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EROSION PARAMETER IDENTIFICATION IN OVERLAND FLOW AREAS:

APPLICATION OF A GLOBAL AND LOCAL SEARCH ALGORITHM by

Vicky Lynn Freedman

A Thesis Submitted to the Faculty of the

SCHOOL OF RENEWABLE NATURAL RESOURCES

In Partial Fulfillment of the Requirements

For the Degree of

MASTER OF SCIENCE

WITH A MAJOR IN WATERSHED MANAGEMENT

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 9 9 6

UMI Number: 1378991

UMI Microform 1378991

Copyright 1996, by UMI Company. All rights reserved.

This microform edition is protected against unauthorized copying under Title 17, United States Code.

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2

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

APPROVAL BY THESIS DIRECTORS

This thesis has been approved on the date shown below:

' Dr. Mariano Hernandez

Cooperating Scientist, USDA-ARS

H

a

Cn—V

Dr. D. I^hillip Guertin

Associate Professor of Watershed Management

Dr. Richard H. Hawkins

Professor of Watershed Management

Date

3

ACKNOWLEDGMENTS

I would like to express my gratitude to those who contributed to the completion of this work. My sincerest thanks to my major advisor, Dr. Vicente L. Lopes, who provided both guidance and economic support throughout the duration of this work and my academic career at the U of A, and contributed to my understanding of hydrology, hydraulics and erosion mechanics. I am grateful and indebted to Dr. Mariano Hernandez, who volunteered his time as my thesis director while Dr. Lopes was on sabbatical, and guided me through optimization theory and the modeling of hydrological processes. I would also like to express my gratitude to Dr. D. Phillip Guertin, who served as my major advisor and editor of this manuscript in Dr. Lopes absence, and to Dr. Richard H.

Hawkins, whose valuable comments contributed to my understanding of modeling hydrological processes and the completion of this manuscript.

I would also like to thank Mary Kidwell at the USDA-ARS for providing me with rainfall simulator plot data, Nick Mokhotu for providing Kendall watershed data, and

Carolyn Audilet of the Bio-Sciences East Computer Lab for allowing me access to computers even when the lab was closed Finally, many thanks to all the friends and family members who supported me throughout this research, but in particular, my sincerest gratitude to my husband, Mario, without whose love, support and sacrifice, this work would have not been possible.

for my husband, Mario

5

TABLE OF CONTENTS

ABSTRACT

INTRODUCTION

Problem Statement

Objectives

Approach

Benefits

LITERATURE REVIEW

Models of Soil Erosion

Rainfall Simulators and Rainfall Simulator Plots in Soil Erosion Research

Erosion Processes

Erosion Induced by Raindrop Impact

Flow-Induced Erosion

Hydraulic Roughness

Soil Erodibility

Erosion Parameters

Automated Techniques

Objective Function

SIMPLE LEAST SQUARES ESTIMATOR

MAXIMUM LIKELIHOOD ESTIMATION

31

32

32

33

25

26

28

29

30

21

21

23

23

13

14

16

17

17

19

6

HETEROSCEDASTIC MAXIMUM LIKELIHOOD ESTIMATOR 34

Parameter Optimization

THE SIMPLEX ALGORITHM

Global Optimization

THE SHUFFLED COMPLEX EVOLUTION -

UNIVERSITY OF ARIZONA ALGORITHM

37

38

40

Difficulties In Parameter Optimization

INTERDEPENDENCE BETWEEN MODEL PARAMETERS

INDIFFERENCE OF THE OBJECTIVE FUNCTION

TO VALUES OF INACTIVE PARAMETERS

40

42

43

43

44 DISCONTINUITIES IN THE RESPONSE SURFACE

PRESENCE OF LOCAL OPTIMA DUE TO THE

NON-CONVEXITY OF RESPONSE SURFACE

Data Calibration

METHODS

The Model

The Hydrologic Component

The Infiltration Component

The Erosion Component

Entrainment by Raindrop Impact

Entrainment by Flow

47

49

50

51

52

44

44

47

47

7

Sediment Deposition

The Data

Determining Values of Hydraulic Parameters

True Parameter Values

Treatment of Systematic Error

Parameter Identification

Convergence Criteria

Methodology Used for Comparison

RESULTS AND DISCUSSION

Synthetic Data Case

RESPONSE SURFACES

THE ERROR-FREE DATA CASE:

THE TWO-PARAMETER PROBLEM

THE ERROR-FREE DATA CASE:

THE THREE-PARAMETER PROBLEM

Analysis of the Results of the Synthetic Data Case

ESTIMATION OF ACTIVE AND INACTIVE PARAMETERS

EFFECT OF ALGORITHM AND OBJECTIVE FUNCTION ON

ACTIVE AND INACTIVE PARAMETERS

EFFECT OF ALGORITHM AND OBJECTIVE FUNCTION

ON FLOW-INDUCED EROSION EQUATIONS

60

61

62

63

54

54

57

58

64

64

65

70

80

86

74

75

77

8

OBJECTIVE FUNCTION

BEST VALUE OF OBJECTIVE FUNCTION

AND AVOIDANCE OF LOCAL MINIMA

87

88

DATA SET VARIABILITY 94

EVALUATION OF FLOW-INDUCED EROSION EQUATIONS 100

OPTIMIZATION PROBLEM SELECTED FOR

NATURAL DATA STUDIES

Analysis of Plot Data

ESTIMATION OF HYDRAULIC PARAMETERS

101

102

103

TRUE VALUES, INITIAL VALUES

AND PARAMETER BOUNDS

SEDIMENT GRAPHS FOR RAINFALL SIMULATOR PLOTS

PROCESSES OF DEPOSITION AND ENTRAINMENT

ACTIVATION OF EROSION BY HYDRAULIC SHEAR

Analysis of Watershed Events

PARAMETER ESTIMATES

CONCLUSIONS AND FUTURE RESEARCH

Summary and Conclusions

Recommendations for Future Research

APPENDIX A

LITERATURE CITED

103

106

118

119

120

120

123

123

126

128

165

9

LIST OF FIGURES

Figure 4.1. Kf -

T

c

Contour Plots for Equation 1

67

Figure 4.2.

Kf

-T

c

Contour Plots for Equation 2

68

Figure 4.3.

Kf

-T

C

Contour Plots for Equation 3

Figure 4.4

Kj - Kf Contour Plots for Equation 1

69

71

72

Figure 4.5. Ki - Kf Contour Plots for Equation 2

Figure 4.6.

Kj - Kf Contour Plots for Equation 3

73

Figure 4.7.

Optimization procedures associated with the highest estimation error for each parameter. Results from 3-parameter problem and 2-parameter

(K rt

) optimization problems

83

Figure 4.8.

Optimization procedures associated with the highest estimation error for each parameter. Results from 2-parameter problems (Kj-Kf) shown above for t c

= 0.502 and x c

= 0.0 84

Figure 4.9.

Comparison of response surfaces for HMLE and SLS criteria

(Equation 2, dry run, 15% slope, p = 0.50)

89

Figure 4.10. Comparison of response surfaces for HMLE and SLS criteria

(Equation 2, wet run, 10% slope, p = 0.25)

90

Figure 4.11. Comparison of response surfaces for SCE-UA and Simplex algorithms.

(Equation 2, dry run, 10% slope, HMLE, p = 0.50)

91

Figure 4.12. Comparison of response surfaces for SCE-UA and Simplex algorithms.

(Equation 2, dry run, 10% slope, SLS, p = 0.25)

92

Figure 4.13. Number of best estimates associated with each initial moisture condition according to average percent estimation error for all four optimization procedures

99

Figure 4.14. Hydrographs and sediment graphs for plot number 31

108

Figure 4.15. Hydrographs and sediment graphs for plot number 34

109

Figure 4.16. Hydrographs and sediment graphs for plot number 56

Figure 4.17. Hydrographs and sediment graphs for plot number 59

Figure 4.18. Hydrographs and sediment graphs for plot number 63

Figure 4.19. Hydrographs and sediment graphs for plot number 66

Figure 4.20. Hydrographs and sediment graphs for plot number 102

Figure 4.21. Hydrographs and sediment graphs for plot number 105

Figure 4.22. Hydrographs and sediment graphs for plot number 120

Figure 4.23. Hydrographs and sediment graphs for plot number 121

10

110

Ill

112

113

114

115

116

117

Table 3.1.

Table 4.1.

Table 4.2.

Table 4.3.

Table 4.4.

Table 4.5.

Table 4.6.

Table 4.7.

Table 4.8.

Table 4.9.

Table 4.10.

Table 4.11.

Table 4.12.

Table 4.13.

11

LIST OF TABLES

Soil Characteristics for Rainfall Simulator Plots and

Kendall Watershed

Hydraulic Parameter and Soil and Plot Characteristics for the Synthetic Data Set

Parameter Bounds and Starting Values for Synthetic Data Set

Error Statistic s for 2-parameter problem

(Kj-Kf)

Error statistics for 2-parameter Problem (Krx c

)

56

Error Statistics for 3-parameter problem

Optimization procedures associated with highest estimation error for 3-parameter optimization problem

(Kj-Krx c

)

Optimization procedures associated with highest estimation error for 2-parameter optimization problem

(Kf

-x c

)

Optimization procedures associated with highest estimation error for 2-parameter optimization problem (Kj-Kf; t c

= 0.502)

79

81

81

82

Optimization procedures associated with highest estimation error for 2-parameter optimization problem

(Kj-Kf; x c

= 0.0)

Average estimation error for different initial moisture conditions for 3-parameter optimization problem

Average estimation error for different initial moisture conditions for 2-parameter optimization problem (Krt c

)

Average estimation error for different initial moisture conditions for 2-parameter optimization problem

(Kj-Kf; x c

= 0.502)

Average estimation error for different initial moisture conditions for 2-parameter optimization problem

(Kj-Kf; x c

= 0.0)

82

95

96

97

98

65

66

78

79

12

Table 4.14.

Table 4.15.

Table 4.16.

Table 4.17.

Optimized hydraulic parameters for the rainfall simulator plots

Parameter bounds and starting values for rainfall simulator plots

Erosion parameter estimates for rainfall simulator plots

Hydraulic parameters for selected Kendall Watershed events

Table 4.1B.

Table 4.17.

