank et al., 1967; Pielke, 1973) with their organization and evolution of the convective field and attempted to show its relationship to the peninsular and synoptic-scale forcing. To predict flooding, the meteorological systems described above are not enough. Many key components, such as rainfall-runoff processes, surface and subsurface hydrological conditions, and land-use properties must be included in the analysis. The antecedent climatic and hydrological factors with their available reservoir storages affect the possible flooding in a basin (Dunne, 1983). The soil moisture, soil saturation, and groundwater level (shallow groundwater tables) are important factors in the flooding events in South Florida. Saturated soils are not required precursor for severe flooding. For instance, flash floods commonly occur when runoff is conveyed to stream channels so rapidly causing unsaturated overland flow, regardless of soil moisture content. In areas others than South Florida, snow and snowmelt have a strong influence in flooding Atlantic Multidecadal Oscillation (AMO) The Atlantic Multidecadal Oscillation (AMO) is an ongoing natural phenomenon of the sea surface temperature variation in the North Atlantic Ocean. The series of long-duration changes includes cool and warm phases that may last for 2 years to 4 years at a time. The affected areas cover most of the Atlantic between the equator and Greenland and some area of the North Pacific. In North America and Europe, Both air temperatures and rainfall are affected associated with changes in the frequency of North American droughts and is reflected in the 12

33 frequency of severe Atlantic hurricanes. The AMO also has influence to human-induced global warming. Drought over North America has been correlated to the Atlantic Multidecadal Oscillation (AMO) and the Pacific Decadal Oscillation (PDO). Recent research suggests that when the AMO is in its warm phase (Positive AMO), these droughts tend to be more frequent and severe. However, Florida and the Pacific Northwest tend to receive more rainfall in positive AMO. (McCabe, 24) The AMO has a strong effect on Florida where receives more rainfall in its warm phase and droughts and wildfires are more frequent in the cool phase. The numbers of tropical storms that become hurricanes are more often during warm phases of the AMO. Since the mid-199s, Florida has been in a warm phase and severe hurricanes have become much more frequently. (NOAA) 2.3 Extreme events, trends, and Seasonality Analysis The rare and extreme, catastrophic flooding events are the most difficult to understand by traditional methods or direct observation; since the gauge record from a river does not provide enough data to characterize the frequency of the event. According to Hirschboeck (2), examining the like hood of clusters of floods within multiple year periods dominated by a particular set of climatic conditions conducive to flooding is more successful than attempting to assess the occurrence of floods on an annual scale. Differences in paleoflood patterns in the magnitude and frequency of the largest floods in the United States show enormous implications for flood-frequency forecasting and river management (Hirschboeck, 2). A research was conducted in Hudson Bay River in the Canadian Arctic showing a decreasing trend in the streamflow that was linked to various large scale atmospheric phenomenons (Dery and Wood, 24 and 25). Seasonal flood peaks, annual runoff volumes 13

34 and trends from 187 to 22 were analyzed in a study conducted in Sweden (Lindstrom and Berstrom, 24). A statistical methodology was applied in a study that was conducted to explore the seasonality and multi-modality of floods as well as to identify any regularity in the spatial distribution of the corresponding extreme rainfall event. Non parametric tests were used to link rainfall and flood events. A pooling based framework in the rainfall regime was applied to identify hydrological homogeneous pooling groups (Cunderlik and Burn, 22). Seasonal linkage between streamflow and precipitation has been studied in previous research in the United States. The increments in winter and spring streamflow were investigated (Lettenmaier, et al., 1994) as well as the analysis of trends in selected quantities of flow discharge for 295 stations (Lins and Slack, 1999). One of the results found on these previous works was that the conterminous United States is getting wetter but less extreme (Lins and Slack, 1999; Douglas et al., 2; Burn and Elnur, 22). The two most frequently used statistical methods to capture flood seasonality are the annual maximum model and the peaks over threshold sampling model. An evaluation of these techniques has been performed which demonstrates the peaks over threshold sampling model outperform the annual maximum model in most analyzed cases providing more information on flood seasonality (Cunderlik et al., 24). The shape, timing, and peak flow of a streamflow hydrograph are significantly influenced by spatial and temporal variability in rainfall and watershed characteristics such as land use. Studies have been carried out to investigate linkages between climate indices, streamflows, and precipitation. A composite analysis approach was used to identify linkages between extreme flow and timing measures and climate indices for a set of 62 hydrometric stations in Canada (Sharif and Burn, 29). The methodology involves examining the data record for the extreme flow and timing measurements for each station for the years associated with the 1 largest values and the 1 lowest values of various climate indices from the data record. In each case, a re-sampling 14

35 approach was applied to determine if the 1 values of extreme flow measures differed significantly from the series mean. The results show that several stations are influenced by the large scale climate indices. The trends in extreme hydrological events, both high and low flow events were analyzed investigating four extreme flow measures: High Flow Magnitude (HFM), High Flow Timings (HFT), Low Flow Magnitude (LFM), and Low Flow Timings (LFT). 2.4 Dependence Analysis Dependence indicates the extent to which the value or condition of one variable can be determined solely from knowledge of the value or condition of other potentially dependent variables regardless spatially or temporally separated. If two variables are positively dependent, then one takes a high value and the other is more likely to take a high value. If the two variables are negatively dependent, then one takes a high value and the other is more likely to take a low value. If two variables are independent, then the value of each one has no effect on the probability distribution of the other. (Hawkes, 25) A correlation have been made by paleohydrologic research between small chances in mean temperature and precipitation over past few thousand of years and large changes in flood frequency and magnitude on river systems (Chatters and Hoover, 1986; Knox, 1993; Ely et al., 1993; Ely, 1997). In the Lower Colorado River basin, a relationship between floods and climatic variations appears to hold true for at least the past 5 years (Ely, 1997). A study of national trend and variation in the United Kingdom flood regime was performed by using extensive peaks over threshold and annual maxima data from 89 gauging stations, two annual series representing flood size and flood occurrence were examined by applying three main tests to identify trends. The tests included linear regression, normal scores linear regression, and Spearman s correlation. The study showed evidence that urbanization can affect flood regime, and also progressive changes in the United Kingdom flood regime can be observed (Robson et al., 1998). 15

36 In British Columbia, a mountainous watershed had been studied by analyzing 175 storms and classifying them into 4 categories: winter rainfalls, snowstorms, summer rainfalls, and storms (mixture of rain and snow). The spatial distribution had been analyzed by comparing the statistical correlation coefficients of the values (precipitation, duration, average intensity, maximum hourly intensity, and relative start time of the storm) of various storm features at each station to values at a selected base station in the area. The Temporal distributions of the storms were evaluated using the well known Huff methodology (Huff, 1967). By recognizing the variability of hyetographs, the temporal distribution of precipitation could be expressed as isopleths of probabilities of dimensionless accumulated storm depths and durations. A later analysis used historical large storm events in the same area to examine the results from earlier study shown that they were spatially and temporally identical (Loukas and Quick, 1996). In Northern Italy, a compound measure was built to illustrate the mean delay between combined flood and rainfall seasonality. The relative performance of four combined flood and rainfall seasonality descriptors was evaluated in the research (Castellarin et al., 21). The hydrological response of a watershed varies with the dynamics of the storm precipitation. Several studies have investigated the linkage between precipitation and the streamflow data by correlating the data using Mann-Kendall test and linear regressions techniques (Lins and Slack, 1999; Lettenmaier et al., 1994; Lins, 1997; Karl et al., 1995). The Man-Kendall test was applied to the Mackenzie Basin in northern Canada to identify the trends and variability in the hydrological regime. Data was collected from 54 flow gauging stations and 1 meteorological stations. Trends were evaluated by different seasons and winter month flows shown strong increasing trends (Burn and Elnur, 22; Abdul and Burn, 26). By using the same concept, streamflow records were collected from 36 gauging stations, 5 major river basins of Minnesota, USA to present the increasing trend (Novotny and Stefan, 27). 16

37 Discharge trends and the dynamics of South American rivers flowing into the southern Atlantic seaboard were analyzed using this methodology (Andrea and Depetris, 27). The annual and seasonal precipitation in Hanjiang Basin as well as the water levels and the streamflows in Yangtze River Basin in China were investigated by following these techniques (Hua et al., 27; Zhang et al., 26). Trends in precipitation and streamflow at different timescales for 5 years of data and the linkage with recent watershed and channel changes were investigated in the semiarid Rio Puerco Basin in northwestern New Mexico. The Rio Puerco Basin was divided in to five hydrologic units. A 5 years data was collected and analyzed to identify seasonality, variability, trends and other characteristics using the non-parametric Mann-Kendall test (Molnar and Ramirez, 21). 2.5 Applications of Joint Probability Analysis Joint probability indicates the chance of two or more conditions occurring at the same time. For the sake of the interest, some overall outcome of interest should depend on the combined occurrence of these conditions. To apply the method in the study, the dependence between the two or more conditions should be neither independent nor fully dependent. Hawkes (25) suggests that environmental variables such as rainfall, sea level, wave height, and river flow are therefore suitable candidates for joint probability analysis of flood and other risks. The joint probability analysis results can be narrowed down to the following three forms: 1. Joint exceedance contours (equal probability lines or can be expressed as return period) of the source variables simultaneously exceeding values indicated by any point on the contour. 2. A long time series of the relevant source variables which is based on measurements. 3. A very long-term simulation which is based on the joint probability density of the relevant source variables. 17

38 The results can be used to predict responses and risks of interest. The traditional design approach to coastal defenses may not satisfy the requirement since the flooding is mainly caused by a combination of two extreme events. Despite of taking these two variables are independent, many studies consider the variables as dependant and focus on the joint distribution analysis. To avoid the drawback of deterministically and empirically based design, joint probability studies of wave heights and periods or sea surface elevations of two points in the sea take a major role in coastal defense engineering (Ferreira and Guedes, 22; Battjes and Groenendijk, 2; Memos and Tzanis, 2; Prevosto et al., 2 and Song et al., 24). A later study focuses on wave height and water level was performed which presents infinite solutions and best solution for a specific failure condition. (Yeh et al., 26) The joint distribution of two variables can be constructed by two approaches. The first one is the analytical approach. If the distributions of two dependant variables are known and their dependent correlation is found, the joint distribution model can be built by statistical derivation. The second one is called desk study approach. HR Wallingford (2) and Rodríguez et al. (1999) used sea water height and water level to present the cases. The problem when applying the desk study approach in ocean defense engineering is that the insufficient field data. (Hawkes et al, 2 and 25) used Monte Carlo to generate the data. Li and Song (26) use a third-generation wave model and a 3D flow model to simulate long term data. The probability based concept has been presented by Plate and Duckstein (1998) on hydraulic Engineering and on coastal engineering (Liu et al., 2). Based on the assumption of the two events occurring simultaneously, all applications mentioned previously have been studied. This research use the same concept in inland structures the joint probability of occurrence of these two variables. 18

39 In brief, the fundamental process is to explore meteorological, climatic, and hydrological synergism with their respecting drainage basin factors as well the long-term climatic and shortterm meteorological processes to fully understand the causes of flood. Streamflow data has the advantage over rainfall data that the complex variability of precipitation, evaporation, transpiration, land use, topography, and other physical characteristics of the region are reflected in the streamflow records (Sharif and Burn, 29). The common statistical approach used by different authors relies on finding the existing links between hydrologic events and flooding in a particular area. In the research, a methodology that starts with a wide data collection, continues with a detail processing and ends with an analysis that delivers a better understanding of the linkage between rainfall and flooding events regarding to the seasonality. To improve the warning systems currently in used, the dependence analysis need to be performed first and the joint probability analysis is applied in the study area. 19

40 3 METHODOLOGY The advantage of streamflow data over rainfall data in data analysis is that the complex variability of precipitation, evaporation, transpiration, subsurface flow drainage, land use, topography, and other physical characteristics of the region are included in the records. The common approach used in different studies are based on using statistical methods to find the existing links between hydrologic events and floods in urban and rural areas. By using the same concept, this study starts from wide data collection and continues with a detail processing and ends with a clear understanding of the relationship between rainfall, streamflow, storm event, and groundwater tables in the pilot watersheds. All methodologies processed in the study are based on statistical methods to improve the warning systems currently in use in South Florida. 3.1 Data Processing The limitation of available data sets and applied methodologies drive data process a fundamental important task. Any long term missing data set, negative values, and values due to control or maintenance need to be checked before further analysis Streamflow, Rainfall, and Storm Events All streamflow data used in this study including daily means and annual peaks are collected from the U.S. Geological Survey (USGS) real-time data. Current data are recorded at 15 minutes to 6 minutes intervals and are transmitted to USGS offices every 1 to 4 hours by satellite, telephone, and radio telemetry. The USGS Real-Time Water Data depicts streamflow conditions as a percentile that is the period of record for the 2

41 current day of the year. Stations have at least 3 years of record are used. In South Florida, majority streamflow stations are not ranked because data records are either less than 3 years or the reported parameters do not including streamflow; for example, only stage is recorded. Since the analysis methodologies applied in the following sections required different data selection approaches and the available data and station location are not always satisfied the needs, the primary selected thirty- one streamflow stations and their paired rainfall stations contribute a total sixty-two stations in South Florida. The overall data length starts from January first, 1998 to April thirtieth, 21. The reason to extend the data is to complete the last dry season. All station locations are based on the data availability and evenly distributed locations. The twelve-year meteorological events are collected for the following analysis. (Appendix B) Table 3 shows the collected storm events in year 23 including type, name, date, and short description. The time of occurrence used in the analysis is based on the date of formation or dissipation if available. 21

42 Table 3 Meteorological events in year 23. Meteorological Event Year Type Name Date Description Tropical Storm Tropical Storm Ana April 2, 23 Swells. Bill June 3, 23 It hits southern Louisiana, producing over 7 inches (178 mm) of rainfall in portions of the western Florida Panhandle. Hurricane Claudette July 8, 23 Gulf of Mexico. 23 Tropical Depression Seven July 25, 23 It drops light precipitation. Hurricane Erika August 14, 23 Tropical Storm Tropical Depression Grace August 3, 23 Henri September 6, 23 Precursor disturbance produces heavy amounts of precipitation across the state. Moisture drops about 1 inch (25 mm) of rain in the Florida Keys and over 3 inches (75 mm) in northern Florida. It hits Clearwater, producing 9.9 inches (231 mm) in Hialeah which leads to minor flooding. Hurricane Isabel September 13, 23 Rip currents. Figure 1 presents a general plot for streamflow (Blue), threshold (Red), and storms event (Green) at streamflow station C-4A from Year 1998 to Year 29. Zero values are preserved. Negative values, indicate reverse flow, tidal, or backwater, are skipped. A minimum zero value is set to y-axis for all streamflow plots. Threshold (Red) is set to be the ninety percent of the wet and dry seasonal streamflow peak. 22

