Hydrology and Fish Population Dynamics in the Okavango Basin

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Title:
Hydrology and Fish Population Dynamics in the Okavango Basin Managing for Uncertainty in a Data Poor Environment
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1 online resource (176 p.)
Language:
english
Creator:
Cathey,Anna M
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Agricultural and Biological Engineering
Committee Chair:
Munoz-Carpena, Rafael
Committee Co-Chair:
Kiker, Gregory
Committee Members:
Allen, Michael S
Cohen, Matthew
Brown, Mark T
Wolski, Piotr

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Subjects / Keywords:
botswana -- delta -- fish -- hydrologic -- hydrology -- model -- modeling -- okavango -- sensitivity -- uncertainty -- watershed
Agricultural and Biological Engineering -- Dissertations, Academic -- UF
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Agricultural and Biological Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
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Abstract:
The Okavango Basin is a large watershed in southern Africa that empties into the Okavango Delta, which is an inland delta that spreads out onto the Kalahari sands of Botswana. Hydrologic and ecological models are being used in the area to create environmental flows as well as explore development and climate change scenarios. The uncertainty in these models is inherent. However, their usefulness can be evaluated through sensitivity and uncertainty analysis (SA/UA). Uncertainty analysis is useful for measuring model reliability and sensitivity analysis apportions the total model uncertainty to individual inputs and processes. This work is threefold involving a SA/UA on two existing hydrologic models in the area (the Pitman rainfall runoff model in the basin and the ORI reservoir model in the Delta) as well as the development and analysis of a fish population model for the Okavango Delta based on the annual flood pulse that the Delta receives.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Anna M Cathey.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Munoz-Carpena, Rafael.
Local:
Co-adviser: Kiker, Gregory.

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lcc - LD1780 2011
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UFE0043260:00001


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1 HYDROLOGY AND FISH POPULATION DYNAMICS IN THE OKAVANGO BASIN: MANAGING FOR UNCERTAINTY IN A DATA POOR ENVIRONMENT By ANNA McDANIEL CATHEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PAR TIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011

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2 2011 Anna McDaniel Cathey

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3 To my parents for their boundless belief in my abilities and to my fianc for his loving patience a nd support

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4 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Rafael Muoz Carpena for his guidance, intellectual support, and ability to push me to my limits. Dr. Muoz Carpena passion for life, work, and his students is contagious and I always came away from my interactions with him with new ideas and a heightened sense of enthusiasm. I would also like to thank my co advisor, Dr. Greg Kiker for his ideas, encouragement, and willingness to think outside the box. Dr. Muoz Carpena and Dr. Kiker worked excelle ntly together able to work with both of them. I would also like to thank Dr. Mark Brown, a committee member and the Principle Investigator of the National Science Fou ndation Integrated Graduate Education and Research Traineeship ( IGERT ) which funded two years of my PhD. The IGERT program is what drew me to pursue a PhD and Dr. Brown made the program well worth it. Additionally, I would like to also my committee membe rs: Dr. and thoughtfulness enabled me to purse a difficult and diverse topic in a remote location where scant data is available. My peers have also been integral in thi s experience. My lab mates have been invaluable for sharing ideas and I would like to thanks Maria Librada Chu Agor, Matteo Convertino, Gareth Lagerwall, David Kaplan, Stuart Muller, and Zuzanna Zajac. The IGERT has also been a constant source of support and encouragement throughout this work and I would like to thank Dina Liebowitz, Deb Wojcik, Lisa Seales, Gregory Parent, Andrea Gaughan, and Mike Murray Hudson. I am grateful for my family without whom I never would have been the person I am today. You provided the encouragement and the love that enabled me to get

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5 through all of this. And to my friends everywhere, thank you for being a friend! Finally I would like to thank John Linhoss, my fianc You have made this a wonderful journey t to see what the rest of the trip brings!

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6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREVIATIONS ................................ ................................ ........................... 11 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUTION ................................ ................................ ................................ ....... 14 Overview ................................ ................................ ................................ ................. 14 Motivation ................................ ................................ ................................ ............... 20 Objectives ................................ ................................ ................................ ............... 21 2 UNCERTAINTY, SENSITIVITY, AND IMPACTS FOR SCENARIO MODELING USING THE PITMAN HYDROLOGIC MODEL IN THE OKAVANGO BASIN ......... 24 Abstra ct ................................ ................................ ................................ ................... 24 Background ................................ ................................ ................................ ............. 25 Methods ................................ ................................ ................................ .................. 29 The Pitman Model ................................ ................................ ............................ 29 Global Sensitivity and Uncertainty Analysis ................................ ...................... 31 Input factor selection ................................ ................................ .................. 33 Morr is Method global sensitivity analysis ................................ ................... 39 Fourier Amplitude Sensitivity Test (FAST) variance based global sensitivity analysis ................................ ................................ .................. 40 Climate change uncertainty ................................ ................................ ........ 41 Results ................................ ................................ ................................ .................... 42 Morris Method ................................ ................................ ................................ .. 42 FAST Global Sensitivity and Uncertainty Analysis ................................ ........... 44 Climate Change Uncertainty ................................ ................................ ............. 46 Summary ................................ ................................ ................................ ................ 48 3 OBJECTIVELY DEFINING UNCERTAINTY IN THE FACE OF DATA SCARCITY USING A RESERVOIR MODEL OF THE OKAVANGO DELTA, BOTSWANA ................................ ................................ ................................ ........... 65 Abstract ................................ ................................ ................................ ................... 65 Background ................................ ................................ ................................ ............. 66 Methods ................................ ................................ ................................ .................. 70

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7 The Study Site: Okavango Delta, Botswana ................................ ..................... 70 The Okavango Research Institute Model ................................ ......................... 71 Global Sensitivity and Uncertainty Analysis ................................ ...................... 74 Mont e Carlo Filtering ................................ ................................ ........................ 78 Selection of Prior Input Probability Density Functions ................................ ...... 79 Results ................................ ................................ ................................ .................... 82 Modified Morris Global Sensitivity Analysis ................................ ...................... 82 FAST Variance Based Global Sensitivity Analysis ................................ ........... 83 Mont e Carlo Filtering ................................ ................................ ........................ 84 FAST Variance Based Global Uncertainty Analysis ................................ ......... 85 Summary ................................ ................................ ................................ ................ 86 4 A FISH POPULATION MODEL FOR THE OKAVANGO DELTA, BOTSWANA: HOW THE FLOOD PULSE DRIVE POPULATION DYNAMICS ........................... 102 Abstract ................................ ................................ ................................ ................. 102 Background ................................ ................................ ................................ ........... 103 Methods ................................ ................................ ................................ ................ 107 Fish Data ................................ ................................ ................................ ........ 107 Model St ructure ................................ ................................ .............................. 109 Model Optimization ................................ ................................ ......................... 113 Global sensitivity and uncertainty analysis ................................ ............... 113 Monte Carlo filtering ................................ ................................ ................. 114 Probability density functions ................................ ................................ ..... 115 Results ................................ ................................ ................................ .................. 120 Global Sensitivity Analysis ................................ ................................ .............. 120 Monte Carlo Filtering ................................ ................................ ...................... 122 Global Uncertainty Analysis ................................ ................................ ............ 123 Summary ................................ ................................ ................................ .............. 123 5 CONCLUSIONS ................................ ................................ ................................ ... 141 APPENDIX A OKAVANGO RESEARCH INSTITUTE MODEL COMPUTING GLOBAL SENSITIVITY AND UNCERTAINTY ANALYSIS SCRIPT AND INSTRUCTIONS 147 B FISH MODEL DATA AND CODE ................................ ................................ ......... 149 LIST OF REFEREN CES ................................ ................................ ............................. 164 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 176

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8 LIST OF TABLES Table page 2 1 Input factor probabili ty density functions (PDF's) for the two regional simulations. [Low 15%, Medium 30%, High 45% of calibrated values.] ....... 51 2 2 Comparison of the three and five region uncertainty analyses for two objective functions ................................ ................................ .............................. 52 2 3 Ranking of importance of input variables ................................ ........................... 52 2 4 Stationary versus climate change confidence in tervals (CI). ............................. 53 3 1 Okavango Research Institute model input factors. ................................ ........... 90 3 2 Uncertainty analysis statistics of aerial flood ing extents after Monte Carlo Filtering. ................................ ................................ ................................ .............. 93 3 3 Width of 95% Confidence Interval for Average Flooding Extents (km2). ........... 93 4 2 Lis t of inputs and initial distributions. ................................ ................................ 129 4 1 Natural mortality (M) and growth coefficients (k) for selected tilapiine species. 130

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9 LIST OF FIGURES Figure page 1 1 Location Map ................................ ................................ ................................ ...... 23 2 1 Model conceptualization. ................................ ................................ ................... 53 2 2 (a) three and (b) five regional simulations of the Okavango Basin. ................... 54 2 3 Morris Method three region Global Sensitivity Analysis (GSA) results for the coefficient of efficiency (ceff) of monthly flow at Mohembo. .............................. 55 2 4 Morris five region sensitivity analysis for the ceff of monthly flow at Mohembo. Abbreviations are defined in table 2 1. ................................ ............ 56 2 6 Fourier Amplitude Sensitivity Test (FAST) uncertainty analysis for annual mean monthly flow at the outlet of each of the regions. ................................ ...... 58 2 7 Uncertainty ............ 59 2 8 FAST three region sensitivity analysis for the ceff of the monthly outflow at Mohembo. ................................ ................................ ................................ ......... 60 2 10 Cobweb plot of best fit model trajectories. ................................ ....................... 62 2 11 Map of the model structure and the identification of the most important processes. ................................ ................................ ................................ ........ 63 2 12 Total sensitivity of ceff for outflow at Mohembo for (a) 3 and (b) 5 regional scenarios. ................................ ................................ ................................ ........... 64 3 1 Okavango Basin and Delta Locatio n Map. ................................ ....................... 94 3 2 The Okavango Research Institute (ORI) model. Adapted from Wolski et al., 2006. ................................ ................................ ................................ .................. 9 5 3 3 Diagram of the ORI reser voir model nodes and links. Adapted from Wolski et al., 2006. ................................ ................................ ................................ ......... 96 3 4 Morris method GSA results for the inundation of the entire Delta. .................. 97 3 5 FAST GSA first and grouped higher order (R) proportional results for the inundation of the entire Delta. ................................ ................................ .......... 98 3 6 FAST first order and higher order, and total order GSA results for the inundat ion of the entire Delta. ................................ ................................ .......... 98

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10 3 7 Frequency histograms for values for a) v 27 and b) v 23 resulting in behavioral outputs. New triangular distributions bas ed on these results are also shown. ... 99 3 8 FAST Global Uncertainty Analysis (GUA) grouped by upstream, middle, eastern, and western reservoirs. ................................ ................................ .... 100 3 9 Comparison of original GUA and Monthe Carlo filtering (MCF) GUA: Nqoga1, Maunachira1, a nd Mborga. ................................ ............................. 101 4 1 Site location. The Okavango Delta, with fish sampling sites marked with stars. ................................ ................................ ................................ ................. 131 4 2 The conceptual model. ................................ ................................ ................... 132 4 3 A demonstration of the response in recruitment as a result of the change in annual maximum flood. ................................ ................................ ................... 132 4 4 An example of the dome shaped double logistic curve (Allen et al., 2009) for catch vulnerability. L low = 2, SD low = 2, L high = 45, SD high = 10. ....................... 133 4 5 The best fit model simulation for the initial inverse optimization. Coefficient of efficiency of 0.64. ................................ ................................ .......................... 134 4 6 First order sensitivity for the coefficient of efficiency of the modeled fish density compared to the measured catch per unit effort (CPUE) ...................... 135 4 7 Fir st and higher order sensitivities for coefficient of efficiency of modeled fish density compared to measured CPUE from first inverse optimization. ...... 135 4 8 Scatter plots of values for the flo od coefficient, k, M u L high L and b creating behavioral outputs. ................................ ................................ ........... 136 4 9 The effect of the flood and the flood coefficient on recruitment. (flood coef = 15) ................................ ................................ ................................ .................... 137 4 10 Important inputs whose behavioral distributions are significantly different from the non behavi oral distributions. ................................ ............................ 138 4 11 All GUA results. (a) unfiltered (b) Monte C arlo (MC) filtered. ....................... 139 4 12 Uniform and MC Filtered GUA outputs with a coefficient of efficiency greater than 0. ................................ ................................ ................................ .............. 140

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11 LIST OF ABBREVIATION S ATTZ Aquatic terrestrial transit ion zone ceff Coefficient of efficiency CPUE Catch per unit effort FAST Fourier amplitude sensitivity test FPC Flood pulse concept GSA Global sensitivity analysis GSA/UA Global sensitivity and uncertainty analysis GUA Global uncertainty analysis MC Monte C arlo OAT One at a time ORI Okavango Research Institute PDF Probability density function

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12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of D octor of Philosophy HYDROLOGY AND FISH POPULATION DYNAMICS IN THE OKAVANGO BASIN: MANAGING FOR UNCERTAINTY IN A DATA POOR ENVIRONMENT By Anna McDaniel Cathey August 2011 Chair: Rafael Muoz Carpena Co chair: Greg Kiker Major: Agricultural and Biologic al Engineering The Okavango Basin is a large and exceptionally pristine transboundary catchment in southern Africa that empties into the Okavango Delta, an i nland delta that spreads onto the Kalahari sands of Botswana and is evapotranspirated before ever reaching the sea. Because of its remote location and large size, the Okavango is a data scarce area which leads to concerns about model reliability. Though the uncertainty in models cannot be taken away, models can be given pedigrees to evaluate their on a determination of usefulness. Two tools that can be used to develop this pedigree are global sensitivity and uncertainty analysis (GSA/UA) Uncertainty a nalysis is useful for measuring model reliability and sensitivity analysis apportions the total model uncertainty to individual inputs and processes In this work a methodology is used to o ptimize computational resources, conduc t a robust GSA/ UA and poten tially reduce input/output uncertainty using Monte Carlo Filtering This work is threef old involving a GSA/UA o n two existing hydrologic models in the area (the Pitman basin model and the Okavango Research Institute (ORI) delta model) as

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13 well as the develo pment and analysis of a fish population model for the Okavango Delta based on the flood pulse concept Result s show that important areas and processes of the Pitman model include the headwaters as well as infiltration, temporal rainfall, and groundwater in puts. Important aspects of the ORI model include a keystone reservoir at the head of the Delta where water is apportioned between downstream reservoirs as well as the volume threshold inputs. In the fish model, the flood pulse is shown to be an important driver. The input that relates the flood pulse to fish recruitment is not highly sensitive but is very important for achieving the best fit model simulations. Monte Carlo Filtering is successfully used to reduce and refine input/output uncertainty in th e modeling applications. These results will aid ongoing efforts in the Basin and the Delta which are currently using hydrologic and ecological models to develop environmental flows, explore development and climate change scenarios, and apportion water.

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14 CHAPTER 1 INTRODUTION Overview The Okavango Basin, located in southern Africa, is characterized as a large (530,000 km 2 ), endorheic, and physiologically diverse basin that delivers a unique and ecologically important flood pulse to the Okavango Delta. The the majority of the contributing area begin in the more humid and steep Angola. The Basin is formed by two major Rivers the Cubango and the Cuito River (Figure 1 1) The Cubango River forms part of the boundary between Namibia and Angola before converging with the Cuito River to form the Okavango River. The Okavango River then flows through the Caprivi Strip in Namibia where the topography is flatter and the climate is drier than in the headwaters. After flowing through Namibia, the Okavango River empties to the Okavango Delta, entirely contained within Botswana, which like Namibia is also dry and extremely flat. Here, on the edge of the Kalahari Desert, is the efore reaching the sea. Average rainfall in the upstream Angolan portion of the Basin is as high at 1,300 mm yr 1 while the two downstream basin nations, Namibia and Botswana, are much more arid with rainfall averaging approximately 560 mm yr 1 in Rundu Namibia (Mendelsohn and Obeid, 2004). Angola also has more topography, with elevations in this portion of the Basin ranging between 1,200 and 1,800m, while the Delta is exceptionally flat, with elevations ranging between 900 and 1,000m (Mendelsohn and O beid, 2004). The eastern headwaters also have different hydrologic characteristics from the western headwaters with the western headwaters demonstrating more

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15 consistent base flows. The northwestern portion of the Basin is underlain by volcanic and metamo rphic rock while the northeastern portion is underlain by Kalahari sands. Thus, the western headwaters are hydrologically more variable while the eastern headwaters have higher base flows and less seasonal variability (Hughes et al., 2006). Rainfall i n the Angolan headwaters occurs between December and March. It takes two months for the flood pulse to reach the top of the Delta and another four to five months for it to attenuate through the Delta, finally reaching the distal ends in August. Flooding extents in the Delta range from 6,000 12,000 km 2 (Wolski et al., 2006) and the entire alluvial fan is approximately 40,000 km 2 (Gumbricht et al., 2005). The Delta is extremely smooth and flat with maximum local relief generally less than 2 or 3 m (McCar thy et al., 2003; Gumbricht et al., 2001). Over 70 fish species (Kolding, 1996) live in the Okavango Delta and there are three main fishing industries: subsistence, small scale commercial, and recreational (Merron 1991, Mosepele and Kolding 2003). Accor ding to Mosepele (2001), 65% of the people in northern Ngamiland depend on fish as a source of livelihood. Monitoring and preserving the health of this resource is therefore important for both economic and ecological reasons. In the Okavango Delta no stu dies have evaluated how fish populations respond to the flood pulse. However, there have been studies that show how the annual flood pulse produces other ecologic responses, such as nutrient enrichment, bursts in primary productivity, and zooplankton abun dance (Hoberg et al., 2002; Merron, 1991). The Okavango Integrated River Basin Management Project (IRBM, 2009) is a transboundary association between Angola, Namibia, and Botswana that promotes an

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16 adaptive approach to management in the Okavango Basin. A daptive Management (AM) can be loosely defined as managing in an uncertain world with a built in plan for reducing uncertainty by changing management policies on an as need basis (Walters & Holling, 1990). AM explicitly calls for the use of models and the acknowledgement of uncertainty (Johnson et al., 1997; Walters, 1986; Walters & Holling, 1990). Models measurements (Saltelli et al., 2008). However, models can be giv en pedigrees that and Monte Carlo (MC) filtering are three tools that bring transparency to the modeling process and judge model quality. Uncertainty analysis quanti fies model reliability precision, and accuracy Sensitivity analysis identifies the key factors that contribute to input/output uncertainty and may also improve the description of uncertainty. Together thes e three methods add to transparency in modeling and apply value to model results for their intended purposes (Saltelli et al., 2008) There are several approaches for conducting a GSA/UA on mathematical models ranging from simple one at a time (OAT) varia tional methods to more sophisticated global techniques. With OAT techniques the variation of the model output is explored through changing a single input at a time. This traditional sensitivity analysis method is limited since it explores a prescribed (an d usually small) parametric range, can only consider a few inputs, and cannot account for interactions between inputs (Saltelli et al., 2008).

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17 When model responses are non linear and non additive, which is the case with most complex models, OAT techniques are not sufficient and global techniques, which evaluate the input factors concurrently throughout the entire parametric space, should be used (Saltelli et al., 2008). Different types of global sensitivity methods may be selected based on the objective of the analysis and the model structure (Cacuci, 2003; Saltelli et al., 2008). A model evaluation framework around two global techniques, a screening method (Morris, 1991) and a quantitative variance based method (Saltelli, 1999), can be quite useful when fa ced with a large number of inputs. This two step process has been used in several recent model investigations (Muoz Carpena et al., 2007; Muoz Carpena et al., 2010; Fox et al., 2010; Jawitz et al., 2008; Chu Agor et al., 2011). The screening method cul ls the inputs so that only the most important ones go on for further analysis in the quantitative variance based FAST sensitivity and uncertainty analyses. The least important inputs are set to constants and ignored for further analysis. This reduces mod el complexity and focuses effort on the most important aspects of the model. The Method of Morris (Morris, 1991) is an OAT sensitivity method, but because it explores the entire parametric space, it is also considered a global method. The Morris method i s qualitative and is less computationally intensive than other quantitative methods, thus its use as a screening tool to identify the most sensitive inputs. Morris of t order

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18 of the results and represents the spread of the importance that each input has based on th e values chosen for the other inputs. Once the most sensitive factors are identified using the Modified Method of Morris, a quantitative sensitivity analysis may be conducted on those most sensitive factors using the Fourier Amplitude Sensitivity Test ( FAST) (Cukier et al., 1978; Koda et al., 1979). FAST uses Fourier analysis to decompose the variance of model outputs into first order variances for each input. The Extended FAST technique (Saltelli et al., 1999) allows for the additional quantification of higher levels of variance. These higher levels of variance describe the interactions between inputs. Extended FAST defines S i as a measure of global sensitivity. Monte Carlo (MC) Filtering can be used to calibrate a model as well as decrease input/ output uncertainty (Saltelli et al, 2008; Rose et al., 1991). In MC filtering, model outputs are classified as either behavioral ( B ) or non behavioral ( ) based on whether or not they exceed some predefined threshold. The inputs for each of t hese model runs are assigned the same status. Two new prior distributions are formed for each input based on the behavioral, non behavioral status for each model run. If these distributions are found to be significantly different then the behavioral dist ribution can be used in place of the original distribution as the prior input. This has the effect of truncating the PDF and decreasing the model uncertainty. The remote location of the basin and the recent civil war in Angola (1975 2002) has resulted in diverse physiology, climate, and hydrology. Most of the stream gauge data is intermittent and the basic physiologic data are on rough scales and/or in discrete

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19 sampling locations. T herefore, issues of data scarcity and resulting uncertainty are compounded in this area. When data for the inputs show no apparent distribution, such as normal or triangular, its PDF can be set to uniform (Muoz Carpena et al., 2007). The uniform distrib ution allows for equal probability of selection across the defined range. Because of the data scarcity in this location this is generally the approach to This conservative approach inherently introduces additional u ncertainty. In response to this, this work will show that MC Filtering can be used to objectively redefine and truncate prior within their predefined ranges and decrease input/output model uncertainty. This is especially useful because one of the l argest critiques of the GSA/UA process is the subjectivity through which the are defined. In data poor areas such as the Okavango Basin this problem is exacerbated and the objective truncation of the is particularly useful. Equifinality is the principle that given a model with multiple inputs it is possible to get exactly the same output with a variety of input sets (Beven, 2001, Beven and Freer, Equifina lity is an important concept in hydrologic modeling due to the large number of inputs. It is especially crucial in reference to predictive modeling. Predictive modeling often involves changes in state, such as development and climate change scenarios. W hen the state of the system changes, internal processes and relationships change. Following this, if a model is getting the right answer for the wrong reason under one state, it is possible that it will no longer get the right answer when internal process es change. Traditionally, the goal of inverse optimization techniques is to find the optimal set of inputs for a given model to match some measured data (Mertens et al., 2006).