Parameter bounds and starting values for selected Kendall Watershed events

Erosion parameter estimates for selected Kendall Watershed events

Tables A1 - A3 6. Parameter Optimization Results for Synthetic Data Study

104

105

107

122

122

122

129-164

13

ABSTRACT

Two optimization algorithms and two objective functions were applied to determine erosion parameters for a physically-based, event-oriented model designed to simulate the processes of sedimentation for small watersheds. Three different flowinduced erosion equations were also tested with the four optimization procedures to examine the predictive capabilities of the equations. Synthetic error-free data as well as data contaminated with correlated and random error provided the means for determining the effectiveness of the four optimization procedures studied. After selecting the most effective optimization procedure and flow-induced erosion equation, the model was tested using sediment data from rainfall simulator plots and a small experimental watershed.

The results from the rainfall simulator studies indicated that a structural problem may exist within the model. The agreement between simulated and observed responses for the watershed events studied indicated that the model was capable of describing sedimentation processes when they occurred on a larger scale.

14

INTRODUCTION

Soil erosion is one of the major hazards threatening land productivity. The loss of

sediment and associated nutrients through runoff and soil erosion can reduce productivity and lead to vegetation losses and further increases in the rates of soil erosion (Gifford and

Busby, 1973). The transport of sediment from hillslopes into adjacent water bodies can also negatively impact reservoir capacities, water-based outdoor recreation, and fisheries.

Thus, the ability to predict soil erosion under current and alternate land-use conditions is important in managing land and water quality.

Physically-based erosion models are potentially capable of providing information on the amount, timing and sources of sediment production. However, a major problem in the application of physically-based models is in parameter identification (Lopes, 1987,

Blau et al., 1988; Page, 1988). Parameters, which can be defined as coefficients, are usually represented in an erosion model as the soil's ability to withstand erosion and are termed soil erodibility parameters. Considerable research has been conducted to relate measurable physical and chemical properties to soil erodibility parameters (Romkens et al.,

1977; Meyer and Harmon, 1984; Musaed, 1994). However, parameter evaluation is often accomplished by some manual or automated calibration procedure Although a manual procedure is subjective and may not generate an optimum parameter set, it usually produces parameter values that can be related to some physical properties of the watershed. Automated calibrated procedures, although more objective, have experienced problems such as convergence to local minima and producing parameter values that

15

effectively minimize the objective function but conceptually have no meaning

(Hendrickson et al., 1988).

Despite the inability of automated optimization algorithms to find an unique optimal parameter set, their use is widespread in conceptual rainfall-runoff modeling

(Dawdy and O'Donnell, 1965; Johnston and Pilgrim, 1976; Pickup, 1977; Sorooshian and

Gupta, 1983; Gupta and Sorooshian, 1985, Hendrickson et al., 1988). As physicallybased erosion models replace the empirically-based Universal Soil Loss Equation (USLE)

(Wischmeier and Smith, 1978), the use of automated techniques in erosion modeling has increased (Lopes, 1987; Page, 1988; Blau et al., 1988, Luce and Cundy, 1994). However, although physically-based models are conceptually superior to empirical models, as with empirical models, their accuracy is still dependent on the accuracy of their input parameters. Unless the best set of parameter values associated with a given calibration data set can be found, a reasonable degree of confidence cannot be placed in the accuracy of model predictions.

Different reasons have been cited for the inability of automated optimization algorithms to find unique optimal parameter values including parameter interaction

(Lopes, 1987), parameter insensitivity (Blau et al., 1988) and optimization procedures that are not powerful enough to do the job (Duan et al., 1992). The problem may lie in either the model structure, selected objective function, optimization method used, or some combination of these factors (Duan et al., 1992).

16

Problem Statement

Improved estimates of soil erosion are needed in order to address concerns about nonpoint pollution and land productivity. For example, long term estimates generated by the Universal Soil Loss Equation are inadequate for making erosion predictions under alternate land uses or on an event basis Subsequent advances in erosion technology have led to the development of physically-based models that conceptually divide erosion into two separate processes: 1) erosion caused by raindrop impact and 2) erosion induced by overland flow However, in order to apply such models, parameters that describe the soil's susceptibility to erosion must be identified.

The accuracy of model predictions is dependent on the sensitivity and accuracy of its input parameters The ability of an automated optimization procedure to determine input parameters is dependent on model structure, selected objective function, and optimization algorithm employed. This study compared optimal parameter values determined by two different optimization procedures and two different objective functions for three flow-induced erosion models. An assessment of the optimization algorithms' capacity to determine the erosion parameters for each of the models established the appropriateness of both the model and optimization procedure for use in erosion prediction when using the Watershed Erosion and Sediment Yield Program (WESP)

(Lopes, 1987).

17

Objectives

The goal of this study was to determine the adequacy of optimization algorithms and objective functions in identifying unique, optimal parameter values. The specific objectives of this study were:

1. Determine the sensitivity of the estimation procedure to calibration data variability

(dry, wet and very wet runs), and whether or not the selected objective function could reduce this sensitivity.

2. Assess both the algorithm's and model's sensitivity to variable and steady-state rainfall intensities

3. Evaluate the capabilities of the selected flow-induced models to predict erosion under fully dynamic conditions.

4. Assess the ability of the selected flow-induced erosion equation in reproducing sediment graphs with physically, realistic parameter values.

5. Identify problems in model structure that inhibited the identification of unique parameter values.

Approach

In order to study the behavior of WESP in conjunction with the different optimization procedures and flow-induced erosion models, synthetic sediment concentration data, which were generated utilizing soil and rainfall characteristics based on rainfall simulator plot data, were utilized so that true values of parameters were known.

18

Error-free data were used to verify that algorithms and objective functions were capable of finding true parameter values when no error was present. Since correlated error may be common in sediment concentration data, two levels of correlated error were introduced into the synthetic data set utilizing a first order Markov chain model.

With the synthetic data, two and three parameter problems were posed with the objective of finding the best parameter sets to reproduce a given sediment graph. By fixing the value of one of the three parameters, parameter sensitivities and interactions were better evaluated when compared to a three parameter optimization problem. Slope gradient effects and variable antecedent moisture conditions were also identified. Given that WESP is a fully dynamic simulation model, the parameters identified from events with a single rainfall intensity were compared to parameters identified from events with variable rainfall intensity rates.

This study utilized three data types; 1) synthetic data as previously described, 2) rainfall simulator sediment concentration data from the United State Department of

Agriculture - Agricultural Research Service (USDA-ARS) Water Erosion Prediction

Project (WEPP) field experiments, and 3) sediment yield data collected from the USDA-

ARS Kendall Watershed at Walnut Gulch. Three different equations were proposed to describe flow-induced erosion. Two optimization algorithms, one based on a local search procedure (Simplex method) and one specifically designed to find a global minimum

(SCE-UA), were used for parameter identification. Two objective functions, the sum of the least square (SLS) and heteroscedastic maximum likelihood estimator (HMLE), were

used to find the minimum error. Utilizing results from the synthetic data study, the most successful flow-induced erosion equation, optimization algorithm and objective function were used for identifying erosion parameters for the plot and watershed studies.

An analysis of parameter estimation error for both active and inactive parameters provided a criterion for identifying the optimal optimization procedure and flow-induced erosion model under different hydrological conditions. Other considerations included relative efficiency in avoiding local optima, continuity and shape of the response surface configurations, and algorithm's ability to attain the best value of the objective function.

19

Benefits

Parameter identification is of paramount importance in hydrological and erosion modeling. Without accurate parameter estimation methods, confidence cannot be placed in model predictions. Many contributions have been made to parameter identification within a conceptual rainfall-runoff framework. However, research is only recently emerging into the use of automated techniques of parameter estimation for physicallybased erosion models. The major benefits of this research are increased insight into parameter identification and the further development of the WESP model. Other models may also benefit by incorporating automated optimization techniques into the parameterization process.

The derivation of an optimum parameter set in erosion modeling may depend heavily upon the calibration procedure utilized. This study contributes to an

20

understanding of the limitations and benefits obtained from the selection of objective functions and search algorithms. An assessment of the sensitivity of the procedure (and/or the model) to variable and steady-state rainfall intensities also contributes to experimental design of future soil erosion studies utilizing rainfall simulators.

21

LITERATURE REVIEW

Decisions on how to control erosion and remediate erosion damage demand a knowledge of erosion risk under existing and alternative land management practices.

Physically-based erosion models are potentially capable of providing this information provided that they can be properly parameterized for any given watershed. The first section of this literature review presents a brief history of important contributions to erosion modeling research. The controlling variables and parameters in erosion modeling are then described. In the final section, techniques of parameter optimization are presented within both an erosion and conceptual rainfall-runoff modeling framework.

Models of Soil Erosion

Many models used in soil erosion studies are empirical and based on defining the most important factors controlling the soil erosion process through the use of observation, measurement, experiment and statistical techniques Zingg (1940) was the first to develop an equation that related erosion to slope steepness and length. Later developments included the addition of a climactic factor (Musgrave, 1947), and a crop factor that took into account the protective nature of different crops (Smith, 1958). This factor approach was later incorporated into the Universal Soil Loss Equation (Wischmeier and Smith,

1978), where the factors affecting the soil erosion process (rainfall erosivity, soil erodibility, topography and land use and management) were quantified. Although empirical models such as the USLE have been widely used to predict soil erosion, the

22

factors are unique to the experimental conditions from which they were derived and should not be used under different conditions.

Empirical models deal with erosion prediction and parameter identification differently than physically-based erosion models. Based on physical laws and theoretical principles, physically-based erosion models attempt to represent the processes of erosion by mathematical equations that represent the erosion processes of soil particle detachment, transport and deposition (Lopes and Ffolliott, 1994). Two different approaches have been used in physically-based erosion modeling. The first approach assumes steady state conditions even though the processes of sediment detachment and transport are known to be unsteady (Meyer and Wischmeier, 1969; Foster and Meyer, 1972; Komura, 1976;

Meyer et al. 1983, Rose, 1985). The second approach models the processes of erosion without steady-state assumptions. A kinematic-wave approximation to the dynamic flow equations is commonly used to model the hydraulics of the erosion processes, even though simplifying assumptions are required, such as constant and uniform rates of rainfall intensity and infiltration. The erosion processes are generally modeled using the continuity equation for sediment transport and empirical relationships for detachment by raindrop impact and hydraulic shear (Bennett, 1974; Singh, 1983; Lopes, 1987)

23

Rainfall Simulators and Rainfall Simulator Plots in Soil Erosion Research

Rainfall simulators have been used for soil erosion research and are designed to simulate precipitation occurring from a natural rainstorm over small areas. The use of rainfall simulators and rainfall simulator plots is advantageous because hydrological and soil erosion characteristics can be measured in a controlled environment. Researchers can also make measurements and observations during a simulated storm that may be difficult or impossible during a natural rainstorm (Meyer, 1994).