43 Figure 1 Plot for streamflow (Blue), threshold (Red), and storms event (Green) from year 1998 to year 29. To link the precipitation data to streamflow and storm event are essential concept in understanding hydro-meteorological flood. Rain gages are classified in four recording types including, manual, operational maintenance with multiple sources, telemetry (radio network), and Campbell Scientific 1 (CR1). The rain gage data provided by South Florida Water Management District (SFWMD) has gone through the interval QA/QC (Quality Assurance and Quality Control) process. However, the rain gage data is reviewed since it is essential and critical data set that is used for data improvement. Gage reliability can also be established by comparing the gage observed precipitation data with calibrated NEXRAD data available from the district. Assessment of cumulative plots of rainfall coming from rain gage and NEXRAD data can also be used for preliminary assessment of rain gage data quality. A small window is presented in Figure 2 to better illustrate the relationship around rainfall, streamflow, wet and dry threshold, and storm event. The streamflow station C-4A and its paired rainfall station SOUTH BA show more rain and higher flow peak and threshold in wet season. 23

44 However, this trend does not hold for all stations and the flowing analysis can better present the idea. All thirty-one stations twelve years plots are included in Appendix C. Figure 2 A window for streamflow (Upper blue), threshold (Red), storm event (Green), and paired rainfall (Lower blue) in year Geospatial Data Analysis and Tools Spatial analysis is required for processing geospatial data related to this study. The geospatial data used in the current study is assembled in the state plane coordinate system. The details of the spatial and projection system are provided in Table 4. 24

45 System Parameters Coordinate System Table 4 Details of the Spatial Coordinate System. Description or Value NAD 1983 HARN State Plane Florida East FIPS 92 Feet Projection Transverse Mercator False Easting False Northing. Central Meridian -81. Scale Factor Latitude Factor Linear Foot US Foot Spheroid GCS North American 1983 Datum D North American 1983 The Geographic Information System (GIS) data files are downloaded from Florida Geospatial Data Library (FGDL). The streamflow data, rainfall data, and groundwater data are obtained from United States Geological Survey (USGS), South Florida Water Management District (SFWMD) and National Climate Center (NCDC). Since the data management related to Excel software, the longitude and latitude are obtained from Hawths tool to project the site correctly if necessary. All geospatial data from Florida Geospatial Data Library (FGDL) or constructed for this study including vector and raster data are listed on Appendix D. There are eight interpolation methods in Spatial Analyst Tools including Inverse distance weighted, Kriging, and Trend. ArcMap also provides six interpolation methods in Geostatistical Wizard. A summary of commonly used interpolation techniques and their applications is listed in Appendix E. There are two interpolation methods in common. One such method is Inverse Distance Weighting Method (IDWM), a deterministic interpolation methods directly based on the surrounding measured values or on specified mathematical formulas. Equation (3.1) is the general 25

46 formula where z is the estimated value at point z, z i is the z value at known point I, d i is the distance between point I and point, s is the number of known points used in estimation, and k is the specified power. (Chang, 27) (3.1) The second is Kriging which assumes that the distance or direction between sample points reflects a spatial correlation that can be used to explain variation in the surface. The geostatistical techniques not only have the capability of producing a prediction surface, but also provide some measure of the certainty or accuracy of the predictions. Kriging is most appropriate when samples have a spatially correlated distance or directional bias in the data. It is wildly used in soil science and geology. The formula is listed below (3.2) wherez(s i )is the measured value at the i th location, λ i is an unknown weight for the measured value at the i th location, s is the prediction location, and n is the total number of measured value. (ArcGIS Resource Center) (3.2) According to Peter P. Siska and I-Kuai Hung s research Assessment of Kriging Accuracy in the GIS Environment, Kriging has better performance in Root Mean Squad Error measurements (RMSE). Equation (3.3) is the formulation for RMSE where P i is the forecast values of the parameter in question, O i is the corresponding verifying value (observed or analyzed), and n is the number of verifying points (grid points or observations) in the verification 26

47 area. In this section, the performance of two methods, Inverse Distance Weighting (IDW) and Kriging, need to be checked to determine the method for further use in this report. The Root Mean Squared Error (RMSE) is applied to evaluate the model performance. An experiment using the simulation resulted in a RMSE value of.487 inches with Inverse Distance Weighting (IDW) and a.48 inches with Kriging. In this research, Kriging was applied interpolation mythology unless specified in the section. (3.3) 3.2 Hydro-meteorological Extremes and Seasonality Analysis The term Hydro-meteorological flood used in this research refers to flash flood caused by atmospheric and hydrological natures. This section uses basic statistics to investigate the relationships between streamflow, precipitation, storm event, and seasonality in South Florida. Since the available long term precipitation data in South Florida is not available, the limited twelve years streamflow and rainfall data analysis from year 1998 to year 29 are performed in this section. Critical rainfall stations, such as Miami International Airport (MIA) and West Palm Beach International Airport (WPB), have available long term data and are used in the advanced analysis. In South Florida, the water year starts from October first to September thirtieth next year. The dry season starts from November first to April thirtieth next year. The wet season starts from May first to October thirty-first. In general more rainfall and higher streamflow events occur in wet season. 27

48 The annual streamflow peaks and their occurrence season will be presented. The annual rainfall peaks and their occurrence season will be presented and the relationship between annual streamflow peaks (extreme events) and the relationship with hurricane events will be built Annual Streamflow and Rainfall Extremes Analysis The analysis performed in this section focuses on the relationship between annual streamflow and rainfall extremes and their links to seasonality. The annual extremes from twelve years historical data are selected by using MATLAB program and all peaks regarding to their occurrence season are signed with weight one for dry season and weight two for wet season. The regular calendar is used (Not a water year). To present the wet and dry season-based results for streamflow and rainfall extremes, the interpolation tool in ArcGIS is used to present a better view of entire South Florida. For comparison purpose, each set has three graphs. A total twelve years analysis starts from year 1998 to year

49 Annual streamflow peak Annual rainfall peak Figure 3 Graphical representation of streamflow and rainfall annual peaks. By using the average method, data are divided into two groups with six years each. An overall average is prepared in the end to present the seasonal trend in six years interval. The average analysis applied to both streamflow and rainfall data in the end of each exercise Seasonal Streamflow Peaks with Threshold and Hurricane Analysis The analysis performed in this section is based on how streamflow peak relates to hurricane event. In the study, threshold is set to be ninety percent of seasonal peak. Those selected over threshold peaks are classified into two categories: hurricane related and nonhurricane related. Figure 4 shows the selection methodology. 29

50 Non-hurricane related event Hurricane related event Non-hurricane related event Figure 4 Selection of streamflow peaks with threshold and hurricane. To generate the graphical presentation, for one station in a given year, a weight three is signed if hurricane related peaks are greater than non- hurricane related peaks. A weight four is signed if hurricane related peaks are equal to non- hurricane related peaks, and a weight five is signed if hurricane related peaks are equal to non- hurricane related peaks. 3.3 Dependence Analysis The essential elements of the joint probability extremes assessment are the distribution of each variable, the extreme values of each variable, and the dependence for each variable-pair. The dependence measurement between two variables can be established by statistical method, such as correlation, and subjective assessments from scatter plot. If two variables are positively dependent, then one takes a high value and the other is more likely to take a high value. If the two variables are negatively dependent, then one takes a high value and the other is more likely to take a low value. If two variables are independent, then the value of each one has no effect on the probability distribution of the other (Hawkes, 25 and 28). 3

51 3.3.1 Correlation Coefficient and Lag Time Analysis The strength of the linear association between two variables is quantified by the correlation coefficient. The general formulation is as follow: (3.4) Where n is the number of pairs of data, x and y are two variables, i is value at the i th location. To bring the study into the details of dependence analysis, the accumulated rainfall preceding a peak discharge event is calculated based on the following equation: (3.5) Where i is the year under analysis, l is the position where the streamflow peak is occurred in the 365 days of year i, n is the time from peak discharge to the beginning of rainfall in days (Lag Time), and R i is accumulated rainfall for year i. The lag time calculation is based on the interval between preceding rainfall occurrence and the peak discharge. The lag time is a measurement of how fast the runoff moving in a basin which can be significantly reduced as the watershed runoff characteristics change by urbanization, vegetation change, existence of control structures, or hydro-meteorological processes. Figure 5is a schematic representation of changes in peak flow and accumulated rainfall. The lag time equals to zero when streamflow peak and preceding rainfall occur in the same day. 31

52 Figure 5 Schematic representation of peak discharge and Accumulated Rainfall. The lagged-time approach can be descript as analysis of the basin response time which is one of the main factors in the flood generating processes. To investigate a basin s character, the analysis of the correlations between the available historical rainfall and annual peak flow data was applied in the study for different lag times. The peak flow for each year at critical points of interest (e.g., outlet and conveyance structures, etc.,) was obtained from the South Florida Water Management District s corporate environmental database DBHYDRO database. The accumulated rainfall was calculated from several days going backward (lag time) from the time of occurrence of the annual peak flow. The number of days or lag times was selected from day zero (the same day when the peak flow occurred) to day tenth. 32

53 3.3.2 Lag Time and Basin Characteristic Generally, a basin is characterized by its hydrological characteristics. The response time between rainfall and peak discharge is a representation of how a watershed responds to a rainfall event. There are many factors influence the response time of a basin, such as level of urbanization, type of vegetation, condition of soil saturation, steepness of the terrain, distribution of drainage, geological condition, seasonality, rainfall density, and rainfall distribution. The results from pervious analysis can be concluded as levels of correlations and their corresponding lag times. High correlations with different lag times in a basin stand that basins in a watershed are basically rainfall-runoff phenomena domain instead of other physical processes in the basin. Low correlations with different lag times in a basin state that basins in a watershed are affected mainly by other physical processes. 3.4 Validity of Design Discharge and Rainfall The peak discharge and rainfall for the design conditions of each selected outlet structure was collected from SFWMD. Rainfall-discharge curves were plotted for each outlet structure for a lag time equal to the duration of the rainfall used in the design conditions. The rainfall from the design conditions (Q d ) was used to determine the discharge from the historical data. The agreement between discharge/rainfall from the historical data and discharge/rainfall from the design conditions can provide guidelines regarding the adequacy of the existing design conditions of critical structures The criterion for the evaluation of the design conditions is based on the following cases: CASE A: Rainfall used for design conditions and the discharge versus accumulated rainfall from historical data are known. Using the discharge versus accumulated rainfall plot for a given lag time equal to the duration used for the rainfall from the design conditions, the design 33

54 discharge was obtained from the graph and compared with the discharge for the design conditions (refer to Appendix B). The following conditions may apply: Q>Q d, discharge (Q) from historical data is greater than design discharge (Q d ) (relevant to design conditions). The design condition is exceeded. Rainfall abstractions are overestimated. The condition may suggest inadequate design for discharge and revision of design is required. Q<Q d, discharge (Q) from historical data is less than design discharge (Q d ). The design condition is not exceeded. The design is conservative. Rainfall abstractions are underestimated. This condition may suggest adequate design. Q=Q d, discharge (Q) from historical data is equal to design discharge (Q d ). The design condition agrees with historical data. This condition may suggest adequate design. Figure 6 Criteria for the evaluation of design conditions for each structure using discharge as the comparison parameter. 34

55 CASE B: Rainfall for design conditions and discharge versus accumulated rainfall from historical data are given. Using the peak discharge versus accumulated rainfall plot for a given lag time equal to the duration of rainfall from the design conditions, the design rainfall for 24 hours and 72 hours was compared with the accumulated rainfall from historical data as described in Appendix B. The following conditions may apply: R>R d, rainfall (R) from historical data is greater than design rainfall (R d ) from design conditions. The design condition is not exceeded. The rainfall required to produce design discharge is greater than rainfall from design conditions. Rainfall abstractions are underestimated. This condition may suggest need for modifications to design storm values. R<R d, rainfall (R) from historical data is less than design rainfall (R d ) from design conditions. The design condition is exceeded. The rainfall required to produce design discharge is less than rainfall from design conditions. Rainfall abstractions are overestimated. This condition may suggest need for modifications to design storm values. R=R d, rainfall (R) from historical data is equal to design rainfall (R d ) from design conditions. The design condition agrees with historical data. The design storm values are appropriate. This condition may rarely happen. 35

56 Figure 7 Criteria for the evaluation of design conditions for each structure using rainfall as the comparison parameter. Conclusions regarding the likelihood of exceedance of the design conditions based on the analysis of the historical data were made. Critical structures are evaluated to determine if they are functioning at their assumed design conditions. The recommendation is to conduct a detailed analysis to determine new design conditions based on the existing drainage basin and canal system configuration. 3.5 Rainfall Distributions The designed synthetic rainfall distributions used in South Florida are typically based on the Natural Resource Conservation Service (NRCS) Type 2 distribution. The values of synthetic rainfall distribution used in the application phase of the decoupled HEC-RAS/MODFLOW model in C-4 basin are taken from the guidelines outlined in the District Environmental Resource Permit Information Manual Volume IV (SFWMD, 29) for evaluation in this research. These rainfall 36

57 distributions are similar to the Soil Conservation Service (SCS) (now the Natural Resources Conservation Service (NRCS)) Type 2 distributions (SFWMD, 29). To evaluate the adequacy of the SCS Type 2 rainfall distributions, a comparison is made between the rainfall distribution curves for 24 and 72 hour durations (FDOT, 24) and the rainfall distributions derived from the historical data. A long term data sets provide better understanding of rainfall distribution. A comparison can be made when plot the synthetic rainfall distributions and historical rainfall distributions all together for evaluating the difference and deviation. Table 5 and Table 6 provide values for 24-hour Duration Rainfall Curve and 72-hour Duration Rainfall Curve from Florida Department of Transportation (FDOT, 24). 37