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20 This is especially problematic when considering the issue of equifinality. Th ough this work does employ inverse optimization style techniques, it does not seek an optimal parameter set. Instead, it looks for evidence of these models as useful tools in the presence of equifinality. This is done in two ways (1) through identifying best fit model input sets and running these through non stationary scenarios and (2) through identifying key inputs which do not display equifinality. Through these two methods, insights into the confidence in model behavior can be gained even in the pres ence of equifinality. Motivation As a transboundary Basin, there are conflicting management agendas in the Okavango watershed. The IRBM transboundary project (2009) advocates for an adaptive approach to management in the Basin. Adaptive Management explic itly calls for the acknowledgement of uncertainty and the use of models. Uncertainty regarding physiographic, ecologic, and hydrologic data in the Basin is especially high because of its large size, remote location, and recent civil war in Angola. Severa l models have been calibrated in the basin and are being used to assess climate change and development scenarios. Hughes et al., (2006) calibrated the Pitman rainfall runoff model in the Okavango Basin. Uncertainty analysis has been conducted on the Pitm an model in several basins in South Africa (Hughes et al., 2010) however; no GSA/UA has been conducted for this model in the Okavango Basin. The ORI (Okavango Research Institute) reservoir model has been calibrated in the Okavango Delta (Wolski et al., 20 06) however; no GSA/UA has been conducted for this model in the Okavango Delta. Climate change and development scenario modeling has been conducted using both the Pitman and ORI models (Andersson et al., 2006; Wolski, 2009). Conducting a

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21 GSA/UA on these m odels will provide a transparent qualification of their use for scenario modeling, enable model simplification through identifying unimportant inputs, and provide a deeper understanding of model structure and interactions. Additionally, quantitative studie s show how the annual flood pulse produces an ecologic response in nutrients and zooplankton (Hoberg et al., 2002) and qualitative observations and conceptual models have been proposed for how the flood impacts fish population dynamics (Hoberg et al., 2002 ; Merron, 1991). A major recommendation in the Environmental Flow Module Special Report for Fish (Mosepele, 2009) is for the development of a quantitative relationship between the flood pulse and fish population dynamics in the Delta. A fish population m odel has yet to be developed for the Delta. Conducting GSA/UA and MC filtering during the development of a fish model will prevent over parameterization, reduce input/output uncertainty, and allow for an investigation of evidence of the flood pulse as a d river despite issues of equifinality. Objectives The objective of this work is to investigate the hydrology and fish population dynamics of the Okavango Basin. Two hydrologic models will be audited : one of the Basin and one of the Delta, both of which h ave previously been developed and calibrated in the area (Pitman, 1977, Hughes et al., 2006, Wolski et al., 2006) and are currently being used to test development and climate change scenarios. For this research these two models will be analyzed for sensit In order to investigate the influence of the annual flood pulse on fish population dynamics in the Delta, a fish population model where recruitment is influenced by inundation area will also be developed. Th is model will be c alibrated, tested,

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2 2 will add depth of meaning to the climate change and development scenarios that are being simulated using the existing hydrologic model s. It will also corroborate the existing conceptual models and qualitative observations that describe fish population dynamics as being driven by the food pulsed hydrology of the Delta. Chapter 2. Conduct an audit of the Pitman rainfall runoff model of the Okavango Basin 1) Analyze the Pitman model for the most sensitive regions and inputs for focusing future research 2) Quantify the uncertainty of the Pitman model 3) Assess the uncertainty of the Pitman model in a non stationary climate change scenario. Chapter 3. Conduct and audit of the ORI reservoir model of the Okavango Delta 1) Analyze the ORI model for the most sensitive regions and inputs for focusing future research 2) Quantify the uncertainty of the Pitman model 3) Using Monte Carlo Filtering attempt to reduce th e input/output uncertainty Chapter 4. Develop and analyze a fish population model for the Okavango Delta which is driven by the annual flood pulse 1) Develop a mathematical fish population model that is seamlessly integrated with ORI hydrologic model 2) Simplify unimportant inputs to constants. 3) Quantify the uncertainty of the Fish model 4) Using Monte Carlo Filtering attempt to reduce the input/output uncertainty 5) Look for proof that the flood puls e is a driver for fish population dynamics

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23 Figure 1 1 Location Map

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24 CHAPTER 2 UNCERTAINTY, SENSITI VITY, AND IMPACTS FOR SCENARIO MODELIN G USING THE PITMAN HYDROLOGI C MODEL IN THE OKAVA NGO BASIN Abstract Understanding and impr oving the reliability of models in watersheds in the presence of uncertainty is crucial for applying hydrologic models in a responsible management context. The Okavango Basin is a large and remote watershed (530,000 km 2 ) that delivers an annual flood pul se to the Okavango Delta, an internationally renowned wetland. Sparse data sets and large gaps in knowledge regarding the diverse basin physiography and hydrology are a major concern in the Basin. An important step toward sustainably developing the water resources of the Okavango Basin is through an exploration of the potential hydrologic scenarios and inherent uncertainty of their modeled outcomes. This work presents two tiered regionalized examination of the Pitman hydrologic model for the Okavango Bas in in southern Africa through a state of the art global sensitivity and uncertainty analysis (GSA/UA). The Pitman model is a monthly rainfall/runoff model that is widely used in southern Africa especially in ungauged basins. In this work, the 24 sub basi ns are grouped into two regional arrangements to compare the significance of the choice of arrangement. The Morris GSA method is first used to qualitatively rank the importance of the inputs and then the variance based Fourier Amplitude Sensitivity Test ( FAST) GSA/UA method is used to quantitatively identify the most sensitive inputs in terms of contributed variance to simulated outputs. Model uncertainty was investigated under a climate change scenario, which predicts increasing temperature and rainfall in the next thirty years. Results show ed that the rate of infiltration, temporal rainfall distribution, groundwater runoff, and the evaporation coefficient are the most important inputs in the model.

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25 Additionally the western headwater was shown to be the most important region. Results also show ed that the choice of regional arrangement does not make a significant difference in the model outcome. Introducing climate change conditions to the watershed increase d uncertainty by approximately 18%. These res ults and conclusions are especially useful within the context of Adaptive Management when interpreting model uncertainty, determining the usefulness of model predictions in a non stationary condition, and focusing monitoring and management efforts. Backgro und Hydrologic modeling is widely used in management programs for predicting the impacts of development and climate change on water supply for both human and environmental needs (Beckers et al., 2009; Hughes, 2002; Jayakrishnan et al., 2005; Koster et al., validated because of the ubiquitous nature of uncertainty in observations and measurements (Beven and Binley, 1992). However, they can be given pedigrees that judge their quality as sensitivity and uncertainty analysis (GSA/UA) are tools used during model analysis to judge model fitness under different applications. Uncertainty analysis quantifies the overall uncertainty of a model and sensitivity analysis identifies the key factors that contribute to that uncertainty. Using these tools, a modeler can assign the level of confidence to a model for its intended purpose. (Saltelli et al., 2008) The Okavango Bas in is a large and remote watershed (530,000 km 2 ) that delivers an annual flood pulse through the Okavango River from Angola, through Namibia, and then into Botswana, which contains the Okavango Delta (Figure 1 1) The Okavango Basin has a distinct and het erogeneous physiological and climatic

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26 organization. Rainfall rates, geology, and topography vary considerably throughout. Rainfall in the Angolan headwaters is 1,300 mm yr 1 but only 560 mm yr 1 in downstream Namibia and Botswana (Mendelsohn and Obeid, 2 004). The Angolan headwaters are also mountainous, with elevations ranging between 1,200 and 1,800m, while the Botswana portion of the Basin is extremely flat, with elevations ranging between 900 and 1,000m (Mendelsohn and Obeid, 2004). The eastern headw aters also behave differently from the western headwaters with the western headwaters having more consistent base flows. Rainfall is evenly distributed within the Basin from the east to the west (Wilk et al., 2006). However, the soil/substrate types is d ifferent and can explain this difference in the base flow behavior (Hughes et al., 2006). The western portion of the Basin is underlain by volcanic and metamorphic rock while the eastern portion is underlain by Kalahari sands. Thus, the western headwater s are more hydrologically variable while the eastern headwaters have higher base flows and less seasonal variability. As a transboundary basin, each of the three countries that contain portions of the Okavango River values the resource in a different w ay. Angola is upstream basin state and the major land holder. Namibia owns a sliver of land in the middle. Botswana contains the terminal end and the famed Okavango Delta, which is an international Ramsar site (Ramsar, 2011). The different values that the three countries place on the Okavango Basin and their different physiographic environments create challenges for management of the resource. As the three countries develop and face climate change they must find ways to equitably distribute the river f lows. Hydrologic models can be useful in this circumstance because they can simulate hydrologic response to proposed

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27 development and climate change. However, without an uncertainty analysis managers cannot contextualize the uncertainty of these models cou ld result in unjustified confidence in modeled predictions Additionally, the remoteness of the basin and the recent civil war in upstream Angola (1975 2002) has resulted in sparse data sets and a high level of uncertainty regarding the diverse basin ph ysiography and hydrology. Most of the gauge data is intermittent and the basic physiographic data are on rough scales and/or discrete sampling locations (Hughes et al, 2006). Therefore, issues of data scarcity and uncertainty are compounded in this loca tion. The Pitman model (Pitman, 1973; Hughes et al., 2006) (Figure 2 1 ) was developed in southern Africa specifically for ungauged catchments (Hughes, 1995). It is widely used in the area and has been calibrated and validated in the Okavango Basin (Hughes et al., 2006). An uncertainty and sensitivity analysis for the Pitman model has been conducted for four basins in South Africa (Hughes et al., 2010). The results of this analysis show that uncertainty and sensitivity in the Pitman model varies from basi n to basin depending on physiography and climate. This current research builds upon these results and involves a GSA/UA of the Pitman rainfall runoff hydrologic model specifically for the Okavango Basin emphasizing the issue of data scarcity regarding ba sin physiography. This GSA/UA is specifically applied with management purposes in mind such as interpreting model uncertainty under stationary conditions and climate change scenarios, identifying gaps in knowledge, and focusing monitoring efforts. The O kavango Integrated River Basin Management Project (IRBM, 2009) promotes and adaptive approach to management in the Okavango Basin. Adaptive Management

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28 (AM) can be loosely defined as managing in an uncertain world with a built in plan for learning by doing (Walters & Holling, 1990). AM explicitly calls for the use of models, the acknowledgement of uncertainty and the use of management actions to reduce that input/output uncertainty (Johnson et al., 1997; Walters, 1986; Walters & Holling, 1990). The publ ication AM for Water Resources Project Planning (NRC, 2004) offers six elements of successful AM programs: (1) management objectives are routinely revised (2) a model of the system is developed and revised (3) a range of management choices are investigated (4) outcomes are monitored and evaluated (5) mechanisms for learning are incorporated into future decisions (6) a collaborative structure for stakeholder participation and learning is constructed and revised. GSA/UA is useful in three of these steps. In step (2) models can be revised through inputs identified in sensitivity analysis. The unimportant inputs can be set to constants to avoid over parameterization and simplify the model structure (Saltelli et al., 2008). For this work, inputs will be defi nes as any variable that is not a time series. In step (3) uncertainty analysis can be used to evaluate management outcomes and assign value to the model. In step (5) mechanisms for learning can be based on gaps in knowledge that are identified through s ensitivity analysis. (Saltelli et al., 2008) In this work, a global sensitivity and uncertainty analysis (GSA/UA) of the Pitman model in the Okavango Basin will be conducted to explore the uncertainty of the model under stationary and non stationary con ditions as well as identify the inputs that are most responsible for that uncertainty. The specific objectives of this work are threefold. This work will (1) quantify the uncertainty of the model results using the FAST GUA method (2) identify the most i mportant or most sensitive inputs and regions using the

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29 Morris and FAST GSA methods and (3) investigate the change in uncertainty in the model when it is run under a non stationary climate change scenario Two objective functions were analyzed in this pr ocess: the mean annual flow and the Nash Sutcliffe goodness of fit by comparing the modeled and measured monthly flow. These results yield important products for manage rs. (1) Confidence in model results was assigned so that managers can use the model along with additional lines of evidence in an informed manner. (2) Gaps in knowledge were identified allowing for strategic monitoring to close those knowledge gaps. (3) The uncertainty of the predictive capacity of the model under a climate change scenario was assessed so that confidence in climate change scenarios can be assigned. Methods The Pitman Model The Pitman model (Pitman, 1973) is a semi distributed monthly time step rainfall runoff watershed model developed in southern Africa (Figure 1 1 ). Since its inception the model has undergone a number of revisions, the most recent being the addition of more explicit surface water/groundwater interactions (Hughes, 20 04). The model has also been integrated into SPATSIM (Spatial and Time Series Information Modeling), a GIS platform to facilitate spatial analyses (Hughes et al., 2002; Hughes and Forsyth, 2006). The Pitman model is one of the most commonly used rainfall runoff models in southern Africa. Data requirements for running the Pitman model include monthly rainfall and evaporation time series, basin and sub basin delineations, and physical characteristics for each sub basin such as soil transmissivity and storat ivity, slope, interception and soil

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30 absorption rates. Additionally, optional anthropogenic inputs and lags include abstractions by irrigation, dams, and reservoirs. Storage processes such as interception, soil moisture holding capacity, and groundwater s ystems are simulated as simple water storage tanks within each sub basin. Kapangaziwiri (2008) provides a more detailed description of the general model. The Pitman model has been set up and calibrated in the Okavango River (Hughes et al., 2006). This calibration will be used as a baseline for the following sensitivity and uncertainty analysis. In the model, the Okavango River Basin, above the Delta, was divided into 24 sub basins. Seventeen of these sub basins have stream gauges at their outlet. How ever, because of the civil war in Angola, most of these records were discontinued prior to 1975 and only two of the downstream most gauges, which are located in Namibia and Botswana, have continuous contemporary data. The model was calibrated over the per iod of 1960 and 1972 and validated between 1991 and 1997. Rain gauge measurements were used in calibration (Hughes et al., 2006). Because of the civil war in Angola (1975 2002), which contains the majority of the watershed, rain gauge readings were disco ntinued. TRMM (Tropical Rainfall Measuring Mission) SSM/I (Special Sensor Microwave Imager) remotely sensed rainfall measurements are available for the entire watershed from 1991 to 1997 (Wilk et. al, 2006) and were used for model validation (Hughes et al ., 2006). The Hargreaves equation (Hargreaves and Allen, 2003) was used to calculate actual evapotranspiration from the water equivalent of extraterrestrial radiation, temperature, and the difference between mean monthly maximum and minimum temperatures. Soil parameters for the Basin were obtained from FAO (Food and Agricultural Organization of the United Nations) data. Geologic

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31 and topographic inputs are derived from USGS data (Persits et al., 2002). Water abstractions are based on population density a nd are assumed to be negligible. The Pitman model was manually calibrated by Hughes et al. (2006) to surface flows in the Basin where gauges are. Calibration parameters included surface runoff, soil moisture storage, groundwater recharge, and soil moistu re evaporation. The three objective functions that were used to minimize the difference between observed and simulated follows during calibration include the coefficient of determination (ceff) and the mean monthly percentage of error (Hughes et al., 2006 ). The calibrated Pitman model simulated outflow at the downstream Mukwe sub basin with a ceff of 0.851 and a mean monthly error of +1.7% (Hughes et al., 2006). Global Sensitivity and Uncertainty Analysis There are various approaches for conducting sensit ivity analyses on mathematical models ranging from simple one at a time (OAT) variational methods to more sophisticated global techniques. With OAT variational derivative techniques the variation of the model output is examined through the variation of on e model input at a time. Sometimes, as an alternative, a crude variational approach is selected in which, incremental ratios are taken by moving factors one at a time from the base line by a fixed amount (for example, 5%) without prior knowledge of the fac tor uncertainty range. These traditional sensitivity analysis methods are limited since they explore a fixed parametric range, can only consider a few inputs, and do not take into account interaction between inputs (Saltelli et al., 2005). When model resp onses are non linear and non additive, as with most complex models, these simple derivative techniques can be misleading and global techniques that evaluate the input factors of the model concurrently over the entire parametric

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32 space (described by probabil ity density functions) should be used. Different types of GSA/UA methods can be selected based on the objective of the analysis (Cacuci et al., 2003; Saltelli et al., 2000; Saltelli 2004). This study proposes a model evaluation framework (Muoz Carpena et al., 2007; Muoz Carpena et al., 2010; Fox et al., 2010; Jawitz et al., 2008; Chu Agor et al., 2011) around two such modern global techniques, a screening method (Morris, 1991) and a quantitative variance based method (Cukier et al., 1978; Saltelli et al., 1999). This two step method only considers inputs and does not investigate structural uncertainty or sensitivity. The screening method allows for the identification of the most important inputs and a resulting reduction of inputs, based on this importan ce, for further investigation the quantitative variance based Extended Fourier Amplitude Sensitivity Test (FAST). This two step process is useful because of the computational time that is required in using the FAST method. Both the Morris and FAST method s are pre and post processed in the software Simlab (2011). The proper use of global sensitivity methods can yield three main products for the Okavango behavior), (2) a ranki ng of importance of the parameters for future data collection and (3) the identification of the type of influence of the important parameters (first or higher order interactions) (Saltelli, 2008). These products can aid managers through assigning confide nce in model results and strategically allocating monitoring efforts. Through uncertainty analysis, alternative model scenarios can also be investigated. Equifinality is the principle that given a model with multiple inputs it is possible to get exactly the same output with different input sets (Beven, 2001, 2006; Beven and Freer, 2001). The risk of this is that the model could be

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33 modeling due to the larg e number of inputs. It is especially crucial in reference to predictive modeling. Predictive modeling often involves changes in state such as development and climate change scenarios. When the state of the system changes, i.e. non stationary, internal p rocesses and relationships change. Following this, if a model is getting the right answer for the wrong reason under one state, it is possible that it will no longer get the right answer when internal processes change. Therefore, any predictive model tha t investigates a change in state should incorporate an analysis that quantifies the change in uncertainty that accompanied that change in state. Thus, uncertainty analysis assigns confidence in model results for management purposes. Input f actor s election Ideally, different probability distributions would be produced for each of the model inputs in each of the 24 sub basins based on measured physical variability. However, due to the lack of data in the area there is little basis for defining different PDF each of the sub basins. Alternatively lumping the Basin as a whole and producing one PDF for each of the inputs would disregard its physical heterogeneity. Therefore, a regional approach was taken. The Basin was split into regions based on geol ogy, regions. Two regional approaches were taken in order to compare the importance of the regionalization concept: a) one with three regions and b) one with five region s (Figure 2 2 ). In the three region approach the basin was separated into eastern headwaters, western headwaters, and southern receiving waters. The eastern and western headwaters are geologically different from each other and both receive higher rainfal l rates than the southern receiving waters. This regional approach mimics the

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34 same regionalization that is referred to by Hughes et al. (2006) in the calibration of the model. In the five region approach the southern region was subdivided into the middle basins, Omatako basins, and southern basins. These subdivisions reflect the difference in rainfall and topography in the area. The subdivision of the southern region in the five regional approach also reflects the fact that the Omatako River is hydrolog ically unique and though it is hydrologically connected, it is now considered a fossil river and there is no record of it contributing flow to the system (Crerar, 1997). Pitman model (Table 2 1). The basin is a fairly pristine system, so inputs regarding anthropogenic disturbances such as dams and reservoirs were not considered. Uniform c onservative approach when there is a lack of data available for describing specific types of distributions (Muoz based on experiments, literature values, physical bounds, or expert opinion. For exam ple, Kapangaziwiri and Hughes (2008) developed an estimation method for inputs into the Pitman model based on physiological basin characteristics that are largely referenced Information System (AGIS, 2007). Howev er, compared to South Africa, the Okavango Basin is located in a remote area and there is significantly less data available and fewer experts in the area. Additionally, many of the inputs are empirical (though physically relevant) and are used as scaling factors or describe the shape of relationships among system components. As a result of the lack of data and the empirical nature of the inputs, the true physical ranges of by

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35 some standard percentage of the calibrated value. For this reason the inputs were categorized as having low, medium, and high uncertainty. The inputs that were identified as having low uncertainties were bounded by 15% of the calibrated value, medium 30%, and high 45% of the calibrated values. Each input for each sub basin within a region is varied independently from its calibrated value within its assigned uncertain range based on the regional value assigned by the matrix. In an area where more da ta is available cross correlation tests could be done to show dependence between data in each sub basin. However, here no such data exists and inputs are assumed independent. The results of the GSA/UA are largely dependent upon the define the input factors and the rationale behind each PDF Monthly rainfall in the Pitman model is disaggregated into four sub monthly inputs that use the rainfall distribution input factor ( RDF ) to describe the temporal heterogeneity of the input distribution. Lower RDF's create even temporal rainfall distributions and higher values result in flashier and heavier rainfall event s within each month. In the calibrated model, RDF was set to 0.7 for each sub basin. Hughes et al. (2003) used values in the Kafue Basin that ranged between 0.6 and 1.28. The Kafue Basin is located approximately 800 km northeast of the Okavango Basin in Zambia. The Okavango Basin covers a large area going from mountainous temperate headwaters to flat semi arid receiving waters. It is unlikely that monthly rainfall is distributed the same way in the headwaters as it is in the southern portions of the b asin. The uncertainty for this input was set to medium.