Rainfall simulator plots are designed to represent a micro-watershed. Like a watershed, the rainfall simulator plot can be represented by more than one element, where an element can be defined as either a plane or channel. Each element may represent changes in soil characteristics, hillslope characteristics or variations in land use. Channel elements can receive sediment inflows from upstream and lateral planes and channels.

Rainfall simulator plots can also be modeled as a single plane, assuming that erosion by hydraulic shear is driven by the hydraulics of broad shallow overland flow.

Erosion Processes

The erosion processes involve the detachment, transport and deposition of soil particles by the erosive forces of raindrops and surface flow. Conceptually, hillslope erosion has been traditionally divided into two phases based on the characteristics of overland flow: interrill and rill erosion. Interrill erosion is the result of detachment induced by raindrop impact and transport by broad shallow surface flow. As the surface

24

flow moves downslope, its flow depth increases and concentrates in rills. Soil detachment occurs in rills when the hydraulic shear of the flowing water is sufficient to overcome the binding forces between individual soil particles. These concentrated flow areas transport the detached sediment from both rill and interrill areas.

An alternative approach to modeling erosion on a hillslope is to assume that concentrated flow areas do not develop when an area is small. In this case, it is assumed that sediment entrainment is induced by two processes: raindrop impact and hydraulic shear of broad shallow overland flow.

Many physically-based erosion models conceptualize the erosion process as one of entrainment of soil particles and the detachment or deposition of sediment as a function of the flow's ability to carry the sediment load (Foster and Meyer, 1972). This is known as transport-capacity approach and basically describes a balance between entrainment and deposition rates of the sediment in flow (Nearing et al ., 1994). When the transport capacity of the flow is exceeded, deposition will occur. If the transport capacity is not reached, then entrainment of detached particles will occur given the available sediment supply.

Another approach to erosion modeling is simultaneous sediment exchange. It is based on a concept of a continuous exchange of particles between the flow and soil surface and does not consider the capacity of the flow to entrain soil particles. Lopes

(1987) developed a model that calculated rates of detachment and entrainment of sediment by flow, detachment and entrainment of soil by raindrop impact, and the deposition of

25

sediment. In this model, net entrainment and detachment occur when the rates of entrainment and detachment exceed the rates of deposition.

Erosion Induced by Raindrop Impact

Generally, the hydrologic variables driving the soil erosion processes in areas where entrainment by raindrop impact predominates are obtained by applying overland flow equations Young and Wiersma (1973) found that the detachment capacity of overland flow was negligible compared to that of raindrop splash, due to the low magnitude of the shear stresses caused by thin sheet flow. Kirkby (1980) found that when rates of erosion are high, soil loss from areas where entrainment caused by raindrop impact is usually low compared to losses from erosion caused by hydraulic shear.

However, erosion induced by raindrop impact can dominate in rangelands or where slope angles are low and slope lengths are short (Nearing et al., 1989).

A common model for entrainment by raindrop impact describes the rate of sediment transport as a non-linear function of rainfall intensity. Other models describe detachment by raindrop impact as a linear function of the rainfall excess rate and rainfall intensity. Such relationships are usually developed based upon extensive rainfall simulation studies on a variety of different soils (Nearing et al., 1989).

26

Flow-Induced Erosion

Entrainment induced by broad shallow overland flow occurs when hydraulic forces overcome the resistance threshold for the soil. The balance between the erosive power of the flow and erosion resistance of the soil determines the entrainment rate. The hydraulic variables driving the soil erosion process are often obtained by equations developed from observations in large channels

When employing the hydraulics of channels Hernandez (1992) found that parameter identification was a problem in rainfall simulator plots where well-defined drainage patterns did not exist. This implies that erosion caused by hydraulic shear may occur even in the absence of a well-defined rill or channel. Govers (1992) suggested that the hydraulics of overland flow are different from those of channel flow. In areas where broad shallow overland flow predominates, the hydraulics can be obtained from overland flow equations.

Flow-induced detachment is often described as a linear function of flow shear stress. The positive intercept on the shear stress axis is called the critical shear stress of the soil. Although many models describe flow-induced detachment as a linear function of hydraulic shear, flume studies have shown this relationship to be non-linear (Nearing et al.,

1994). In the Water Erosion Prediction Project (WEPP) model by Lane and Nearing

(1989), the critical shear stress is described as a mathematical entity that results from the linearization of the model. Nearing et al. (1994) warns that it should not be physically interpreted as a threshold level of shear stress. However, mathematically, threshold

27

parameters can be difficult to optimize due to parameter insensitivity (Johnston and

Pilgrim, 1976). A misrepresentation of the physical processes in the model can cause problems in parameter identification.

Many flow-induced models incorporate existing transport formulas that were developed based on experimental work in channels. The bedload formula of Yalin (1963) has been frequently used (Dillahah and Beasley, 1983; Kahnbilvardi et al., 1983; Park et al., 1982). Yalin's formula is of the excess-shear type, and is based on the theoretical assumption that bed-load discharge rate is a function of the range of particles in saltation rather than their number. Foster and Meyer (1972) first proposed the use of the Yalin equation for overland flow areas and Alonso et al. (1981) confirmed its ability for predicting erosion in shallow flow areas. The Water Erosion Prediction Project (WEPP) incorporates the Yalin formula into the erosion component of the model (Foster et al.,

1989).

The total load formula of Yang (1973) based on the theory of stream power has been frequently used. Bagnold (1966) first proposed the concept of stream power, which is based on a balance of energy rather than a balance of forces, to determine the entrainment rate Bagnold defined stream power as the product of bed shear stress and mean flow velocity. Sediment discharge, however, is not usually a sole function of shear stress. Consequently, Yang (1973) introduced the concept of unit stream power, which is the amount of energy dissipated per unit time and per unit weight of the flow and is equal to the product of slope and mean velocity. The total sediment concentration (not

28

sediment discharge) is therefore directly related to unit stream power. Moore and Burch

(1986) and Loch et al. (1989) have since demonstrated that the Yang formula is a good predictor of flow's transport capacity in overland flow areas.

Even though such formulas are based on physical principles, they have been calibrated utilizing experimental data. Model predictions may be erratic when these formulas are used to describe the transport capacity of flow in overland flow areas. This is due to the fact that these areas are very shallow and slopes may be much greater than those encountered in channels (Govers, 1992) where sediment transport formulas such as

Yalin (1963) and Yang (1973) have been developed. However, Govers (1992) found that equations based on shear stress, unit stream power and effective stream power could be used in some cases to effectively predict the sediment transport capacity of overland flow.

Hydraulic Roughness

The identification of the magnitude of the hydraulic roughness coefficient is pertinent to modeling flow-induced erosion. Laminar flow over rough surfaces is usually characterized by a high friction factor due to turbulent friction losses around protrusions causing roughness (Phelps, 1975). In general, the resistance coefficient depends on the

Reynolds number (Re) of the flow. The Darcy-Weisbach friction factor is most commonly used in hydrologic modeling, but Manning's and Chezy's coefficients may be used as well.

29

Soil Erodibility

Soil erodibility is defined as the resistance of the soil to both detachment and transport (Morgan, 1986). Soil erodibility, along with rainfall characteristics, topography, cover and management, is a major determinant of soil erosion and is a function of the chemical and physical properties of a soil. Particle size, aggregate stability, shear strength, infiltration capacity, organic matter and chemical content are widely accepted as the soil variables most strongly influencing a soil's erodibility.

Physically-based soil erosion models incorporate soil erodibility parameters into that part of the model dealing with soil entrainment and transport (Romkens et al., 1977;

Meyer and Harmon, 1984; Musaed, 1994). Before the advent of physically-based erosion models, several researchers have related measurable physical and chemical properties to indices of soil erodibility for agricultural soils (Bennet, 1939, Barnett and Rogers, 1966;

Wischmeier and Mannering, 1969; Wischmeier et al., 1971) However, the regression relationships developed require data that are not readily available for rangeland soils.

Simpler relationships based on texture, organic matter and volumetric water content have been developed to evaluate soil erodibilities for physically-based models (Alberts et al.,

1989; Flanagan, 1991)

Qualitatively, texture can be used as index of erodibility. In general, fine-textured soils are usually cohesive and difficult to detach. The small particles of fine textured soils are easy to transport unless the aggregates are large. Coarse-textured soils easily detach, but the large particles are difficult to transport. Medium-textured soils are both easily

30

detached and transported and are thus classified as highly erodible soils (Wischmeier and

Mannering, 1969).

Erosion modeling, however, requires a quantification of soil erodibility. Since the mechanisms differ for flow-induced and raindrop induced entrainment, erodibility parameters for each process are distinct. Considerable research has been dedicated to studying the separate processes of erosion and soil erodibility parameter identification

(Meyer et al., 1975, Young and Onstad, 1978; Hussein and Laflen, 1982, Van Liew and

Saxton, 1983, Bradford et al., 1987). A wide range of single parameters and combinations of parameters have been identified with varying degrees of success. When regression relationships are inappropriate, soil erodibilities can be identified by optimization.

Erosion Parameters

In order to model soil erosion by water, it is important to understand the controlling variables and parameters in the soil erosion process. Generally, physicallybased erosion models are structurally defined with a set of equations based on the physical laws representing the governing processes. In order to apply an erosion model to any given watershed, the relationships have to be made specific for that watershed Numerical values are defined for the equation's parameters that control the model's operation so that predicted sediment yields match observed sediment yields. This procedure is called model calibration. Model calibration for erosion modeling is made even more difficult by the fact

31

that erosion components are driven by hydrological models that contain their own parameters that also have to be identified.

Two different approaches have been used to determine parameter values in physically-based erosion models. The first approach assigns parameter values based on an assumption that the model parameters have a physical meaning. Values are determined based on a knowledge of the erosion processes or on measurable properties in the watershed. The second approach utilizes an automated optimization algorithm where parameter values are determined based upon a comparison between observed and simulated sediment yields in terms of an objective function. Computers are generally used because the number of iterations involved in solving the optimization problem. Although parameters identified by automated optimization algorithms are more objective and reproducible than estimates made based on the subjective judgment of a hydrologist, automated techniques may generate unrealistic parameter values that minimize the differences between simulated and observed sediment yields, but conceptually have no meaning (Hendrickson et al., 1988).