58 Table 5 24-hour Duration Rainfall Curve (Source: FDOT, 24). Time (hrs) P cumulative / P total

59 Table 6 72-hour Duration Rainfall Curve (Source: FDOT, 24). Time (hrs) P cumulative / P total Joint Probability Analysis Joint probability analysis (JPA) is a proper method for analyzing the joint distribution of two correlated random variables. In the research, the two variables are annual peak discharges and accumulated rainfall amounts preceding the occurrence of annual peaks. Generally, peak discharges and accumulated rainfall amounts may follow different probability distributions. The theory of a multivariate distribution of correlated random variables with different marginal is applicable in theory. In practice, an alternative method is to apply the multivariate normal distribution to describe the joint probability distribution of correlated random variables (Yue, 2). This method normalizes the different marginal distributions using a transformation 39

60 technique and then applying joint distribution of the normalized variables using the multivariate normal distribution. Bivariate normal distributions can be applied to develop joint probability distributions. However, transformation of the data (e.g., Box-Cox transformation method) may be required to transform the original data to near the normal distribution to satisfied the assumption of samples are normally distributed. The transformation parameters can be estimated using optimization approaches by maximizing the maximum likelihood function. Once joint probability distributions are obtained, the conditional distributions can be used to obtain the associated return periods of annual streamflow peaks and accumulated rainfall amounts preceding the peak event. A brief description and formulation of bivariate normal distribution are given next. The description and derivation of probability density function are derived based on two variables (e.g. X and Y). In the research, the first variable X is the accumulated rainfall, and the variable Y is the annual peak discharge. Bivariate Normal Distribution is a joint distribution of the random variables X and Y, when the joint density takes the form: (3.6) For < x,y < +, where µ x = E[X], µ y = E[Y], σ 2 x = Var[X], σ 2 y = Var[Y], and ρ is the correlation coefficient between X and Y. An equivalent representation of equation (3.6) is given in (3.7), where X = (X,Y) T is a vector of random variables, T represents the transpose, µ = (µx, µy) T is a vector of constant, and 4

61 is a two by two non-singular matrix such that its inverse -1 exists and the determinant, where: (3.7) The shorthand notation used to denote a multivariate (bivariate being a subset) normal distribution is X ~ N(µ, ). In general, represents what is called the variance covariance matrix. When X = (X 1, X 2,, X n ) T and µ = (µ 1, µ 2,, µ n ) T it is defined as: (3.8) The obtained results from the proposed method provide additional information which cannot be obtained by single variable storm frequency analysis, such as the joint return periods of the combinations of annual discharge peaks and accumulated rainfall amounts, and the conditional return periods of one variable if the other is given. Also, the results can meaningfully contribute to design of flood control structures. For a given storm event return period, it is possible to obtain various occurrence combinations of storm peaks and amounts, and vice versa (Yue, 2). The return periods exceeding threshold values of variables X and Y can be calculated if cumulative distribution functions (F(.)) are available for X and Y variables individually or jointly. (3.9) 41

62 (3.1) The return periods based on cumulative distributions for X and Y are given in equations (3.11) and (3.12). (3.11) (3.12) The return periods for two variables, X and Y exceeding specific threshold values can be calculated using equations (3.13) and (3.14). (3.13) (3.14) The adequacy of the design criteria can be evaluated using the return periods obtained from historical data using joint probability distributions. The exceedance probability is estimated using the following equation: (3.15) 42

63 3.6.1 The Probability of Exceedance Definition of the probability of exceedance becomes more complicated when two or more variables are considered. In this case, the likelihood that rainfall and discharge are exceeded at the same time requires the establishment of joint probability. The probability of exceedance was calculated, as a probability of a joint critical value that is going to be exceeded based on the collected data available. Joint cumulative probability functions for accumulated rainfall from a given lag time corresponding to the design conditions (Duration = one day and three days) versus peak discharges were generated for all structures. The probability of exceedance was calculated and reported for different critical scenarios providing the chance of flooding based on the historical data analysis. The analysis is performed based on the following expressions: (3.16) (3.17) (3.18) (3.19) Where P ocurrence is the probability of occurrence of a rainfall and discharge less than a given design condition, P exceedance is the probability of exceedance of a rainfall and discharge greater than a given design condition, R is accumulated rainfall from historical data, Q is discharge from historical data, R d is rainfall for design conditions, and Q d is discharge for design conditions. If P exceedance > P ocurrence, the design parameters of structure need to be reviewed. 43

64 Individual probabilities for a given rainfall and discharge were determined using the cumulative joint distribution function for each structure under scrutiny. Recommendations to evaluate the existing conditions of a basin were given based on the calculated individual probabilities. For rainfall and peak discharge values in the upper limits (extreme events) of the historical data, the individual probability should be low otherwise the drainage condition of the basin (canal and outlet structures) would have to be reviewed (Figure 8). Guidelines were elaborated for each analyzed basin for possible changes in design criteria. Discharge and stage for the design conditions of each selected outlet structure was collected from the SFWMD. A stage-discharge curve per year was plotted for each outlet structure and the probability of exceedance was calculated, as a probability of a critical value that is going to be exceeded based on the collected data available. Comparing the discharge and water surface elevation for the design conditions with the stage-discharge curves, conclusions were made defining the threshold values for a given structure. The collection of events that exceed the design conditions and the analysis of the stage-discharge curve will convey information regarding the reduction of any conveyance capacity and the threshold rainfall amounts that lead to flooding (Figure 8). 44

65 Rainfall, stream flow and upstream water surface elevation data from for each selected outlet structure YES NO Lack of Information Stage -discharge curve by year for each selected outlet structure Design conditions of each selected outlet structure YES NO Selection of critical parameters (discharge and upstream water surface elevation) Lack of Information Probability of exceedance of critical parameters Figure 8 Probability of exceedance of discharge and stage knowing the design conditions for each selected outlet structure. The existing land-use patterns in the upstream drainage area (using data from SFWMD ArcHydro database) for each basin were investigated to assess the pervious and impervious area ratios. The area that is contributing to the runoff and their link with the lag time and the peak discharges for the selected control structures needs to be evaluated. Geospatial analysis showing patterns, trends in land-use and their connections to flooding for each basin should be investigated Box-Cox Transformation Many statistical tests and analyses are based on the assumption of normality which brings tests to be simple, mathematically tractable, and powerful compare to tests that do not fall into the normality assumption. However, data sets are not approximately normal in the reality. The appropriate transformation technique can yield a data set that approximately follow a normal 45

66 distribution which increases the applicability and usefulness in statistical tests and analysis that based on the normality assumption. Box-Cox transformation (Box and Cox, 1964) is a particularly useful family of power transformation which has been widely used since it is first proposed and has fulfilled the basic assumptions of linearity, normality and homoscedasticity simultaneously (Sakia, 1992). The transformation is defined as: (3.2) Where Y is the response variable with a value greater than zero and λ is the transformation parameter. For λ =, the natural logarithm (log base e) of the data is taken and the formula is defined as: (3.21) In the current study, the sample data are annual peak discharge (Q) and accumulated lagged rainfall values (R). The following formulas are modified from equations (3.2) and equation (3.21): (3.22) The original variable rainfall and streamflow values can be obtained by using the back transformation listed below: 46

67 (3.23) The transformation parameter lambda (λ) is a value of lambda (λ) corresponding to the maximum correlation which is the optimal choice for lambda (λ) Box-Cox Transformation Parameter Estimation The parameter lambda (λ) is estimated using an optimization approach. The formulation for the optimization of the parameter is given by the following equations (15-19). The objective is to maximize the log-likelihood function (3.24) and all constraints are listed from (3.25) to (3.28). Maximize (3.24) Subject to: (3.25) (3.26) 47

68 (3.27) (3.28) A nonlinear optimization solver is required for optimal solution (i.e., the optimal value) of transformation parameter lambda (λ), with upper and lower limits specified. Optimal estimate of the transformation parameter can be obtained using genetic algorithms (GA). Genetic algorithms (GA) are required for the solution as the transformation parameter values. Gradient-based optimization solvers may not always provide global optimal solution and have tendency to produce local optimal solutions. The transformation parameter lambda (λ) can be obtained directly from the software Mathworks MATLAB 21a Normality Tests It is important to know that the power transformation is a useful technique but not necessary to fulfill the expected normal distribution requirement. Therefore, normality tests are applied to determine whether or not a data set is well-modeled by a normal distribution or to compute how likely an underlying random variable is to be normally distributed. Depending on one's interpretations of probability, there is a form of model selections and can be interpreted several ways. In terms of statistics, descriptive statistics measures a goodness of fit of a normal model to the data, frequentist statistics tests data by comparing to the null hypothesis, and Bayesian statistics computes the likelihood that the data come from a normal distribution with given parameters or other distributions under consideration. The null hypothesis in a default position is capable of being proven false by using a test of observed data in practice of science 48

69 involves formulating finding hypotheses, and statements. In the following subsections, a graphical method probability plot is performed first and three frequentist tests are applied including the Pearson's chi-squared test, the Jarque-Bera test and, the Lilliefors test Probability Plot The probability plot is a graphical technique for evaluating if two data sets follow a given distribution, such as the normal distribution or Weibull distribution. Two data sets could be both empirical observations, one empirical set against a theoretical set, or both theoretical sets. Probability Probability plot (P P plot) and Quantile Quantile plot (Q Q plot) are two plots commonly referred. Probability-Probability plot (P P plot) or Percent-Percent plot is a probability plot for examining how closely two data sets agree by plotting two cumulative distribution functions against each other. Quantile Quantil plot (Q Q plot) is more commonly used for comparing two probability distributions by plotting their quantiles against each other. If the data is plotted against a theoretical distribution, the points should form approximately a straight line. Departures from this straight line indicate departures from the theoretical distribution. If the two distributions are similar, the points in the Q Q plot will be approximately lay on the line where y = x. However, if two distributions are linearly related, the points in the Q Q plot will be approximately lay on a line, but not necessarily on the line y = x. Normal probability plot is a special case of Q Q plot which uses the graphical technique for assessing whether or not a data set is approximately normally distributed. The data is plotted against a theoretical normal distribution. Therefore the points should form an approximate straight line. Departures from this straight line indicate departures from normality. The vertical axis is the ordered response values and the horizontal axis is the normal order statistic medians or 49

70 means. The following formula presents a calculation method. For each data i = 1,,n, find z i such that: (3.29) The sample size can affect the relationship between the points and the theoretical normal distribution line. The points are expected to be very close to the reference line with a sample size greater than 1. On the other hand, smaller samples will see a larger variation Pearson's Chi-Squared Test A chi-squared test (also referred to as chi-square test or x 2 test) is a statistical hypothesis test in which the sampling distribution fits a chi-squared distribution when the null hypothesis is true. Pearson's chi-squared test is used to assess the tests of goodness of fit and the tests of independence. A test of goodness of fit establishes the difference between an observed frequency distribution from a theoretical distribution. A test of independence assesses whether paired observations on two variables are independent. The formula for calculating value of the teststatistic is listed below: (3.3) Where x 2 is Pearson's cumulative test statistic which asymptotically approaches a distribution, O i is an observed frequency in i th position, E i is a theoretical frequency asserted by the null hypothesis, and n is number of data. 5

71 Jarque-Bera Test The Jarque Bera test is named after Carlos Jarque and Anil K. Bera (1987) which performed a test of the null hypothesis that the sample comes from a normal distribution with unknown mean and variance to against the alternative that does not come from a normal distribution. The Jarque-Bera test is a two-sided goodness-of-fit test suitable when a fullyspecified null distribution is unknown and its parameters (µ and σ) must be estimated. The test statistic is defined as: (3.31) where n is the sample size, s is the sample skewness, and k is the sample kurtosis. For large sample sizes, the test statistic has a chi-square distribution with two degrees of freedom. The formations of sample skewness (s) and sample kurtosis (k) are listed below: (3.32) (3.33) where and are the estimates of third and fourth central moments, is the sample mean, and is the estimate of the second central moment, the variance. The Jarque-Bera test uses the chi-square distribution to estimate critical values for large samples and uses a table of critical computed values by using Monte-Carlo simulation for sample sizes less than 2 and significance levels between.1 and.5. 51

72 Lilliefors Test The Lilliefors test is an adaptation of the Kolmogorov Smirnov test named after Hubert Lilliefors (Lilliefors, 1967). The test is applied to test the null hypothesis that data come from a normally distributed population, when the null hypothesis does not specify the type of normal distribution; for example, the expected value and variance of the distribution. The test primary performs an estimation of the population mean and the population variance. By using the estimations, the maximum discrepancy between the empirical distribution function and the cumulative distribution function (CDF) of the normal distribution can be obtained secondary. Finally, if the maximum discrepancy is statistically significant, the null hypothesis is rejected. If the null hypothesis had taken only one normal distribution, the maximum discrepancy is made smaller than what it would be since the hypothesized CDF moves closer to the data by using the estimation based on the data. Therefore, the test assuming its probability distribution is stochastically smaller than the Kolmogorov Smirnov distribution which makes it more complicated compare to the Kolmogorov Smirnov test Densities Given a bivariate normal distribution of annual peak discharge (Q) and accumulated Q rainfall (I), the marginal distribution of Q is N(, Q ) and R is N( R, R ), the conditional density of Q given R = R th is a normal distribution with mean ( given by the following equations: Q R ) and variance ( 2 Q R ) are (3.34) 52

73 (3.35) The conditional distributions of Q and R are also normally distributed with modified means and standard deviation values as given by the following equations. (3.36) (3.37) The conditional period, T Q R is estimated using the equation given below. (3.38) The conditional densities and the return periods are estimated using the Box-Cox transformed data Validity of Joint Distribution The observed and the theoretical joint probabilities based on the peak discharges (Q) and accumulated rainfall amounts (I) can be evaluated to assess the validity of the bivariate normal distribution model fitted. The theoretical probabilities from the fitted model and the observed non-exceedance probabilities can be easily obtained and can be compared. A high correlation between the observed and the theoretical values generally indicates the validity of the sampled data used for fitting the joint distribution in the bivariate normal distribution model. Alternatively, a conceptually simpler approach would be using generated random vectors (joint variables) which are based on the fitted bivariate normal distribution parameters. Therefore, the generated values 53

74 can be compared with observed data. The summary statistics from the observed data can be compared with generated values. 3.7 Threshold Analysis: Lead Times The preceding accumulated rainfall amounts of a peak discharge event is calculated based on different lag times using historical data for each year (see Figure 9) based on the following equation. (3.39) Where i is year under analysis, l is temporal position where the peak is located in the 365 days of year i, lt is lead time (accounted in time intervals backwards from the time of occurrence of peak), n is time from peak discharge to the beginning of rainfall (Lag Time), and R i is accumulated rainfall for year i. The lag time is calculated as the interval between rainfall occurrence before the lead time and the peak discharge in response to such rainfall. The lag time can be significantly reduced as the watershed runoff characteristics change by urbanization, vegetation change, existence of control structures, or hydro-meteorological processes. Figure 9 is a graphical presentation for lead times method. 54