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36 In the model, interception (Figure 2 1 ) is defined for the two different vegetation types ( PI1 and PI2 ). Vegetation type 1 represents non forested land cover and vegetation type 2 is forested. Pitm an (1973) asserts that interception in southern Africa can ranges between 0 and 8 mmday De Groen (2002) considers a range of 2 5 mmday but cites that established thresholds for South Africa are 1 to 2 mmday and as much as 7 mmday when litter interception is included. In the calibrated model the interception rates are set to 1.5 for vegetation type 1.0 and 4.0 for vegetation type 2. This is a physically based input but is only allowed to range across two vegetation types. The uncertainty for these two inputs was set to medium. Evapotranspiration is based on the ratio between potent ial and actual evaporation at different levels of soil moisture ( R ), the area of the sub basin covered by type 2 vegetation ( AFOR ), a factor that scales the evapotranspiration for vegetation type 2 ( FF ), and the riparian strip factor ( RSF ). R determines t he shape of a linear relationship between actual and potential evaporation loss at different moisture storage levels. The value of R is bounded between 0 and 1. Actual evaporation is calculated according to Eq. 1 where PE is potential evaporation, S is cu rrent soil moisture storage, and ST is the maximum soil moisture storage. R is a physical parameter, but there is little data for the area and the uncertainty level was set to medium. (1) AFOR represents the percent of the bas in covered by type 2 vegetation. The GLC2000 land cover map from the Global Vegetation Modeling Unit (GLC, 2000) was used to derive these parameters. Because AFOR is based on remotely sensed data, its uncertainty was set to low. FF allows type 2 vegetati on to have greater ET than type 1

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37 and is set to 1.3 in all of the sub basins. This categorizes the entire Okavango Basin by two types of vegetation. Its uncertainty is set to medium. RSF determines the water loss due to evapotranspiration in areas adjac ent to the channel. There is a great deal uncertainty associated with RSF as it is a fairly empirical input with very little data available for verification. Its uncertainty level was set to high. In the Pitman model, infiltration is governed by the amount of monthly rainfall and the inputs ZMIN and ZMAX ZMIN and ZMAX provide boundary conditions for a triangular distribution of absorption rates. If rainfall is greater than the absorption rate ccurs. ZMIN and ZMAX are manually fitted to approximate the Kostiakov (1932) equation to calculate infiltration. The Kostiakov equation is an empirical model that assumes infiltration decreases in time during a rainfall event according to a power function The uncertainty for these inputs was set to high because of the empirical nature of the equation, the differences in describing the soils in the area. Soil moisture st orage is represented by the maximum depth of water storage ( ST ). If ST (mm) is filled, then the additional rainfall becomes runoff. The uncertainty for ST was set to medium because there is some physical evidence for relative differences in these values based on the geology of the region but there is still very little data regarding actual values. Hortonian overland flow is calculated according to two inputs: the runoff generated at the maximum soil moisture ( FT ) and a power function that allows a nonli near relationship between runoff and soil moisture ( POW ). Kapangaziwiri (2008) give values for FT (mm month 1 ) ranging between 0.4 and 43.4 for

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38 basins with low drainage density and 0.4 to 72.3 in basins with high drainage density. The model was calibrat ed with values ranging from 0 to 38 with a great deal of variation between regions. Because this range is so large, the uncertainty for FT was set to high. POW simulates the curve that represents how runoff slows as moisture in the soil decreases. Value s for POW in the calibrated model vary between 2.5 and 4.0. There is little data describing accurate values for this input and its uncertainty was set to high. Groundwater recharge in the Pitman model is one dimensional and governed by the rate of recharg e at the maximum soil moisture storage ( GW ) and a power function that describes the non linear relationship between recharge and soil moisture ( GPOW ). GW is the groundwater equivalent to FT and GPOW is the groundwater equivalent to POW Xu and Beekman (2 003) compiled literature values on ranges of groundwater recharge in southern Africa. They found maximum rates ranging from 4.2 to 420 mm month 1 Because this range is so large, the uncertainty for GPOW and GW was set to high. There are a number of ad ditional groundwater accounting parameters in the model: drainage density ( DDENS ), transmissivity ( T ), storativity ( S ), the depth of the aquifer below the channel at which groundwater ceases to flow ( RWL ), the groundwater slope ( GWS ), and the riparian str ip factor ( RSF ). Kapangaziwiri (2008) states that estimates of drainage density from 1:250,000 maps are three times greater than those from 1:50,000 maps. T (m 2 d 1 ) is equal to the soil permeability time the aquifer thickness. Milzow et al., (2009) cit es that the aquifer thickness in the Okavango Delta varies between 70 and 400m. S (m 3 ) is equal to the product of specific storage and aquifer thickness. According to Singhal and Gupta (1999) storativity in unconfined

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39 aquifers generally ranges between 0. 05 and 0.30. Milzow et al., (2009) use a specific yield (or drainable porosity) of 0.05 in a MODFLOW model of the Okavango Delta. The calibrated values of S ranged between 0.001 and 0.05 throughout the basin. The uncertainty for DDENS T S RWL and GW S are set to medium. Morris Method global sensitivity a nalysis The GUA/GSA of the Pitman model begins with a qualitative screening assessment using the Morris Method (Morris, 1991). The PDF for each model input ( X i p discrete levels. A matrix of model runs is then formed by input space for each PDF. Though the sensitivity calculation is one at a time derivative, it may actually be considere d global because it samples throughout the multivariate parametric space. The elementary effects ( F i ) are obtained according to the equation (2) Morris proposes two sensitivity indices: i and i i represents the average magnitude of change in the model output resulting from varying each input within its PDF. This is essentially the direct importance or significance of the input. Because the model is run in batch with varying inputs sets, each i depends on all of the other inputs and can vary in each model run. Thus, i is the st andard deviation of i and estimates higher order relationships such as nonlinear and interaction effects. The Modified Method of Morris (Campolongo et al., 2007) has a number of improvements over the original Method of Morris. It allows for an analysis of models with multiple outputs, allows factors to be grouped, and has a more effective sampling

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40 strategy at no additional computation cost. Furthermore, the enhanced sensitivity index * is approximately as good as indices which are based on variance met hods (Campolongo et al., 2007). Fourier Amplitude Sensitivity Test (FAST) v ariance b ased g lobal s ensitivity a nalysis Once the most sensitive factors are identified using the Modified Method of Morris, a more quantitative sensitivity analysis was conducte d on only the most sensitive factors using the variance based Fourier Amplitude Sensitivity Test (FAST) (Cukier et al., 1978; Koda et al., 1979). FAST uses Fourier analysis to decompose the variance of a model output into variances for each input. The Ex tended FAST technique (Saltelli et al., 1999) allows for the quantification of higher levels of variance. These higher levels of variance describe the interactions of inputs. V(Y) is the summation of the first order variance in each input and also the r esidual which is the variance attributed to all interactions. Thus, V(Y) describes the total variance of a single input including higher levels of variance (Eq. 3). (3) FAST also defines S i as a measure of global sensitivity. S i is the ratio of the variance that is attributed to a single input divided by the total model variance. In a model in which there are no interactions the sum of all S i s across all inputs is equal to one. In models where there are interactions this sum will be greater than one. For the extended FAST GSA/UA, a model is run for C = Mk iterations, where k is the number of inputs, an M Since the extended FAST method uses a randomized sampling procedure, it provides an extensive set of outputs that can be used in the global, Monte Carlo type uncertainty

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41 percentile statistics can be derived for each output of interest. Climate c hange u ncertainty The previous methods are used to show how the model behaves under stationary conditions. However, models are often used to model non stationary scenarios such as climate change where the baseline conditions change. Under these conditions, the original validation may no longer be valid. This section presents a method to investigate the change in model uncertainty under a non stationary climate change scenario. The best fit model simulations (ceff > 0.70) are inspec ted to identify the variety of input scenarios that provide a good model fit in the stationary simulation. A threshold ceff of 0.70 is chosen which allows 0.31% of the runs (466) to be included as best fit simulations. This number was determined to be ap propriate for both the number of passable simulations. These simulations are chosen from the FAST three region model and are selected only from those runs that have a c eff of greater than 0 in each of the three regions. The three region version is chosen over the five region version because the model was originally calibrated using the three region approach. The prediction of increasing temperatures in the area is fair ly well established in the area. However, there is less agreement concerning future precipitation. the Okavango Basin will increase between 2.3 and 3C and rainfall will increase bet ween 0 and 20% within the next 30 years. work to downscale for the Okavango. O ther studies in the area show a variety of rainfall predictions ranging from drier to wetter conditions (Andersson et al., 2006; Murr ay

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42 Hudson et al., 2006; Milzow et al., 2008 ; Todd et al., 2008; and Wolski and Murray Hudson, 2008). These predictions generally range from a 10% increase to a 15% decrease in rainfall. or this work, a scenario that involves an increase in temperature of 2.6C and an increase in rainfall of 15% will be considered. Additionally, to address the drier scenarios, and for symmetry, a scenario that involves an increase in temperature of 2.6C and an decrease in rainfall of 15% will be considered The uncertainty is compared between the original scenario and the two climate change scenario s The goal of this exercise is not to calculate the uncertainty of the climate change scenario s but rather to investigate the difference between t he uncertainty of the model under stationary versus non stationary scenario s. S hifting conditions into drier and wetter conditions allows an investigation into whether the model becomes more disorganized under wetter conditions and more organized under dr ier conditions or if there is a linear relationship between the amount of water in the system and the degree of uncertainty. Results The Morris and FAST GSA/UA methods were run using a three region approach that includes the western headwaters, eastern he adwaters, and southern basins as well as a five region approach in which the southern basin is subdivided into middle, Omatako, and southern regions (Figure 2 2 ). The objective functions are the mean annual flow and the ceff of the monthly flows. Morris M ethod In the Morris Method three region approach the importance of both direct effects (x axis) and indirect effects or interactions (y axis) is evident (Figure 2 3 ). The western

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43 region contributes the most important inputs. Also of interest are the inpu ts ZMAX and RDF which are important in all three regions. These inputs relate to rainfall and interception (Table 2 1). The inputs that are less important are identified and set as constants in the next quantitative and more computationally intensive var iance based sensitivity analysis. A cutoff of 0.2 on the direct effect axis was used to separate the important inputs from the less important. This cutoff excludes the inputs that are clustered around the origin and for clarity are not labeled. Though th is cut off is based on visual inspection, the low level of importance shown for some of the input factors that are still included later in the following variance based analysis demonstrates that the cutoff is satisfactory. In the Morris Method five regio n approach, as with the three region approach, both the direct and indirect effects are important as evidenced by the even spread on both the x and y axes (Figure 2 4 ). Similarly, ZMAX and RDF continue to be important in most of the five regions. A cutof f at 0.1 on the direct effects axis is used to distinguish important inputs for further analysis in the FAST GSA/UA ( Figure 2 5 ). Again, as with the three region approach, the unimportant inputs are set to constants in future analyses. The low level of i mportance shown for some of the input factors that are still included in the following analysis demonstrates that the cutoff is satisfactory. The regional approaches are compared to identify how regionalization affects input importance ( Figure 2 5 ). The addition of the Omatako and middle regions does have an effect on the GSA results; however this effect is not as overwhelming as might be expected given the proportional size of the new regions. Only two inputs from the

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44 Omatako region and one from the mid dle region fall above the cutoff for important inputs in the five regional approach. FAST Global Sensitivity and Uncertainty Analysis After the Morris Methods narrowed down the most important inputs, the variance based FAST analysis was then used to quan titatively compute the uncertainty and sensitivity in the three and five regional approaches. The same two objective functions were considered: the mean annual flow and the ceff. Again the ceff is the preferred objective function because it is able to ta ke into account the monthly variation in flow. However, when conducting the uncertainty analysis the use of the ceff becomes problematic because the uncertainty is measured between regions. This comparison is not practical for three reasons. The first re ason is that measured data is required to compute the ceff and in the eastern headwaters and Omatako region there is a limited amount of measured flow data. During calibration, 12 to 69 monthly measured flow values were available for calibration in the mi ddle reservoirs and none were available in the Lower Omatako ( Hughes et al., 2006) The second problem is that the sub basins must confluence in order to compute a ceff for an entire region. In the eastern headwaters and the middle region the sub basins do not confluence. Finally, given the calibrated inputs (Hughes, 2006), the eastern headwater region underestimates monthly flows and results in a poor model fit and a low ceff. In the equation for the ceff the numerator and denominators are both squared and as a result the poorer fit runs have exaggerated values for the ceff. Because of these issues, the uncertainty analysis is conducted using the mean annual flow as the objective function when comparing regions. But when the uncertainties of the regi onal approaches are compared at the

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45 southern terminus of the River, the ceff is used as the objective function. For the GSA these issues are not a problem and the ceff is the objective function. For the GUA the mean annual flow is presented as regular and also normalized by the regional catchment areas. This is done because the contributing area for the southern region is much larger and actually contains the water from the eastern and western Regions and so it has larger flows and as a result larger unce rtainties associated with those flows (Figure 2 6 ). Through normalization the differences in the flow variability due solely to the sizes of the regions are negated. Normalizing the three regions by their areas shows the southern region becoming relativ ely more certain and the eastern region relatively less certain (Figure 2 6 ). Normalizing the five regions by their areas shows the southern region becoming more certain and the eastern and western region becoming less certain (Figure 2 6 ). Omatako shows very little normalized uncertainty, which is likely due to the small quantity of water that it contributes due to its arid climate. The three and five region 2 7 ). This analysis shows that the addition of two regions does not add to the uncertainty of the model results and are therefore somewhat superfluous from a management perspective. The statistics of the uncertainty analysis confirms the similarities (Table 2 2 ). This tabl e can also be used to compare the results of this study to those of Hughes et al. (2010). In Hughes study the ceff that the Pitman model produced for the uncertainty analysis ranged between 0.21 and 0.83 in various South African Basins. However, here the mean is 0.23 to 0.25. The comparatively poor fit that is achieved in the Okavango Basin is likely due to selection of probability distributions that were

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46 defined in this study and the catchment size. This study assigned uniform distributions to the inpu ts while Hughes et al. (2010) used triangular distributions. Additionally, this is a much larger watershed with 24 sub basins while the Hughes study focused on smaller watersheds (174 898 km 2 ) with a maximum of three sub basins. The quantitative FAST GSA was also conducted for both of the regional simulations (Figures 2 8 and 2 9 ). In each of these simulations ZMAX in the western region is the most important input. Additionally, the western RDF, southern RDF, and southern ZMAX show a high level of impor tance in each regional simulation. Both of the regional simulations display a relatively large degree of interactions and higher order indexes which is in agreement with the Morris results. Climate Change Uncertainty Hydrologic models are frequently used to predict conditions under non stationary scenarios. In these situations the original validation that was performed under the stationary conditions may not be valid under the non stationary condition because of changing internal relationships This iss ue is possibly exacerbated when there are equifinality concerns because there may be many equally probably input sets that are able to simulate equally good outputs and no way solution When the state of the system changes and internal relationships are modified, what was once an equinfinal and good fit result may no longer be so. Investigating the degree of equifinality and th e change in uncertainty between stationary and non stationary scenario s (such as climate change) gives insights into the usefulness of the Pitman model as a predictive tool. For this analysis, the best fit model simulations from the FAST three region GSA/UA were isolated to identify the variety of input scenarios that provide a good

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47 model fi cobweb plots, input trajectories are shown that map the top 30 model simulations (Figure 2 10 ). The lines in this figure demonstrate the various trajectories that best fit inp ut sets may take as well as the input vales themselves. The lack of pattern or trend shows that there are many varying input sets that are able to create a good model fit and thus a high level of equifinality. This is expected given the data scarcity, t he wide input and the uncertainty was compared between the original scenario and two cl imate change scenario s In the climate change scenario s rainfall was increased and decreased by 15% and temperature was increased by 2.6 C throughout the basin. The goal of this exercise was not to calculate the uncertainty of the climate change scenari o, but rather how the uncertainty of the model changes under possible non stationary condition s Results show an average increase in flow between the original simulation and the wet climate change scenario by 28% and an average decrease in flow between t he original simulation and the dry climate change scenario by 50 % and Th e uncertainty analysis show surprising results. Dividing the 95% confidence interval by the average annual flow gives a lower relative uncertainty for the wet scenario (16.9%) than for the current scenario (18.9%). And the dry scenario has the h ighest relative uncertainty (30. 5%) (Table 2 4) This demonstrates that the relative uncertainty of the model decreases just slightly under wetter conditions but increases more significantl y under drier conditions

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48 Summary Evaluating model uncertainty and identifying gaps in knowledge are important for the Adaptive Management process. Doing so gives managers a measure of confidence in model results and identifies areas for strategic monit oring whereby model results can be improved. This two step and multi tiered regionalization GSA/UA of the Pitman model highlights the most important and least important inputs, quantifies model uncertainty, compares regional groupings, and explores the un certainty of a non stationary model scenario. The conclusions are useful when interpreting model uncertainty, determining the usefulness of model predictions, identifying gaps in knowledge, and focusing monitoring efforts. Through the sensitivity analysis the most important and least important inputs were identified within each region. The inputs are qualitatively ranked free of regional specification based on the ceff (Table 2 3 ). The ranking is based on the summation of the Morris * and sensitivity indexes in both regional simulations for each input. The most sensitive inputs are GPOW RDF ZMAX FF and GW The least sensitive inputs are T S RWL DDENS and GWS Input importance is also mapped on the model structure to identify th e most important processes (Figure 2 11 ). This map shows that rainfall distribution, catchment absorption, evapotranspiration, and groundwater recharge are the most important processes and groundwater runoff is the least important process. This map is es pecially useful in the NRC (2004) Adaptive Management processes step 2: developing and revising the model. Using the GSA results, the most sensitive inputs (i.e. GPOW RDF ZMAX FF and GW ) should be focused on first through additional research and monit oring. This will close the data

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49 equally good model fits. Another point of interest is the large number of important inputs from the western region in both the three an d five region simulations (Figure 2 12 ). This is likely the result of the distribution of rainfall in the northern portion of the basin and the larger size of the western headwater region relative to the eastern headwater region. The majority of the rain falls in the headwaters and this initial input drives much of the system. In the calibrated model, the eastern and western headwaters receive on average between 1000 and 1280 mm of rainfall yr 1 while the southern region receives between 540 and 640 mm y r 1 Additionally, the western region is much larger than the eastern region and occupies 22% of the total basin the 11% that is occupied by the eastern region. Because of the importance of the headwater regions, monitoring efforts that seek to improve m odel results should focus on these areas. The uncertainty analysis that is normalized by the regional areas shows that the flows at the downstream outlet are more certain than those in the headwater regions. This is not necessarily intuitive as one might initially think that uncertainty would be compounded in the downstream direction. This effect may be due to the more arid climate in the southern portion of the watershed. In this region there is less input of water and more evapotranspiration. The sou thern reaches of the rivers are losing reaches with the sum of the calibrated mean annual flow in the eastern and western headwaters summing to 10,128 m 3 x10 6 and the mean annual flow at the southern outlet at Mohembo only 8,465 m 3 x10 6 (Hughes et al., 2006) Thus, since there are fewer important hydrologic inputs in the southern portion of the basin, the certainty of the

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50 mean annual flow, when normalized by the area of the contributing basin, is increased in the downstream direction. This reinforces how fo cusing monitoring efforts in the headwaters will strategically decrease model uncertainty. there are multiple input sets that can simulate the same goodness of model fit. This issue of equifinality was further analyzed to evaluate the effects that it may have on using the model in a predictive capacity for a wet and dry climate change projection It was shown that, after isolating the best fit model simulations, the relative u ncertainty of the average annual flow decreased slightly under the wet condition but increased more significantly under the dry condition. A manager may interpret the increase in uncertainty under the dry condition and being significantly different from the currently Data scarcity presents many challenges for modelers and policy makers when calibrating, interpreting model results, and applying models in predicative and dynamic capacities. However, there are ways of managin g for these challenges. Adaptive Management is specifically designed for determining a course of action within the context of uncertainty. The tools presented here are couched within the context of Adaptive Management and quantitatively analyze model un certainty in the light of practical management issues. This work identifies which inputs should be focused on to reduce input/output model uncertainty ( GPOW RDF ZMAX FF and GW ) and also shows how the uncertainty of the model changes under a wet and dr y scenario. It is predictions that forecast drier conditions.

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51 Table 2 1. Input factor probability density functions ( PDF's ) for the two regional simulations. [Low 15 %, Medium 30%, High 45% of calibrated values.] Parameter Description Uncertainty RDF ( ) Rainfall distribution function Medium PI1 (mm m o 1 ) Interception for veg type 1 Medium PI2 (mm m o 1 ) Interception for veg type 2 Medium AFOR ( ) Percent area cover ed by veg type 2 Low FF ( ) Evaporation scalar for veg type 2 Medium ZMIN (mm m o 1 ) Minimum infiltration rate High ZMAX (mm m o 1 ) Maximum infiltration rate High ST (mm) Maximum soil moisture storage Medium POW ( ) Power function for unsaturated runoff H igh FT (mm m o 1 ) Maximum unsaturated zone runoff High GW (mm m o 1 ) Maximum groundwater runoff High R ( ) Actual versus potential evaporation Medium GPOW ( ) Power function for groundwater runoff High DDENS (km km 2 ) Drainage density Medium T (m 2 d 1 ) T ransmissivity Medium S Storativity Medium GWS ( ) Groundwater slope Medium RWL (m) Rest water level Medium RSF (%) Riparian strip factor High ( ) Denotes unitless input

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52 Table 2 2 Comparison of the three and five region uncertainty analyses for two objective functions (1) the ceff for monthly flow at Mohembo and (2) the Mean Annual Flow in million cubic meters ( MCM ) at Mohembo. Mean Median SD SEM Skew Kur Coefficient of Efficiency 3 Region 0.226 0.411 9.77e 3 4.98e 3 3.16 13.6 5 Region 0.252 0.405 7.88e 3 4.02e 3 3.20 15.2 Mean Annual Flow 3 Region 8399 8327 1941 16.5 0.472 0.290 5 Region 8365 8242 1791 13.6 0.398 0.275 SD = standard deviation; SEM = standard error of the mean; Skew = skewness, kurt = kurtosis Table 2 3 Ranking of importance of input variables Rank ( ) Input Uncertainty of PDF ( ** ) Important 1 RDF Medium 2 ZMAX High 3 GW High 4 FF Medium 5 GPOW High Medium Importance 6 ST High 7 R Medium 8 PI2 Medium 9 FT High 10 ZMIN High 11 POW High 12 AFOR Low 13 PI1 Medium Less Important 14 DDENS Medium 15 S Medium 16 T Medium 17 RSF High 18 GWS Medium 19 RWL Medium (*) Importance desi gnated by the Morris Method through a total sum of first and higher order interactions. Important greater than 10, 10 < Medium importance < 1, Low importance < 1. (**) Low 15%, Medium 30%, High 45% of calibrated values.

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53 Table 2 4 Stationary versus climate change confidence intervals (CI). Units in flow (MCM yr 1 ). Climate change conditions both include an increase in temperature. Stationary Wet Climate Change Dry Climate Change Lower 95% CI 7,373 9,923 4,216 Average 8,108 10,822 4,851 Upper 9 5% CI 8,897 11,747 5,694 CI / Average 18.8% 16.9% 30.5% Figure 2 1 Model conceptualization.

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54 Figure 2 2 (a) three and (b) five regional simulations of the Okavango Basin.

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55 Figure 2 3 Morris Method three region Global Sensitivity Analysi s ( GSA ) results for the coefficient of efficiency ( ceff ) of monthly flow at Mohembo. Abbreviations are defined in table 2 1.

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56 Figure 2 4 Morris five region sensitivity analysis for the ceff of monthly flow at Mohembo. Abbreviations are defined in ta ble 2 1.

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57 Process Input Three Regions Five Regions West East South West East South M iddle Omatako Rainfall RDF Interception PI1 PI2 Infiltration ZMIN ZMAX Evapotranspiration (ET) RSF AFOR FF Actual/Potential ET R Unsaturated storage POW FT Unsat. GW storage ST Groundwater (GW) storage GW GPOW T S RWL G W discharge DDENS Interbasin flow GWS Figure 2 5 importance. Gray shading indicates agreement in importance/unimportance between the three and five region approaches.

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58 Figure 2 6 Fourier Amplitude Sensitivity Test (FAST) uncertainty analysis for annual mean monthly flow at the outlet of each of the regions. Flow is shown as (a) three region regular, (b) three region normalized by catchment area, (c) five regio n regular, and (d) five region normalized by catchment area.

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59 Figure 2 7

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60 Figure 2 8 FAST three region sensitivity analysis for the ceff of the monthly outflow at Mohembo. First order sensitivity in grey, higher orders in black, and the total sensitivity as the entire bar.

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61 Figure 2 9 FAST five region sensitivity analysis for the ceff of the monthly outflow at Mohembo. First order se nsitivity in grey, higher orders in black, and the total sensitivity as the entire bar.

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62 Figure 2 1 0 Cobweb plot of best fit model trajectories. Input values are normalized (0 1) by the ranges of their assigned probability density functions ( ) Each line represents one of the top 30 best fit lines from the three region simulation. These runs are chosen only from those that have a ceff greater than 0 in all regions.

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63 Figure 2 1 1 Map of the model structure and the identification of the most i mportant processes. Black fill indicates the most important inputs, dark grey is medium importance, light grey is low importance, and white indicates no importance (boundary conditions or outputs). Structure adapted from Kapangaziwiri (2008) and Hughes e t al. (2006).