Automated Techniques

Parameter estimation from data and prior information is an important area of research. With the advent of the digital computer, research into the use of automated techniques for hydrological modeling has increased The automated optimization technique is comprised of three parts; 1) the objective function, 2) the optimization

algorithm, and 3) the calibration data. The following presents a discussion of each of these elements.

32

Objective Function

For any method of optimization, there exists some objective measure; i.e. objective function, as to how closely the sediment yield data simulated by the model compares with the actual measured values (Gottfried and Weisman, 1973). The selected objective function will affect the values of the fitted parameters because each criterion of best fit places a different emphasis on the differences between measured and calculated values

(Sorooshian and Dracup, 1980; Sorooshian et al., 1983).

SIMPLE LEAST SQUARES ESTIMATOR

When automatically calibrating an erosion model for a particular watershed, a nonlinear programming algorithm is used to minimize the objective function, F:

F =

f(e,,e

2

,03,., e„)

(2.1)

where 0 n

= model parameters. The most frequently used objective function is the simple least squares (SLS) criterion

(2.2)

33

where Ct,aim is the simulated sediment concentration at time

t ,

c^s is the observed sediment concentration at time

t,

and n is the total number of data points.. The simple least squares criterion assumes that model residuals are uncorrelated and homoscedastic.

MAXIMUM LIKELIHOOD ESTIMATION

Objective functions based on maximum likelihood (ML) theory have been demonstrated to provide more reliable estimates of parameter values than the SLS criterion (Sorooshian and Dracup, 1980; Sorooshian and Gupta, 1983; Sorooshian et al.

1983). This phenomena is due to the fact that if the objective function accounts for stochastic properties of the model errors, then it is easier for the optimization method to search for the best parameter values (Sorooshian and Gupta, 1983).

The selection of the objective function has been somewhat arbitrary in the erosion literature (Lopes, 1987; Page, 1988; Blau et al., 1988). The simple least squares criterion is usually employed and implies that model residuals are assumed to be uncorrelated and homoscedastic. However, violations in the aforementioned assumptions often occur in erosion modeling, where residuals are heteroscedastic and autocorrelated. If the stochastic nature of the residuals is not considered, then unsatisfactory parameter estimation can result. Biased parameter estimates will lead to unsatisfactory model predictions

The accuracy of the maximum likelihood procedure can be highly dependent on the available information. If the data set is of sufficient length and well represents the

34

variability, then it will be expected to produce good parameter estimates. However, if the representability of the data is questionable, then the ML approach may not produce good estimates of the parameters. To minimize such effects, Sorooshian (1981) updated the procedure so that the lack of information can be expressed through the use ofBayes' theorem. Accordingly, Bayesian theory, can be used to update the parameters using newly acquired data, if available.

HETEROSCEDASTIC MAXIMUM LIKELIHOOD ESTIMATOR (HMLE)

Stream discharge measurements are usually affected by non-homogenous variance

(Aitken, 1973; Sorooshian and Dracup, 1980). This means that with increasing sediment concentrations and yields, an increase in error variance can also be expected. Sorooshian and Dracup (1980) proposed the use of the Hetereoscedastic Maximum Likelihood Error estimator (HMLE) for the case where errors are assumed to be uncorrelated and heteroscedastic (non-homogeneous variance).

The HMLE estimator is the maximum likelihood, minimum variance, asymptotically unbiased estimator when the variance of errors in the observed data is assumed to be related to the magnitude of the data (Sorooshian, 1981; Sorooshian and

Dracup, 1980). The errors in the data are assumed to be Gaussian with a zero mean. The

HMLE estimator is defined as:

35

min HMLE f l w

.

t = l

where w, is the weight assigned to time t and is computed as:

(2 4)

where/ t is the expected true sediment concentration at time

t

and

X

is the unknown transformation parameter that stabilizes the variance. The expected true sediment concentration can be approximated by either Ct,

S im or Ct,

0 bs (Sorooshian et al., 1983) but it is currently recommended that measured sediment concentration values be utilized since it is a more stable estimator (Sorooshian et al., 1993). However, Gupta (1984) warns that u t i l i z i n g m e a s u r e d v a l u e s m a y c r e a t e b i a s i n t h e e s t i m a t e o f

X.

Sorooshian (1981) proposed a two-stage method for non-linear models. In the first stage, given a set of model parameters, the residuals of the model are obtained. In the second stage, values of {©} obtained in the first stage are used to compute the most probable value of

X.

This procedure is repeated until a satisfactory value of

X

has been found. The optimal value of

X

is one that satisfies the following equation:

(2.3) where

X

must be solved for iteratively. Once the value for

X

is obtained, it is used to solve

Equations (2.3) and (2.4) to compute the HMLE objective function.

36

Duan (1991) developed an equivalent and more stable procedure for estimating X by rearranging Equation (2.5) to give the following;

R = ^ - l

R a

( 2 . 6 ) where

n

R d =Z

W

.(

C t.ob

S

-

C

.,sim) i=l

j

(2.7)

R

„ = iw,(c, obs

- c t sim

)

2 a t i=l

(2.8)

(2.9) a d where aj = the logarithmic average of the observed sediment concentration and the HMLE function is computed as:

1 R d

HMLE = rr^ r—r exp[(2^ - l)a d

]

(2.10) where A, = 1 is used an initial value. An iterative procedure is then used to estimate X such that R = 0 in Equation (2.6).

37

Parameter Optimization

When an automated technique is employed, the algorithm will attempt to minimize the objective function by varying the parameter values. If the objective function is minimized, it is considered to be a successful trial, and those values are usually retained.

To track values retained during the optimization procedure, contour plots can be generated where coordinate axes correspond to parameter values. Contours represent equal values of the objective function on the response surface. If a two-parameter model is used, the procedure can be generalized by the following. Search methods begin by defining a straight line that passes through a point representing starting values for the parameters The objective function is then evaluated at different points along that line.

When an optimum value is encountered, a new search direction is defined and the process is repeated until a minimum value is found. The manner in which the search directions are defined and how each line is searched depends on the method of optimization. The minimum or optimum value of the objective function corresponds to optimized values of the parameters (Ibbitt and O'Donnell, 1971).

Direct search methods have been used to identify parameters in erosion models.

Lopes (1987) and Blau et al. (1988) utilized the Simplex method for parameter identification, a direct search method developed by Nelder and Mead (1965).

Convergence problems were encountered by Lopes (1987) due to parameter interaction in the Water Erosion Simulation Project (WESP) model used in this study. Parameter insensitivity was identified as a difficulty by Blau et al. (1988) for a transport capacity

model developed by Shirley and Lane (1978). Although the problems identified in these studies were not directly related to the constraints of the Simplex method, such a procedure is not designed to handle the presence of multilocal optima (Gottfried and

Weisman, 1973) which are commonly encountered in the calibration of erosion and hydrological models.

38

THE SIMPLEX ALGORITHM

The Simplex method belongs to a group of techniques that are defined as hill climbing methods. They determine a path toward an optimum by evaluating the objective function at several points rather than by calculating derivatives (Gottfried and Weisman,

1973). These iterative search techniques (which also includes Rosenbrock's (1960) technique and Hooke and Jeeves' (1961) pattern search method) have been found to be superior to gradient techniques in hydrologic modeling. This is because values of the derivatives of the model equations with respect to its parameters cannot be explicitly obtained due to the presence of threshold-type parameters in the model (Johnston and

Pilgrim, 1976; Moore and Clark, 1981). Johnston and Pilgrim maintained that the response surface would have discontinuous derivatives and that these would cause the optimization algorithm to prematurely terminate. Gupta and Sorooshian (1985) compared the Simplex method to the derivative based Newton method and found the Simplex method to be more efficient.

39

In the Simplex method, an essential role is played by the geometric figure called a simplex that is defined as a set of

n +

1 points in //-dimensional space. In the case of

n =

2, the corresponding figure is an equilateral triangle; when

n =

3, it is a tetrahedron. The method can be viewed as the moving, shrinking, and expanding of the simplex toward a minimum. To find the minimum error value, the Simplex method searches the parameter space using an initial simplex, using

n

corners of the eight corner points in the space of reasonable values. The function is evaluated at each of the vertices. The point with the highest errors is replaced by a point with lower error to form a new simplex To determine the location of this new point, the worst point is reflected through the centroid of the other two points. If the function evaluated at the new point does not reduce the error, then the new point is generated by contraction toward the centroid. If neither of these approaches finds a point with lower error, then the entire simplex is contracted towards the point having the lowest error. This iterative process is continued until convergence to a minimum is found. The stopping criterion suggest by Nelder and Mead

(1965) is:

(2.11) where f

(xj)

= function of the observed data, f

(xo)

= function of the simulated data, and

e

is some small preset number (Kowalik and Osborne, 1968).

40

Global Optimization

Parameter identification for hydrologic and erosion modeling can be formulated as a global optimization problem where the objective function is concave and possesses many local minima in the region of interest Global optimization methods, however, will normally use some local procedure, which limits their ability to converge to a global minimum. Duan et al. (1992) suggests that automatic calibration procedures in current use for hydrological models are incapable of finding the globally optimal parameter estimates due to problems such as parameter interaction, non-convexity of the response surface, discontinuous derivatives and presence of multilocal optima.

THE SHUFFLED COMPLEX EVOLUTION -

UNIVERSITY OF ARIZONA (SCE-UA) ALGORITHM

Duan et al. (1992) developed a new global optimization procedure called Shuffled

Complex Evolution - University of Arizona (SCE-UA). The SCE-UA algorithm searches through the parameter space using the simplex geometrical shape. However, the points are periodically shuffled to avoid a convergence to a local minimum. The algorithm begins by randomly selecting s number of points where s=pxm and p = the number of complexes and

m

= the number of points in each complex. After computing the function value at each point, the points are sorted in order of increasing function value and are stored in an array that is partitioned into a number of communities (complexes) selected by

41

the user. By randomly choosing

n

+ 1 points from each complex, a simplex is formed according to a trapezoidal probability distribution defined as: p

=

2 0 ^ 1 - i ) m(m +1)

( 3 2 5 ) where the point with the highest probability is pi = 2/m +1 and the point with the lowest probability is p m

= 2/m(m+l).

New points replace points with the greatest error using two iterations of the

Simplex method described earlier and the generation of random points within the feasible space. After evolving each simplex

2n

+ 1 times, the simplex is then dissolved and the updated points are returned to the complex where new «+l points are randomly selected to form a new simplex in the same manner as previously described. After a certain number of generations, new complexes are formed with the updated points by shuffling, a nonrandom action. In this way, the sharing of information about the search space is accomplished. This entire process is repeated until a minimum is reached (Duan et al.,

1992).