75 Figure 9 Schematic representation of occurrence of peak discharge and accumulated rainfall along with definition of lead time. 3.8 Partial Duration Series Analysis and Peaks over Thresholds (POT) There are two basic approaches for extracting extreme data while performing a univariate extreme value analysis. These approaches can be described as: 1. Epochal Method: this method extracts the maximum or minimum value for each year or in equal length intervals as seasons. (classical approach) 2. Peaks over threshold Method: this method extracts all points above a given threshold; the numbers of extracted points are not equal in each selected interval. The peak over threshold (POT) approach enables the analysis using all the data exceeding a sufficiently defined threshold, and it is more effective than the classical approach, which uses only the largest value (or smallest value) in each of a number of comparable sets. In peak over 55

76 threshold (POT) approach, it is important to choose a threshold that maintains the independency of the events. Lower thresholds would introduce a strong bias in many instances. One of the difficulties in an extreme value study is that the limited length of the available records. When an annual maximum time series is used, the fitting of the parameters for a good analysis should based on at least a 3 years long time-series. However, the available length of dataset is much shorter in most cases especially in South Florida. The uncertainty associated with statistical analysis may increase when an analysis is performed with a short set of data. In order to reduce the uncertainty, daily streamflow data is used within a Peaks over Threshold (POT) framework, in which the basic idea is to use more than one extreme event per year to increase the available information with respect to the epochal method that uses annual maximum flood data. Annual maximum series or partial duration series (PDS) (Langbein, 1949 and Stedinger, 2) are generally applied in the flood frequency analysis. In case of annual maximum series, the extreme value is selected each year. Selection of extreme events in such a setting ensures that the selected events are independent. The partial duration series is also referred as peak over threshold (POT) approach. This approach requires identification of events that are independent and the values associated with the all extreme events within a year exceed a specific pre-fixed threshold. The selection of threshold is based on a criterion such as: (3.4) Where M th is the threshold value obtained based on the minimum value of all annual extremes values obtained each year, i. 56

77 (3.41) Historically, the conception of independence has played a prominent role in the determination of POT events. If events are generally selected from an independent class, less information is required and calculations are correspondingly easier. Analysts need to resort to non-parametric tests for randomness to decide whether or not a given sequence is truly random (or independent) and identically distributed. In statistical literature, a truly random process refers to a process that can produce independent and identically distributed samples. If an observed value in the sequence is influenced by its position in the sequence or by the observations which precede it, the process is not truly random. The randomness is related with the property of the data, and it is essential to fulfill the theoretical bases of many classical statistical tests. Even though the observations are not truly random in many practical applications, analyst can still perform those simple and can obtain theoretical results with a certain degree of confidence if the level of randomness can be explained and reached. In the research, tests are carried out to ensure the independence of selected events for the POT analysis. A non-parametric test referred to as Runs test is used to assess serial randomness: whether or not observations occur in a sequence in time. Tests are performed for the selected peaks over threshold (POT). The test is based on the null hypothesis that the values come in random order, against the alternative that they do not. The test is based on the number of runs of consecutive values above or below the mean of the selected data. The Runs test is performed for partial duration series selected for different lag times between them, the times that separate the selected peak discharges are 1, 2, 3, 4, 5, 55, 6 days. For C-4 and the C-6 basins, the peak 57

78 discharges can be considered as independent events when they are separated by 5 days to 6 days. The selected gap time for the POT selection for different thresholds is 6 days (Figure 1). The selection process of the peaks over threshold is described in the following flow chart. (Figure 11) Figure 1 Schematic representation of the selection process for POT in the partial duration series analysis. 58

79 Figure 11 Steps for the selection of POT in the partial duration series analysis. 59

80 4 CASE STUDY 4.1 Background of Case-study Region The objective of this study is to evaluate the spatial and temporal precipitation events, streamflow events, and their link to extreme metrological events at outlet structures in South Florida and focus on providing multivariate statistics results. The primary limitation of the study is available data in the study region. To provide insightful and meaningful results, the research starts from short data sets with wide sit selection and narrow down to stations with long tern data sets to insure the methodologies applied in the research reach the degree of confidence. 4.2 Homogeneous Rainfall Areas The selected study area is in South Florida for its high population and high urbanized characters. The total study area covers twenty counties. (Figure 12) Lake Okeechobee, the largest freshwater lake in the state of Florida, is located In the middle of Figure 12covering Glades, Okeechobee, Martin, Palm Beach and Hendry counties. In southern area, the third largest national park Everglades National Park preserves nearly 1.5 million acre throughout Miami-Dada, Monroe, and Collier counties. 6

81 Figure 12 South Florida map with counties. The site selection is based on climatologically homogeneous rainfall regions which were developed for the South Florida Water Management District (SFWMD). (Pathak et.al, 29) Figure 13 presents streamflow stations (red circle) with letter Q and their paired rainfall stations (blue square) with letter R. Table 7 lists thirty-one stations full name and their associated abbreviation used on Figure 13. A full description including all stations full name, location, type, and latitude and longitude is listed on Appendix A. 61

82 Figure 13 Streamflow stations and rainfall stations used in the study. 62

83 Table 7 List of streamflow stations and rainfall stations. NO. Streamflow Station NO. Rainfall Station Q R1 PALMDALE_R Q R2 KISS.FS2_R Q R3 WRWX Q R4 KRBNR Q R5 3A-SW_R Q R6 S336_R (Different Rain Area(8)) Q R7 S335_R Q R8 DANHP_R Q R9 951EXT_R Q R1 COCO1_R Q R11 S79_R Q R12 FORTMYERWS Q R13 JDWX Q14 C-4A R14 SOUTH BA_R Q15 C6.NW36 R15 MIAMI.AP_R Q16 G136_C R16 G136_R Q17 S14_TOT R17 T14_R Q18 S18C_S R18 S177_R Q19 S29_S R19 S29_R Q2 S34_C R2 S124_R Q21 S38C_C R21 S125_R Q22 S39_S R22 WCA1ME Q23 S46_S R23 SIRG Q24 S49_S R24 C24SE Q25 S5A+S5AW R25 WPB AIRP_R Q26 S6 R26 S7_R Q27 S65_S R27 S65_R Q28 S77_S R28 S77_R Q29 S8_S R29 S8_R Q3 S97_S R3 BLUEGOOS_R Q31 S99_S R31 SCOTTO 4.3 C-4 Basin and C-6 Basin The selected study area for joint probability analysis is emphasized on the C-4 and C-6 basins. A map of selected streamflow stations and rainfall stations are shown in Figure 14. Since 63

84 precipitation data is essential for the study, rain gauge locations are located near to critical flood control structures that are selected in the analysis. Figure 14 Location of selected rain gauges and streamflow stations in C-4 and C-6 basins. The C-4 basin is located in northeastern Miami-Dade County. The basin is drained primarily by the C-4 Canal (Tamiami Canal). The C-4 Canal begins in the west at the L-3/L- 31N being the primary conveyance, and is connected to three other primary canals: C-2, which makes an open connection at SW 117th Avenue, C-3 (Coral Gables Canal), which makes an open connection just east of the Palmetto Expressway, and C-5 (Comfort Canal), which branches from 64

85 the C-4 at Blue Lagoon north of Coral Gables via gated culvert S-25A. Normal flow is west to east with discharge to tidewater via Structure S-25B after connecting to the C-6 Canal (Miami Canal) (PBS&J, 24). Canal stages and discharges are maintained by SFWMD for C4.coral (see Table 8). The C-6 basin is located in eastern Miami-Dade County. C-6 begins at S-31 at the intersection of L-3 and L-33. The L-33 borrow canal is aligned north-south along the east boundary of WCA-3B and connects to the west end of C-6 at S-32. Normal flows are from C-6 to the borrow canal. Flow in the C-6 canal is to the southeast with discharge via S-26 to Biscayne Bay. S-32A is always closed and acts as a divide between the C-6 and C-4 basins. During periods of low natural flow, water is supplied to the C-6 basin from WCA-3B via S-31 as needed to maintain the optimum stage in C-6 and to recharge well fields at Hialeah and Miami Springs. Water is subsequently diverted from C-6 to the C-7 and C-9 basins as needed to maintain the optimum stages in the canals in those basins and to recharge well fields near C-9. C-4 and C-5 discharge to tidewater in C-6 downstream of S-26 (PBS&J, 24). The C-6 Canal (Miami Canal) drains a watershed area of approximately 7 square miles and is located immediately north and east of the C-4 Canal Basin area (see Table 8). Basins C-2, C-3, C-4, C-5, and C-6 cover a watershed area that is approximately 225 square miles (PBS&J, 24). Frequent flooding during higher than normal rainfall events is experienced in these basins with water overflowing the canal banks and that will induce property damage. Information of the structures in C-4 and C-6 are summarized in Table 8. 65

86 Table 8 Basin information and associated structures for C-4 and C-6 basins. Canal C-4 (Tamiami) C-6 (Miami) Drainage Basins C-4, C-3, C-2, C-5 C-4, C-6, C-5 Length (miles) Drain Area (mi 2 ) Associated Structures S-32A, S-25B, S-336, G-119, S-25A, S S-26, S-31, G-72 S-32, S-32A, S-25B, S Description of Canals and Control Structures There are several secondary canals located in the C-4 basin including Northwest Well field Recharge Canal, North line Canal, NW 41st Street Ditch, Snapper Creek Canal, NW 97th Avenue Canal, Westbrook Canal, and FEC Canal and Northwest Canal. Total six major control structures with significant impacts are located on the drainage system of C-4 and C-6 basins including S- 25A, S-25B, G-119, S-336, Control Structure #3, and S-38 (BS&J, 24). These six control structures are manually controlled gated culvert and gated spillway for preventing saline intrusion or providing aquifer recharge. C-4 and C-6 basins are highly urbanized area with plenty of control structures Land Use of C-4 and C-6 Basins The land use was calculated based on the classification from FGDL (Florida Geographic Data Library) for year 24 in the C-4 and C-6 basins and for two main categories as it can be seen in the following tables: Table 9 Classification of Land use for C-4 basin. Land Use Area Percentage (%) Urbanized Area 45 Non-Urbanized Area 55 Table 1 Classification of Land use for C-6 basin. Land Use Area Percentage (%) Urbanized Area 62 Non-Urbanized Area 38 66

87 The C-4 and C-6 basins are typical examples of South Florida hydrogeology which exhibiting high urbanized areas in man-delineated basins where topographic differences are negligible. The high groundwater table is presented which dominated by the behavior of the unconfined and highly transmissive Biscayne Aquifer. According to previous study, the intersection of the canals with the groundwater table feeds an important part of the baseflow (PBS&J, 24). The design rainfall used during the application phase of the explained methodology is based on the synthetic rainfall events from the guidelines outlined in the District Environmental Resource Permit Information Manual Volume IV (SFWMD, 29). The temporal rainfall distribution patterns are based on the SFWMD rainfall and are similar to the Natural Resource Conservation Service (NRCS) Type 2 distribution. The performance of the C 4 and C-6 structures were evaluated under 1 and 25 years storms according to the storm total volume for the C 4 and surrounding basins (SFWMD, 21). Based on the revisions of the basis of review, the allowable peak discharge value for C-4 and C-6 basins is corresponding to the peak discharge rate after development with a 25 year event design storm. 67

88 5 RESULTS 5.1 Computer Software The data analysis and the methodology applications performed in the study heavily relay on computer software for exploratory data and statistical analysis. The following software with specifications listed below are used in the research. Microsoft Office Excel 27 Mathworks MATLAB 21a ARCMAP 9.3.1(Environmental Systems Research Institute (ESRI)) Hawth s Analysis Tool for GIS Microsoft Office Excel 27 was the primary software used to perform data analysis in streamflow, rainfall, and extreme events relationships including seasonal domain and trends. The software ARCMAP 9.3.1was used to analyze the temporal and spatial relationships and the Hawth s Analysis Tool was used to obtain the station locations if necessary. The dependence analysis nad joint probability analysis were performed using Mathworks MATLAB 21a. 5.2 Application of Annual Extremes Analysis Application of Annual Streamflow Peak Analysis The analysis performed in this section focuses on the relationship between annual streamflow peak and seasonality. The annual peaks from twelve years historical streamflow data are selected by using MATLAB program and all peaks regarding to their occurrence season are 68

89 assigned with weight one for dry season and weight two for wet season. The regular calendar is used. (Not water year) To present the wet and dry seasonal domain for streamflow, the interpolation tool is used to present a better view of entire South Florida. Figure 15 to Figure 18 lists streamflow station with dry (Gray) and wet (Black) seasonal domain from year 1998 to year 29. The tabular result is attached in Appendix F (Part I). Year 1998 Year 1999 Year 2 Figure 15 Annual streamflow peak in wet (Gray) and dry (White) (Continued). Year 21 Year 22 Year 23 Figure 16 Annual streamflow peak in wet (Gray) and dry (White) (Continued). 69

90 Year 24 Year 25 Year 26 Figure 17 Annual streamflow peak in wet (Gray) and dry (White) (Continued). Year 27 Year 28 Year 29 Figure 18 Annual streamflow peak in wet (Gray) and dry (Gray). To take an overall long term view, an average method is applied in this stage. Data are divided into two groups with six years each. An overall average is prepared in the end. From the graphical illustrations (Figure 19), annual streamflow peaks are most likely occurring in the northwest region in South Florida. 7

91 Year 1998 to Year 23 Year 24 to Year 29 Year 1998 to Year 29 Figure 19 Average analysis of Annual streamflow peak in wet (Black) and dry (Gray) Application of Annual Rainfall Extreme with Wet and Dry Analysis The analysis completed in this section applies the same mythology and technique as pervious section. The occurrence of rainfall extreme is signed a weight one to dry season and weight two to wet season. Again, each graph presents one year (Not water year). The tabular result is attached in Appendix F (Part II). For comparison, each set has three graphs. A total twelve years analysis starts from Figure 2 to Figure 23 show that rainfall extremes in southeast region occur in wet season (Gray). Central location (lake Okeechobee) and the East side has rainfall annual peaks happen in dry (White) season more often. All rainfall extremes are happened in wet season for year 1999 and year 27. Since the analysis uses interpolation technique to generate graphical representation, results for year 1999 and year 27 are not available. 71