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64 Figure 2 1 2 Total sensitivity of ceff for outflow at Mohembo for (a) 3 and (b) 5 regional scenarios. a b

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65 CHAPTER 3 OBJECTIVELY DEFINING UNCERTAINTY IN THE FACE OF DATA SCARCITY USING A RESERVOIR MO DEL OF THE OKAVANGO DELTA, BOTSWANA A bstract Sensitivity and uncertainty analysis are tools that give mangers quantifiable confidence in model results and give modelers and researchers guidance for optimizing models. The Okavango Delta in Botswana is a large and unique inland Delta and one of the most pristine large wetlands in Africa. The Okavango Research Institute (ORI) hydrologic model is a linked reservoir model that is overlaid by a GIS grid model that was developed to simulate the affect that the annual flood pulse has on the Delta. T his model has been calibrated and validated in the Delta however; a formal sensitivity and uncertainty analysis has not yet been conducted on the model. The objective of this research is to conduct a global sensitivity and uncertainty analysis (GSA/UA) on the ORI hydrologic model of the Okavango Delta in order to identify the most important model inputs, to quantify the uncertainty in the model output, and objectively define prior probability distributions through Monte Carlo Filtering. This research app lies two GSA/UA methods to the ORI model: the modified Method of Morris and the extended FAST technique. The Morris method is a qualitative GSA screening technique and the extended FAST method is a quantitative variance based GSA/UA technique. One of the largest criticisms of the GSA process is the rather arbitrary methods for establishing probability distributions ( ). Through this work, Monte Carlo Filtering is shown to be an objective method for redefining prior and reducing model uncertainty Results indicate that the most important inputs are the volume thresholds, which determine the volume at which water will begin to spill out of each reservoir, followed by

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66 the floodplain porosity, which is important for infiltration, and the island exti nction coefficient, which is used to simulated transpiration. There is one reservoir, Nqoga1, which is highly interconnected and may be considered a keystone reservoir. Monte Carlo filtering was used as an objective method for reassigning the and r educing model uncertainty. These results identify Nqoga1 as a critical area for monitoring and managing within the Delta, refine and reduce input/output model uncertainty, and present a general method, useful beyond this work, for objectively developing input especially where data is scarce. Background Because models are simplifications of reality they are inherently uncertain. Ignoring model uncertainly undermines the value of a model for its use in the decision making process (Beven, 2006a). Un derstanding this uncertainty is important for several reasons. Understanding uncertainty allows decision makers to acknowledge the reliability of models when weighing the risk of various decisions (Saltelli et al., 2008). In addition, identifying sensiti ve factors allows for the optimized allocation of resources for future data collection leading to the refinement of the input values and a reduction of model uncertainty. Sensitivity analysis can also provide an understanding of how input factors interact within the model structure. In a complex model, these interactions may not be obvious and may have important impacts on the model output due to their non additive nature. (Saltelli et al., 2008) Adaptive Management (AM) acknowledges uncertainty through a cyclical process that adapts and improves based on continuous monitoring and assessment of goals and is advocated for complex problems (Walters and Holling, 1990). Modeling is an integral part of the AM process as models are able to simulate large scal e

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67 experiments and scenarios. As a cyclical process, through AM models are continuously improved based on new information. Global Sensitivity and Uncertainty Analysis (GSA/UA) and Monte Carlo Filtering (MCF) are two tools that can be used to systematicall y understand and reduce the input/output reliability (Scott, 1996; Saltelli et al., 2008). GUA quantifies model uncertainty and GSA apportions the total model uncertainty to each of th e inputs. One of the biggest criticisms of GSA/UA is the rather arbitrary setting of prior MCF can be used to objectively redefine prior to make the results of the uncertainty analysis more realistic when there is no physical or empirical dat a to base the on otherwise. This is particularly an issue in data scarce areas where there is little data available for defining the Using MCF to redefine prior based on realistic or acceptable model results gives an objective way to s et prior This is particularly useful for management purposes because only realistic model results are investigated. Redefining these allows a GUA that more accurately represents the calibrated values and gives insights into the inner working of the model. This paper presents the work of a GSA/UA and MCF on a reservoir model in the Okavango Delta, Botswana. The Okavango Basin is a large transboundary watershed in southern Africa. Within the Basin, the Okavango River feeds the Okavango Delta, which is a large inland delta. The Delta is an alluvial fan that spreads out on the edge of the Kalahari Desert. The water from the Basin is evapotranspirated in the Delta and never reaches the sea. This inland Delta and its hydrology are particularly important

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68 and unique because it supports a diversity of wildlife and people who otherwise exist in an extremely dry environment (Kgathi et al., 2006). From 1975 to 2002 Angola, the largest landholder in the basin, has been in a civil war and as a result e conomic development in the area has been limited (Kgathi et al., 2006). Current peace in the area has removed previous barrier for development and commercial expansion and the impacts on natural resources that may follow may be imminent. Additionally, cl imate change projections show various scenarios for the future of the area which could alter the current flooding patterns of the Delta (Milzow et al., 2010, Wolski 2009). As policy makers in the area face management decisions, the availability of a hydro logic model that can simulate flows within the Delta with some known degree of certainty will give managers the appropriate confidence in predictive modeling scenarios. The relatively empirical approach of the reservoir model has persisted due to the Del The Delta is located in northwest Botswana, no roads traverse the area, and trips to the internal portion are possible only by air or boat. Furthermore, the system is quite large. Flooding extents in the Delta range from 6,000 12,000 km 2 (Wolski et al., 2006) and the entire geologic alluvial fan is approximately 40,000 km 2 (Gumbricht et al., 2005). The area also is extremely smooth and flat. Maximum local relief is generally less than 2 or 3 m (McCarthy et al., 2003; Gumbricht et al., 2001) and the standard deviation of transects are approximately 5 6 /1 km/km (Gumbricht et al., 2005). Therefore, as in most wetlands, flow direction and inundation area is determin ed by very slight changes in elevation. Traditional techniques that measure changes in elevation over this scale do

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69 not capture the shallow relief. Furthermore, temporal variations in flow are caused by poorly understood and hard to model processes such as hippopotami movements, sedimentation, and vegetation. Overall, data scarcity is a significant issue in this area. A reservoir modeling approach has often been used to simulate flows in the Delta (Dincer et al., 1987; SMEC, 1990; Scudder et al., 1993; Gieske, 1997; WTC, 1997; Wolski et al., 2006). A review of these models can be found in Kiker et al. (2008) and Wolski et al. (2006). In the ORI model, flows in the Delta are routed through linked reservoirs. The ORI links the reservoir model to a GIS gr id model which floods individual I km 2 pixels. The inundated status of each pixel is based on probabilities gathered from historical satellite imagery of the flooding extents. This model was developed specifically for the Okavango Delta and the outputs inc lude reservoir flooding extents as well as outflows from the distributaries. The ORI model has been calibrated and tested (Wolski et al., 2006); however, neither a formal uncertainty nor sensitivity analyses have been conducted on the model. Such an unce rtainty and sensitivity analysis will give reliability in the model providing managers with confidence in model results. This process will also identify key areas and processes for managing and monitoring. Outputs from the ORI model have been used by poli cy makers in the area (DEA, 2008). The Okavango Delta Management Plan (2008) references the use of this model for predicting the impacts of abstractions, seismic activity, climate change, and channel clearing. This model is currently being used to predict climate change scenarios (Wolski, 2009). provide confidence in model results, and identify areas where future work may enhance

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70 model performance. This is done by conducting a GSA/UA and MCF on the model. Prior probability density functions ( ) will be developed to represent the range of plausible values for any given input. Inputs factors here will be defined as non time series variables in the model. The GSA/UA will us e a framework in which the Modified Morris GSA (Morris, 1991; Campolongo et al., 2007) screening method will be employed followed by the quantitative extended FAST GSA/UA technique (Cukier et al., 1978; Koda et al., 1979; Saltelli et al., 1999). Monte Car lo Filtering (Saltelli et al., 2008) will be used to objectively redefine and refine model uncertainty. Through this methodology, the sensitivity and uncertainty of the model will be quantified, model uncertainty will become more accurate and precis e, and a deeper understanding of the model structure and reliability will be gained. For the purposes of this research and in of the Delta as well as the individual r eservoirs will be the objective function. This takes into account not only the entire Delta, but also the individual reservoirs and their behaviors. The inundation area is an appropriate objective function because in the absence of good topographic data i t is a proxy for volume, and is an important social and ecological factor with regards to floods and droughts. Methods The Study Site: Okavango Delta, Botswana The Okavango Basin is a large transboundary [maximum flooding extents 12,000 km 2 (Wolski et al. 2006)] watershed located in southern Africa that is shared between three countries: Angola, Namibia, and Botswana (Figure 3 1). River flow originates in the Angolan portion of the Basin where rainfall ranges from 1600 mm yr 1 in the north to 600 mm yr 1 in the south. Ninety five percent of the runoff generating area is contained

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71 within Angola (Hughes et al, 2006). The two downstream basin nations: Namibia and Botswana are much more arid with rainfall averaging approximately 450 mm yr 1 Angola is also more mountainous, with elevations ranging between 1,200 and 1,800m, while the Botswana portion of the Basin is extremely flat, with elevations ranging between 900 and 1,000m. Water flows in a delayed annual flood pulse through the Okavango River from An gola, through Namibia, and then into Botswana, which contains the Okavango Delta. Rainfall in Angola occurs between December and March. It takes two months for this flood pulse to reach the top of the Delta and another four to five months to attenuate th rough the Delta, finally reaching the distal ends from August through October (Wolski et al., 2006). This water never finds the sea and instead spreads out over the flat alluvial fan that is the Okavango Delta and is then evaporated on the border of the K alahari dessert. The Okavango Research Institute Model model created specifically for the Okavango Delta (Figure 3 2 and 3 3). The ORI model represents flood duration, floo d frequency, flooding extents, and outflow from the Boro River at Maun (Figure 3 2). The model operates on a monthly time step. Flow is input at the top of the Delta according to gauge data. The volume of water in each reservoir is then simulated by the continuity equation where V is the volume of water, I is inflow, ET is evapotranpiration, P is precipitation, Q is outflow, and Q inf is infiltration to groundwater (Wolski et al., 2006). (1)

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72 The inundated area is related to the volum e of water through a power relationship. In this equation A is the inundated area, V is the volume of water, and n and b are empirical coefficients which are based on topographic data. (2) Upstream reservoirs may have several outle ts and can feed more than one downstream reservoir. Model inputs for each reservoir include area, topography, evaporation, and flow variables (Table 3 1 and Figure 3 3). Some of these inputs are constant throughout the model (extinction depths, porositie s, and n in the area/volume relationship), and some vary between the reservoirs ( b in the area/volume relationship, delay, rainfall ratio parameter, and groundwater reservoir areas), and some vary between the reservoir connections (flow resistance and volu me threshold). Outputs from the reservoir model include monthly inundation area and outflow from the Boro distributary. Groundwater flow and infiltration processes are represented by a series of sub reservoirs including the surface reservoir, two floodpla in groundwater reservoirs, and two island groundwater reservoirs. Water in the surface reservoir infiltrates into the floodplain groundwater reservoirs. Water in the floodplain groundwater reservoir then flows to the island groundwater reservoirs. The s econd island and second floodplain groundwater reservoirs are activated based on the volume of the flood. Actual evaporation is based on potential evapotranspiration. Transpiration is not simulated with specific regard to types of plants. Evapotranspirat ion in non inundated areas uses the concept of an extinction coefficient which simulates a linear decrease in the rate of evapotranspiration with depth from the surface. There is little meteorological

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73 data available for the interior of the Delta. However a weather station in Maun, on the border of the Delta, has time series data from 1970 to the present. The model uses the Penman Montieth equation (Allen et al., 1998) to calculate the reference crop transpiration. These calculations were adjusted based on weather station measurements made inside the Delta and also with measurements from an eddy covariance system located at Maun and Mxaraga (Wolski et al., 2006). Rainfall is input over the inundated areas of the Delta based on an inverse distance wei ghted relationship between two weather stations in the area. The rainfall ratio parameter is used to vary the rainfall in the reservoirs based on this relationship. In the model, rain within the Delta is not simulated as falling on the non inundated area s and cannot directly raise the water table The ORI model links this reservoir model to a GIS model to simulate the spatial distribution of the flood. The GIS model is based on satellite imagery of flooding extents. A Gaussian probability density functio n (PDF) was assigned to each pixel describing the likelihood of the pixel being inundated given the size of a flood. Then, given a volume of water in the reservoir model each pixel is assigned an inundation/non inundation status based on the more likely p robability. The model was calibrated manually through trial and error adjustments of the inputs (Wolski et al., 2006). The objective functions for calibration were the flooding extents and outflow from the Boro River. The estimation of flooding exten ts is derived from satellite imagery available from McCarthy et al. (2003). The Delta was gridded and in each image cells were assigned a flooded/non flooded status. Outflow from the Boro River is gauged in the river.

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74 Results from the ORI model show good correlation with observed data. The model produces a monthly inundation area that compares to observed data with a root mean squared error (RMSE) of 528 km 2 and a correlation coefficient of 0.90 for the entire Delta (Wolski et al., 2006). Additionally, the outflow at Maun shows a RMSE 0f 11.8 M m 3 /month and a correlation coefficient of 0.91 when compared to observed flows (Wolski et al., 2006). Global Sensitivity and Uncertainty Analysis There are a number of different approaches available for conducti ng a sensitivity analysis on models ranging from simpler one at a time (OAT) variational methods to advanced global techniques (Salteli et al., 2008). With OAT techniques the variation of the model output is investigated through changing a single model in put at a time. This traditional sensitivity analysis method is limited since it usually explores a small parametric range, can only consider a few inputs, and does not account for interactions between inputs (Saltelli et al., 2008). When model responses a re non additive and non linear, as with many complex models, simple OAT techniques are not appropriate and global techniques that evaluate the input factors of the model concurrently over the whole parametric space (described by probability density functio ns) are more appropriate. Different types of global sensitivity methods can be selected based on the objective of the analysis (Cacuci, 2003; Saltelli et al., 2008). This study uses a model evaluation framework around two such modern global techniques, a s creening method (Morris, 1991) and a quantitative variance based method (Saltelli et al., 1999). This two step process has been used in the examination between input output relationships in several environmental investigations (Muoz Carpena et al., 2007; Muoz Carpena et al., 2010; Fox et al.,

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75 2010; Jawitz et al., 2008; Chu Agor et al., 2011). The screening method allows an initial reduction in the number of inputs to use in the more computationally intensive quantitative FAST sensitivity and uncertainty analyses. Global sensitivity methods can yield four main products for the Okavango application. (1) GSA can be used to explore model behavior and give an assurance of the absence of errors (Saltelli et al., 2008). (2) GSA provides a ranking of the import ance of the inputs. This is useful in prioritizing future data collection and calibration techniques (Saltelli et al., 2008). (3) GSA also indentifies the type of influence of the important inputs (first order or higher interactions) (Saltelli et al., 200 8). (4) An uncertainty assessment of the model can also be used as the basis for the risk assessment of proposed management scenarios. Modified Method of Morris The GSA/UA of the ORI model begins with a qualitative screening assessment using the Method o f Morris (Morris, 1991). The Morris method is less computationally intensive than other quantitative methods and is therefore used to screen for the most sensitive or important inputs. The least important inputs are then set to constants and the most impo rtant inputs are further examined using the FAST quantitative GSA/UA. Each model input (X i PDF which is divided into p levels. An input matrix is formed that describes the region of experimentation with k factors and p levels. Values are selected from this matrix for model simulations according to an efficient sampling trajectory. When performing the Morris method, the model is run for n(k+1) iterations, where k is the number of inputs and n may vary between 4 and 10.

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76 Th e probability distribution function (PDF) of elementary effects (F i ) is obtained through randomly sampling values in the parameter space (X 1 X 2 i ) and then computing according to Eq. 3 (3) Morris proposes two sensitivity indic i representing the the standard deviation of F i which estimates higher order relationships such as nonlinear and interaction effects. This stand ard deviation represents the spread of the importance that each input is responsible for based on values chosen for the other inputs. The Modified Method of Morris (Campolongo et al., 2007) has a number of improvements over the original Method of Morris. It allows an analysis of models with multiple outputs, allows factors to be grouped, and has a more effective sampling strategy at no additional computation cost. Under the Morris method elementary effects are local OAT measures but this method may actua lly be considered global because * is an average of the elementary effects. Therefore, the sensitivity index * is approximately as good as indices based on variance methods. On the Morris scatter plots the x axis ranks the direct or first order of inpu t factor importance and the y axis ranks the higher order interactions of the input factors. These higher order interactions represent the range of the importance of an input that depends on the values for the other inputs. Quantitative v ariance b ased FAS T g lobal s ensitivity and u ncertainty a nalysis Once the most sensitive factors are identified using the Modified Method of

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77 Morris, a more quantitative sensitivity analysis is conducted on the most sensitive factors using the Fourier Amplitude Sensitivity Test (FAST) (Cukier et al., 1978; Koda et al., 1979). FAST uses Fourier analysis to decompose the variance of model outputs into first order variances for each input. The Extended FAST technique (Saltelli et al., 1999) also allows for the quantification of higher levels of variance. These higher levels of variance describe the interactions between inputs. V(Y) describes the summation of the first and higher order interactions as the total variance of a single input according to Eq. 4. (4) V is then the summation of the first order variance for all of the inputs ( V i ) and also the residual which is the variance attributed to all interactions (V i, j, etc ). FAST also defines S i as a measure of global sensitivity. S i is the ratio of the variance that is attributed to a single input divided by the total model variance. In a model in which there are no interactions the sum of all S i where there are interactions this sum will be greater than one. One limitation of this variance based technique is that it cannot be used on factors that are interdependent (Crosetto and Tarantola, 2001). When performing the extended FAST GSA/UA, a model is run for iterations, where M is a numb f is the number of inputs (Saltelli et al., 1999). Both the Modified Method of Morris and Extended FAST rely upon sampling values from for each uncertain input factor and running the model based on these sampled data sets. SimLab v3.2.6 (SimLab, 2011) software was used to sample input and construct sets of model inputs. The model is then run multiple times

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78 based on the input sets created by SimLab. The model results are combined into SimLab with the inputs to obtain to Method of Morris statistics and FAST indexes. Monte Carlo Filtering Monte Carlo Filtering (MCF) is an additional method that can be used to decrease input/output uncertainty (Saltelli et al., 2008; Rose et al., 1991). The Okavango is an area where data is particularly scarce and as a result the prior are wide and rather arbitrary. MCF can be used to objectively redefine and truncate prior to reduce and refine input/output uncertainty. In MCF, a threshold is established such that model outputs are classified as either behavioral ( B ) or non behavioral ( ). Based on whether or not the output exceeds the behavioral threshold, two subsets of the input set are defined as X i / B and X i / To deter mine if X i / B is statistically different from X i / the two sided Smirnov test is used on each input factor independently. The null hypothesis (H o ) is that the distribution of the inputs producing B is equal to the distribution of inputs produc ing Following, hypotheses with and as the for input factor X i are : ( 5 ) ( 6 ) The Smirnoff test statistic is: ( 7 ) F is the cumulative distribution function for input factor, X i sup is the maximum vertical distance between the behavioral and non behavioral cumulative distributions, and is the absolute value. A large value for implies that and

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79 are statistically different from each other and also that X i is an important factor in defining the behavior of the model. (Saltelli et al., 2008) Selection of Prior Input Probability Den sity Functions A careful and methodical selection of the prior for each input is especially important because it can be the most subjective step in this process and has significant impacts on the results. Probability distributions are determined bas ed on literature values and expert opinion. When the data for the inputs shows no apparent distribution such as normal or triangular, a uniform PDF allows for equal probability of selection across the defined range (Muoz Carpena et al., 2007). Because o f the lack of data that would suggest otherwise and in the spirit of conservatism, uniform distributions are used as the default prior distribution type. In an area where more data is available cross correlation tests could be done to show dependence betw een data. However, here no such data exists and inputs are assumed independent. The area volume relationships for the reservoirs, n and b i were based on 2 foot contour topographic data. The inputs n and b i are empirically fit according to Eq. 2. I n the calibrated model, n is set to 25 and b i ranges between 0.62 and 0.76 depending on the reservoir, i Because of the absence of literature values regarding these inputs and the empirical nature of their derivation, the for n and b i are each desi gnated as uniform, and vary by some percentage around the calibrated value. The uniform PDF for n is set to Uniform ( 20%) of the calibrated model. Setting for b i is more problematic since, as a power exponent, varying b i by 20% of the calibrated values makes the model fail occasionally. As a result, the uncertainty associated with these inputs is set to a more limited distribution: Uniform ( 10%) of the calibrated value. In

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80 the absence of literature values or other expert opinion the default for the remainder of the inputs are assumed to range between 20% of the calibrated value. The volume threshold input factors, v az where a is the upstream reservoir and z is the downstream reservoir, det ermine at what level water flows from one reservoir to another. These inputs are empirically derived and were used to calibrate the model (Wolski et al., 2006). There are 20 volume threshold inputs. Due to the absence of literature values describing ran ges of probably values for these inputs, the are set to uniform, varying 20% of their calibrated values. Each reservoir is divided into an area of floodplain and island so that groundwater flow between the two areas can be represented. There are 16 floodplain, fa i and 16 island area inputs, ia i The for the area for floodplains and islands within each reservoir is set to uniform, varying 20% of the calibrated values. The island and floodplain extinction depths, idet and fdet are important for evapotranspiration, empirically represent rooting depth, and one value for each is used to represent the entire Delta. The calibrated value was 20m for the floodplains and 5m for the islands. There are no literature values for extinction depths in the Okavango Delta or surrounding areas. However, literature values of modeled and measured values in for extinction depths in other areas vary between 1.5 and 60m (Shah et al. 2007, Niswonger et al., 2006, and Owe and Brubaker, 2000) This large range limits the capability of generating a probability distribution based on good physical data. Consequently, uniform distributions of 20% of the calibrated values are assigned for the island and floodplain extinction depths.