Duan et al. (1992) tested the performance of the SCE-UA along with three other global search procedures on the model SIXPAR; the adaptive random search (ARS) method, a combined ARS/simplex method and a multistart simplex (MSX) method.

Results show that both the MSX and SCE-UA methods were effective in finding the globally optimal parameters. However, the SCE-UA method was found to be three times more efficient. The efficiency of the SCE-UA method was confirmed for the Sacramento

42

Soil Moisture Accounting model (SAC-SMA) (Sorooshian et al., 1993). In this study, the

SCE-UA achieved a 100% success rate in locating the global minimum while the MSX method had little success with more than twice the number of iterations.

Luce and Cundy (1994) compared parameter values found by the SCE-UA procedure and the (local search) Simplex method utilizing a physically-based model for studying runoff and erosion from forest roads. The authors report that both methods were successful in converging to unique optimal parameter sets for infiltration and overland flow parameters. In only three out of the 84 cases were the hydrographs improved by using parameter values estimated with the SCE-UA algorithm. These were cases where the error surface was flat and the Simplex method terminated prematurely.

Difficulties in Parameter Optimization

Despite using a systematic approach, different sets of observed data can produce very different parameter sets (Johnston and Pilgrim, 1976). Different initial parameter values can also generate distinct sets of optimum parameter values (Page, 1988). Since the accuracy of measurements is never perfect, it cannot be expected that parameter values be identical to their true values. Inevitably, random errors occur It is desirable, however, that they be close to their true values. Although each series of measurement will obtain different values for the parameters, it is desirable that these estimators fluctuate around their mean values and that they do not vary extremely from one series of measurements to another (Schmidt, 1982).

43

Several factors can contribute to large fluctuations in parameter estimates, some of which have been reported in the erosion modeling literature (Lopes, 1987, Page, 1988;

Page et. al, 1989; Blau et al., 1988). Ibbitt and O'Donnell (1971) and Johnston and

Pilgrim (1976) have outlined the following reasons for the inability to obtain unique and conceptually realistic parameter sets for conceptual rainfall-runoff models which also play a major role in erosion modeling.

INTERDEPENDENCE BETWEEN MODEL PARAMETERS

When model parameters interact, the change in the value of one parameter can be compensated by changes by in one or more of the other parameters. For a two-parameter model, a long flat-bottomed valley in the response surface results and optimization methods will make little or no progress along the floor of a valley toward its lowest point.

INDIFFERENCE OF OBJECTIVE FUNCTION

TO VALUES OF INACTIVE PARAMETERS

The objective function (and thus the simulated model output) is not affected by the changes in the value of a parameter This may be caused by parameter redundancy or it is not activated by the calibration data set When this occurs, zero gradients occur in some areas of the response surface and optimization methods make no progress towards a minimum.

DISCONTINUITIES IN THE RESPONSE SURFACE

Local-type direct search methods for are not designed to handle the presence of discontinuous derivatives. In such cases, the optimization method may terminate before encountering the true optimum

44

PRESENCE OF LOCAL OPTIMA DUE TO

NON-CONVEXITY OF THE RESPONSE SURFACE

Local optima are defined as points on the response surface that have lower values of the objective function than any surrounding points, but have greater values than another point in another region of the response surface. The optimization method may therefore terminate at a point that is not the global minimum

Furthermore, even the most complex models may not completely represent the physical processes of erosion. Therefore, it is possible that some of the difficulties in identifying a unique set of parameter values may be due to model structure.

Data Calibration

The selection of trial model parameters is made during calibration. Data used in the calibration of an erosion model should be representative of the factors influencing the erosion processes. However, calibration is rarely straight-forward. Data come from various sources with different degrees of accuracy and levels of representativeness.

45

Some researchers have attempted to use longer periods of data for calibration to account for a wide variety of conditions in the watershed. Sorooshian et al (1983) argue that it is not the length of record that is most important, but the information contained within it. They propose that the most important aspect of the calibration phase is considering the stochastic properties of the data, which relates to the appropriate selection of the objective function. Parameter estimation developed within a framework of maximum likelihood theory can aid in the selection of the appropriate objective function, which can smooth the response surface and make it more concentric. This improved concentricity increased the chances for convergence to the true parameter set (Sorooshian andDracup, 1980).

Sorooshian et al. (1983) compared the performance of the HMLE and SLS criterion in the calibration of a soil moisture accounting model of the U.S. National

Weather Service River Forecasting System (SMA-NWSRFS). The model was calibrated using daily records of variable length and then tested for a 6 year period They found that

SLS technique was better able to provide the closest reproduction of the observed hydrograph for the calibration period. However, the HMLE estimator was found to provide the best model performance for the forecasting period.

The likelihood function plays a critical role in both classical and Bayesian theories of inference. In classical theory, it is used to construct maximum likelihood estimators

(MLEs) which have desirable asymptotic properties. In Bayesian theory, it is used to update the prior distribution using newly acquired data. Wilson and Haan (1991)

46

developed a calibration procedure that combines site measurements of erodibility with those parameters already identified in the Water Erosion Prediction Project (WEPP) database. Assuming a normal distribution of interrill erodibility and a log-normal distribution of rill erodibility, theoretical relationships were derived to estimate parameters using Bayesian estimation theory. When tested, results showed that the technique worked well in combining site-specific information with prior information represented by regression equations (Wilson et al., 1991).

METHODS

The Model

The Water Erosion Simulation Program (WESP) (Lopes, 1987) was used in this study. The erosion component of the model was modified to incorporate three different flow-induced erosion equations for overland flow which were coupled with the hydrological component of the WESP model.

47

The Hydrological Component

Woolhiser and Ligget (1967) described the movement of water over a plane using

a kinematic approximation of the spatially-varied, unsteady and one-dimensional flow equations:

dh dq

3t Sx q = ah m

(3.1)

(3.2) where h = depth of flow [L], q = the discharge per unit width [L

2

T'

!

], r = the rainfall excess rate [LT

1

], x = the distance downslope [L], t = time [T], and a and m are parameters related to the slope and roughness of the flow. For normal flow conditions,

Manning's equation yields m = 5/3 and a = (1/n) S

0

'

2 where n = Manning's roughness coefficient and S

0

= the slope of the plane [L/L] By substituting Equation (3 .2) into

Equation (3.1):

— + amh m

"

dt

oh

= r

3x

(3.3)

48

In order to solve the kinematic wave equations, the depth at the upstream boundary must be defined. For an uppermost plane, the boundary conditions are: h(0,t) = 0 for t > 0 (3.4)

For planes where runoff is being contributed by other planes, the boundary conditions are

(Woolhiser et al., 1990): h(0,t) = a u M

L u > t ) m i , w u aw

(3.5) where Lu = the length of the contributing plane, h u

(Lu,t) = the depth at the lower boundary of the contributing plane at time t, w„ = the width of the contributing plane, a u

= the slope roughness parameter for the contributing plane, m u

= the exponent for the contributing plane, and a, m and w refer to the receiving plane. The initial conditions are: h(x,0) = 0 for x > 0 (3.6)

In the kinematic approximation, the friction slope is assumed to be equal to the plane slope (Sf = S

Q

). This translates into an assumption of the water surface slope being equal to the plane slope (Lighthill and Whitham, 1955, Henderson, 1963; Woolhiser and

Ligget, 1967). If the kinematic flow number is greater than 10, then solutions to the kinematic wave equations provide good approximations to the shallow water equations.

The kinematic wave equations are solved numerically by a four-point implicit finite difference method:

49

(3.7) where 0 and co and are weighting factors for space and time respectively.

The Infiltration Component

The rainfall excess rate (r) is calculated in WESP by subtracting the difference between rainfall intensity and infiltration rates. When rainfall begins on an infiltrating soil, there is always an initial period where the infiltration rate (f) is equal to the rainfall rate (i) and the rainfall excess (r) is zero. The maximum infiltration rate (f c

) is described as a function (f) of the initial water content (0;) and the amount of water already infiltrated in the soil:

(3.8)

The Green and Ampt (1911) infiltration equation is used in the WESP model.

Two parameters are important to the infiltration model; the saturated hydraulic conductivity (Ks) and the net capillary drive (Ns):

(3.9)

Ns = <p(S_-S,)<|> (3.10) where <|> = soil porosity, S max

= 0s/<|> = maximum relative saturation, 0

S

= saturated water content [L

'VL

3

], Sj = 0/4)

= initial relative saturation, and

<p

= the soil matric potential [L],

50

The Erosion Component

WESP calculates the sediment concentration in broad shallow flow areas by

applying the sediment continuity equation in combination with the overland flow equations. Because the hydraulic conditions of overland flow are often totally different from those of channels and it is assumed that in small watersheds well-defined rills do not develop, the overland flow equations were also used to solve the sediment continuity equation for flow-induced detachment and transport in this study. The sediment continuity equation normally used for one-dimensional flow on hillslopes is (Bennett, 1974)

M

+

* S L e

,

+ e i

dt dK '

1

where c = the sediment concentration [ML

3

], ei = the input sediment flux to the flow p . , , )

[ML"

2

T"'] by raindrop impact, ef = the flow-induced input sediment flux to the flow

[ML"

2

T"'], and dispersion terms have been neglected. The first term in the continuity equation represents the rate of storage of sediment within the flow depth. The second term represents the change in sediment load with distance.

The WESP model (Lopes, 1987) utilizes a simultaneous sediment exchange approach. WESP represents the erosion/deposition process on hillslopes as two separate processes of sediment entrainment and deposition. For broad shallow flow areas, sediment entrainment is carried out by raindrop impact and hydraulic shear. Entrainment and deposition can occur simultaneously at different rates and the resultant sediment concentration is determined by the relative magnitude of these two processes. Thus,

51

<|>(x, t) = e f

- d + ej

(3.12) where <i> = sediment flux to the flow [ML"

2

!"

1

], ef = flow-induced sediment entrainment

[ML"

2

T"'], d = rate of sediment deposition [ML"

2

!'

1

] and e, = rate of sediment entrainment by raindrop impact [ML"

2

!""

1

].

Entrainment by Raindrop Impact

One raindrop-induced entrainment equation was selected for use in this study.

Notwithstanding the successes of other raindrop-induced erosion equations, the equation selected has been proven to be effective over a wide range of conditions and is currently used in the Water Erosion Prediction Project (WEPP) (Lane and Shirley, 1982). Ulman

(1994) also demonstrated its success in describing rain-drop induced erosion on forest roads.