92 Graphical presentation is not available since all station peaks are wet seasonal domain. Year 1998 Year 1999 Year 2 Figure 2 Annual rainfall extreme in wet (Gray) and dry (White) (Continued). Year 21 Year 22 Year 23 Figure 21 Annual rainfall extreme in wet (Gray) and dry (White) (Continued). 72

93 Year 24 Year 25 Year 26 Figure 22 Annual rainfall extreme in wet (Gray) and dry (White) (Continued). Graphical presentation is not available since all station peaks are wet seasonal domain. Year 27 Year 28 Year 29 Figure 23 Annual rainfall extreme in wet (Gray) and dry (White). By using the average method, data are divided into two groups with six years each. An overall average is prepared in the end. From the graphical presentations (Figure 24), annual rainfall extremes are more likely occur in the southern area in wet season. 73

94 Year 1998 to Year 23 Year 24 to Year 29 Year 1998 to Year 29 Figure 24 Average analysis of Annual rainfall extreme in wet (Gray) and dry (White) Application of Streamflow Peaks with Threshold and Hurricane Analysis To generate the graphical presentation, a weight assigned method is applied to classify the difference. For one station in a given year, a weight three is signed if hurricane related peaks are greater than non- hurricane related peaks. A weight four is signed if hurricane related peaks are equal to non- hurricane related peaks, and a weight five is signed if hurricane related peaks are equal to non- hurricane related peaks. Graphical presentations through Figure 25 to Figure 28 illustrate a twelve years analysis. The tabular result is attached in Appendix G. 74

95 Year 1998 Year 1999 Year 2 Figure 25 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and non-hurricane related (Orange) (Continued). Year 21 Year 22 Year 23 Figure 26 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and non-hurricane related (Orange) (Continued). 75

96 Year 24 Year 25 Year 26 Figure 27 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and non-hurricane related (Orange) (Continued). Year 27 Year 28 Year 29 Figure 28 Streamflow peaks with hurricane related (Blue), equal event (Yellow) and non-hurricane related (Orange). Except year 1999 and year 2, all ten years analysis shows that non-hurricane related peaks domain the south region. This conclusion is based on wet and dry seasonal threshold. In other words, hurricane events bring amount of rainfall and extreme high peak; however, peaks over threshold are happening more often compare to limited hurricanes in a given year. 76

97 Moreover, hurricane related peaks domain the west region; therefore, less peaks appear regarding limited hurricanes. To briefly conclude the funds from all four analyses, annual streamflow peaks are most likely occurring in the northwest region in South Florida in dry season unlike annual rainfall extremes which are more likely occur in the southern area in wet season. In third stage, hurricane events bring amount of rainfall and extreme high peak; however, peaks over threshold are happening more often compare to limited hurricanes in a given year. Moreover, hurricane related peaks domain the west region in South Florida; therefore, less peaks appear regarding limited hurricane events. 5.3 Applications of Dependence Analysis The dependence analysis is the required primary process before a joint probability analysis can be performed. The selected streamflow stations and their nearest rain gauge stations are evenly located in the meteorologically homogeneous precipitation areas with daily data from year 1998 to year 29. The annual peak flow was selected for those 12 years and the accumulated rainfall was calculated for different lag times. Please note that for C-4 and C-6 basins, longer type of datasets were selected and will be used for further analysis. Table 11 summaries the correlations for different lag times from to 1 days. The best correlation and their corresponding lag times are listed on the right of the table. The average lag times from day zero to day ten are listed in the bottom of the table. A summary of best correlation and lag times is listed in Table

98 Table 11 Summaries of Peak Discharge and Accumulated Rainfall Correlations and the Corresponding Lag Times. Streamflow Correlations in Time Lags Stations G136_C S14_TOT C-4A S65_S S99_S S97_S C6.NW S18C_S S77_S S46_S S5A+S5AW S38C_C S49_S C4.Coral Average Correlations

99 Table 12 Best Correlation and the Lag Time Occurrence in Days. Streamflow Stations Best Correlation Lag Times (Days) G136_C S14_TOT C-4A S65_S S99_S S97_S.91 7 C6.NW S18C_S S77_S S46_S.47 2 S5A+S5AW S38C_C.28 1 S49_S C4.Coral Average Correlations

100 Figure 29 Average Correlations for Accumulated Rainfall and Peak Discharge Lag Time from Day Zero to Day Tenth. Figure 29 presents the increasing trend in the average correlations with the lag time increase which illustrates most of the selected streamflow stations have the typical rainfall-runoff response of watersheds. To better understand the basins, the rearranged Table 13 lists best correlation in orders from high to low with a threshold value of.5 to distinguish high and low correlations. A highly urbanized basins with structures as C4.Coral (Best Correlation =.959) and C6.NW36 (Best Correlation =.927) with high rainfall-peak discharge correlations demonstrating an evident response of the basin to rainfall. Natural basins or controlled structures such as S65 (Best Correlation =.383) and S77 (Best Correlation =.299) present very low correlations between rainfall and peak discharge showing that the dependence of the hydrologyhydraulic behavior of the basin on other factors, such as variations in groundwater table, storage 8

101 capacity, soil moisture conditions as well as operational rules implemented for hydraulic structures. Table 13 Classification of Streamflow Stations Based on Rainfall and Discharge Best Correlations. Classification High peakdischarge lagged rainfall correlation Low peakdischarge lagged rainfall correlation Streamflow Stations Basin Best Correlation Lag Time (days) C4.Coral C C6.NW36 C S-65D S97_S C-23 / Upper St. Lucie Estuary.91 7 C-4A South Shore S5A+S5AW S-5A Estero Bay S18C_S C-111 South East Lake Tohopekaliga Fisheating Creek Barron River Telegraph Swamp East Collier Arbuckle Creek S46_S C-18/Corbett.47 2 S14_TOT Conservation Area 3A S49_S St. Lucie North Fork G136_C S S99_S C S65_S S65A S77_S East Caloosahatchee / Lake Okeechobee Everglades National Park / Conservation Area 3A.36 S38C_C C-14 West High Correlations and Rainfall-runoff Domain Basins Figure 3 shows the results for stations C4.CORAL and C6.NW36. High correlations from both stations represent that both stations are rainfall-runoff domain. Station C4.CORAL has a quick increasing trend from day zero to day one. The increasing trend slows down after day one. The station C6.NW36, on the other hand, has a subsequent reduction of the time of concentration 81

102 after day two. Both stations have fast runoff response state that stations are located in highly urbanized basins. Figure 3 High Accumulated Rainfall-Peak Discharge Correlations for Different Lag Times from Day Zero to Day Tenth. A clear relationship between rainfall and runoff can be established for the data collected for two basins. Once the time of concentration is reached (day one for C4.CORAL and day two for C6.NW36), the correlation stay high for the basin showing that the peak event is controlled primarily for the rainfall-runoff physical phenomena instead of other physical processes in the basin, such as operational protocols, base flow, or groundwater fluctuations Low Correlations and Non rainfall-runoff Domain Basins The low rainfall-peak discharge correlations are directly related to basins affected mainly by groundwater table fluctuations, soil moisture, base flow, and other basin physical features. Figure 31 are results from station S77_S and station ID The first station is located in East Caloosahatchee / Lake Okeechobee basin. The second station is located in Everglades National Park / Conservation Area 3A. Both stations are in rural area with low correlations that can also be affected by the land use of the basin, large rural natural areas with high bed friction factors, and storage capacity. 82

103 Figure 31 Low Accumulated Rainfall-Peak Discharge Correlations for Different Lag Times from Day Zero to Day Tenth. Accumulated rainfall-peak discharge correlations for different lag times from day zero to day tenth are plotted for all twenty-three stations and listed in Appendix H. Both stations represent a weak dependence in relationship between rainfall and peak discharge can be identified by low correlation. This is a consistency with the drainage behavior of natural and agricultural areas Variation of Accumulated Rainfall and Peak Discharge for Different Lag Times with Hurricanes The variation of accumulated rainfall and peak discharge for different lag times between day zero to day tenth are performed in this section. From year 1998 to year 29, a total twelve years analysis with hurricane related data (blue circle) and non-hurricane related data (red circle) is performed for all paired stations. Figure 32 shows the result for Station S97_S. 83

104 6 4 Lag time:days R 2 = R 2 = R 2 = Lag time:1days R 2 =.331 R 2 = R 2 = R 2 =.846 R 2 =.597 R 2 =.628 Lag time:2days Lag time:3days R 2 =.86 R 2 =.633 R 2 = R 2 =.865 R 2 =.649 R 2 =.698 Lag time:4days 6 4 R 2 =.85 R 2 =.65 R 2 =.839 Lag time:5days Lag time:6days 8 Lag time:7days 8 Lag time:8days 6 4 R 2 =.899 R 2 =.896 R 2 = R 2 =.91 R 2 =.894 R 2 = R 2 =.9 R 2 =.888 R 2 = Lag time:9days 8 Lag time:1days 6 4 R 2 =.9 R 2 =.891 R 2 = R 2 =.97 R 2 =.913 R 2 = Figure 32 Variation of Accumulated Rainfall and Peak Discharge for Different Lag Times for Station: S97_S. The following observations can be made from the scatter plots with high correlation at Stations: S97_S, C6.NW36, S49_S and C4.Coral from historical data from year 1998 to year 29: Positive correlations can be seen presenting a direct relationship between rainfall and runoff in the basin. For peak events the high urbanized basins are controlled for a runoff response to a rainfall impulse. Positive correlations are identified for lag times between and 1 day depending on the basin, demonstrating a quick response of runoff observed for the peak event. The increase in rainfall produces an increment in the peak discharge which demonstrates that peak discharge (maxima) are related to meteorological events instead of base flow or other phenomena. 84

105 The following observations can be made from the scatter plots with low correlation at Stations: S65_S, , G136_C and S14_TOT from historical data from year 1998 to year 29: Negative correlation presents on day zero presents a non direct relationship between rainfall and runoff in the basin. For peak events the rural basins are controlled by control operations, antecedent moisture conditions (AMC) and base flow. Positive correlations are presented for lag times greater than two days which demonstrate a slow response of runoff under extreme meteorological rainfall events. The increasing rainfall does not necessarily produce an increment in the peak discharge. Clusters can be found around small rainfall values with high peak discharges, producing an increment in the peak discharge, which demonstrate maximum that peak discharges are more related with base flow, groundwater levels, and impact of friction factors in the drainage of water or operational protocols. The variation of accumulated rainfall-peak discharge correlations for different lag times from day zero to day tenth are plotted for all stations and are listed in Appendix I. 5.4 Comparison of Synthetic Rainfall Distributions and Historical Rainfall Distribution in Different Durations A narrowed sit stations with long term available data set are used in this section. The historical precipitation data are collected from the National Climatic Data Center (NCDC) website. Three rain gauge stations include the Miami International Airport (MIAMI.AP_R), the West Palm Beach Airport (WPB AIRP_R) and the Boca Raton (BOCA_R). For rainfall distribution analysis, 42, 35, and 12 years of historical data are used to calculate the 24 hour 85

106 duration rainfall curves, and 29, 27, and 1 years of historical data are used for the 72 hour duration rainfall curve. The available data sets follow the order of listed rain gauge stations. Figure 33 shows the location for all three rain gauge stations. Table 14 lists rain gauge stations and their available data lengths. Figure 33 Map of Selected Rain Gauges for Rainfall Distribution Analysis. Table 14 Selected Rain Gauges for Rainfall Distribution Analysis. Rain Gauge Start Date End Date Miami International Airport February 1, 1939 December 31, 29 West Palm Beach Airport January 1, 197 December 31, 29 Boca Raton August 1, 1948 July 31,

107 Long term historical rainfall data helps the better understanding of the rainfall distributions. Three rain gauge stations are selected including Miami International Airport, West Palm Beach Airport, and Boca Raton. Figure 34 represents the synthetic and the historical rainfall distribution curves in a twenty-four hours duration. Deviations occur from two to twenty hours from the beginning of the storm. An early peak with high intensity produces fast and high peak discharge which leads potential flooding. Rain Gauge: Miami International Airport Rain Gauge: West Palm Beach Airport Rain Gauge: Boca Raton Figure 34 Comparison of Synthetic and Historical Rainfall Distributions in 24 Hours Duration. 87

108 Figure 35 shows the comparison around the synthetic rainfall distribution curve, the historical rainfall distribution data, and the average curve in seventy-two hours duration in three stations. Deviations in the precipitation occur from ten to twenty hours. The historical rainfall distribution has lower intensity in the early storm duration. From forty to sixty hours, the intensity of the storm is higher for the historical rainfall distribution. The distribution of historical rainfall data is conservative at the beginning of the storm, but the intensities raised around twenty to thirty hours and surpass synthetic rainfall distribution curves. On the right of Figure 35, rain gauge Boca Raton presents a huge deviation of the precipitation ratio from twenty to sixty hours compare to synthetic distribution. Therefore, the historical rainfall distribution at Boca Raton station is less conservative compare to station Miami International Airport and station West Palm Beach Airport. 88

109 Rain Gauge: Miami International Airport Rain Gauge: West Palm Beach Airport Rain Gauge: Boca Raton Figure 35 Comparison of Synthetic and Historical Rainfall Distributions in 72 Hours Duration. 89

110 5.5 Applications of Joint Probability Analysis in C-4 and C-6 Basins In the study, the analysis emphases in C-4 and C-6 Basins for their high population and high urbanized characters. The design conditions and control levels of structures for basins C-4 and C-6 are listed in Table 15. The selected structures and their paired three rain gauge stations and for basins C-4 and C-6 and the period of record for which data records were collected and analyzed. Table 16 summarized the available data length for structure and their paired rainfall stations. Table 15 Design Conditions and Control Levels for Basins C-4 and C-6. Location C-6 Miami Canal at S-26 C-4 Tamiami Canal at S-25B Peak Q (cfs) Design Q(cfs) Wet Season Control Level (ft NGVD) Average Canal Level (ft NGVD) Optimum Stage Drought Management Control Level (ft NGVD) Proposed Minimum Canal Operation (ft NGVD)