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81 The reservoir time constant input factor, k az is equivalent to surface flow resistance. As with the volume thresholds, there are 20 reservoir time constant inputs in the model. These values vary between 0.02 and 5. This input factor was used to origina lly calibrate the model (Wolski et al., 2006). In the absence of literature data regarding values for the reservoir time constant in the Okavango Delta, the prior are designated as uniform, varying 20% of the calibrated value for each r eservoir. Soil porosity is represented as two input factors in the model: island ( ipor ) and floodplain ( spor ) soil porosity. In the model, these factors are constant throughout the Delta, are not reservoir dependant, and are each set to 0.3. Literature v alues show soil porosity varying between 0.29 and 0.59 in the Okavango Delta (Wolski and Savenije, 2006) and between 0.43 and 0.49 in Kalahari Sands (Wang et al., 2007). Based on these literature values, the PDF for both the island and floodplain soil por osities are set to be uniform varying between 0.29 and 0.59. The delay input factor, delay i is an empirically derived discrete switch that slows the water flow and is turned on in four of the reservoirs in the calibrated model. This input was used in c alibrating the model. For the sensitivity analysis, the delay inputs which are turned on the calibrated model are set to turn on or off with a probability of 50%. Rainfall is input into the model using meteorological data collected from Maun and Shakawe. An inverse distance weighted coefficient was used to allow for proportional differences in rainfall amounts in each of the 16 reservoirs, m In the calibrated model, values for m range between 0 and 0.9. To investigate the full range

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82 behaviors that this input may have on the model these inputs are assigned uniform that range between 0 and 0.9. Results Modified Morris Global Sensitivity Analysis With 114 inputs, the model was run 1150 times using the Morris method (Table 3 1). Two general conclusio ns were drawn from the Morris method sensitivity analysis (Figure 3 4). First, because the inputs are scattered on both the and * axes, both direct (first order) and indirect (higher order) effects are important. Second, a ranking of input significance on the inundated area output is determined. The results showed that there are nine inputs which stand out as the most se nsitive, each falling above 300 on the direct and indirect axis of the Morris graph (Figure 3 4). These most sensitive inputs are: the island extinction coefficient ( idet ), the floodplain soil porosity ( fpor ), the exponential coefficient in the area volum e relationship for Xudum ( b 5 ), and the volume thresholds for the linkages between the Nqoga1 and: Boro ( v 26 ), Manuachira1 ( v 27 ), Xudum ( v 25 ), Thaoge ( v 24 ), Nqoga2 ( v 23 ), and Selinda ( v 29 ). From the strong sensitivity of these six volume thresholds, the im portance of this type of model input is apparent. There are a total of 20 volume thresholds in the model. In the Morris analysis, six of the 20 volume thresholds stood out as highly sensitive inputs and they all flow out of the Nqoga1 reservoir (Figure 3 4). There are a total of seven volume thresholds that flow out of Nqoga1. The one volume threshold that originates from Nqoga1 but is not highly sensitive is linked to the Boro reservoir where there are duplicate links to the Nqoga reservoir. This doub le link in the Boro, which splits the flow between Nqoga1 and the Boro, is unique in the Nqoga1 reservoir and may explain why this volume threshold did not display the same level of sensitivity.

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83 Nqoga1 is a keystone reservoir that has important influence in determining how the flood is routed between the reservoirs and is responsible for much of the sensitivity in the model. The identification of Nqoga1 as a keystone reservoir and the volume thresholds as important inputs has management implications. This area should be focused on for monitoring and managing because it is likely that changes in this area could affect how the water is apportioned to the downstream reservoirs. FAST Variance Based Global Sensitivity Analysis The subset of the most sensitive i nputs identified in the Morris method was selected for further investigation in using the Extended FAST method (Table 3 2). For this GSA/UA, the model was run 8,289 times, while varying the nine most sensitive inputs that were identified in the Morris met hod ( idet fpor b 5 v 26 v 27 v 25 v 24 v 23 and v 29 ), throughout the parameter space that was defined by the probability distributions. All other inputs were set to constants. As with the Morris analysis, the output of interest is the flooding extents for each reservoir as well as for the entire Delta. The FAST GSA corroborated the Morris results showing that while the higher order interactions within the model are important, the first order interactions dominate 3 5 and 3 6). The higher order interactions account for approximately 20% of the totally model sensitivity. In the FAST GSA the first order interactions are all relatively similar while the higher order interactions vary somewhat between the inputs. The fact that each of the important volume thresholds flowing out of the Nqoga1 reservoir has similar levels of importance shows that, for management purposes, this entire reservoir should be monitored for changes and protected from disturbance.

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84 Monte Carlo Filter ing Using the results from the FAST GSA, MCF was conducted to objectively reduce input/output model uncertainty. For MCF, the FAST results were filtered to find simulations that matched within 50% of the calibrated output (average inundation area) for ea inundation was within this range. No single GSA result produced a simulation in which all of reservoirs were within 50% of the calibrated values. The inputs were examined to find the source of this issue. The important volume thresholds, v 23 v 24 v 25 v 27 and v 29 are all for spillover from Nqoga1 into various reservoirs. All prior for these inputs were set to 20% based on the empirical nature of the model structure and the lack of data providing physical evidence. The original calibrated values for these thresholds range from 580 to 700. For each GSA simulation, each of the volume thresholds was randomly set within the same range of 20% and water was diverted betw een the reservoirs. It was determined that no behavioral solutions were found because the were too large and when the volume thresholds in v 24 v 25 v 26 and v 29 were set to opposite ends of their distributions, the water was spilling into one of th e reservoirs and drying out the others. For this reason, the for the important volume thresholds were truncated to 10% and the GUA was rerun. Originally, 50% was chosen as the behavioral threshold goal but this resulted in a minimum number of be havioral solutions even after the truncation of the in important volume threshold Increasing the behavioral threshold to 60% of the calibrated value increased the number of behavioral solutions thus, increasing the likelihood of finding statistica l differences between behavioral and non behavioral distributions. The truncation of the important volume threshold resulted in 103 out of 8,289

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85 simulations meeting the behavioral criteria. The two sided Smirnov test showed that the behavioral dis tributions for v 23 and v 27 were significantly different from the non behavioral distributions. These are two highly linked reservoirs (Figure 3 3). The new filtered for v 23 and v 27 displayed somewhat triangular distributions (Figure 3 7). New no rmal were assigned with v 23 T ~ (585, 645, 715) and v 27 T ~ (522, 616, 638). The FAST GSA/UA was rerun with the new prior distributions. FAST Variance Based Global Uncertainty Analysis The FAST MCF GUA results were normalized by the maximum inunda tion area to be able to compare the results across reservoirs. The results revealed a range of responses to the uncertainty analysis amongst the internal reservoirs (Figure 3 8). The upstream reservoirs and Maunachira1 demonstrated the least amount of un certainty. This uncertainty increases in the middle and downstream eastern reservoirs and reaches its maximum in the western reservoirs. This trend mirrors the physical dynamics in the Delta where the upstream areas are more confined, permanently flooded and display little variation, while the downstream areas are seasonally flooded and exhibit more variation in their annual flooding extents. Furthermore, in the FAST results some of the reservoirs display a Gaussian distribution (Mborga and Khwai) and s ome have long tails (Selinda). After MCF, confidence intervals, standard deviations, standard error of the mean, skewness, and kurtosis for the flooding extents for each reservoir and the entire Delta were constructed to examine the variance (Table 3 2). This data shows the varying levels of uncertainty in the reservoirs. As was shown in Figure 3 8, the upstream reservoirs vary minimally and the downstream reservoirs displaying much more uncertainty. As can be seen in the standard deviation of the total Delta, the uncertainty

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86 in the total Delta is not the summation of the uncertainty in the individual reservoirs. This may be due to the fact that the inputs for this uncertainty analysis have little to do with the water balance and are more related to wher e the water is moving around within the different reservoirs. In the various simulations the total volume of surface water is changed less than which reservoirs the water is routed to. The MCF GUA was compared to the original GUA (Figure 3 9 and Table 3 3). MCF reduced input/output uncertainty in all of the reservoirs except two (Panhandle and Nqoga1) (Table 3 3). These two reservoirs are both upstream from where the effects of the important volume thresholds should be felt. Reduction of uncertainty d ownstream from Nqoga1 varied between 13 and 189%. Thus, MCF was used to objectively truncated the prior and reduce the input/output model uncertainty improving the precision of the results. Additionally, the MCF produced modeled results that more c losely represented the calibrated ranges (Figure 3 9). After the MCF, the results cluster around the calibrated values more so than before this additional analysis. Therefore, MCF enhanced both the accuracy and the precision of the uncertainty analysis. Summary This GSA/UA of the ORI reservoir model provides important insights into the sensitivity and uncertainty of the ORI model and into reservoir models in general. These insights are useful for modelers when in interpreting and defining model uncertain ty and for policy makers through providing insights into model reliability and identifying important areas of the system. Managers may use the results of a sensitivity analysis to strategically monitor and identify and optimize model reliability. The Ad aptive Management process is a

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87 cyclical process that is constantly modeling, monitoring, learning and improving. This work has identified important areas for study that will optimize the learning process. The greatest hindrance to modeling the hydrology of the Delta, as shown by this work, is the lack of topographic data. This sensitivity analysis highlights the importance of volume thresholds in the ORI reservoir model and particularly the keystone reservoir Nqoga1. Future monitoring efforts, which ar e aimed at improving model performance, should be focused on understanding the topography of this area and how water is routed from this area to the distributaries. Managers should be aware that any changes in the Nqoga1 area may have dramatic downstream consequences. Additionally, the MCF reduced input/output uncertainty which improves model reliability and adds confidence to model results which is helpful for managers when making decisions based on these results. This exercise was useful for modelers because it reduced input/output model uncertainty and also identified the importance of model inputs. During this sensitivity analysis a number of inputs were shown to be relatively insensitive and have little impact on the model outputs. Insensitive inp uts include k az ia i and m i In future modeling efforts these inputs may be ignored or set to constants in order to simplify modeling and calibration without compromising accuracy. Conversely, the inputs: v az idet and fpor were found to be particularl y sensitive. Additionally, Nqoga1 was identified as a keystone reservoir. Future calibration efforts should start by adjusting these inputs within their likely bounds. In particular v az was found to be quite sensitive and future efforts may focus on ob taining physically determined values for v az Doing so

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88 may not only improve the model but will also result in a more physically representative model. Ivanovi and Freer (2009) point out that it is especially difficult to objectively test the uncertainty of inputs for empirical models and that most methods proposed thus far have been subjective (Beven, 2006b; Beven et al., 2007; Goldstein and Rougier, 2009). Pa ppenberger and Beven (2006) also make the point that uncertainty analyses may be too subjective to be of practical use. In this work MCF was used in conjunction with GSA/UA to explore how can be objectively set and input/output model uncertainty red uced. This is especially an issue in data poor areas. MCF was used to objectively redefine prior based on behavioral outputs and decrease input/output uncertainty. This is a function of the structure of the reservoir model and the uniform distribu tion of the inputs. In the GUA, changing the volume thresholds shifted water from one reservoir to another in a physically unrealistic manner. When the thresholds were given uniform distributions and assigned values on opposite ends of their distribution s, one reservoir would fill up while the others remained empty. Thus, the narrow uncertainty bounds clustered around the origin in the original GUA for these inputs. Changing these input based on the behavioral distributions narrowed their ranges a nd forced the majority of the inputs toward the calibrated values. This reduced input/output uncertainly and it moved the outputs closer toward the calibrated valued (Figure 3 9). Therefore, this MCF method successfully redefined the behavioral so that the model outputs more closely resembled the calibrated values in precision and accuracy.

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89 In conclusion, this work has technical and management implications by presenting an objective way of defining prior reducing model uncertainty, and identi fying important areas for study. As the ORI model is used and refined in the future, these points can optimize how the model is improved. The MCF method for defining prior is a generic tool that can be used to improve any GSA/UA application, especi ally in data poor areas.

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90 Table 3 1. Okavango Research Institute model input factors. Input Units Prior PDF CV Description Spatially Uniform fdet m U(4,6) 5 Floodplain extinction depth idet m U(16,24) 20 Island extinction depth fpor % U(0.29,0.5) 0 .3 Floodplain soil porosity ipor % U(0.29,0.5) 0.3 Island soil porosity n Unitless U(20,30) 25 Area/volume coefficient (Eq. 2) 1 b 1 Unitless U( 20%*) 0.62 Area/volume coefficient (Eq. 2) 2 b 2 Unitless U( 20%*) 0.71 3 b 3 Unit less U( 20%*) 0.76 4 b 4 Unitless U( 20%*) 0.73 5 b 5 Unitless U( 20%*) 0.76 6 b 6 Unitless U( 20%*) 0.76 7 b 7 Unitless U( 20%*) 0.72 8 b 8 Unitless U( 20%*) 0.72 9 b 9 Unitless U( 20%*) 0.73 10 b 10 Unitless U( 20%*) 0.73 11 b 11 Unitless U( 20%*) 0.73 12 b 12 Unitless U( 20%*) 0.73 13 b 13 Unitless U( 20%*) 0.71 14 b 14 Unitless U( 20%*) 0.72 15 b 15 Unitless U( 20%*) 0.72 16 b 16 Unitless U( 20%*) 0.73 1 v 1,1a Mm 3 U( 20%*) 500 Volume threshold 2 v 1,1b Mm 3 U( 20%*) 2000 3 v 2,1 Mm 3 U( 20%*) 580 4 v 2,2 Mm 3 U( 20%*) 700 5 v 2 ,3 Mm 3 U( 20%*) 700 6 v 2,4 Mm 3 U( 20%*) 400 7 v 2,5 Mm 3 U( 20%*) 700 8 v 2,6 Mm 3 U( 20%*) 700 9 v 2,7 Mm 3 U( 20%*) 650 10 v 3 Mm 3 U( 20%*) 250 11 v 5 Mm 3 U( 20%*) 35 0 12 v 6,1 Mm 3 U( 20%*) 450 13 v 6,2 Mm 3 U( 20%*) 500 14 v 6,3 Mm 3 U( 20%*) 210 15 v 7,1 Mm 3 U( 20%*) 300 16 v 7,2 Mm 3 U( 20%*) 300 17 v 7,3 Mm 3 U( 20%*) 300 18 v 8 Mm 3 U( 20%*) 300 Volume threshold 19 v 15 Mm 3 U( 20%*) 700 20 v 16 Mm 3 U( 20%*) 700

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91 Table 3 1. continued Input Units Prior PDF CV Description 1 k 1,1a Months U( 20%*) 0.3 Resistance to flow (residence time) 2 k 1,1b Months U( 20%*) 5 3 k 2,1 Months U( 20%*) 1.5 4 k 2,2 Months U( 20%*) 1 5 k 2,3 Months U( 20%*) 4 6 k 2,4 Months U( 20%*) 0.12 7 k 2,5 Months U( 20%*) 3.8 8 k 2,6 Months U( 20%*) 1.5 9 k 2,7 Months U( 20%*) 1 10 k 3 Months U( 20%*) 3 11 k 5 Months U( 20%*) 0.1 12 k 6,1 Months U( 20%*) 0.09 13 k 6,2 Months U( 20%*) 0.02 14 k 6,3 Months U( 20%*) 0.02 15 k 7,1 Mo nths U( 20%*) 1.7 16 k 7,2 Months U( 20%*) 0.8 17 k 7,3 Months U( 20%*) 1 18 k 8 Months U( 20%*) 5 19 k 15 Months U( 20%*) 0.1 20 k 16 Months U( 20%*) 0.3 1 delay 1 Binary D(0,1) 1 Delay input for units 2 delay 2 Binary D(0,1) 1 3 delay 3 Binary D(0,1) 1 4 delay 4 Binary D(0,1) 1 1 m 1 % U(0.1,0.9) 0.1 Rainfall ratio input 2 m 2 % U(0.1,0.9) 0.5 3 m 3 % U(0.1,0.9) 0.8 4 m 4 % U(0.1,0.9) 0.4 5 m 5 % U(0.1,0.9) 0.8 6 m 6 % U( 0.1,0.9) 0.8 7 m 7 % U(0.1,0.9) 0.5 8 m 8 % U(0.1,0.9) 0.4 9 m 9 % U(0.1,0.9) 0.4 10 m 10 % U(0.1,0.9) 0 11 m 11 % U(0.1,0.9) 0 12 m 12 % U(0.1,0.9) 0 13 m 13 % U(0.1,0.9) 0 14 m 14 % U(0.1,0.9) 0.9 15 m 15 % U(0.1,0.9) 0.9 16 m 16 % U(0.1,0.9) 0

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92 Table 3 1. continued Input Units Prior PDF CV Description 1 fa 1 km 2 U( 20%*) 184 Floodplain groundwater reservoir area 2 fa 2 km 2 U( 20%*) 100 3 fa 3 km 2 U( 20%*) 152 4 fa 4 km 2 U( 20%*) 138 5 fa 5 km 2 U( 2 0%*) 321 6 fa 6 km 2 U( 20%*) 348 7 fa 7 km 2 U( 20%*) 100 8 fa 8 km 2 U( 20%*) 140 9 fa 9 km 2 U( 20%*) 92 10 fa 10 km 2 U( 20%*) 1 11 fa 11 km 2 U( 20%*) 100 12 fa 12 km 2 U( 20%*) 100 13 fa 13 km 2 U( 20%*) 100 14 fa 14 km 2 U( 20%*) 206 15 fa 15 km 2 U( 20%*) 84 16 fa 16 km 2 U( 20%*) 300 1 ia 1 km 2 U( 20%*) U( 20%*) U( 20%*) 41.4 Island groundwater reservoir area 2 ia 2 km 2 U( 20%*) 7.8 3 ia 3 km 2 U( 20%*) 18 4 ia 4 km 2 U( 20%*) 133.2 5 ia 5 km 2 U( 20%*) 176 6 ia 6 km 2 U( 20%*) 141 7 ia 7 km 2 U( 20%*) 25.2 8 i a 8 km 2 U( 20%*) 18 9 ia 9 km 2 U( 20%*) 47.4 10 ia 10 km 2 U( 20%*) 0.6 11 ia 11 km 2 U( 20%*) 6 12 ia 12 km 2 U( 20%*) 6 13 ia 13 km 2 U( 20%*) 6 14 ia 14 km 2 U( 20%*) 16 0.8 15 ia 15 km 2 U( 20%*) 48.6 16 ia 16 km 2 U( 20%*) 50 U = uniform continuous distribution, D = uniform discrete distribution, CV = calibrated value *PDF Varies between 20% of the calibrated model

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93 Table 3 2. Uncertainty ana lysis statistics of aerial flooding extents after Monte Carlo Filtering Reservoir Mean Median Lower 95% CI Upper 95% CI SD* SEM* Skew* Kur* Panhandle 788.9 789.1 785.3 792.2 2.1 0.0 0.2 0.9 Nqoga1 492.7 492.7 489.7 495.8 1.9 0.0 0.1 1.0 Nqoga2 488.2 499.3 178.9 693.2 157.1 1.7 0.5 0.7 Thaoge 340.1 239.1 83.3 821.7 246.5 2.7 0.9 0.1 Xudum 775.5 780.5 86.8 1883.0 550.5 6.1 0.6 0.5 Boro 912.5 791.1 373.4 1621.5 414.1 4.6 0.4 1.2 Maunacharia1 483.6 483.8 477.0 488.8 3.5 0.0 0.3 0.4 Maunacharia 2 319.7 306.8 188.0 474.6 81.6 0.9 0.3 0.5 Selinda 243.1 78.0 10.9 872.2 274.5 3.0 1.2 0.3 Mborga 455.4 426.4 340.4 411.9 143.1 1.6 0.6 0.1 Khwai1 220.7 206.6 117.8 360.3 69.8 0.8 0.5 0.4 Total Delta 5504.8 5485.0 4920.0 6171.0 357.5 3.9 0.3 0.2 CI = confidence interval, SD = standard deviation; SEM = standard error of the mean; Skew = skewness, kurt = kurtosis Table 3 3. Width of 95% Confidence Interval for Average Flooding Extents (km2). Reservoir Original Prior PDF MCF PDF % Difference Pan handle 5.5 6.9 21.2% Nqoga1 5.4 6.1 12.5% Nqoga2 815.1 514.3 45.2% Thaoge 1863.3 738.3 86.5% Xudum 2210.0 1796.1 20.7% Boro 1428.5 1248.1 13.5% Maunacharia1 411.0 11.8 188.8% Maunacharia2 526.5 286.6 59.0% Selinda 1621.1 861.3 61.2% Mborga 884.3 71.6 170.0% Khwai1 384.4 242.5 45.3% Delta 2167.8 1251.0 53.6%

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94 Figure 3 1. Okavango Basin and Delta Location Map.

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95 Figure 3 2. The Okavango Research Institute ( ORI ) model. Adapted from Wolski et al., 2006.

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96 Figure 3 3. Diagram of the OR I reservoir model nodes and links. Adapted from Wolski et al., 2006.

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97 Figure 3 4. Morris method GSA results for the inundation of the entire Delta. Volume/area coefficient for Xudum ( b 5 ), island extinction coefficient ( id et ), floodplain porosity ( fpor ), volume threshold between Nqoga1 and: Nqoga2 ( v 23 ), Thaoge ( v 24 ), Xudum ( v 25 ), Boro ( v 26 ), Maunachira1 ( v 27 ), and Selinda ( v 29 ). Labels for the less important inputs are omitted for clarity.

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98 Figure 3 5. FAST GSA first a nd grouped higher order (R) proportional results for the inundation of the entire Delta. Volume/area coefficient for Xudum ( b 5 ), island extinction coefficient ( idet ), floodplain porosity ( fpor ), volume threshold between Nqoga1 and: Nqoga2 ( v 23 ), Thaoge ( v 24 ), Xudum ( v 25 ), Boro ( v 26 ), Maunachira1 ( v 27 ), and Selinda ( v 29 ). Figure 3 6. FAST first order and higher order, and total order GSA results for the inundation of the entire Delta. Volume/area coefficient for Xudum (b 5 ), island extinction coeffic ient (idet), floodplain porosity (fpor), volume threshold between Nqoga1 and: Nqoga2 (v 23 ), Thaoge (v 24 ), Xudum (v 25 ), Boro (v 26 ), Maunachira1 (v 27 ), and Selinda (v 29 ). The height of the entire bar is the total sensitivity.

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99 a) v 27 b) v 23 Figure 3 7. Frequency histograms for values for a) v 27 and b) v 23 resulting in behavioral outputs. New triangular distributions based on these results are also shown.

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100 Figure 3 8. FAST Global Uncertainty Analysis ( GUA ) grouped by upstream, middle, eastern, an d western reservoirs. Values are normalized by the maximum flooding extents in each reservoir for comparison. Refer to Figures 2 and 3 for the exact locations.

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101 Figure 3 9. Comparison of original GUA and Monthe Carlo filtering ( MCF ) GUA: Nqoga1, Mau nachira1, and Mborga. In each graph the calibrated average inundation for each reservoir is also indicated.