If hydraulic shear is considered to be negligible in raindrop-induced entrainment areas and uniform rainfall intensity is assumed in the area of interest, then (Lane and

Shirley, 1982):

(3.13) where Kj = raindrop induced erodibility parameter [MTL"

4

], i = rainfall intensity [LT"

1

], and r = rainfall excess rate [LT

1

] This expression relates soil particle entrainment to rainfall erosivity and the erodibility of the soil The transport in broad shallow flow areas is related to the ratio of the rainfall excess rate to the rate of rainfall intensity, which can

be interpreted as a normalized runoff intensity for sediment transport by broad shallow flow (Lopes and Lane, 1988).

52

Entrainment by Flow

Three different equations describing erosion by hydraulic shear were evaluated in this study. All of the equations have been presented in Govers (1992), but have only been evaluated under steady state conditions. Since WESP is both time variant and spatially varied, the equations were implemented as fully dynamic equations. Each of the equations can be represented by the generic form of: e f

= K f p

( x ) b

( 3 . 1 4 ) where Kf = a flow-induced erodibility parameter [dimensions equation dependent], p = an index used to distinguish between coefficients for each equation, x = a variable specific to the equation, and b is an exponent with a value =1.5 (Lopes, 1987; Hernandez, 1992).

The first equation relates entrainment by flow to excess effective stream power: e f l

= K

f l

(n e

)

, J

(3.15) where O e

= excess effective stream power. Bagnold (1980) defined the concept of excess effective stream power as: n c

= n - n < ( 3 . 1 6 ) where Q = the effective stream power and Q c

= the critical stream power. These have been defined as:

53

(3.17)

(3.18) where x = yhS = the hydraulic shear [ML"'T

2

], y = the fluid specific weight [ML"

2

T"

2

], h

= the flow depth [L], u = the mean flow velocity [LT"

1

],

T

c

= pu,

2t

= critical hydraulic shear [ML"

1

!"

2

], p = fluid density [ML"

3

], and U.

t

= mean critical shear velocity [LT'

1

].

The second equation used in this study relates entrainment by flow to the shear stress of the flow: e f 2

— K f 2

( x t c

)

(3.19)

The third equation presented includes the effect of particle size on the transport capacity of the flow: v

l

>

/J

' where D = the effective particle diameter [L], The effective particle diameter was

(3.20) determined by:

D

_ ' n d j

(3.21) where mi = the weight percentage of sand, silt and clay, d; = the geometric mean of sand, silt and clay, and <? and In represent the exponential and natural logarithm operators.

54

Sediment Deposition

The rate of sediment deposition (d) in WESP is determined by a relationship defined by Mehta (1983), which states that deposition is a linear function of the sediment concentration and the effective particle fall velocity: d = P T w

V s c (3.22) where P = a constant [dimensionless], T w

= the top width of the flow [L], V s

= the effective particle fall velocity [LT

1

], and c = the sediment concentration [ML"

3

]. For deposition in overland flow areas, P was assumed to equal 0.50 (Davis, 1978).

The Data

Three sets of data were used in this study . 1) synthetic data 2) data that were collected from rainfall simulator plots set up by the USDA-ARS WEPP team at different sites across the western United States and 3) Kendall watershed at the Walnut Gulch

Experimental Watershed.

The synthetic data used was generated based on the experimental procedures and soil characteristics of the rainfall simulator plots. Three different slopes (5, 10 and 15%) were incorporated into the data set so that the effect of slope on the ability of the optimization procedure to find an optimal parameter set could be evaluated.

For the rainfall simulator plots, three rainfall simulation treatments using a V-Jet

80100 nozzle were applied to 3 .05 x 10.5 m plots. When the initial conditions were dry, rainfall was applied at a rate of 60 mm/hr for 60 minutes. The wet antecedent moisture

55

treatment was applied twenty-four hours after the dry treatment, at a rainfall rate of 60 mm/hr for 30 minutes. The very wet antecedent moisture treatment was applied when no surface water was evident on the plot by visual inspection, approximately 30 minutes after the wet treatment, at intensities of 60 mm/hr and 130 mm/hr during a 30 minute period

(Simanton et al., 1985). The synthetic data were assumed to have a similar treatment.

Ten rainfall simulator plots from four different areas in the Western United States were selected for this analysis, based on a criterion of a minimum slope of 7% so that the parameters would be sufficiently activated. For both the synthetic and rainfall simulator data, sediment graphs were used to compare measured and simulated data. Characteristics of each rainfall simulator plot are given in Table 3 .1.

Table 3.1. Soil Characteristics for Rainfall Simulator Plots and Kendall Watershed

Plot No.

&

Effective

Particle Slope

Watershed % Sand % Silt % Clay Soil Type Diameter (mm) (%)

31 16.7 14.2 69.1 Gravely

Sandy Loam

1.209E-01 10.2

34 16.7 14.2 1.209E-01 10.0

56

59

63

5.0

5.0

7.8

25.5

25.5

28.7

69.1 Gravely

Sandy Loam

69.5 Very Gravely

Fine Sandy

Loam

69.5 Very Gravely

Fine Sandy

Loam

63.5 Fine Sandy

Loam

1.0175E-04

1.0175E-04

7.9060E-05

8.1

7.5

8.6

66

102

7.8 28.7

15.1 36.0

63.5 Fine Sandy

Loam

48.9 Loam

7.9060E-05 8.8

1.0810E-04 11.2

105

120

15.1 36.0 48.9 Loam

44.2 33.4 22.4 Clay

1.0810E-04 9.8

1.0643E-05 11.2

121

Kendall

44.2 33.4 22.4 Clay

62.7 23.0 14.2 Sandy Clay

1.0643E-05 11.6

8.1047e-06 9.4

56

57

Kendall Watershed is located in the eastern part of the Walnut Gulch Experimental

Watershed. The watershed has gentle hillslopes covered by grasses, an average slope of

9.4%, and is dominated by soils of a sandy clay texture (see Table 3 .1). Because the runoff is small in relation to its rainfall depth due to its gentle slope, sandy soils and grass stands, Kendall Watershed has not developed a well-defined channel. Thus Kendall

Watershed was selected for this study because it can be modeled as a single plane.

Three rainfall-runoff events from the years 1975-1977 were selected for this study.

The events were chosen based on their maximum rainfall duration that produced measurable amount of runoff and sediment. Because the sediment graphs were unavailable for these events, parameters for the erosion equation were optimized based on the total sediment yield for each event.

Determining Values Of Hydraulic Parameters

For the natural data studies, values of the hydraulic parameters had to be determined before optimizing for the erosion parameters. The SCE-UA algorithm and the

SLS objective function were used to determine the values for hydraulic roughness

(Manning's n), net capillary drive (Ns), and saturated hydraulic conductivity (Ks). The

Nash-Sutcliffe coefficient (r

2

) was used as a measure of goodness-of-fit between simulated and observed of both the runoff rate and sediment concentration values:

58 r

2

= 1 -

Z( x

.,,

B

-x

Mto

)

Z(x, ob5

- x)

2

(3.23)

where X tobs

= the measured value, Xt, sim

= the simulated value, X= the average observed value, and n = the number of observations. When the simulated and observed correspond well, the values of the coefficient will lie between 0.5 and 1.0, where 1.0 represents a perfect comparison (Nash and Sutcliffe, 1970).

True Parameter Values

Soil parameter values that were determined for the Water Erosion Prediction

Project (WEPP) model resulting from the rainfall simulator plots at Walnut Gulch provided a basis for determining the true parameter values of the synthetic data set. Since both WEPP and WESP describe entrainment by raindrop impact with the same equation, the raindrop-induced soil erodibility parameter (K;) was assumed to be have the same value. The value for critical shear stress (x c

) was also assumed to equivalent between the synthetic and rainfall simulator data sets. However, the values of the flow-induced erodibility parameter (Kf) had to be altered to fit each of the equations describing entrainment by hydraulic shear.

WEPP describes detachment by hydraulic shear on bare soil as (Foster et al.,

1989):

59

D r

= K f

(x-x c

) 1-- v V

(3.24) where D r

= the flow detachment rate [ML'

2

!"

1

], G = the sediment load [ML"

2

!'

1

] and T c

= the transport capacity of the flow [ML"

2

T'']. This equation is similar to Equation 2 of this study with the exception of the value of the exponent and the relationship describing detachment utilizing the transport capacity approach. To determine the true parameter value of Kn for the synthetic data, the WEPP K r value [TL

1

] was multiplied by the length of the plot to evaluate Kn [T] for b = 1. To fully activate the Kf parameter, the WESP model was run using the synthetic data developed for the very wet run with a 15% slope for b = 1. The

Kf for each equation was adjusted so that the sediment yield at b = 1.5 was equal to the sediment yield at b = 1 for Equation 2.

For both the rainfall simulator plot studies and the watershed events, the true values of Kfp were equation dependent whose true values were not known. The optimized values of x c

, however, were compared to true values of the critical shear stress as determined by the Shields diagram For the rainfall simulator plot studies, the optimized values of K; assumed the value that was determined in the WEPP field experiments in the rainfall simulator studies. For the watershed events, a value of Ki could be determined by a regression equation developed by WEPP. However, because WESP does not incorporate adjustments in soil entrainment and transport due to plant and rock cover, Kj was manually calibrated for the selected events.

60

Treatment of Systematic Error

The error model used to synthetically generate observed sediment concentration data was a first-order autoregressive model known as a Markov model (Lipschutz, 1968).

This model assumes that the additive errors are autocorrelated to a lag-one by a simple linear relationship given by: e = pe,_

1

+ii t

(3.25) where s t

= additive errors at time /, p = the first-lag autocorrelation coefficient that measures the degree of systematic error (-1 <p< 1), and r)t = the purely random component of measurement error, which is assumed to have a Gaussian distribution with a zero mean, constant variance, and is independently and identically distributed for all

t.

The variance of the independent variables a

2 is defined as. a

2

= 1 - p

2

(3.26) where the standard deviation of the errors was set equal to 20% of the standard deviation of error-free sediment concentration values. This error exceeded the error generally encountered in hydrologic data series according to Sorooshian (1980). Two different levels of correlated error were created by fixing the value of p, the serial correlation coefficient, at 0.25 and 0.50. This error model was chosen based upon the high probability of correlated error in sediment concentration data that is also known to be present in streamflow measurements (Sorooshian and Dracup, 1980)

61

Parameter Identification

Erosion parameters were fitted to produce an optimal parameter set that corresponded to the actual sediment concentration graphs of both the natural and synthetic data. Search and optimization algorithms were used to find the best values of the parameters Kj, Kfpand x c

. For the synthetic data, both two and three parameter problems were posed with data that was error free, as well as with data with two levels of correlated error. In the first case, the value of Kj was fixed, and Kf

P and x c were determined by optimization. In the second case, Ki and Kf p were optimized with x c

= 0 and x c

^ 0. In the third case, all three erosion parameters were determined by optimization.