111 Table 16 List of Structures and Three Paired Rain Gauge Stations paired with Available Data Length. Structure Id Rainfall Station Start Date End Date C4.CORAL C6.NW36 G93 S25B MIAMI.AP_R 1-Jan-6 31-Dec-9 MIAMI.FS_R 1-Jan Dec-9 S335_R 1-Jan Dec-9 MIAMI.AP_R 1-Jan-6 31-Dec-9 MIAMI.FS_R 1-Jan Dec-9 S335_R 1-Jan Dec-9 MIAMI.AP_R 1-Jan Dec-9 MIAMI.FS_R 1-Jan Dec-9 S335_R 1-Jan Dec-9 MIAMI.AP_R 1-Jan Dec-9 MIAMI.FS_R 1-Jan Dec-9 S335_R 1-Jan Dec Optimal Box-Cox transformation parameter Estimation The Box-Cox transformation is applied to normalize the data if data does fit the anticipated normal distribution. In this section, the parameter lambda (λ) is estimated using an optimization approach (Genetic algorithms) in the study. The gradient-based optimization solver do not always produce global optimal solutions and have the tendency to provide local optimal solutions. Figure 36 and Figure 38 are Box-Cox Transformation Parameter for Peak Discharge for C4.CORAL and paired rainfall station MIAMI.FS_R. 91

112 Figure 36 Optimal Box-Cox Transformation Parameter for Peak Discharge for C4.CORAL and MIAMI.FS_R. Figure 37 Optimal Box-Cox Transformation Parameter for Peak Discharge for C4.CORAL and MIAMI.FS_R. (Lag Time = 3) Probability Plots and Box-Cox Transformation The probability plots for peak discharge before and after the Box-Cox transformation are shown below. These plots are used for visual check of goodness-of-fit to Normal distribution. 92

113 Figure 38 is the probability plots for peak discharge and accumulated rainfall (lag =3) for streamflow station C4.CORAL and rainfall station MIAMI.FS_R before and after Box-Cox transformations. Figure 38 Probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL- MIAMI.FS_R before and after Box-Cox transformations Joint Cumulative Probability Plots The bi-variate cumulative probability distribution for peak discharge and accumulated rainfall for C-4 structure is shown below. 93

114 Figure 39 Joint cumulative probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R. Figure 4 Joint exceedance probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R. 94

115 Joint Return Period (Years) Exceedence Probability Inches 3.Inches 4.Inches 5.Inches 6.Inches 7.Inches 8.Inches 9.Inches 1.Inches 11.Inches 12.Inches 13.Inches 14.Inches 15.Inches 16.Inches 17.Inches 18.Inches Peak Discharge (ft 3 /sec) Figure 41 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Inches 3.Inches 4.Inches 5.Inches 6.Inches 7.Inches 8.Inches 9.Inches 1.Inches 11.Inches 12.Inches 13.Inches 14.Inches 15.Inches 16.Inches 17.Inches 18.Inches Peak Discharge (ft 3 /sec) Figure 42 Joint return period plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL- MIAMI.FS_R Results for Basins C-4 and C-6 Figure 43to Figure 46 present cumulative probability function (CDF) of accumulated rainfall and peak discharge for 24 and 72 hour rainfall duration for structures C4.Coral and C6.NW36. Based on the methodology, the CDF for a bi-variable distribution can be analyzed by the following table. 95

116 Rainfall Duration (hr) Table 17 Probability of occurrence and exceedance for structures C6.NW36 and C4.Coral. Structure Rainfall Threshold (R T ) Discharge Threshold (Q T ) P ocurrence P exceedance P o (R<R T,Q<Q T ) P e (R>R T,Q>Q T ) 24 C6.NW C6.NW C4.Coral C4.Coral Figure 43 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 24 hour rainfall duration for structure C6.NW36. 96

117 Figure 44 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 72 hour rainfall duration for structure C6.NW36. Figure 45 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 24 hour rainfall duration for structure C4.Coral. 97

118 Figure 46 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for 72 hour rainfall duration for structure C4.Coral. Figure 47 to Figure 49 present the variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C4.CORAL. Figure 5 part a), b) and c) show the rainfall-peak discharge correlations for different lag times from to 1 days for C4.Coral paired with three different rain gauge stations along the basin. Figure 51 to Figure 53 present the variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C6.NW36. Figure 54 part a), b) and c) show the rainfall-peak discharge correlations for different lag times from to 1 days for C6.NW36 paired with three different rain gauge stations along the basin. Figure 55 to Figure 58 present the variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure G93. Figure 59 part a), b) and c) show the rainfall-peak discharge correlations for different lag times from to 1 days for G93 paired with three different rain gauge stations along the basin. 98

119 Figure 6 to Figure 63 present the variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure S25B. Figure 64 part a), b) and c) show the rainfall-peak discharge correlations for different lag times from to 1 days for S25B paired with three different rain gauge stations along the basin. The performance of structures G93 and S25B were analyzed by using the criteria explained in Table 18 and Table 19. Table 18 shows the comparison between design conditions and historical data for structure G93 (Miami.FS.R) using Qd and Rd as the design parameters. The first part of the table show the rainfall obtained from the graph by given for the design discharge. Rainfall is greater than design rainfall therefore the design conditions are not exceeded, the design is conservative, adequate and it does need to be reviewed. For the second part of Table 18 discharge is less than design discharge and the conclusions are the same. Table 19 shows the comparison between design conditions and historical data for structure S25B (Miami.FS.R) using Q d and Rd as the design parameters. The first part of the table show the rainfall obtained from the graph for the design discharge. Rainfall is greater than design rainfall therefore the design conditions are not exceeded, the design is conservative, adequate and it does need to be reviewed. For the second part of Table 19 discharge is less than design discharge and the conclusions are the same. 99

120 Figure 47 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C4.CORAL and rainfall station Miami.AP.R. Figure 48 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days structure C4.CORAL and rainfall station Miami.FS.R. 1

121 Figure Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C4.CORAL and rainfall station S335_R. (a) (b) (c) Figure 5 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for C4.Coral. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R. 11

122 Figure 51 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C6.NW36 and rainfall station Miami.AP.R. Figure 52 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C6.NW36 and rainfall station Miami.FS.R. 12

123 Figure 53 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure C6.NW36 and rainfall station S335_R. (a) (b) (c) Figure 54 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for C6.NW36. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R. 13

124 Figure 55 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure G93 and rainfall station Miami.AP.R. Figure 56 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure G93 and rainfall station Miami.FS.R. 14

125 Figure 57 Variation of accumulated rainfall and peak discharge for 24 and 72 hour for structure G93 and rainfall station Miami.FS.R. Table 18 Comparison between design conditions and historical data for structure G93 (Miami.FS.R). Linear Least Square Equation: Q=m*R+b Duration Structure (hr) Slope Intercept R Q (m) (c) d (cfs) (in) 24 G93-Miami.FS_R G93-Miami.FS_R Duration (hr) Structure 24 G93-Miami.FS_R Linear Least Square Equation: Q=m*R+b Slope Intercept Q R (m) (c) d (in) (cfs) 8.5 (1 34 years) 72 G93-Miami.FS_R (25years) 37 15

126 Figure 58 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure G93 and rainfall station S335_R. (a) (b) (c) Figure 59 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for G93. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R. 16

127 Figure 6 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure S_25B and rainfall station Miami.AP.R. Figure 61 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure S_25B and rainfall station Miami.FS.R. 17

128 Figure 62 Variation of accumulated rainfall and peak discharge for 24 and 72 hour for structure S_25B and rainfall station Miami.FS.R. Table 19 Comparison between design conditions and historical data for structure S_25B (Miami.FS.R). Linear Least Square Equation: Q=m*R+b Duration Structure (hr) Slope Intercept R Q (m) (c) d (cfs) (in) 24 S_25B-Miami.FS_R S_25B-Miami.FS_R Linear Least Square Equation: Q=m*R+b Duration Structure (hr) Slope Intercept Q R (m) (c) d (in) (cfs) 24 S_25B-Miami.FS_R (1 years) S_25B-Miami.FS_R (25years)

129 Figure 63 Variation of accumulated rainfall and peak discharge for different lag times from to 1 days for structure S_25B and rainfall station S335_R. (a) (b) (c) Figure 64 Rainfall-Peak Discharge Correlations for different lag times from to 1 days for S25_B. a) Rainfall Station: Miami.AP-R b) Rainfall Station: Miami.FS-R c) Rainfall Station: S335-R. 19

130 Table 2 presents the correlations for different lag times for the evaluated structures in C-4 and C-6 basins. It can be seen that structure C4.Coral has a better correlation with rain gauge station S335-R. Analyzing the drainage conditions for C-4 basin it can be observed that any rainfall measured by S335-R will drain directly through Tamiami Canal, having a fast response in the discharge for the structure C4.Coral. Peak flows at Structure C6.NW36 have a slightly better correlation with S335-R (rain gage station) than with the rain gauge station Miami.FS-R. Structures G93 and S25B present a better correlation with rain gauge station Miami.FS-R For all structures in basin C-4 and C-6, the correlations between accumulated rainfall and discharge are large from a lag time of 1 day, this rainfall-runoff respond is a characteristic of a high urbanized area. Table 2 Best Correlation and the corresponding lag time for the evaluated structures in C-4 and C-6 basins. Streamflow Station C4.CORAL C6.NW36 G93 S25B Best Correlation Lag Times (Days) Rain Gauge Station S335_R.97 4 MIAMI.FS_R.71 5 MIAMI.AP_R S335_R MIAMI.FS_R.79 3 MIAMI.AP_R.75 3 S335_R MIAMI.FS_R.88 5 MIAMI.AP_R S335_R MIAMI.FS_R.91 4 MIAMI.AP_R.84 2 Average Correlations

131 5.5.5 Peaks over Threshold (POT) Analysis The peaks over threshold analysis or the partial duration series analysis is carried out using the methodology developed in this study. The inter-event time between any two events is adopted as 6 days. The optimal Box-Cox parameter values of peak discharge and accumulated rainfall are obtained using Genetic Algorithms. The optimal values are shown in Figure 65 and Figure 66. The joint cumulative density function is shown Figure 67. The bi-variate joint excedence probability distribution for peak discharge and accumulated rainfall is shown in Figure 68. The joint probability distributions conditioned on specific values of rainfall are shown in Figure 69 and the return periods are shown in Figure 7. Figure 65 Optimal Box-Cox transformation parameter for peak discharge for C4.CORAL- MIAMI.FS_R. 111

132 Figure 66 Optimal Box-Cox transformation parameter for accumulated rainfall (lag=3) for C4.CORAL- MIAMI.FS_R. Figure 67 Joint cumulative probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R. 112

133 Exceedence Probability Figure 68 Joint exceedence probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R Inches 3.Inches 4.Inches 5.Inches 6.Inches 7.Inches 8.Inches 9.Inches 1.Inches 11.Inches 12.Inches 13.Inches 14.Inches 15.Inches 16.Inches 17.Inches 18.Inches Peak Discharge (ft 3 /sec) Figure 69 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL-MIAMI.FS_R. 113

134 Joint Return Period (Years) Inches 3.Inches 4.Inches 5.Inches 6.Inches 7.Inches 8.Inches 9.Inches 1.Inches 11.Inches 12.Inches 13.Inches 14.Inches 15.Inches 16.Inches 17.Inches 18.Inches Peak Discharge (ft 3 /sec) Figure 7 Joint return plots for peak discharge and accumulated rainfall (lag =3) for C4.CORAL- MIAMI.FS_R. 5.6 Joint Probability Analysis with Cool Phase and Warm Phase The analysis applied in the section is using the defined year duration where cool phase starts from year 197 to year 1995 and warm phase starts from 195 to 21. The basic concept is to approach the joint probability analysis from a global perspective to analyze the difference between cool phase and warm phase. In this section, three streamflow stations and their paired rainfall stations are added where the data of rainfall station are collected from United State Historical Climatology Network (USHCN). All three stations are selected from natural basins unlike C-4 and C-6 basins used in pervious which are under certain operation. The stations C4.CORAL and C6.NW36 (Table 21) are divided into cool phase and warm phase for comparison purpose (Table 22). The new three stations (Table 23) with long term data sets and the duration of cool phase and warm phase are listed in Table 24. The dependence analysis for undisturbed watershed is listed on Appendix K. 114

135 Table 21 Data length for selected paired stations in C-4 and C-6 basins. Streamflow Station Rainfall Station Start End Length (Years) C4.CORAL MIAMI.AP_R C6.NW36 MIAMI.AP_R Table 22 Duration of cool phase and warm phase for stations in C-4 and C-6 basins. Phase Start End Length (Years) Cool Warm Table 23 Data length for new selected paired stations. Streamflow Station Rainfall Station Start End Length (Years) 233 Saint Leo Inverness Tallahassse Table 24 Duration of cool phase and warm phase for new stations. Phase Start End Length (Years) Cool Warm From Figure 71 to Figure 76, the variation of accumulated rainfall and peak discharge for different lag times from day zero to day tenth in cool phase and warm phase are listed. The natural watershed has the capability to increase the rainfall and streamflow response time. The lower correlations form day zero to day two in cool phase can be observed, and in the warm 115

136 Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) phase, the correlations are raised in day one. In warm phase of AMO, the higher rainfall is received in Florida which contributes the faster response time. 6 Lag Time (Days) = Streamflow Station: 2313 Paired Rainfall Station: Inverness (Y-axis:Annual Peak Streamflow / X-axis: Accumulatived Precipitation) 6 Lag Time (Days) = 1 6 Lag Time (Days) = 2 6 Lag Time (Days) = 3 4 r = r = r = r = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = r = r = r = r = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) 4 r = r = r = Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Figure 71 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2313 in cool phase. Streamflow Station: 2313 Paired Rainfall Station: Inverness (Y-axis:Annual Peak Streamflow / X-axis: Accumulatived Precipitation) 1 Lag Time (Days) = 1 Lag Time (Days) = 1 1 Lag Time (Days) = 2 1 Lag Time (Days) = 3 r = r = r = r = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = 7 1 r = r = r = r = Precipitation (Inches) Lag Time (Days) = 8 1 r = Precipitation (Inches) Lag Time (Days) = 9 1 r = Precipitation (Inches) Lag Time (Days) = 1 1 r = Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Figure 72 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2313 in warm phase. 116