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102 CHAPTER 4 A FISH POPULATION MO DEL FOR THE OKAVANGO DELTA, BOTSWANA: HO W THE FLOOD PULSE DRIV E POPULATION DYNAMIC S Abstract The hydrology of the annually flood pulsed Okavango Delta in Botswana is theorized to be a main driver for fish population dynamics. A common hypothesis explaining this phenomenon is the Flood Pulse Concept which describes a lateral connection between the channel and floo dplain and a biological response to episodic floodplain inundation corresponding to the release of terrestrial nutrients. This work tests that hypothesis in the Okavango Delta by developing and applying a quantitative model of fish population dynamics wit h the flood pulse as the major driver. Because of their commercial importance, tilapia and in particular Oreochromis andersonii Tilapia rendalli and Oreochromis macrochir are used as indicator species. The model runs on an annual time step where age co horts are tracked over time. Density dependant recruitment, mortality, and vulnerability to catch were simulated. Model results are compared to commercial gill net data. In this data poor environment there is a high level of uncertainty and resulting eq uifinality and so an optimal input set is not sought. However, evidence for the flood as a driver of fish population dynamics is explored. This work uses an unbiased inverse optimization routine that searches throughout the parametric space for the best model solutions. For model evaluation, prior probability density functions ( ) were established for each input. Global sensitivity analysis was then used to identify the inputs that were the most important in determining the model outcome. It can be argued that the definition of prior is arbitrary. Therefore, Monte Carlo filtering was used to redefine and truncate some of the important prior Model results give a best fit of a coefficient of efficiency of 0.64

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103 and a coefficient of determ ination (R 2 ) of 0. 64 The most important inputs relate to mortality, catch vulnerability, and growth. There is a high level of equifinality in the inputs that set the baseline population for the fish initially suggesting uncertainty in model predictions However, the flood coefficient, which is the only input that directly relates the inter annual dynamics of fish recruitment to the flood pulse converges to an optimal value. At this optimal value modeled recruitment is almost doubled between low and h igh flood years. This provides evidence that the flood is indeed a major driver of fish population dynamics in the Delta. Background The Okavango Delta in Botswana is a large inland delta located in an arid climate that experiences an annual flood pulse from its upstream watershed (Figure 4 1). Flooding extents in the Delta range from 6,000 12,000 km 2 (Wolski et al., 2006) and the entire alluvial fan is approximately 40,000 km 2 (Gumbricht et al., 2005). The Okavango Delta is home to over 70 fish spec ies (Kolding, 1996) and there are three main fishing industries in the Delta: subsistence, small scale commercial, and recreational (Merron 1991, Mosepele and Kolding 2003). Mosepele (2001) states that 65% of the people living in northern Ngamiland (which encompasses the entire Delta) depend on fish for a portion of their livelihood. Monitoring and preserving the health of this resource is important both economically for the people who fish here and ecologically for the fish and other wildlife that depend on the annual flow of water during the dry season. Environmental flows are criteria that describe the degree to which a natural hydrologic regime must be maintained in order to preserve valuable features of an ecosystem (Tharme and King, 1998; King et a l., 1999, Tharme, 2003). Environmental

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104 Flows have been developed for the Okavango Delta and a special report was published specifically addressing flows for fish communities (Mosepele, 2009). A major recommendation in the report is for the development of a quantitative relationship between the flood pulse and fish population dynamics in the Delta so that managers can understand feedbacks between the hydrology of the De lta and fish populations Fish population models are often used to track populations an d predict responses to management decisions in other locations (Walters and Martel, 2004; Rogers et al., 2010; Walters et al., 2008). A fish population model has yet to be developed for the Okavango Delta. The Flood Pulse Concept (Junk et al., 1989) (FPC) describes an ecological response to flood pulsed hydrology where nutrient availability is linked to the inundation of the floodplain. In the FPC the aquatic/terrestrial transition zone (ATTZ) is the area where the aquatic environment meets the terrestria l environment. On an incoming and inundates the floodplain, the ATTZ has high inputs of nutrients from terrestrial sources such as vegetation and detritus. This leads to high primary productivity in the ATTZ. The FPC, the inputs of nutrients, and high primary productivity in the ATTZ iterates with each flood and it is hypothesized that fauna can adapt to take advantage of the increased food availability (Junk et al., 1989 ). In the Okavango Delta no quantitative studies have been conducted to specifically show how fish respond to the flood pulse. However, there have been studies in the Okavango that show how the annual flood pulse produces an ecologic response in other e cological aspects (Hoberg et al., 2002; Merron, 1991). Hoberg et al.

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105 (2002) provide a food web conceptual model for ecological responses to the annual which results in a release of nutrients. During the rising flood there was a burst in nutrients, primary production, and phytoplankton. Concentrations of nitrogen rose from 1.5 to 3.5 mg l 1 1 Primary production reached its 1 d 1 and maximum chlorophyll a l 1 (Hoberg et al., 2002). The authors went on to state that resting zooplankton eggs hatched when they were submerged by the floodwater and fed on the abundant phytoplankton and ot her food sources provided by the burst in primary production. Peak concentrations of zooplankton went from 0.1 to 10 mg DW l 1 during the rise of the flood and reached up to 90 mg DW l 1 at the extreme near shore edges. At the same time a qualitative an (Hober g et al., 2002). The tilapiine species O. andersonii T. rendalli and T. sparrmanii were observed following the rising flood into the study area. Juveniles of the same species were als o seen with an increasing frequency just after the peak of the flood. Gut analysis of the fishes showed that smaller fish fed on more zooplankton indicating very few fry were observed with the conclusion that they migrated out of the area before the connection with the main river system was lost. In another study, Merron (1991) related spawning period to the flood pulse in the Okavango Delta. He proposed that t he higher the magnitude of the annual flood, the longer the water is retained on the floodplain, leading to a longer spawning period and greater overall production of fish. Additionally, Mosepele et al. (2009) proposed that

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106 survivability for smaller fishe s is increased in dense floodplain vegetation types because the vegetation provides protection from predators. Thus, there is a stated management need for a model that links the flood pulse to fish population dynamics q uantitative data shows that variou s aspects of the ecology of the Delt a other than fish are driven by the flood pulse and there are qualitative indications that the fish population dynamics are affected by the flood pulse In response to these issues and needs, this work produced a fish population model of the Okavango Delta that is driven by the annual flood pulse. The model was inverse calibrated to objectively investigate the range of best fit model input sets. Traditionally, the goal of inverse optimization techniques is to find the optimal set of inputs for a given model to match some measured data (Mertens et al., 2006). Beven and Binley (1992) and Beven (1993) comment on the limitations of the concept of an optimal parameter may be many input sets that all lead to an equally acceptable mode simulation. These inputs may come from different places in their defined parametric space and there may be no way to distinguish between the equally ac ceptable input sets. This work did not seek an optimal parameter set, but instead look ed for evidence of the flood as a driver for fish population dynamics despite signs of equifinality. This work use d global sensitivity and uncertainty analysis (GSA/UA ) in two ways. First GSA was used to identify unimportant inputs that could be set to constants without drastically affecting the outputs thus simplifying the model (Muoz Carpena et al., 2007; Muoz Carpena et al., 2010; Fox et al., 2010; Jawitz et al., 2008; Chu Agor et al., 201) One of the largest critiques of GSA/UA is the rat h er arbitrary methods for setting

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107 the prior probability distributions ( ). Therefore, secondly, this work use d Monte Carlo (MC) Filtering (Saltelli et al., 2008) to object ively redefine the prior within their predefined defined physical bounds based on realistic model results. The objective of this work was to produce and calibrate a mechanistic fish population model for the Okavango Delta that is driven by its ann ual flood pulse. Because of their commercial importance, tilapia and in particular O. andersonii T. rendalli and O macrochir were used as indicator species. Evidence for the flood as a drive for fish populations dynamics was sought and the computer mo del was used to test the conceptual model that fish population dynamics are driven by the flood pulse. Methods Fish Data Daily c ommercial catch data were available from January 1996 to December 2005. Fish were caught in gill nets and the daily catch per unit effort (CPUE) was recorded. The CPUE is defined as the number of fish caught per gill net per day. Because a relatively standard gear is used throughout th is catch, CPUE can be used as an indicator of fish density using the assumption that a co nstant fraction of the stock was captured per gill net day (q) Brogstro e m (1992) investigated this assumption by comparing the effect of population density on gillnet catchability for brown trout in lakes. Though he did not find a linear relationship, h e did find a relationship that approached a linear shape in the values of the data corresponding to this work. H owever in Brogstr oe data that was used in this work were commercial catch da ta which can cause discrepancies. T he fisher s do not fish randomly and may set their nets where they believe the fish densities will be the highest which can cause CPUE data to not reflect the overall

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108 abundance (Walters and Martell 2004) Fish were als o caught in the northern portion of the Delta in the Panhandle where flooding is more confined and the effect of the annual flood is less pronounced (Figure 4 1). T he fish data that were available only include fish counts Since a ge, weight, and length d ata which are necessary for simulating monthly spawning dynamics were not collected an annual model was deemed to be the most appropriate. CPUE approximates density. And so, f or the purposes of this model, the objective function was the coefficient of efficiency (Nash Sutcliffe, 1970) between the modeled maximum annual density and the measured maximum annual CPUE. In order to compute this from the modeled data, the fish abundance must be expressed as density (e.g. fish km 2 ) This was done using in t he Okavango Research Institute (ORI) hydrologic model of the Delta (Wolski et al., 2006). The ORI model is a linked reservoir model that simulates flooding extents in the Delta. One of the reservoirs in the model explicitly represents the Panhandle, wher e the fish data were collected. The Panhandle is not the reservoir that was of focus for model calibration Additionally, this area is more confined and permanently inundated than the lower Delta which experiences greater dynamics in annual inundation ar ea Because of these reasons, in each year the total modeled vulnerable fish population was divided by the average across years of the minimum flooding extents simulated in the Panhandle. This produce d an overall better model fit than dividing by the ann ual minimum flooding extents. The fish that were caught were not recorded to the level of species. Instead, tilapias (of the family Chiclidae) were lumped into one category. According to Mosepele et al. (2003) Cichlidae is the principal family of fish in both subsistence and commercial

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109 gill net fisheries. The three species with the highest ind ices of relative importance for the commercial gill net fishery are all tilapia and include in order of importance: (1) O. andersonii (2) T. rendalli and (3) O. macrochir For the purposes of this work, these three fish species were used as a representative species to base parameters such as maximum age and growth functions, with particular importance placed on O. andersonii Model Structure This work describ es the development of an annual time step fish population model driven by the flood pulse in the Okavango Delta (Figure 4 2). Fish were divided into annual age classes which were tracked over time. The model structure use d the Beverton and Holt (1957) st ock recruitment relationship (Eq. 1) to estimate recruitment, where [Fish] is the number of recruits per year [t] [Fish] is the population fecundity per year (a function of the stock size), and [ ] and [F ish 1 ] are paramete rs (1957). / describes the maximum recruitment at a high stock and describes the maximum recruitment/stock at a low stock. (1) is given by Eq. 2 where [Fish t 1 ] is the number of fish per age class, e [ ] is fecundity (number of eggs hatched), and [ ] is the number of broods per season. (2) Three constants are required to parameterize the Beverton and Holt equation: one point along the Beverton and Holt curve where recruitment is constant ( ), survival from natural mortality ( ) [ ] and the Goodyear Com pensation Ratio ( ) [ ]

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110 (Goodyear, 1977). The steady state condition of the Beverton and Holt equation is given as Eq. 3: (3) Thus, R t (constant recruitment, used for calibration of the number of fish in the population ) can easily be found by Eq. 4: (4) CR represents the maximum compensatory increase in juvenile survival as the stock size is decreased. and can be derived from CR from the steady state equation (Eq. 5) at low spawner abundance. is the juvenile survival ratio ( ) (Walters and Martel, 2004) (Eq. 5) such that: (5) And is th e density dependent parameter which can be derived from (Walters and Martel, 2004) (Eq. 6): (6) The fish population response to the flood was simulated through recruitment. In order to relate r ecruitment to the flood a modification was made to the Beverton and Holt density dependant recruitment relationship such that the number of fish was positively proportional to the annual maximum inflow (Eq. 7). In this equation, is the maxi mum inflow in a given year [ M m 3 yr 1 ] is the average of the annual maximum inflows [ M m 3 yr 1 ] and c t is a scaling coefficient [ yr M m 3 ] This did not change the actual population within the fish in the matrix, but did change the populat ion for the

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111 calibration of and and calculations for recruits in each time step (Eq. 8, 9, 10, and 11). In these equations the ( )* represents factors a ffected by the flood coefficient but which are not actually changed in the matrix. (7) (9) (8) (10) (11) The result is a change in the carrying capacity of recruits and rate of recovery between high and low flood years (Figure 4 3). In each time step, after recruitment is calculated, mortality is calculated. Each age experiences mortality according to the Beverton and Hol t ( [t 1 ] 1959) relationship (Eq. 12). In this represents instantaneous total mortality [ yr 1 ] The exponential function results in an exponential decrease in abundance with age is divided into two parts: natural mortality ( ) and fishing mortality ( ) where (12) Various studies point to an allometric relationship between body weight and mortality (Peterson and Wroblewski, 1984; McGurk, 1986; Lorenz en, 1996). de Graaf et al. (2005) showed relationship can also be related to body length through the von Bertalanffy (1957) length/weight relationship

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112 where is the mortality at unit length ( yr 1 g 0.3 ), is body length in cm, and [ g cm b ] and [ ] are coefficients (Eq. 13). (13) Length at a given age is calculated according to the von Bertalanffy equation, where is the length at age (yr) and is the growth coefficient in Eq. 10 (unitless), and is the asymptotic length (Eq. 14). (14) Not all fish are equally vulnerable to cat ch because of the size selective gear. Once the modeled fish population is calculated, the vulnerability to catch of each age class is computed. A dome shaped double logistic model (Allen et al., 2009) is used where is the vulnerability (un itless) at age (yr 1 ), is the average length (cm) at age is the lower length (cm) at 50% vulnerability, is the standard deviation of the distribution for is the upper length (cm) at 50% vulnerability, and is the standard deviation of the distribution for (Eq. 15). (15) The measured maximum annual CPUE generally occurs at the annual low flood when the fish are most concentrated in a smaller area. In order to get a similar measure of density, the vulnerable fish were divided the minimum area of inundation in the Panhandle where the fish were caught. This measure was considered the modeled annual maximum CPUE. For each year, the modeled annual maximum CPUE was compared to the measured annual maximum CPUE using the coefficient of efficiency (ceff) (Nash and Sutcliffe, 1970) which was the objective function of the model.

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113 Model Optimization were developed for each input based on literature values and dat a when available. The model was run iteratively sampling inputs from the using the extended Fourier amplitude sensitivity test (FAST) sampling routine (Cukier et al., 1978; Koda et al., 1979; Saltelli et al., 1999) with SimLab software (SimLab, 2011 ). This sampling routine is an unbiased method that samples throughout the parametric space and is able to highlight the variety of input sets that result in good fit model simulations. Sensitive inputs were i dentified and uncertainty was measured. MC filtering (Saltelli et al, 2008) was used to filter out unacceptable model simulations as defined by the ceff and prior distributions were refined based on the inputs that generated the acceptable simulations. I nsensitive inputs were set to constants. The GSA was rerun with the new prior distributions and varying only the important inputs. Global s ensitivity and u ncertainty a nalysis Global sensitivity and uncertainty analysis (GSA/UA) is used to apportion the variation of model outputs onto the model inputs based on prior distributions. The extended FAST GSA/UA method (Cukier et al., 1978; Koda et al., 1979) uses Fourier analysis to decompose the variance of a set of model outputs into first order variances for each input. For this method, the model is executed C Nk times, where k is the number of inputs and N al., 1999). The extended FAST technique (Saltelli et al., 1999) allows for the additional computation of higher levels of variance which descr ibe the interactions between the inputs (Eq. 14). Here, describes the total variance of a single input including first

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114 and higher levels of variance. (1 6 ) FAST also defines as an index of global sensitivity. is the ratio of the variance that is ascribed to a single input divided by the total model variance. In a model where there are no interactions, the sum of the models where there are interactions this sum is greater than one. Note that inputs used in this method must be independent and are assumed so for this work. Monte Carlo f iltering Using the results of the FAST GSA/UA, MC filtering (Saltelli et al., 2008) was used to filter o ut the unrealistic outputs or non behavioral results and redefine the prior distributions based on the realistic outputs. MC Filtering divides the outputs into ) based on a threshold that is defined by the user. The / status is mapped back to the inputs and two subsets of each model input, were defined as and based on their behavioral /non behavioral status. The behavior al subset contains elements and the non behavioral subset contains elements such that where is the number of model simulations. The and are then assigned. The two sided Smirnov test is performed to check the significance of the difference between the two distributions and In the Smirnov test the test statistic is defined by Eq. 15: (1 7 ) The null hypothe sis for this test is = That is, the null hypothesis tests if the distribution of the inputs that create behavioral outputs the same as the distribution of the inputs that create non behavioral outputs. The null hypothesis is

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115 importance of that input for driving the behavior of the model (Saltelli et al, 2008). For this work, the two sided Smirnov test was used to determine if the behaviora l and non behavioral input distributions were significantly different. If the null hypothesis was rejected, the prior distribution was reassigned based on Probability d ensity f unctions Inherent to these methods is the importance of the se lection of the prior Regardless of the internal mathematics within the model, when conducting a GSA, assigning a wider PDF to an y input makes it more sensitive or important in determining the model output This is because there is a wider range to sample from. Accordingly, much effort was devoted to assigning appropriate for each input based on best knowledge for each factor. In an area where more data is available cross correlation tests could be done to show dependence between data. Howe ver, here no such data exists and inputs are assumed to be independent. Model inputs and their are shown in Table 4 2 When the data for the inputs shows no apparent distribution such as normal or triangular, the PDF can be set to uniform (Muoz Ca rpena et al., 2007). The uniform distribution allows for equal probability of selection across the defined range. Within the model, mortality was simulated through the allometric relationship between fish length and mortality for each age class (Eq. 15) M ortality was first given as a constant, where is total instantaneous mortality (yr 1 ), is instantaneous fishing mortality (yr 1 ), and is instantaneous natural mortality (yr 1 ). Estimates of the t s in the Okavango Delta in the 1990 s was approximately 5,000

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116 s and about 40 total full (Mosepele, 2001; Kgathi et al., 2005). Because of the small scale of commercial and s ubsistence fishing in the Okavango Delta and the low efficiency of the gear fishing pressure in the Delta is fairly light (Mosepele and Kolding, 2003; Kgathi et al. 2005). Thus, for the purposes of this research was considered negligible a nd thus, Z ~ M T he prior distribution for was defined based on literature values (Table 4 2 ). Values of for tilapiine species were compiled from a number of studies including the three indicator species in the Okavango Delta. These values range between 0.67 and 1.39 for the indicator fishes in the Okavango Delta. Based on these data, the PDF for was set to Uniform (0.67, 1.39). The allometric relationship between mortality and fish body length (de Graaf et al., 2005) calculates a decreasing rate of mortality with increasing body length. The four inputs required for calculating mortality according to the allometric relationship are the mortality at unit weight (y 1 g 1 ), the von Bertalanffy paramete rs and parameters, and asymptotic length (cm), The original value for was given as a constant (Lorenzen, 1996). More recently, it has been determined that should be estimated for each species (de Graaf et al., 2005; Lorenzen, 2001). de Graaff et al. (2005) used values of ranging between 1 and 4.5 for species with survival rates between 50 and 80%. Based on the values of chosen for the PDF survivability for these fishes ranges between 25 and 50%. The PDF for was set so that ranges similar to the values that were defined for through the literature could be achieved. Based on this analysis the PDF for was set to Uni form (3, 8)

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117 Mosepele and Nengu (2003) show values for the weight [g] length [cm] parameters and parameters for the three indicator species specific to the Okavango Delta. For O. andersonii is given as 0.004 and is 3.424, for O. macrochir is 0.014 and is 3.106, and for T. rendalli is 0.026 and is 2.911. Based on Mosepele and for and were established: Uniform (0.004, 0.026) and Uniform (2.911, 3.424). Mosepele and Nengu (2003) calculated asymptotic lengths for the three indicator species in the Okavango Delta L for O. andersonii was found to be 53cm, for T. rendalli it was 47cm, and for O. macrochir it was 40cm. From these values, the PDF for L was set to Uniform (40, 53). A number of studies in southern Africa and the Okavango Delta investigate t he growth co efficient for various tilapiine species including the three indicator species used in this study (Table 4 2 ) In these studies, k varies from 0.25 to 1.0 from. Based on these data, the PDF for was set to Uniform (0.25, 1.0). The Goodyear compensation ratio (CR) (Goodyear, 1977) describes the rate at which juvenile survival changes following a depletion in stock High values of CR allow juvenile survival to increase substantially as the stock declines due to fishing, resulting in high compensation According to Walters et al. (2008), w hen recruitment compensation is not especially strong the CR is less than 20. Walters et al. (2007) state a analysis of a variety of fish (mostly pelagic species) values for CR ranged from 1.4 to 123.5 with an average of 18.6. A meta analysis of stock recruitment data by Goodwin et al. (2006) showed CR is in the range of 5 to 100. Most specific ally,

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118 between 3 and 50. Based on these analyses, the PDF for CR was set to Uniform (3, 50). A dome shaped double logistic vulnerability catch curve (Allen et al., 2009) was u sed to model the vulnerability of fish at different lengths and corresponding age classes (Figure 4 4). This mimics the size selective nature of the gill nets. The vulnerability curve was parameterized from inspection of a smaller dataset of commercial cat ch from the Okavango Delta where fish length was available (Mosepele, 2009). A histogram of caught fish length was produced and a corresponding catch curve was developed with for each of the parameters. The for the lengths and standard deviat ion at the upper end of the curve were set to include a wider and higher distribution to account for the mortality that is occurring and also to test for a logistic vulnerability shape versus a dome shape. Based on inspection, the were set as follow s: is Uniform (23, 25), is Uniform (1, 3), is Uniform (28, 60), and is Uniform (10, 30). The model calculates recruitment per time step based on the number of mature fish in that year. Several studie s have investigated the age at which cichlids become mature in southern Africa. Dudley (1974) measured the total length and sexual maturity of O. andersonii in the Kafue floodplains. He found that that during the years of his study, no fish under 26 cm w ere mature three out of 64 fish from 26 29 cm were mature and more than 30% of larger males and 40% of larger females were in mature Dudley (1974) also aged the fish with annual ring formations. He concluded that O. andersonii usually spawn after the age of four and very rarely under the age of three.

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119 Similarly, Van der Waal (1976) found that in the Zambezi River the minimum size for sexual maturity in O. andersonii was 25 27 cm. Hay et al. (2000) also measured the minimum size of for sexual maturi ty in O. andersonii in the Okavango River Namibia which they found to be 13 cm for males and 26 cm for females. Based on these literature values, with emphasis on the ring formation as better measure of age than length, and a PDF for age a sexual maturity was set to Normal (4.25, 0.5). Fecundity refers to the number of eggs hatched per brood. According to Mortimer (1960) O. andersonii between 17 and 25 cm in length, lay 349 to 567 eggs in ponds. Additionally, Chandrasoma and De Silva (1981) found intr aovarian egg counts in T. rendalli ranged between 760 and 6,160 in a lake in Sri Lanka. And Marshall (1979) found that O. macrochir can produce 1,000 to 5,000 eggs within their ovaries and may mouthbrood up to 800 eggs in Lake Mcilwaine, Zimbabwe. Accord ing to these data and, the PDF for fecundity was set to Uniform (350, 800). Several sources state that these indicator species may lay more than one brood per season. Skelton (1993) stated that T. rendalli and O. andersonii both raised several broods ea ch summer. Naesje et al. (2004) described that T. rendalli may lay several broods each season in the Kwando River, Namibia. Mortimer (1960) examined O. andersonii for physiological indications of having multiple broods per season. This study did not find physiological indications of multiple broods. However, the same study also observed two instances in ponds where one breeding pair spawned twice in one season. From this literature, the PDF for broods per season in the model was set to Uniform (1, 2).