Sum of the least squares of the error (SLS) (Equation 2 .2) and the heteroscedastic maximum likelihood estimator (HMLE) (Equations 2.6-2.10) were used as the objective functions. The Simplex algorithm (Nelder and Mead, 1965) was used to find the optimal values of the parameters with a single start. This method is quick, but can terminate prematurely if the error surface is flat or pitted The Shuffled Complex Evolution (SCE-

UA) (Duan et al., 1992) was also used for parameter identification. The SCE-UA was designed to find the global minimum for error surfaces with multiple local optima Both methods required a range of reasonable parameter values.

A two-parameter optimization problem (Krx c

) was posed for the natural data sets, which included the rainfall simulator plots and Kendall Watershed rainfall-runoff events.

Only the most successful combination of flow-induced erosion equation, optimization

62

algorithm and objective function in the synthetic data study were used for parameter identification in the natural data sets.

WESP was run to find the optimal parameter set. To this end, the search routines submitted parameter values to WESP, which then ran the model on an event basis.

Simulated and measured sediment concentrations were compared in the objective function

(SLS and HMLE) If the stopping criteria were not met, the search routine submitted new parameter values to the model and the process repeated itself until acceptable values with minimal error were found.

Convergence Criteria

For the Simplex algorithm, the optimization process will terminate if one of the following stopping criteria is met; the prespecified tolerance limit for minimum change in the values of the objective function has been satisfied (function convergence), the coordinates of the simplex have changed by less than the specified amount (parameter convergence), or the maximum number of iterations has been reached. Since the SCE-UA algorithm is based on an extension of the Simplex local-search algorithm, the stopping criteria are the same However, the SCE-UA algorithm allows the user to specify the number of shuffling loops in which the criterion value must change by the prespecified tolerance for function convergence. The SCE-UA algorithm will also terminate if the population of points converges into a sufficiently small space that will not allow the spread of the population in each parameter direction to exceed more than one thousandth of the

63

corresponding feasible parameter range. Any further search would not result in significant improvement of the parameter estimates.

For both the Simplex and SCE-UA methods, the objective function tolerances were set at 0.001. For the SCE-UA method in the two-parameter case, 4 complexes of points were selected. For the three parameter case, 6 complexes were used. The minimum number of shuffling loops in which the criterion value must change by the prespecified tolerance for function convergence was set to 10.

Methodology Used for Comparison

The comparison of the two search algorithms and the two objective functions was carried out by maintaining the same initial conditions, parameter bounds and starting parameter values. For the synthetic data, a comparison of the three different flow-induced erosion equations was performed by evaluating how successful the optimization algorithm and objective function were in arriving at the true parameter values under different antecedent moisture conditions.

To evaluate the performance of the search algorithms and objective functions, the following criteria were used:

1) the relative efficiency with which inactive and active parameters were estimated,

2) the relative efficiency in avoiding local optima,

3) the continuity and shape of the response surface configurations, and

4) the ability to attain the best value of the objective function.

64

RESULTS & DISCUSSION

Synthetic Data Case

The results for the synthetic data study were for a case where precipitation intensities and duration were based on rainfall simulator experiments at Walnut Gulch

Experimental Watershed. As described earlier, soil and hydraulic variables were based on the soil description for Plots 31 and 34 (see Table 4.1). Then, for a specified set of hypothetically true parameter values, sediment concentration values were generated by the

WESP model for three different values of slope (5, 10 and 15%). These data were then input as observed values for optimization. Because preliminary investigations of the rainfall simulator plots indicated that the value of critical shear stress (

t

c

) approached zero, two sets of synthetic data were generated where

t

c

= 0.0 and

t

c

= 0.502.

In order to study parameter interactions and any uncertainties in the processes of erosion, four different parameter optimization problems were posed as described in the

METHODS section of this thesis These parameter optimization problems were studied when no error was present in the data and when the data were contaminated with correlated and random error. To this end, the sediment concentration values were contaminated with error according to the error model outlined earlier. The same initial conditions were used throughout to assure identical response surface configurations.

Table 4.1. Hydraulic Parameter and Soil and Plot Characteristics for the Synthetic

Data Set.

Parameter/V ariable

Effective Particle Diameter

Porosity

Saturated Hydraulic Conductivity (Ks)

Maximum Soil Saturation (Smax)

Initial Soil Moisture Content (Si dry)

Initial Soil Moisture Content (Si wet)

Initial Soil Moisture Content (Si very wet)

Soil Moisture Tension Parameter (v|/ dry)

Soil Moisture Tension Parameter (\|/ wet)

Soil Moisture Tension Parameter (y very wet)

Plot Length

Plot Width

Hydraulic Roughness Coefficient (Manning's n)

Value

0.1209

0.437

5.98

0.92

0.31

0.61

0.87

70

25

15

10.7

3.05

0.04

Units mm

(dimensionless) mm/hr

(dimensionless)

(dimensionless)

(dimensionless)

(dimensionless) mm mm mm m m m

65

RESPONSE SURFACES

To determine the boundaries and starting values of the parameters, the response surfaces for each of the three flow-induced erosion equations were generated. To this end, incremented parameter values were submitted to WESP (without the aid of the optimization algorithms) and the values of the objective function were calculated for very wet runs on plots of 15% slope where it was assumed that the parameters would be most activated. Separate response surfaces were generated for the SLS and HMLE objective functions. Figures 4.1-4.6 show the resulting response surface configurations. Upper and

66

lower parameter bounds were determined by identifying "regions of attraction." Starting values for the parameters were defined as the midpoint between the upper and lower bounds. The true parameter values, starting values and lower and upper parameter bounds used are given in Table 4.2.

Table 4.2. Parameter Bounds and Starting Values for Synthetic Data Set

Parameter

K n

Kn

K fi

T c

Units

True

Values

Lower

Bound

Upper

Bound

Starting

Value

M/L

2

T

1 7188E-04 5.00E-05 4.00E-04 2.25E-04

t

2

/l°-

5 m°-

5

3.7365E-03

1.00E-03 1.00E-02 4.25E-03

T

2

/L

0005

M

0

'

5

3.9685E-05 2.50E-05 1.50E-04 9.00E-05

M/LT

2

0.502 0.3 2.0 1.15

The response surface configurations depicting the relationship between Kf and x c

(Figures 4.1 - 4.3) show two difficulties that are related to the structure of the model. The first difficulty is that of parameter insensitivity that is identified by the shape of the response function. x c is less sensitive than Kf as noted by the relative sensitivity of function value in the two parameter directions Along the Kf axis, the response surface wall is steeper than in the

t

c

(inactive parameter) direction. A second difficulty is due to interactions that exist within the model. This effect is noted in the valley that has formed on the response surface that is inclined along the

Kf axis.

HMLE Criterion

5.00

4.00-

$

to

3.00-

2.00-

1.00-

5.00E-004 1 00E-003

Kf

1.50E-003 2.00E-003

SLS Criterion

5.00

4.00o

a

3.00-

2.00-

—12000 -

— 91.26 i26J»;

1.00-

2.00E-004 6 00E-004 100E-003

Kf

1 40E-003

—r

1

•••'• -i— c

—-

1 80E-003

Figure 4.1. K f

-x c

Contour Plots for Equation 1.

HMLE Criterion

4.00-

3.00-

2.00-

1.00-

4.00E-002

Kf

SLS Criterion

5.00-

4.00-

s

S2

3.00-

2.00-

1.00-

125.00

1.00E-002 3 DOE-002 4.00E-002

Figure 4.2. K f

T c

Contour Plots for Equation 2.

HMLE Criterion

4.00-

3.00-

J2

2.00-

1.00-

•8.13

1J»

2 00E-004

Kf

SLS Criterion

4.00E-004

5.00'

4.00-

3.00-

$

2.00—

1.00-

128.00

200E-004

4 0QE-004

Figure 4.3. Kf

-x c

Contour Plots for Equation 3.

The response surface configurations for

Ki and

K f

(Figures

4.4-4.6) also demonstrate differences in the relative sensitivities of the two parameters. Very small changes in

Kf could produce very large changes in the value of

Kj.

This observation leads to a more serious difficulty associated with the relationship between

K,-K r

.

The vertical, elongated contours that are especially dominant in the contour plot of Equation 2 indicate that for any one value of

Kf, an infinite number of values for

Ki are possible. Another difficulty present in the contour plots of Equations 1 and 3 is the discontinuous response surface associated with extreme values of both

Ki and

Kf.

The response surfaces also demonstrate that the values of the SLS objective function are higher than those of the HMLE criterion for the same parameter values. This result means that the value of the SLS criterion for the same initial conditions will be higher than that of the HMLE objective function, and a direct comparison of their values cannot be made.

THE ERROR-FREE DATA CASE: THE TWO-PARAMETER PROBLEM

For the error-free data case, parameter estimates were considered to be a success if the they were not more than 1% in error of its true value. Based on this criterion, for all three cases of the two-parameter optimization problem, the synthetic error-free data study demonstrated a 100% success rate for both the Simplex and SCE-UA algorithms and for both objective functions in finding the true parameter values for plots with slopes of 10% and 15% for all three flow-induced erosion equations. However, no successful trials

HMLE Criterion

250.00-

200.00-

250.00-

200.00-

5 150 00-

100.00-

50.00-

10000-

50.00-

5.00E-004

Kf

SLS Criterion

I 1 1 i !

Figure 4.4. K; - Kf Contour Plots for Equation 1.

250.00-

O

0 d

O 150 00-

I

50.00—I

HMLE Criterion

2 00E-002

Kf

3 00E-002

SLS Criterion

250.00-

177

D)

Q

Q

a

150.00-

50.00-

2.00E-002

Kf

3 00E-002

Figure 4.5. Ki - Kf Contour Plots for Equation 2.

72

HMLE Criterion

250.00-4

200.00-j

5 150.00H

100.00-J

50.00-]

1.00E-004 200E-004

Kf

3 QOE-004

SLS Criterion

250.00-J

200.00-{

5 150.00-)

IOO.OO

H

50.00-]

1.00E-004

2.00E-004 3 00E-004

Figure

4.6. Kj - Kf

Contour Plots for Equation

3.