137 Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow Station: 233 Paired Rainfall Station: Saint Leo (Y-axis:Annual Peak Streamflow / X-axis: Accumulatived Precipitation) 6 Lag Time (Days) = 6 Lag Time (Days) = 1 6 Lag Time (Days) = 2 6 Lag Time (Days) = 3 4 r = r = r = r = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = r = r = r = r = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) 4 r = r = r = Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Figure 73 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 233 in cool phase. Streamflow Station: 233 Paired Rainfall Station: Saint Leo (Y-axis:Annual Peak Streamflow / X-axis: Accumulatived Precipitation) 15 Lag Time (Days) = 15 Lag Time (Days) = 1 15 Lag Time (Days) = 2 15 Lag Time (Days) = 3 1 r = r = r = r = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = r = r = r = r = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) Lag Time (Days) = Precipitation (Inches) 1 r = r = r = Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Figure 74 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 233 in warm phase. 117

138 Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) Streamflow (cfs) 4 x 14 Lag Time (Days) = Streamflow Station: 2329 Paired Rainfall Station: Tallahassse (Y-axis:Annual Peak Streamflow / X-axis: Accumulatived Precipitation) 4 x 14 Lag Time (Days) = 1 4 x 14 Lag Time (Days) = 2 4 x 14 Lag Time (Days) = 3 3 r = r = r = r = Precipitation (Inches) x 1 4 Lag Time (Days) = r = Precipitation (Inches) 4 x 14 Lag Time (Days) = r = Precipitation (Inches) 4 x 14 Lag Time (Days) = r = Precipitation (Inches) 4 x 14 Lag Time (Days) = r = Precipitation (Inches) 4 x 14 Lag Time (Days) = Precipitation (Inches) 4 x 14 Lag Time (Days) = Precipitation (Inches) 4 x 14 Lag Time (Days) = Precipitation (Inches) 3 r = r = r = Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Figure 75 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2329 in cool phase. 4 x 14 Lag Time (Days) = 3 Streamflow Station: 2329 Paired Rainfall Station: Tallahassse (Y-axis:Annual Peak Streamflow / X-axis: Accumulatived Precipitation) 4 x 14 Lag Time (Days) = 1 r = r = x 1 4 Lag Time (Days) = r = x 14 Lag Time (Days) = 3 3 r = Precipitation (Inches) 4 x 14 Lag Time (Days) = 4 3 r = Precipitation (Inches) 4 x 14 Lag Time (Days) = 5 3 r = Precipitation (Inches) 4 x 14 Lag Time (Days) = 6 3 r = Precipitation (Inches) 4 x 14 Lag Time (Days) = 7 3 r = Precipitation (Inches) 4 x 14 Lag Time (Days) = 8 3 r = Precipitation (Inches) 4 x 14 Lag Time (Days) = 9 3 r = Precipitation (Inches) 4 x 14 Lag Time (Days) = 1 3 r = Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Precipitation (Inches) Figure 76 Variation of accumulated rainfall and peak discharge for different lag times from day to day 1 for station 2329 in warm phase. 118

139 5.6.1 Normality Tests and Box-Cox Transformation The Normality Tests including probability plot, Kolmogorov Smirnov test, Pearson's Chi- Squared test, Jarque Bera test, and Lilliefors test are performed both raw datasets and Box-Cox transformed data sets. Figure 77 shows the probability plots for peak discharge and accumulated rainfall (lag = ) for station 2329 before and after Box-Cox transformations. Data has long trail (S curve) before the transformation and follow the theoretical linear distribution after Box- Cox transformation. Figure 77 Probability plots for peak discharge and accumulated rainfall (lag =) for station 2329 before and after Box-Cox transformations. The natural watershed increases the response time between rainfall and stremflow causes Box-Cox transformation technique difficult to apply in the lower lag times (day zero to day second) since zero values cannot be accepted. Table 25 is the summary of Normality tests after Box-Cox transformation at lag time day three. After applied Box-Cox transformation, all data sets passed Pearson's Chi-Squared Test. 119

140 Table 25 Summary of Normality tests after Box-Cox transformation at lag time day three. Station Phase Data Inverness 233- Saint Leo Tallahassse Cool Phase Warm Phase Cool Phase Warm Phase Cool Phase Warm Phase Kolmogorov Smirnov Test Normality Tests Pearson's Chi-Squared Test Jarque Bera Test Lilliefors Test Rain Not Passed Passed Passed Passed Flow Not Passed Passed Not Passed Not Passed Rain Not Passed Passed Passed Passed Flow Not Passed Passed Not Passed Not Passed Rain Not Passed Passed Passed Passed Flow Not Passed Passed Passed Passed Rain Not Passed Passed Passed Passed Flow Not Passed Passed Not Passed Not Passed Rain Not Passed Passed Passed Passed Flow Not Passed Passed Not Passed Not Passed Rain Not Passed Passed Passed Passed Flow Not Passed Passed Not Passed Not Passed Cool and Warm Phases Analysis with different Lag Times The new three paired stations are located in natural basins which cause short lag times from lay zero to day three hard to applied the analysis since the increasing rainfall amounts do not necessary contribute the increasing streamflow. The analysis applied in the section focus on the comparison of cool phase and warm phase under different basin characters with lag time day three and lag time day five. All results are listed on Appendix L. The following analysis compares streamflow station ID 2329 and paired rainfall station Tallahassse to streamflow station C4. CORAL and paired rainfall station MIAMI.AP_R at lag time day five. Figure 78and Figure 79 are Cumulative probability function (CDF) of the bivariable (Accumulated Rainfall and Peak Discharge) distribution. The cumulative probability of 12

141 peak discharge at 1 cfs receives a 5 inches rainfall is.58 for station C4. CORAL and.68 for station ID The natural basin has lower cumulative probability states that the basin has longer time of concentration and a batter water conservation capability. Figure 8and Figure 81 present the conditional probability of joint exceedance where streamflow station C4. CORAL has steep distributes with high joint exceedance since the annual peaks at station C4. CORAL is low. Figure 82 and Figure 83 present conditional probability of joint return period for station C4. CORAL and station ID Figure 78 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for structure C4. CORAL at lag time =5. 121

142 Exceedence Probability Figure 79 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution for structure ID 2329 at lag time =5. Streamflow Station:C4.CORAL Paired Rainfall Station:MIAMI.AP-R in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in Peak Discharge (cfs) Figure 8 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for station C4.CORAL. 122

143 Joint Return Period (Years) Exceedence Probability Streamflow Station ID:2329 Paired Rainfall Station:Tallahassse in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in Peak Discharge (cfs) x 1 4 Figure 81 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for structure ID Streamflow Station:C4.CORAL Paired Rainfall Station:MIAMI.AP-R in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in Peak Discharge (cfs) Figure 82 Joint return plots for peak discharge and accumulated rainfall (lag =5) for station C4.CORAL. 123

144 Joint Return Period (Years) Streamflow Station ID:2329 Paired Rainfall Station:Tallahassse in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in Peak Discharge (cfs) x 1 4 Figure 83 Joint return plots for peak discharge and accumulated rainfall (lag =5) for structure ID The analysis of cool phase and warm phase comparison at time lag day five is performed with streamflow station ID 2313 and paired rainfall station Inverness. Figure 84 and Figure 85 are results for Cumulative Probability Function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution (lag =5) for structure ID 2313 in cool and warm phases. T a given rainfall 4 inches, the cumulative probability of receiving 4 cfs streamflow is.75 for cool phase and.64 for warm phase. Figure 86 and Figure 87 are results for conditional probability of joint exceedance for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in cool and warm phases. The joint exceedance at peak discharge 2 cfs with 4 inches rainfall is.39 for cool phase and.7 for warm phase. Figure 88 and Figure 89 are joint return plots for peak discharge and accumulated rainfall at lag time day fifth. The joint return at peak discharge 4 cfs with 5 inches rainfall is 39 years for cool phase and is 5 years for warm phase. 124

145 Figure 84 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution (lag =5) for structure ID 2313 in cool phase. Figure 85 Cumulative probability function (CDF) of the bi-variable (Accumulated Rainfall and Peak Discharge) distribution (lag =5) for structure ID 2313 in warm phase. 125

146 Exceedence Probability Exceedence Probability Streamflow Station ID:2313 Paired Rainfall Station:Inverness Cool Phase in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in Peak Discharge (cfs) Figure 86 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in cool phase. Streamflow Station ID:2313 Paired Rainfall Station:Inverness Warm Phase in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in Peak Discharge (cfs) Figure 87 Joint exceedance conditional probability plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in warm phase. 126

147 Joint Return Period (Years) Joint Return Period (Years) in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in. Streamflow Station ID:2313 Paired Rainfall Station:Inverness Cool Phase Peak Discharge (cfs) Figure 88 Joint return plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in cool phase in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 1 in. 11 in. Streamflow Station ID:2313 Paired Rainfall Station:Inverness Warm Phase Peak Discharge (cfs) Figure 89 Joint return plots for peak discharge and accumulated rainfall (lag =5) for structure ID 2313 in warm phase. 127

148 5.7 Peaks over Threshold Analysis Peak Discharge for Different Sets of Discharge Thresholds The analysis performed in this section uses four stations including streamflow stations C4.CORAL, C6.NW36, G93 and S25B in C-4 and C-6 basins. Figure 9 shows the number of peak discharges selected for different thresholds from 7% to 95% of the maximum peak discharge. The decreasing trend applied for all three stations except station G93which has the same number of peaks after 8% threshold states that the station G93 experiences certain degree of operation. Figure 9 Number of peaks for different % of max peak discharge thresholds. Figure 91to Figure 93 present the location of the selected peak over thresholds (POT) for different thresholds from 7 to 95% of the maximum peak discharge for C4.CORAL. While the 128

149 number of peaks increased with lower thresholds, the result could be possibly interfered by the lower threshold. The total number of peaks per year are not necessary stays the same. 7% of max peak discharge threshold 75% of max peak discharge threshold Figure 91 Location of POT events for 7% and 75% of max peak discharge threshold for C4.Coral. 8% of max peak discharge threshold 85% of max peak discharge threshold Figure 92 Location of POT events for 8% and 85% of max peak discharge threshold for C4.Coral. 129

150 9% of max peak discharge threshold 95% of max peak discharge threshold Figure 93 Location of POT events for 9% and 95% of max peak discharge threshold for C4.Coral Rainfall and Annual Peak Discharge Correlations with Different Discharge Thresholds Figure 94 and Figure 96 show the correlations between annual peak discharges and daily rainfall from (same day of annual peak discharge) to 3 lag time days. C-4 basin can be characterized by two types of response, a quick response of the basin (see box 1 in figure below) for a given rainfall event with response times in the order of 1 to 3 days and a delayed response related with the storage capacity of the basin (see boxes 2 and 3 in figure below) with response times of 9 days and 28 days to 29 days (see Figure 94 to Figure 97). Original correlations are reported in Table 26and Table 27, high peak discharge thresholds greater than cfs show good correlations for, 8, 25, 27 and 28 days. Low peak discharge thresholds show good correlations for, 1 and 8 days. C-6 basin can be characterized as a basin with a faster response than C-4. Similarly to C-4 basin, C-6 basin present two types of response, a quick response of the basin (see box 1 in figure below) for a given rainfall event with response times in the order of 1 to 3 days and a delayed response related with the storage capacity of the basin (see box 2 in figure below). Original correlations are reported in Table 26 and Table 27, high peak discharge thresholds greater than 13

151 712.5 cfs show good correlations for, 27 and 28 days. Low peak discharge thresholds show good correlations for and 1 days. Table 26 Correlations for a set of discharge thresholds in C-4 basin. Days Thresholds (cfs) NaN NaN NaN NaN NaN NaN NaN NaN NaN 131

152 Table 27 Correlations for a set of discharge thresholds in C6 basin. Days Thresholds (cfs) NaN NaN 132

153 Figure 94 Correlations between rainfall and annual peak discharges for different discharge thresholds. C4.Coral - Miami.F.S. 133

154 Figure 95 Variation of individual rainfall for different lag times from to 1 days and peak discharge for C4.Coral. 134

155 Figure 96 Correlations between rainfall and annual peak discharges for different discharge thresholds. C6.NW36 - Miami.F.S. 135

156 Figure 97 Variation of individual rainfall for different lag times from to 1 days and peak discharge for C6.NW

157 6 CONCLUSION 6.1 Contributions of the Study A best practice guidance document, including advice on the use of analysis methods is developed in this study, use of dependence data analysis and joint probability analysis for assessment of existing flood control structures needs to be developed for practitioners Dependence Analysis Seasonality in precipitation and streamflow in South Florida is a key factor in the analysis of peak events. Most annual peak events are generally concentrated in the wet season (summer months) with intense rainfall events resulting in runoff excepting in one year of the period of record used for this study. Most of the hurricanes and peak storm events are indirectly linked with peak discharges as it can be seen in the data analysis performed in this project. Classification of events into storms and hurricanes with an analysis of the rainfall and streamflow data was performed to establish a clear linkage between them. Statistical analysis was conducted to link spatial and temporal variability of precipitation events to peak flooding events in South Florida. Dependence analysis using annual peak discharges and accumulated rainfall values provides a reasonable estimate of the response time of the basins. Negative and positive dependencies were observed in many basins at different dependence lag times. Negative or no dependence mainly suggests a rainfall-runoff response that is not typical of a natural watershed. Such response is typical of many managed basins in south Florida. From the correlation analysis 137

158 between maximum rainfall for different lag times from zero to ten days and peak discharges, can be concluded that C-4 and the C-6 basins have a basin response time of approximately two to three days. Basins C-4 and the C-6 present high correlations between maximum discharge and average stage values for different lag times from zero to ten days and peak discharges. Correlations and moving average correlations for the basin C-4 and the C-6 show a similar behavior, with two types of responses. A quick response of the basin for a given rainfall event with response times in the order of one to four days and a delayed response related with the storage capacity of the basins with response times in the order of 25 to 3 days. The C-4 basin has also an intermediate delayed response in the order of 8 to 12 days that was not observed for the C-6 basin. The C-6 basin can be characterized as a basin with a faster response time than C-4. This behavior can be observed for the annual maximum discharge analysis as well as for the peaks over thresholds analysis. Extreme rainfall events are not linked to the annual peak discharges in some basins. This link is not strong even at a lag time of 1 days from the day of maximum peak value of discharge. The inclusion and non-inclusion of extreme precipitation events related to hurricane events affected the conclusions of dependence analysis in South Florida basins. Selection of an appropriate rain gage based on its location within a watershed (or a basin) is critical for the success of a dependence analysis in South Florida. Results for time differences in occurrence and correlations between maximum rainfall and peak discharges demonstrate that maximum peak discharges in the basins are not necessarily related with maximum precipitation events (i.e., annual extremes) in a given year. More than 5% of the annual extremes do not occur within 1 days lag time of peak discharge. Most of the times of occurrence values lie within zero to 1 days. The correlation coefficient values between 138