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120 A ny measured data is inherently flawed, including the gauged inflow. The ORI hydrologic model results give a correlation coefficient of 0.90 between measured and modeled monthly inundated area in the Delta (Wolski et al., 2006). Based on this, a Uniform PD F was assigned to the annual maximum inflow as 10% of the measured values. The flood coefficient ( ) is a scaling factor with the units Fish yr 1 age 1 flooding extent 1 that describes how recruitment changes with the magnitude of the flood (E q. 7). There is no literature to support values for this coefficient and this research is the first investigation into the quantitative effects of the flood on fish populations in the Okavango Delta. A trial and error investigation into the appropriate ranges for this coefficient was conducted prior to the GSA to get a sense of the values that would drive the model into a behavioral fit. Values for this coefficient ranging between 5 and 25 create acceptable model outputs. Therefore, the PDF for the fl ood coefficient was set to Uniform (5, 25). Results With 16 inputs, the model was run 39,952 times for the GSA/UA using the FAST methodology before MC Filtering. In this set of simulations, when comparing annual measured and modeled CPUE, the model achiev ed a maximum ceff of 0.64 and an R 2 of 0. 64 (Figure 4 5). Thus, the model did a reasonable job of predicting the inter annual fluctuations in fish abundance based on flow. Global Sensitivity Analysis According to the FAST GSA results (Figure 4 6), The m ost important factors in this model in order of importance, are: k (growth factor), M u (mortality at unit length), and L high (upper length at 50% vulnerability). These are also highly interactive inputs

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121 (Figure 4 7). Inputs that contributed less than 1 % of the total model sensitivity include: maturity, fecundity, broods per season, L low (Figure 4 7). These unimportant inputs were set to constants in the next round of MC Filtering. Additionally, in future modeling efforts less emphasis may be placed on these inputs because of their relative lack of importance. The flood coefficient, which determines the relationship that flow has on recruitment, was not one of the most sensitive inputs. However, this is the only input that directly relate d recruitment and population size to the flood and cre ated dynamics in the population All of the other inputs are related to the overall size and/or biomass of the population. To get density, which is the objective function, the total population was divided by the ar ea of inundation. Because the fish population data came from the Panhandle which is less dynamic that the rest of the Delta, and for reasons previously mentioned, the population was divided by a constant area each year. Therefore, without the flood coeff icient, the fish population density was not dynamic. T he flood coefficient is the only input that relate d the inter annual variability of the population size to the flood size. A scatter plot that compares the flood coefficient to the objective function (the ceff ) shows that the best model fit converges at a flood coefficient of 15 (Figure 4 8). So, in order to get the best model behavior, the flood coefficient must be approximately 15 At a value of 15, the number of recruits varies between 7,300 and 14,200, approximately doubling the number between low and high flood years (Figure 4 9). This is not the case for the other inputs. For the rest of the inputs, there is generally a high degree of equifinality and the model is able to achieve a good fit with using Thus, there are many ways to get a

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122 baseline population level by manipulating all of the other inputs and a high level of equifinality in this portion of the model. But there is only one way to cha nge the dynamics which i s through the flood coefficient. The flood coefficient converges to an optimal value and there is less equifinality in this portion of the model. This shows that there is evidence that the flood is indeed a driver for the fish pop ulation dynamics despite the equifinality in the baseline population of the model. Monte Carlo Filtering All of the outputs were mapped to their corresponding inputs so that the inputs that created the best fit outputs could be better understood. A thresh old of a ceff of 0.50 was set with any model outcome greater than or equal to 0.50 defined as behavioral and any output less than 0.50 defined as non behavioral. The value of 0.50 was strategically chosen as a threshold as a tool for model optimization. Using this threshold, of the 39,952 runs 329 were shown to be behavioral. This optimized the solutions that fell into the behavioral category while at the same time ensured that there were enough values in the behavioral range to employ the two sided Smir nov test. The two sided Smirnov test showed that of the 12 important inputs, 9 had distributions where the behavioral inputs were significantly different from the non behavioral inputs: L high M u Z, and the flood coefficient. When the behavioral input distributions were shown to be significantly different from the non behavioral input distributions, new Th e 8 significantly different distributions were all skewed and so triangular distributions were chosen to represent (Figure 4 10). The GUA was then rerun to understand how this truncation affected the mo uncertainty.

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123 Global Uncertainty Analysis The model was rerun through the FAST GSA/UA with the unimportant inputs set to constants and the posterior triangular were assigned to the inputs: L high M u k, and Z. The model results showed similar optimization from before the MC Filtering with a maximum ceff of 0.64 and a maximum R 2 of 0.64. Through MC filtering input/output model uncertainty was reduced from and average ceff of 19.4 to 5.4. The minimum ceff was reduced from 306.5 to 202.4. The 95% confidence interval was reduced from 0.10 to 48.4 in the original simulation to 0.59 to 23.01 in the MC Filtered simulation (Figure 4 11 b and Figure 4 12). Summary The objective of this wor k was to test the conceptual model that fish population dynamics in the Okavango Delta is driven by the annual flood pulse through a mathematical simulation. T here is one input in the model (the flood coefficient) that directly drives the inter annual var iability in fish population does not mean that the effect of the flood and the flood coefficient is not important for producing the best fi t model simulations. This input is actually responsible for setting up a ll of the dynamics in the model In order to explain this the density calculation must first be described in greater detail. In order to get a measure of density, the vulnerable fis h were divided by the area of inundation in the Panhandle where the fish were caught. Surprisingly, dividing the fish by the ORI modeled average minimum area of inundation in the Panhandle reservoir provided a better fit than the ORI modeled annual minimu m area of inundation in the Panhandle. There may be several explanations for this apparent discrepancy.

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124 The ORI model was calibrated to the larger Delta and in particular the Boro reservoir and not specifically the Panhandle. So, there may be more model error in the flooding extents in the Panhandle as it was not the focus of the model calibration. The Panhandle is also less dynamic than the larger Delta and more channelized. Therefore, dividing the fish by a constant areal value is somewhat consistent with the physical characteristics of the site. This portion of the model could easily be made to simulate the entire Delta and be made dynamic to investigate the impact of inter annual inundation, if fish data were made available. However, in this model version, in order to get a measure of density, in each year the vulnerable fish were divided by a constant, which is the average minimum area of inundation of the Panhandle. Thus, without the flood coefficient the model would be at steady state. The most important inputs in the model are all related to this steady state or baseline vulnerable population size. All of the other inputs (besides k) exhibit major issues of equifinality (Figure 4 8) They do not converge and are able to vary throughout their PDF ranges while still achieving best fit results. Thus, the average population size can be modeled in various ways. However, the value for the flood coefficient converges toward a single solution (Figure 4 8) Thus, the inter annual variability can onl y be modeled in one way. In order to achieve a model fit that approximates the average population ( ceff of 0.5), you only need to get the baseline population correct. However, in order to get the best model fit, and reach a ceff above 0.5, both the basel ine population and the inter annual variability must be modeled correctly. The fact that the model is able to simulate a ceff of 0.64 (which is greater than

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125 0.5) provides evidence that the flood is an important driver in the system despite the fact that th ere is equifinality in portions of the model. To reduce input/output uncertainty, MC filtering was applied to the initial inverse optimization routine. The important prior distributions were re set to triangular distributions where the behavioral inputs were found to be significantly different from the non behavioral inputs based on the two sided Smirnov test. L high M u k , L and Z were all found to have significantly different behavioral inputs and were assigned triangular distributions. This m ethod reduced model input/output uncertainty by reducing the 95% confidence interval of the coefficient of efficiently between the modeled density and the measured CPUE from a width of 48.5 to 23.6. This is useful and allows the modeler in a data poor env ironment to focus in on likely values within the ranges of physically acceptable The modeler can then work in tandem with the biologist to ensure that the new makes sense in a real biological setting providing converging lines of evidence for a more accurate depiction of the system and its interactions. Research investigating the influence of the flood pulse on fish populations throughout the world has been conducted with a variety of results (Mrona and Gascuel, 1993; King et al., 2003; Ga ff et al., 2004; DeAngelis et al.1997). Much of this research shows that these relationships are complex and difficult to quantify. King et al. (2003) investigated floodplain usage by fish in the Murray Darling Basin, Australia where there is annual inu ndation via snow melt and flood pulse has been theorized to be a major driver for fish populations. Through sampling, these authors noted that floodplain utilization by fish was not as pronounced as expected. They proposed a

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126 more complex system and sugges ted a model based on optimum conditions for floodplain utilization including: temperature, flood pulse predictability, the rate of change in the hydrograph, and inundation duration and area. However, the flood pulse in the Murray Darling Basin may be less predictable than in the Okavango Delta implying that the fish in the Murray Darling may be more opportunistic and less consistent in their behavior. Mrona and Gascuel (1993) investigated statistical relationships between commercial fish catch and the annual flood in the Amazonian floodplain. They found three interesting relationships. (1) There was a positive correlation between catch and the flood peak three years prior, which they speculated to be associated with recruitment. (2) There was a n ass ociation between catch and the water level during its rise two years prior that was possibly associated with competition. (3) There was a n association between catch and severe low water stage two years prior that likely due to increased mortality. They were able to produce a statistical model with three variables that explained more than 83% of the variability in the annual fish abundance. Similar to the Okavango, this system experiences a regular and predictable flood pulse. The Everglades also experie nces an annual flood pulse during the rainy season which has been theorized to be a driver for fish populations (Gaff et al., 2004 and DeAngelis et al., 1997). DeAngelis et al. (1997) constructed a mechanistic model, Across Trophic Level System Simulation Landscape Fish model (ALFISH), that spatially predicts fish abundance based on the flood pulse. This fish model was built on top of a spatially explicit hydrologic model that simulates the annual flood pulse. The model simulates seasonal dynamics in pro duction due to the flooding as well as trophic

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127 interactions. Periphyton, macrophytes, detritus, mesoinvertebrates, and macroinvertebrates are all modeled as well as two fish functional groups: big fish and small fish. As the flood rises, modeled fish mov e into the floodplain in response to increased food availability. Then, as the flood recedes, modeled fish move to find refugia and mortality increases as a result of crowding and predation. Four types of mortality were simulated: background mortality, de nsity dependent mortality, predation by the large fish, and failure to find refugia. The fish life cycle was simulated according to the von Bertalanffy age/length/weight relationships (von Bertalanffy, 1957). Gaff et al. (2004) critique d ALFISH and conc lud ed that inundation area is not the only driver for fish populations and that other parameters may be just as important. They state d that the best model fit that ALFISH is able achieve is an R 2 of 0.88 for water depth and 0.35 for fish density with an i nverse relationship between water depth and fish density However, an R 2 of 0.35 betwee n fish density and water depth reflects empirical data showing that the hydrology only accounts for 20 40% of the variability in the sampled fish population density. T hus the results here that achieve a ceff of 0.64 and an R 2 of 0.77 are relatively promising 2 of 0.35 in the Everglades where fish density was modeled in response to water depth. However, the Ok avango fish model is a bucket model that runs on an annual time step whereas Gaff Therefore, there were fewer data points modeled in the Okavango model making it perhaps a simpler soluti on. Several limitations can be identified in this work. The fish data were from commercial catch and not experimental data and so fisherman preferences, knowledge,

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128 and other human variables may play into the data. The data contained only counts of tila pia fish and was not accompanied with age, length, or weight and so seasonal interpretations of responses to flooding were impractical. The data came from the Panhandle which is more permanently flooded and is less dynamic than the larger Delta. Addition ally, the ORI hydrologic model which simulates the inundation area was not specifically calibrated to the Panhandle reservoir and was more focused on the larger Delta. Finally, the variability of the annual fish population is not exceptionally dynamic. T he maximum annual CPUE only fluctuates between 26 and 39. In heavily fished areas, fish populations are often much more dynamic, lending to more variability to model. In spite of these limitations, this model was still able to simulate fish population dy namics to within a ceff of 0.64. In addition to this the flood coefficient, which is the only input that affect the inter annual variability in the population size, converges toward a single value. At a value of 15, this flood coefficient is responsible for approximately doubling recruitment between low and high flood years. Thus, this mechanistic model corroborates the conceptual model (Merron, 1991; Mosepele et al., 2009) and qualitative observations (Hoberg et al. (2002) made about fish response to th e flood pulse in the Okavango Delta.

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129 Table 4 2. List of inputs and initial distributions. Abbrev. Input Units Distribution 1 Z Natural Mortality yr 1 U(0.67, 1.39) 2 k Growth coefficient yr 1 U(0.25, 1.0) 3 M u Mortality per unit weight yr 1 g 0.3 U(3, 8) 4 a Weight/length parameter g 1 cm b U(0.004, 0.026) 5 b Weight/length parameter unitless U(2.911, 3.424) 6 L Asymptotic length cm 1 U(40, 53) 7 CR Compensation ratio unitless U(3, 50) 8 L low Lower length at 50% vulnerability cm 1 U(1, 3) 9 SD low Standard deviation of L low cm 1 U(23, 25) 1 0 L high Upper length at 50% vulnerability cm 1 U(28,60) 1 1 SD high Standard deviation of L high cm 1 U(10, 30) 1 2 Maturity Maturity yr N(4.25, 0.5) 1 3 Fecundity Fecundity unitless U(350, 800) 1 4 Broods Br oods unitless U(1, 2) 1 5 Inflow Inflow M m 3 yr 1 U(10%) 1 6 Flood coefficient Flood coefficient yr M m 3 U(5, 25)

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130 Table 4 1. Natural mortality (M) and growth coefficients (k) for selected tilapiine species. Species M K Location Reference Oreochromis a ndersonii 1.39 1.0 Okavango Delta, Botswana Mosepele and Nengu (2003) Oreochromis andersonii 0.67 0.25 Okavango Delta, Botswana Booth et al. (1995) Oreochromis macrochir 1.5 1.0 Okavango Delta, Botswana Mosepele and Nengu (2003) Oreochromis macrochir 0. 95 0.42 Okavango Delta, Botswana Booth et al. (1996) Tilapia rendalli 1.22 0.78 Okavango Delta, Botswana Mosepele and Nengu (2003) Serranochromis angusticeps 1.46 1.0 Okavango Delta, Botswana Mosepele and Nengu (2003) Serranochromis robustus 1.21 0.8 3 Okavango Delta, Botswana Mosepele and Nengu (2003) Oreochromis niloticus 0.254 Lake Victoria, Kenya Getabu (1992) Haplochromis anaphyrmus 1.45 0.671 Lake Malawi, Mozambique Tweedle and Turner (1977) Haplochromis mloto 0.92 0.55 Lake Malawi, Mozambiq ue Tweedle and Turner (1977) Lethrinops longipinnus 1.69 0.571 Lake Malawi, Mozambique Tweedle and Turner (1977) Lethrinops parvidens 1.20 0.487 Lake Malawi, Mozambique Tweedle and Turner (1977) Tilapia esculenta 1.75 0.28 Lake Victoria, Kenya Garrod (1963)

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131 Figure 4 1. Site location. The Okavango Delta, with fish sampling sites marked with stars.

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132 Figure 4 2. The conceptual model. Figure 4 3. A demonstration of the response in recruitment as a result of the change in annual maximum flood. The rate of recovery and the carrying capacity increases during smaller floods. Flood 1 >Flood 2 >Flood 3

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13 3 Figure 4 4. An example of the dome shaped double logistic curve (Allen et al., 2009) for catch vulnerability. L low = 2, SD low = 2, L high = 45, SD high = 10. Histogram data from Mosepele, 2009. (a) shows the dome shaped curve and (b) shows the curve approaching a logistic shape. Both shapes are possible within the bounds of the defined

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134 Figure 4 5. The b est fit model simulation for the initial inverse optimization. Coefficient of efficiency of 0.64.

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135 Figure 4 6. First order sensitivity for the coefficient of efficiency of the modeled fish density compared to the measured catch per unit effort ( CPUE ) from the first inverse optimization. Figure 4 7. First and higher order sensitivities for coefficient of efficiency of modeled fish density compared to measured CPUE from first inverse optimization.

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136 Figure 4 8. Scatter plots of v alues for t he flood coefficient, k, M u L high L and b creating behavioral outputs. Note that the flood coefficient converges toward a single value in the highest values for the coefficient of efficiency (best fit model runs)

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137 Figure 4 9. The effect of the flood and the flood coefficient on re cruitment. ( flood coef = 15)

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138 Figure 4 10. Important inputs whose behavioral distributions are significantly different from the non behavioral distributions. These graphs show histograms of the behavioral distributions and their newly defined triang ular prior distributions.

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139 a. b. Figure 4 11. All GUA results. (a) unfiltered (b) Monte Carlo (MC) filtered Because of the scale on the x axis ( 400 1 with 50 unit increments labeled), coefficient of efficiencies greater than 1 are undetectable. Therefore, Figure 12 shows a blowup of results with coefficient of efficiencies greater than 0.

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140 Figure 4 12. Uniform and MC Filtered GUA outputs with a coefficient of efficiency greater than 0.

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141 CHAPTER 5 CONCLUSIONS This research involving the G lobal Sensitivity and Uncertainty Analysis ( GSA/UA ) and Monte Carlo Filtering ( MC F) of the Okavango Research Institute ( ORI ) Pitman, and Okavango Delta fish models provides important insights into the use of these tools not only for model building but als o as tools for reducing input/output uncertainty, measuring the quality of models for management purposes, and corroborating ecological theories in the presence of model equifinality. GSA/UA is a tool that is traditionally used to aid modelers by identify ing important inputs and setting unimportant inputs to constants. This allows the modeler to avoid over parameterization and also enables the modeler to focus on the most important inputs for further study. GSA/UA also often provides important insights i nto model inner workings that may aid in model development. All of these aspects of GSA/UA were accomplished in this work. However, a broader application of the use of these tools allowed additional insights to be garnered from these methods. The GSA/UA and MCF of the ORI reservoir model provided valuable insights into the sensitivity, uncertainty, and inner workings of the ORI model and into reservoir models in general. The importance of volume thresholds was discovered ; the keystone reservoir was ident model uncertainty and increasing predictive capabilities was presented. MCF also decreased model input/output uncertainty and brought the uncertain outputs closer to the calibrated values These insights are useful for both modelers and policy makers in understanding the system and making decisions regarding model scenarios of climate change and development.

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142 The Morris screening coupled with the FAST quantitative two tiered regionalizat ion GSA/UA of the Pitman model highlighted the most important and least important inputs, quantified model uncertainty, compared regional groupings, and explored the uncertainty of alternative nonstationary model scenario s. The most sensitive processes in the model have to do with rainfall distribution, catchment absorption, evapotranspiration, and groundwater recharge. G roundwater runoff is the least important process. The most important regions are the more humid headwater regions where the majority of the flow originates. The importance of each input is also compared to the uncertainty that was defined for the probability density functions ( ) (low, medium, high). Generally, the inputs that were assigned high uncertainties were more important than those with medium or low uncertainties. This because inputs with a wider PDF are able to vary over a larger parameter space. The conclusions are useful especially w ithin the context of Adaptive Management when interpreting model reliability, determining the usefulness of model predictions, identifying gaps in knowledge, and focusing monitoring efforts. Because of the data scarcity in the Okavango Basin, and resulting wide that were set for the inputs in the Pitman model, there are multiple input sets that can simulate the same goodness of model fit. The issue of equifinality was evaluated to understand the effects that it may have on using the model in a predic tive capacity for climate change scenarios. The most practical best fit model simulations were run under a wet and dry climate change scenario. These nonstationary conditions showed that

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143 the relative uncertainty of the model increased under dry condition s but only very slightly decreased under wet conditions. A fish population model driven by the flood pulse was developed for the Okavango Delta. The fish model integrated GSA/UA and MCF in the development stages. The most important factors in this mode l were Mu (mortality at unit length), L high (upper length at 50% vulnerability), and k (growth factor). These were also highly interactive inputs. The least important inputs in the model were maturity, fecundity, the number of broods per season, the floo d coefficient, SD low and L low These inputs were set to constants for model simplification. The flood coefficient was found to be an unimportant input. However, this does not mean that the effect of the flood and the flood coefficient is not important for producing behavioral outputs. The most important inputs in the model are related to the overall population size, whereas the flood coefficient is the only input that relates solely to inter annual population dynamics. A ll of the important inputs (be sides k) show major issues of equifinality. They do not converge and are able to vary throughout their PDF ranges while still achieving best fit solutions. However, the flood coefficient converges toward a single value that produces a best model fit. Th is provides evidence that the flood is an important driver in the system despite equifinality in the model. Furthermore, this phenomenon points to a gap that traditional GSA/UA techniques may miss. This unimportant input is actually crucial for attaining the best model results. After the GSA/UA was conducted and the unimportant inputs were set to constants in the fish model, MC filtering was performed. The prior distributions were re set where the behavioral inputs were found to be significantly dif ferent from the non

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144 behavioral inputs based on the two sided Smirnov test. L high and Z were all found to have significantly different behavioral inputs and were assigned triangular distributions based on the distribution of the behaviora l inputs. This method reduced input/output uncertainty and is useful in a data poor environment allowing the modeler to focus on likely values within the ranges of physically acceptable The modeler can then work with ecologists to ensure that the model makes ecological sense and provide converging lines of evidence for theories of system interactions. Data scarcity presents many challenges for modelers and policy makers when developing, calibrating, interpreting model results, and applying model s in predicative and dynamic capacities. However, there are ways of managing for these challenges. Adaptive Management is specifically designed for determining a course of action within the context of uncertainty. The tools presented here are couched w ithin the context of Adaptive Management and quantitatively analyze model uncertainty in the light of practical management issues. The issue of data scarcity is acknowledged upfront and input/output uncertainty is reduced, model quality is assigned, and e vidence of ecological theory is supported in spite of equifinality. Models are by nature simplifications of reality. Thus, as they are useful in understanding the world in a simpler fashion, they also introduce uncertainty. The human brain in fact vie ws reality through a conceptual filter and every model, measurement, and connection that we perceive is inherently uncertain. GSA/UA is intended to shed light on this uncertainty. There are two major limitations associated with the GSA/UA in the work cont ained in this body of research. This work only investigates the uncertainty and sensitivity of model inputs. It does not address the

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145 uncertainty in model structure or time series data. In addition, there is inherent eloped for these inputs. As was shown in the Pitman model, the width of the PDF has an impact on the importance of the input. In a inputs based on evidence and da some percentage and also given a ranking such as high, medium, and low uncertainty. uncertainty analysis of the uncer tainty analysis could be conducted; however that is beyond the scope of this work. MCF input/output uncertainty. However, this method is based on setting a rather arbitrary threshold for behavioral outputs. Arguments can be made for setting this threshold. For example in the ORI case, the threshold was set by balancing inputs and outputs. The fi sh model the threshold was set to calibrate the model to an optimal range. Nonetheless, the threshold is a cutoff, a line drawn in the sand. A continuum based on the degree of behavior would be a better filter. The Adaptive Management process is a cyc lical process that is constantly modeling, monitoring, learning and improving. Evaluating model uncertainty and identifying gaps in knowledge are important parts of the Adaptive Management process. Doing so gives managers a measure of confidence in model results and identifies areas for strategic monitoring whereby model results can be improved. This work has identified important areas for study that will aid the Adaptive Management process

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146 I mportant inputs have been identified where strategic monito ri ng will produce better models and unimportant inputs have also been identified to aid model simplification Additionally unimportant inputs that drive model behavior into the best fit model ranges were identified. This is a gap in the existing body of kn owledge on GSA/UA. Ivanovi and Freer (2009) state that it is particularly difficult to objectively test the uncertainty of inputs for empirical models. Pappenberger and Beven (2006) also make the point that uncertainty analyses may be too subjective to be of practical use. In t his work MCF was used in conjunction with GSA/UA to explore how can be objectively set and input/output model uncertainty reduced. This is especially an issue in data poor areas. Here, MCF was successfully used to objectively redefine prior b ased on behavioral outputs and decrease input/output uncertainty.