159 extreme rainfall and peak discharges are generally around.5.6. These correlations are typical for all structures in C-4 and the C-6 basins. Similar conclusions can be made about the correlations between maximum stages and peak discharges, and also between maximum rainfall and stages (See Figures 16 to Figure 21). According to the characteristics of the basins in South Florida (SFWMD, 29, 21 and PBS&J, 24), it can be concluded that this is an expected behavior for basins where groundwater storage accompanied by soil moisture conditions play an important role in the runoff generation mechanisms. The response time in the C-4 basin may be directly relevant to the existence of the Pensuco wetland area in that basin. Variation of peak discharges and stages based on historical data show that for the Tamiami canal the control stage has been exceeded several times in the last 44 years. The control stage has been exceeded for high and low discharges demonstrating the necessity to evaluate the storage availability in the basin preceding extreme events. Dependence analysis approach applied to few basins was helpful in assessment of existing design storms and peak discharges Rainfall Distribution Curves Based on Historical Data An extensive analysis have been conducted in this project, to evaluate the adequacy of 24 and 72 hours synthetic rainfall distribution curves and their use in the design of SFWMD structures. The analysis was enhanced by including rainfall data from Miami International Airport, West Palm Beach International Airport and Boca Raton Id rainfall gauge stations in different SFWMD basins. Historical average rainfall distribution associated with annual discharge extremes agree well with SFWMD 72 hour synthetic distribution. However, the 24 hour duration storm distributions do not agree well with synthetic storm distributions. This conclusion is based on evaluation of long-term precipitation data at three locations in SWFWD region. 139

160 6.1.3 Joint Probability Analysis: Flood probabilities are rarely a function of just one source hydrologic variable and therefore understanding the risk posed by the combined effect of two or more extreme hydrological process variables is extremely important. These variables need to be critically evaluated under a joint probability framework so that adequacy of design conditions can be assessed. In order to complete a joint probability analysis, a dependence analysis concerning the main process variables needs to be completed. In the current study, dependence analysis was completed first for 25 basins and then the joint probability effort was focused in C-4 and the C-6 basins using annual peak discharge and lagged accumulated rainfall amounts. The adequacy of design conditions for some of these structures was evaluated using simple dependence approaches. Collected historical data was compared with design conditions for some major structures in basins C-4 and the C-6 to evaluate the adequacy of design conditions and the marginal safety. The joint probability framework provides a valuable tool for assessment of the existing design extremes used for the structures. Evaluation of joint probability of occurrence of extremes requires an appropriate selection of the joint probability distribution function. Transformations of the sampled data are required to fit the bivariate probability distribution functions. In the current study, Box-Cox transformation with optimal estimation of parameters was successful. The joint exceedance probability curves, conditional exceedance and joint return period curves are extremely useful in analyzing the adequacy of designs. The joint probability approach using bivariate Gaussian distributions are extended to include a threshold based criterion related to lead times. The incorporation of lead time in the analysis helps in developing flood control warnings/watches and protocols for operational changes based on joint exceedence conditional probabilities. The approach was also used to 14

161 evaluate the two critical responsible factors (high groundwater table and extreme precipitation) for flooding in a joint probability framework. Cumulative bivariate normal distribution functions can be used to assess the joint exceedence probabilities associated with extreme events. 6.2 Limitation of the Study The location and the distance between paired streamflow station and rainfall station play an important role in the research since the longer distance, the higher the physical factors may involved and effect the results of the analysis. The available data with proper location are not available in South Florida especially the paired data set require both parameters have the same available length. Also, many streamflow structures in South Florida have certain degree of operation which may affect the result. Joint probability analysis requires pair data sets which make the streamflow and rainfall study suitable. However, both parameters have to be highly correlated but still keep independent. During the research, the groundwater was considered to be one of the paired data but the relationship could not be found. 6.3 Recommendations for Future Research Meteorologically extreme events may not always be linked to the peak flooding volumes and therefore a need for extensive evaluation of other processes affecting the peak flooding is required. Peaks over threshold (POT) values should be assessed in a dependence analysis framework. However the requirement for independent events needs to be established. Adequacy of synthetic rainfall distributions used for design need to be exhaustively evaluated by comparing the temporal distributions of historical storm amounts leading to peak discharges to those from synthetic design storms used in design of structures. The threshold values of design parameters 141

162 used for design of flood control structures can be re-evaluated based on historical data for all the basins in the District. Dependence analysis based on historical data concerning two or more parameters affecting the magnitude of flooding extent should be carried out to assess the design criteria used for flooding protection structures. Dependence mapping can be done for a region to analyze the response time of watersheds in a specific region. Runoff response time should be documented for natural and managed basins based on extensive historical data analysis. Multivariate probabilistic analysis can be conducted to analyze the simultaneous occurrence of events. The joint probability approach (JPA) generally recommended by FEMA for surge analysis can be adopted to assess the performance of flood protection structures requires extensive historical data for the analysis. The adequacies of selected statistical distributions to different variables need to be critically evaluated. The joint probability approach using bivariate Normal distributions can be extended to include a threshold based criterion related to lead times. The incorporation of lead time in the analysis will help in developing flood control warnings and (or) watches and protocols for operational changes based on joint exceedence conditional probabilities. The conditional exceedance probability in warm phase is higher than cool phase in the current study; therefore, climate changes have significant inferences on joint probability analysis and a future study should be considered. 142

163 Longitude.) 7 APPENDIX A (A list of thirty-one streamflow stations and their paired rainfall stations with Latitude and NO. STATION_ID Latitude Longitude Q Q Q Q Q Q Q Q Q Q Q Q Q Q14 C-4A Q15 C6.NW Continued on next page. 143

164 NO. STATION_ID Latitude Longitude Q16 G136_C Q17 S14_TOT Q18 S18C_S Q19 S29_S Q2 S34_C Q21 S38C_C Q22 S39_S Q23 S46_S Q24 S49_S Q25 S5A+S5AW Q26 S Q27 S65_S Q28 S77_S Q29 S8_S Q3 S97_S Q31 S99_S NO. SITE_NAME Latitude Longitude R1 PALMDALE_R R2 KISS.FS2_R R3 WRWX R4 KRBNR R5 3A-SW_R R6 S336_R R7 S335_R R8 DANHP_R R9 951EXT_R R1 COCO1_R R11 S79_R R12 FORTMYERWS R13 JDWX R14 SOUTH BA_R R15 MIAMI.AP_R Continued on next page. 144

165 NO. SITE_NAME Latitude Longitude R16 G136_R R17 S14_R R18 S177_R R19 S29Z_R R2 S124_R R21 S125_R R22 WCA1ME R23 SIRG R24 C24SE R25 WPB AIRP_R R26 S7_R R27 S65_R R28 S77_R R29 S8_R R3 BLUEGOOS_R R31 SCOTTO

166 8 APPENDIX B (The list of twelve-year meteorological events is shown below.) Meteorological Event Year Type Name Date Description Hurricane Earl September 3, 1998 Moderate to heavy rainfall peaking at inches (416 mm) where it struck land (Panama City) Outer rainbands produce moderate amounts of rainfall Tropical Hermine September 2, 1998 throughout the state, peaking at Storm inches (359 mm) in Fort 1998 Lauderdale Florida Keys, the hurricane Hurricane Georges September 25, 1998 produced 8.41 inches (214 Tropical Storm Mitch November 5, 1998 mm) of rain in Tavernier It makes landfall near Naples and drops up to 11.2 inches (284 mm) of rainfall in Boca Raton Meteorological Event Year Type Name Date Description Hurricane Dennis August 29, 1999 High surf paralleling the east coast of Florida Hurricane Floyd September 15, 1999 It parallels the eastern coastline about 115 miles (185 km) offshore The storm drops over 1 inches 1999 Tropical Harvey September 21, 1999 (25 mm) of rain, flooding Storm several homes. It produces heavy rainfall Hurricane Irene October 15, 1999 peaking (flooding) at inches (443 mm) in Boynton Beach. Continued on next page. 146

167 Meteorological Event Year Type Name Date Description 2 Hurricane Debby August 23, 2 Tropical Storm Tropical Storm Tropical Storm Tropical Storm Nine September 9, 2 Gordon September 18, 2 Helene September 22, 2 N/A October 3, 2 Its remnants produce heavy rainfall across southern Florida It produces light rainfall in the western Florida Panhandle It makes landfall on Cedar Key, dropping up to 9.48 inches (24 mm) of rainfall (Pensacola). It hits near Pensacola, damaging hundreds of homes from floodwaters Precursor disturbance to Tropical Storm Leslie. It produces 1 2 inches ( mm) of rainfall across southeastern Florida, flooding about 93, houses. Meteorological Event Year Type Name Date Description 21 Subtropical Depression Tropical Storm Tropical Storm Allison June 12, 21 Barry August 6, 21 Gabrielle September 14, 21 Hurricane Michelle November 5, 21 Continued on next page. It moves through Alabama and Georgia, with its outer rainbands producing up to 1.1 inches (357 mm) of rain at the Tallahassee Regional Airport It makes landfall at Santa Rosa Beach, producing heavy rainfall across much of Florida which peaks at 11.7 inches (297 mm) in Stuart. It hits Florida, dropping moderate to heavy rainfall including a peak total of 15.1 inches (384 mm) in Parrish. It passes to the south of the state, dropping up to 4.99 inches (127 mm) of rainfall 147

168 Meteorological Event Year Type Name Date Description 22 Tropical Storm Tropical Storm Tropical Storm Tropical Storm Tropical Storm N/A July 13, 22 Precursor disturbance to Tropical Storm Arthur (Florida Panhandle). Heavy amounts of precipitation peaking at 4.79 inches (122 mm) in two locations Bertha August 4, 22 High surf Edouard September 4, 22 Hanna September 14, 22 Isidore September 26, 22 Hurricane Lili October 3, 22 Tropical Storm Continued on next page. Kyle October 11, 22 The storm crossed the peninsula from East to West dropping up to 7.64 inches (194 mm) of rain in DeSoto City, resulting in some flooding It produces moderate precipitation across the state (Alabama and Mississippi) It hits southern Louisiana, though its large circulation drops rainfall across the state peaking at 9.1 inches (231 mm) (Pensacola) It makes landfall on southern Louisiana, and drops 1.4 inches (26 mm) of rainfall in Pensacola It made landfalls near Charleston, South Carolina, and Long Beach, North Carolina. 148

169 Meteorological Event Year Type Name Date Description 23 Tropical Storm Tropical Storm Ana April 2, 23 Swells Bill June 3, It hits southern Louisiana, producing over 7 inches (178 mm) of rainfall in portions of the western Florida Panhandle Hurricane Claudette July 8, 23 Gulf of Mexico Tropical Depression Seven July 25, 23 It drops light precipitation Hurricane Erika August 14, 23 Tropical Storm Tropical Depression Grace August 3, 23 Henri September 6, 23 Precursor disturbance produces heavy amounts of precipiation across the state Moisture drops about 1 inch (25 mm) of rain in the Florida Keys and over 3 inches (75 mm) in northern Florida It hits Clearwater, producing 9.9 inches (231 mm) in Hialeah which leads to minor flooding Hurricane Isabel September 13, 23 Rip currents Meteorological Event Year Type Name Date Description 24 Tropical Storm N/A August 12, 24 Hurricane Charley August 13, 24 Hurricane Frances September 5, 24 Hurricane Ivan September 16, 24 Hurricane Jeanne September 26, 24 Tropical Storm Continued on next page. Matthew October 1, 24 It moves ashore on Saint Vincent Island, producing light rainfall Strongest hurricane to hit the United States since Hurricane Andrew in 1992 (Windy) Heavy rainfall peaking at inches (42 mm) at High Springs It strikes Orange Beach, Alabama as a major hurricane (Windy) It produced moderate winds and rainfall reaching inches (34 mm) at Kenansville It hits southern Louisiana, with its outer rainbands producing light rainfall across the western portion of Florida

170 Meteorological Event Year Type Name Date Description It strikes Pensacola, moderate Tropical precipitation throughout Florida Arlene June 11, 25 Storm and breaches on barrier islands on the Florida Panhandle Hurricane Cindy July 6, 25 It hits south-central Louisiana Hurricane Dennis July 1, 25 It makes landfall just west of Navarre Beach as a major hurricane, producing moderate rainfall It makes landfall near the Broward/Miami-Dade County 25 Hurricane Katrina August 25, 25 border, heavy rainfall peaking at inches (415 mm) in Perrine. Hurricane Ophelia September 12, 25 Coastline Hurricane Rita September 2, 25 Storm surge Tropical Storm Tammy October 5, 25 Hurricane Wilma October 24, 25 Continued on next page. It hits near Atlantic Beach, resulting in moderate rainfall and light damage It makes landfall near Cape Romano, producing hurricane force winds and moderate precipitation across much of southern portion of the state. 15

171 Meteorological Event Year Type Name Date Description Tropical Storm Tropical Storm Subtropical Storm Tropical Storm N/A June 13, 26 Ernesto August 3, 26 It produced 4.51 inches (115 mm) of rainfall in Sarasota It strikes Plantation Key. Moderate rainfall peaking at 8.72 inches (221 mm) in South Golden Gate (flooding in Palmdale) Andrea May 9, 27 Coastline Barry June 2, 27 Hurricane Humberto September 13, 27 Tropical Storm Tropical Storm Depression Ten September 22, 27 Storm Olga December 13, 27 It drops moderate precipitation across the drought-ridden state that peaks at 6.99 inches (178 mm) in Palm Beach It drop light rainfall on the western Florida Panhandle It made landfall on the Florida Panhandle and shorthly dissapear The remnants drop moderate precipitation in the state Meteorological Event Year Type Name Date Description Tropical Storm Tropical Storm Cristobal July 16, 28 Fay August 18, 28 Hurricane Gustav August 31, 28 Tropical Storm The precursor drops moderate rainfall, causing minor street flooding and little damage. It made a record four landfalls in Florida. Extreme flooding was reported in many counties in central Florida and the Florida panhandle. Hurricane Gustav brushed the Florida Keys, heavy rain up to 4.12 in (15 mm). Ida November 4, 29 It brings rainfall to the panhandle 151

172 9 APPENDIX C (All streamflow and paired rainfall station plots.) 152

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