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147 APPENDIX A OKAVANGO RESEARCH INSTITUTE MODEL COMPUTING GLOBAL SENSITIVITY AND UNCERTAINTY ANALYSIS SCRIPT AND INSTRUCTIONS The following script is used to run the ORI model (Wolski et al ., 2006) in batch on the UF High Performance Computing (HPC) system. Essentially, for the GSA the ORI program is run in batch. The batch is broken up into smaller jobs to run more quickly. Each smaller job has a .job file associated with it. The progra m, Wine is used to run the Windows based Visual Basic program on the HPC Unix machine. File name: Delta.job Begin script: #! /bin/sh #PBS N testjob #PBS o testjob.out #PBS e testjob.err #PBS M acathey@ufl.edu #PBS l nodes=1:ppn=1 #PBS l walltime=30: 10:00 date hostname echo "Changing directory to work area" cd /scratch/ufhpc/acathey/ORISA/MCF/delta400 pwd ## ## Create dummy display ## echo "Picking a unique display" ncpus=`grep vendor_id /proc/cpuinfo | wc l` display=1 while [ $display le $ncpus ] ; do if [ e /tmp/.X${display} lock ] ; then break fi display=`expr $display + 1` done echo "Unique display found: $display"

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148 echo "Firing up dummy X server" Xvfb :${display} > /dev/null 2>&1 & jobs export DISPLAY=:${display} ## ## Actual program commands ## echo "Invoking wine" wine delta.exe echo "Done" date ## ## Clean up after ourselves ## echo "Killing dummy X server" kill TERM %1 # This should already be cleaned up, but just in case rm f /tmp/.X${display} lock rm f /tmp/.X11 uni x/X${display}

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149 APPENDIX B FISH MODEL DATA AND CODE Table B 1. Modeled fish populations and measured data. Measured data from Mosepele, 2009. Date 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Total Modeled Population 19,467 19,866 21,600 23,565 22,352 24,198 23 ,172 23,172 26,164 25,965 Modeled Fish Density 28 28 31 34 32 35 34 33 37 37 Sampled CPUE 26 29 30 31 34 38 31 36 39 34 Modeled yr 1 6,052 7,021 8,541 9,375 6,856 9,432 7,567 7,567 10,837 8,717 Modeled yr 2 4,463 3,937 4,567 5,556 6,099 4,460 5,016 5,0 16 4,922 7,049 Modeled yr 3 2,525 2,977 2,626 3,046 3,706 4,068 4,093 4,093 3,346 3,283 Modeled yr 4 1,742 1,704 2,009 1,772 2,056 2,501 2,007 2,007 2,762 2,258 Modeled yr 5 1,975 1,181 1,155 1,362 1,201 1,394 1,861 1,861 1,361 1,872 Modeled yr 6 869 1 ,341 802 784 925 816 1,151 1,151 1,264 924 Modeled yr 7 740 591 911 545 533 628 643 643 782 859 Modeled yr 8 531 503 401 619 370 362 377 377 437 532 Modeled yr 9 368 361 342 273 421 252 290 290 256 297 Modeled yr 10 201 250 245 232 185 286 167 167 197 174

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150 Table B 2. Monthly sampled catch per unit fish data used for validation (Mosepele, 2009) Date CPUE Date CPUE Date CPUE Date CPUE Jan 96 20.3 Aug 98 23.0 Mar 01 16.1 Oct 03 36.5 Feb 96 26.7 Sep 98 25.0 Apr 01 18.3 Nov 03 26.6 Mar 96 7.8 Oct 98 27 .3 May 01 19.1 Dec 03 23.5 Apr 96 8.7 Nov 98 30.0 Jun 01 23.5 Jan 04 19.1 May 96 13.4 Dec 98 21.3 Jul 01 20.6 Feb 04 10.3 Jun 96 14.6 Jan 99 19.8 Aug 01 20.4 Mar 04 10.6 Jul 96 22.5 Feb 99 15.7 Sep 01 23.8 Apr 04 15.1 Aug 96 15.3 Mar 99 15.4 Oct 01 36 .9 May 04 28.9 Sep 96 26.3 Apr 99 14.3 Nov 01 37.5 Jun 04 24.3 Oct 96 26.3 May 99 21.8 Dec 01 32.0 Jul 04 28.1 Nov 96 22.2 Jun 99 18.2 Jan 02 24.1 Aug 04 38.9 Dec 96 22.8 Jul 99 22.6 Feb 02 22.8 Sep 04 28.3 Jan 97 21.2 Aug 99 21.4 Mar 02 16.1 Oct 04 2 8.2 Feb 97 19.4 Sep 99 31.3 Apr 02 15.7 Nov 04 21.3 Mar 97 16.7 Oct 99 25.7 May 02 19.9 Dec 04 30.3 Apr 97 15.1 Nov 99 19.0 Jun 02 19.1 Jan 05 33.9 May 97 17.7 Dec 99 13.2 Jul 02 19.8 Feb 05 22.2 Jun 97 20.7 Jan 00 21.1 Aug 02 18.4 Mar 05 14.2 Jul 97 19.4 Feb 00 21.4 Sep 02 22.5 Apr 05 14.5 Aug 97 17.9 Mar 00 27.3 Oct 02 21.4 May 05 17.4 Sep 97 23.8 Apr 00 30.7 Nov 02 27.3 Jun 05 21.0 Oct 97 28.7 May 00 14.7 Dec 02 30.7 Jul 05 24.4 Nov 97 26.9 Jun 00 20.7 Jan 03 30.9 Aug 05 16.1 Dec 97 21.7 Jul 0 0 21.3 Feb 03 29.6 Sep 05 29.6 Jan 98 22.4 Aug 00 24.1 Mar 03 21.9 Oct 05 25.7 Feb 98 22.6 Sep 00 24.8 Apr 03 12.2 Nov 05 34.3 Mar 98 14.6 Oct 00 34.0 May 03 11.5 Dec 05 20.3 Apr 98 10.4 Nov 00 33.0 Jun 03 12.1 May 98 13.2 Dec 00 27.0 Jul 03 20.4 Jun 98 15.8 Jan 01 27.4 Aug 03 25.0 Jul 98 19.4 Feb 01 17.1 Sep 03 24.0

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151 Fish Model Code (Visual Basic 6.0 ) Main m odule Public do_hydro_model As Boolean Public do_veg_model As Boolean Public workdir, mapdir As String Public noftsteps As Integer Public ev apdata As Integer Public file_suffix As String Sub Main() docalcflag = True workdir = "H: \ HOORCfishModel \ annual \ annual SA for Cal MC constantkz3" 'CurDir mapdir = workdir & "maps \ evapdata = 2 file_suffix = "_2008" 'file_suffix = "_baseline" 'file _suffix = "_csiro" 'file_suffix = "_had" 'file_suffix = "_ccc" 'file_suffix = "_gldf" 'file_suffix="" 'Call hydro_model.hydro_model_main Call fish_model.fish_model_main End Sub Fish module Public file_flood, file_inflow, file_out, file_fish, d_file, G SA_out As String Public flood() As Integer

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152 Public inflow() As Variant Public total_flood(), maxflood(), minflood() As Double Public recdate(), fl(), infl() As Variant Public unitcode() As Integer Public population_matrix() As Double Public caught_pop() As Double Public y, ts, month, start, nofyears, lifespan As Integer Public z(400000), fecundity(400000), broods(400000), juvsurvival(400000), recK(400000), llow(400000), sigmalow(400000), lhigh(400000), sigmahigh(400000), coef(400000), Linf(400000), deGraafM (400000), inflow10(400000), deGraafa(400000), deGraafb(400000), k(400000) As Double Public runs, G As Variant Public recruits(400000) As Variant Public maturity(400000) As Variant Public Spawners() As Double Public maxinflow(), mininflow(), minet(), maxet( ) As Double Public et() As Double Public stop_error As Integer Public EndSub As Label Sub fish_model_main() DoEvents file_flood = workdir & \ out_area_2008.csv" file_inflow = workdir & \ input_2008.csv" Call getnoftsteps nofyears = Int(noftsteps / 12) this is the number of years to analyse start = 5 MaxLength = 53 lifespan = 10 ReDim recdate(noftsteps) ReDim total_flood(noftsteps), maxflood(noftsteps), minflood(noftsteps) As Double 'flood in km^2 ReDim inflow(noftsteps) As Variant 'inflow in MCM at mohembo ReDim fish_biomass(lifespan, noftsteps) As Double 'biomass is in tons ReDim fish_length(lifespan) As Double 'fish length in cm ReDim population_matrix(lifespan, noftsteps) As Double

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153 ReDim total_pop(noftsteps) As Double ReDim et(noftst eps) As Double ReDim caught_pop(noftsteps), write_pop(noftsteps) As Double ReDim Spawners(noftsteps) As Double runs = 39952 Call read_inflow Call read_hydro_results Call GSA 'Call noGSA End Sub Sub GSA() file_out = workdir & \ Data \ fish_GSA" & ".csv" O pen file_out For Output As #2 file_GSAinputs = workdir & \ Data \ inputs2.sam" Open file_GSAinputs For Input As #3 Input #3, buzz Input #3, buzz Input #3, buzz Input #3, buzz For G = 1 To runs Input #3, maturity(G), fecundity(G), broods(G), recK(G), z( G), llow(G), sigmalow(G), lhigh(G), sigmahigh(G), coef(G), Linf(G), deGraafM(G), inflow10(G), deGraafa(G), deGraafb(G), k(G) maturity(G) = Round(maturity(G)) Next G Close #3 For y = start To nofyears If y > 41 And y < 52 Then Writ e #2, "ts & y + 1954, End If Next y Write #2,

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154 For G = 1 To runs stop_error = 0 For y = start To nofyears Call make_fish For N = 1 To lifespan L = Linf(G) (1 Exp( N k(G))) vulnerable = 1 / (1 + Exp( (L llow(G)) / sigmalow(G))) 1 / (1 + Exp( (L lhigh(G)) / sigmahigh(G))) If vulnerable < 0 Then vulnerable = 0 caught_pop(ts) = caught_pop(ts) + population_matrix(N, ts) vulnerable Next N If y > 41 And y < 52 Then If stop_error = 1 Then Write #2, 99999, Else Write #2, caught_pop(y) / 700, '(minflood(y) sflood(G)), '700, End If End If Next y Write #2, ReDim caught_pop(noftsteps) As Double Next G Close #2 Close #99 End Sub Sub noGSA() file_out = workdir & \ fish_biomass_2008" & ".csv" Open file_out For Output As #2 'ceff = 0.644 matu rity(1) = 4.094159213 fecundity(1) = 437.2761259 broods(1) = 1.208960917 recK(1) = 33.6917796 z(1) = 1.088943345 llow(1) = 24.87692895

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155 sigmalow(1) = 1.328258326 lhigh(1) = 66.68612323 sigmahigh(1) = 10.96894501 coef(1) = 13.07474026 Linf(1) = 40.08541241 d eGraafM(1) = 3.455538356 inflow10(1) = 1.026840991 deGraafa(1) = 0.017888095 deGraafb(1) = 3.069450496 k(1) = 0.4709256 maturity(G) = Round(maturity(G)) 'set headers in excel file Write #2, "", For ts = 1 To lifespan Write #2, "age & ts, Next ts W rite #2, G = 1 For y = start To nofyears Call make_fish Next y For y = start To nofyears For m = 1 To 12 ts = (y 1) 12 + m Write #2, "ts & ts, For N = 1 To lifespan L = Linf(G) (1 Exp( N k(G))) vulnerable = 1 / (1 + Exp( (L llow(G)) / sigmalow(G))) 1 / (1 + Exp( (L lhigh(G)) / sigmahigh(G))) If vulnerable < 0 Then vulnerable = 0 Write #2, population_matrix(N, y) vulnerable, N ext N Write #2, Next m Next y

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156 Close #2 Call write_to_excel End Sub Sub make_fish() ts = y '(y 1) 12 + month ReDim Spawners(noftsteps) As Double 'Average max annual flooding extents in the panhandle = 876 .2 km2, average = 801.6, min = 704.4 'Average max annual inflow for all years = 1529, for fish data years = 1328 'Average min annual inflow for all years = 350.7, for fish data years = 269.8 'Average avg annual inflow for a ll years = 774 'Average max ET for all years = 190.5, avg = 126.1, min = 63.4 Nt1 = 25000 No = Nt1 + (maxinflow(ts) 1529 inflow10(G)) coef(G) Ro = (1 Exp( z(G))) No reca = recK(G) Ro / N o recb = (recK(G) 1) / No If ts = 5 Then 'calculate the age structure for the first year For N = 1 To lifespan If N = 1 Then population_matrix(N, ts) = Ro Else L = Linf(G) (1 Exp( N k(G))) zL = deGraafM(G) deGraafa(G) ^ 0.3 L ^ ( 0.3 (deGraafb(G))) 'ge Graaf et al (2005) Simulation of Nile Tilapia in Ponds, used Mu of 1 population_mat rix(N, ts) = population_matrix(N 1, ts) Exp( zL) End If Next N Else 'calculate the age structure for the rest of the years For N = 1 To lifespan If N = 1 Then For n2 = maturity(G) To lifespan

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157 Spawners(ts) = Spawners(ts) + population_matrix(n2, ts 1) Next n2 On Error GoTo EndSub Spawners(ts) = (Spawners(ts) c oef(G)) fecundity(G) broods(G) population_matrix(N, ts) = reca Spawners(ts) / (1 + recb Spawners(ts)) Else L = Linf(G) (1 Exp( N k(G))) zL = deGr aafM(G) deGraafa(G) ^ 0.3 L ^ ( 0.3 (deGraafb(G))) 'ge Graaf et al (2005) Simulation of Nile Tilapia in Ponds, used Mu of 1 population_matrix(N, ts) = population_matrix(N 1, ts 1) Exp( zL) End If Next N End If Exit Sub EndSub: stop_error = 1 End Sub Sub getnoftsteps() Open file_flood For Input As #1 check number of time steps noftsteps = 1 Do While Not EOF(1) Line Input #1, temp noftste ps = noftsteps + 1 Loop Close #1 End Sub Sub read_inflow()

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158 Dim infl(6) As Variant ReDim maxinflow(noftsteps) ReDim mininflow(noftsteps) ReDim maxet(noftsteps) ReDim minet(noftsteps) Open file_inflow For Input As #11 For t = 1 To noftsteps I nput #11, infl(1), infl(2), infl(3), infl(4), infl(5), infl(6) inflow(t) = infl(2) et(t) = infl(5) Next t Close #11 For y = 1 To nofyears maxinflow(y) = inflow((y 1) 12 + 1) mininflow(y) = inflow((y 1) 12 + 1) maxet(y) = inflow ((y 1) 12 + 1) minet(y) = inflow((y 1) 12 + 1) For m = 1 To 12 ts = (y 1) 12 + m If inflow(ts) > maxinflow(y) Then maxinflow(y) = inflow(ts) End If If inflow(ts) < mininflow(y) Then mininflow(y) = inflow(ts) End If If et(ts) > maxet(y) Then maxet(y) = et(ts) End If If et(ts) < minet(y) Then minet(y) = et(ts) End If Next m

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159 Next y End Sub Sub read_hydro_results() ReDim fl(11, noftsteps) As Variant ReDim recdate(noftsteps) As Variant Open file_flood For Input As #10 Input #10, a, b, c, d, e, f, q, H, i, J, k, L For t = 1 To noftsteps Input #10, buzz, recdate(t), fl(1, t), f l(2, t), fl(3, t), fl(4, t), fl(5, t), fl(6, t), fl(7, t), fl(8, t), fl(9, t), fl(10, t), fl(11, t) total_flood(t) = fl(1, t) '+ fl(4) + fl(5) + fl(6) + fl(11) + fl(2) + fl(3) + fl(9) + fl(7) + fl(8) + fl(10) Next t Close #10 For y = 1 To nofyears 1 maxflood(y) = total_flood((y 1) 12 + 1) minflood(y) = total_flood((y 1) 12 + 12) For m = 1 To 12 ts = (y 1) 12 + m If total_flood(ts) > maxflood(y) Then maxflood(y) = total_flood(ts) End If 'If total_flood(ts) < minflood(y) Then minflood(y) = total_flood(ts) 'End If If m = 1 And total_flood(ts + 12) < minflood(y) Then minflood(y) = total_flood(ts + 12) End If

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160 If m = 2 And total_flood(ts + 13) < minflood(y) Then minflood(y) = total_flood(ts + 13) End If If m = 12 And total_flood(ts) < minflood(y) Then minflood(y) = total_flood(ts) End If If m = 11 And total_flood(ts) < minflood(y) Then minflood(y) = total_flood(ts) End If Next m minflood(y) = (total_flood((y 1) 12 + 11) + total_flood((y 1) 12 + 12) + total_flood((y) 12 + 1) + total_flood((y) 12 + 2)) / 4 Next y End Sub Sub write_to_excel() 'this writes the bucket model to excel Set objExcel = CreateObject("Excel.Application") Set objWorkbook1 = objExcel.Workbooks.Add(workdir & \ fish_biomass_2008.csv") Set objWorkbook2 = objExcel.Workbooks.Open(workdir & \ Data \ results.xlsx") objExcel.Visible = True objWorkbook1.Worksheets("fish_biomass_2008").Range("A1:k645").Copy objWorkbook2.Worksheets("fish_biomass_2008").Range("A1:k645").PasteSpecial Paste = xlValues objWorkbook1.App lication.CutCopyMode = False objWorkbook1.Close Set objExcel = Nothing End Sub

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161 Figure A 1. Fish model flow chart.

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162 Sample input file ( GSAinputs .dat) maturity fecundity broods CR z Llow Lhigh c Linf Mu inflow a b k 4.01 543.74 1.38 13.87 0.79 23.80 1.11 78.04 27.33 12.47 51.42 5.33 1.07 0.01 3.12 0.88 3.96 691.40 1.36 44.79 0.90 23.32 2.95 51.73 25.32 14.53 50.98 4.48 0.91 0.01 3.16 0.87 4.85 753.59 1.16 41.73 0.70 24.22 1.2 4 61.81 17.34 14.24 51.98 3.31 1.01 0.01 3.13 0.79 3.37 614.69 1.07 32.07 1.36 23.07 2.09 45.03 21.81 13.39 51.77 5.50 1.07 0.02 3.26 0.80 4.68 788.02 1.53 37.01 1.05 23.35 1.04 57.02 24.90 13.12 43.52 5.78 1.03 0.01 3.34 0.90 2.83 371.00 1.13 48.81 0.7 0 23.51 1.73 63.32 23.77 12.22 49.13 5.34 1.07 0.02 3.30 0.93 4.56 420.69 1.05 11.61 0.83 23.16 1.22 45.73 26.48 14.10 42.50 3.22 1.00 0.02 2.97 0.95 4.67 699.16 1.55 38.32 1.07 23.45 1.16 53.23 26.58 12.73 44.91 6.48 1.06 0.02 3.42 0.98 4.80 590.31 1.3 5 23.08 0.75 23.72 2.27 62.92 12.88 12.85 51.12 7.69 1.04 0.02 3.30 0.89 4.22 789.63 1.64 35.72 0.87 23.86 1.40 77.37 20.25 13.60 46.73 7.01 1.01 0.02 3.30 0.79 3.66 596.12 1.71 39.81 0.82 23.86 1.89 64.52 21.36 13.36 46.37 4.23 0.96 0.02 2.92 0.88 4.68 675.57 1.53 37.19 1.05 23.36 1.05 56.48 25.14 13.06 43.72 5.88 1.04 0.01 3.35 0.91 4.67 586.71 1.54 38.14 1.07 23.43 1.14 53.77 26.34 12.79 44.71 6.38 1.06 0.02 3.41 1.00 4.20 567.76 1.99 20.76 1.17 23.42 2.83 51.50 24.53 14.52 49.44 6.53 0.98 0.02 3.29 0.85 4.50 595.10 1.99 11.48 0.95 23.63 2.97 73.19 29.60 12.41 52.12 7.87 1.05 0.02 3.39 0.79 4.05 405.71 1.34 29.54 1.18 24.21 1.15 39.43 17.67 14.64 48.70 3.63 0.95 0.02 2.98 0.82 4.38 670.24 1.40 40.67 0.80 23.35 2.60 63.34 21.55 12.98 47.21 5.30 0.9 5 0.01 3.26 0.90 4.29 768.64 1.86 42.20 0.91 23.96 1.79 41.94 16.59 13.98 50.79 3.52 0.91 0.01 3.24 0.99 4.68 474.25 1.54 37.95 1.07 23.42 1.12 54.31 26.10 12.84 44.51 6.28 1.05 0.02 3.39 0.98 3.95 678.62 1.89 28.67 1.00 24.34 1.13 37.51 25.13 12.69 43. 18 3.65 1.05 0.01 3.37 0.99 4.17 577.16 1.58 22.38 1.06 23.84 1.03 52.56 18.53 13.49 49.81 5.33 0.98 0.01 3.11 0.78 4.38 669.16 1.15 40.49 0.80 23.37 2.58 63.88 21.31 13.03 47.41 5.20 0.95 0.01 3.25 0.88

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163 Sample output file timestep age 1 age 2 age 3 age 4 ag e 5 age 6 age 7 age 8 age 9 age 10 ts 49 7255 4719 3148 2124 1440 978 664 452 307 209 ts 50 7255 4719 3148 2124 1440 978 664 452 307 209 ts 51 7255 4719 3148 2124 1440 978 664 452 307 209 ts 52 7255 4719 3148 2124 1440 978 664 452 307 209 ts 53 7255 4 719 3148 2124 1440 978 664 452 307 209 ts 54 7255 4719 3148 2124 1440 978 664 452 307 209 ts 55 7255 4719 3148 2124 1440 978 664 452 307 209 ts 56 7255 4719 3148 2124 1440 978 664 452 307 209 ts 57 7255 4719 3148 2124 1440 978 664 452 307 209 ts 58 72 55 4719 3148 2124 1440 978 664 452 307 209 ts 59 7255 4719 3148 2124 1440 978 664 452 307 209 ts 60 7255 4719 3148 2124 1440 978 664 452 307 209 ts 61 7893 4719 3148 2124 1440 978 664 452 307 209 ts 62 7893 4719 3148 2124 1440 978 664 452 307 209 ts 6 3 7893 4719 3148 2124 1440 978 664 452 307 209 ts 64 7893 4719 3148 2124 1440 978 664 452 307 209 ts 65 7893 4719 3148 2124 1440 978 664 452 307 209 ts 66 7893 4719 3148 2124 1440 978 664 452 307 209 ts 67 7893 4719 3148 2124 1440 978 664 452 307 209 ts 68 7893 4719 3148 2124 1440 978 664 452 307 209 ts 69 7893 4719 3148 2124 1440 978 664 452 307 209 ts 70 7893 4719 3148 2124 1440 978 664 452 307 209 ts 71 7893 4719 3148 2124 1440 978 664 452 307 209 ts 72 7893 4719 3148 2124 1440 978 664 452 307 2 09 ts 73 11163 5134 3148 2124 1440 978 664 452 307 209 ts 74 11163 5134 3148 2124 1440 978 664 452 307 209 ts 75 11163 5134 3148 2124 1440 978 664 452 307 209 . .

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176 BIOGRAPHICAL SKETCH Anna Cathey received her B achelor of Art s in 1997 from the U niversity of Colorado at Boulder in a nthropology. She went on to work as an entomologist for Chatham County mosquito control in Savannah, Georgia. She continued to pursue her education receiving a Master of Science in a gricultural and b iological e ngineering from the University of Georgia in 2005. She then worked in the field of stream restoration for Michael Baker Engineering. She returned to academia to pursue her doctorate in 2007 at the University of Florida