73

74

resulted for plots with a 5% slope. This could be attributed to the assignment of a constant value for the deposition parameter (P) in broad shallow overland flow areas, which may have resulted in too much deposition occurring in areas of lower slope values.

To eliminate this problem, the value of the dimensionless constant |3 in Equation 3 .22 may have to be optimized for events where the slope is less than 10%. Because of the inability of the optimization procedure to estimate the parameter values of the error-free data for slopes of 5%, plots with these values were eliminated from the analysis.

THE ERROR-FREE DATA CASE: THE THREE-PARAMETER PROBLEM

For the three-parameter, error-free synthetic data case, only the SCE-UA algorithm with both criteria demonstrated a 100% success rate in estimating the true values of Ki, Kf, and x c

The Simplex in combination with the SLS criterion was entirely successful for both Equations 1 and 2, but for Equation 3, demonstrated only at 50% rate of success in cases where the parameters were most activated. Equation 1 demonstrated success with the Simplex and HMLE criterion on the very wet run with a 15% slope.

Equation 2 demonstrated the same success on the very wet run on slopes of 10 and 15%.

No successful HMLE events resulted with Equation 3.

The success of the Simplex and HMLE for the very wet runs may indicate a sensitivity of the optimization procedure to a variable rate of rainfall intensity However, it is more likely that Simplex was more sensitive to the degree of activation of the parameters. On the 15% slope, the parameters would be more active than on a 10%

75

slope, which is why Equation 1 found success with only one of the variable intensity rainfall runs.

The KrT c contour plots (see Figures 4.1-4.3) reveal that the SLS criterion has a smoother, more elliptical response surface than that of the HMLE. This may explain why in part, for the three-parameter optimization problem, the Simplex in conjunction with the

HMLE criterion is unable to estimate the true values of the parameters However, the uncertainty involved in the estimation of K; seems to be more pertinent to the estimation problem. For the Simplex and HMLE optimization procedure, with the exception of

Equation 3, a higher estimation error is associated with runs on a 10% slope. More error is also associated with the wet runs than on the dry runs, presumably due to the fact that the dry runs are of a longer duration and better activate the threshold parameter,

T

c

. If t c is not fully activated, then any uncertainty in the erosion processes are then incorporated into the parameter K, when using the HMLE criterion.

Analysis of The Results for the Synthetic Data Case

The parameter values estimated in the synthetic data study have been tabulated in

Tables A1-A36. The following discussion is limited to the cases where error is present in the data, given the success of the error-free data case for the two-parameter optimization problems. For the analysis of the three-parameter problem when error is present, the error-free data case will be considered since the nonsuccesses under ideal conditions will

76

provide a basis for understanding the parameter estimation error when noise is incorporated into the data set.

A tally of the highest error associated with each of the four optimization procedures (Simplex and SLS, SCE-UA and SLS, Simplex and HMLE and SCE-UA and

HMLE) was performed to evaluate the optimization procedures. The highest error determination considered only the absolute differences in percent error, regardless of the magnitude of difference. If more than one procedure was associated with the highest error, then each procedure was counted as having the highest error. This procedure implies that a small difference in the percent error in the synthetic data would translate into a significant difference in error when working with observed data in the field. Such a pattern was noted between the error-free data sets and the correlated error cases.

Equation 1 in the 2-parameter cases, for example, demonstrated the highest error of estimation when no error was present in the data. This error was magnified when estimating the parameter values for the correlated error events

The HMLE was compared to the SLS criterion only with the algorithm with which it was associated For example, the parameter estimation error of the Simplex and HMLE was only compared to the Simplex and SLS, and differences in error between the SCE-UA and HMLE and the SCE-UA and SLS were also considered separately. However, a direct comparison of the highest estimation error associated with each of the four procedures was performed without any special consideration given to the algorithm that employed the selected objective function.

77

The relative amount of estimation error with respect to each flow-induced erosion equation is considered when selecting the best equation for use in the natural data studies.

The selection of the best optimization procedure considered the four evaluation criteria outlined in the methodology presented in this thesis, where the two-parameter cases are evaluated separately from that of three-parameters since there were estimation problems associated with two of the four procedures in the error-free data case.

ESTIMATION OF ACTIVE & INACTIVE PARAMETERS

By incorporating error into synthetic data, uncertainties in the processes modeling erosion are created. Such uncertainties are manifested in the estimation of the parameters that are used to describe these processes. Whereas Kj is related to the erodibility of the soil by raindrop impact; Kf and x c are related to detachment and transport by hydraulic shear. Kf describes the transport capacity of the flow and x

Q refers to the critical shear that the flow must exceed in order for detachment to occur.

Clearly, all four of the parameter optimization problems posed demonstrated that the greatest uncertainty is incorporated into the estimates of Kj (see Tables A1-A36).

Not only were large errors present in the estimates of Ki for all of the correlated error cases, but with the three-parameter, error-free data set as well Excluding the errors in estimation for the error-free, three parameter problem, the average percent error for K; at both levels of correlated error was 92.13% (93 .77% and 90.48% for Kj-Kf and K,-Krt c respectively). This value was much greater than the average estimation error for x c

78

(19.32% and

21.27% for

K r x c and

Ki

-Krt c respectively) and

Kf (10.37% and

4.27% for

Kf

-Tc and

Kj-K f respectively) (see Tables 4.3-4.5).

Table 4.3. Error Statistics for 2-parameter problem (Kj-Kf).

Eq.

No.

1 p Fixed Value of Avg. % Error % Error SD Avg. % Error % Error SD

Tc

K f

K f

Ki Ki

0.25

0.50

0.502 3.43

3.66

2.05

3.07

88.33

94.99

41.27

56.16

2

3

0.25

0.50

0.25

0.50

0.25

0.50

0.25

0.50

0.25

0.50

0.0

0.502

0.0

0.502

0.0

2.19

3.94

8.82

2.04

4.14

1.92

2.25

3.04

7.92

7.90

2.00

3.04

1 07

1.77

9.01

1.90

1 44

1.90

13.33

13.36

96.13

91.59

78.70

88.73

117.22

60.57

101.32

92.83

86.72

128.14

43.43

50.58

48.33

43.61

43.14

37.09

48.16

51.53

47.13

40.33

AVG

4.27 4.50 93.77 45.90

Two factors contributed to the large error found in the estimate of

Ki.

For the

Ki-

Kf estimation problem, the value of x c was fixed. Because the critical shear stress is a threshold parameter and Kf is related to the transport capacity of the flow, then detachment by hydraulic shear cannot vary by a large measure. Therefore any uncertainties or changes to be accounted for in detachment, were noted in the parameter value for Kj The nearly vertical line relationship demonstrated in the response surface configuration of the

K;-K f plots is another contributing factor (see Figures 4.3-4.6).

Kf is

79

more sensitive to changes in the objective function than K;, and presumably for any given value of

Kf, there is more than one possible value of

K;.

Table 4.4. Error Statistics for 2-parameter Problem (KrXc)

Eq.

No.

P

1 0.25

0.50

2 0.25

0.50

3 0.25

0.50

AVG

Avg. % Error

K f

10.65

18.96

5.70

8.44

6.15

12.29

10.37

% Error SD

K f

10.41

24.60

5.92

6.26

8.93

5.65

10.30

Avg. % Error

Tc

25.83

37.55

9.54

14.52

6.46

22.03

19.32

% Error SD

Tc

14.78

31.43

8.09

8.42

8.84

11.80

13.89

Table 4.5. Error statistics for 3-parameter problem.

Eq. p Avg. % Error % Error SD Avg. % Error % Error SD Avg. % Error % Error SD

No.

Kj

Ki

K f

K f

Tc

1 0.25 89.44

0.50 94.71

48.54

57.01

14.11

20.21

14.57

23.54

31.69

38.97

17.44

31.78

2

0.25 96.74

0.50 91.02

50.45

51.06

8.33

8.08

11.59

4.40

12.38

16.86

14.43

10.87

3 0.25 84.03

0.50 86.94

59.16

59.70

6.21

11.32

6.75

6.77

6.89

20.82

6.59

12.26

AVG 90.48 54.32 11.38 11.27 21.27 15.56

80

The parameter estimates resulting from the 3-parameter problem were consistent with those of the 2-parameter cases; the estimate of Ki contained the highest error and the average percent errors for both Kf and x c were nearly the same between the K r x c and

Ki-Kf -x c problems (see Tables 4.3 - 4.5). In general, the error associated with the estimates of Ki was slightly higher in the 2-parameter case than in the 3-parameter problem, while the converse was true for both Kf and x c

. This outcome meets the theoretical expectation since most of the uncertainty is incorporated in K; when x c is fixed, whereas the uncertainty in the 3-parameter problem is incorporated into all of the parameter estimates.

The average estimation error, however, can be misleading. The abilities of each of the flow-induced erosion models to produce parameters close to their true values need to be evaluated separately. The generalizations stated above, therefore, are for describing the tendencies and uncertainties present in the different parameter optimization problems posed in this study.

EFFECT OF ALGORITHM & OBJECTIVE FUNCTION

ON ACTIVE AND INACTIVE PARAMETERS

Tables 4 .6 - 4 .9 and Figures 4 7 - 4 .8 show that in all of the two-parameter cases studied, the Simplex algorithm, in general, provided parameter estimates closer to the true values for both the active (K f and Ki) and inactive (x c

) parameters. There were only two exceptions to this rule that occurred in the Krx c optimization problem, where the estimate

Table 4.6. Optimization procedures associated with highest estimation error for 3-parameter optimization problem

(Ki-Krt c

).

Parameter p

0.25

Ki

K f

Tc

Ki

K f

Tc

0.50

Simplex SCE-UA Simplex SCE-UA and and and and

SLS SLS HMLE HMLE

5

4

3

2

4

2

4

1

1

7

7

8

1

6

6

2

6

4

9

8

9

8

2

4

Table 4.7. Optimization procedures associated with highest estimation error for 2-parameter optimization problem (Kpx c

).

Parameter p

0.25 K f

Xc

K f

T c

0.50

Simplex SCE-UA Simplex SCE-UA and and and and

SLS SLS HMLE HMLE

3

2

6

6

0

3

5

7

8

8

4

2

7

6

3

4

Table 4.8. Optimization procedures associated with highest estimation error for 2-parameter optimization problem (Ki-Kf; x c

= 0.502).

Parameter p

0.25

Ki

K f

K;

K f

0.50

Simplex SCE-UA Simplex SCE-UA and and and and

SLS SLS HMLE HMLE

1

1

3

2

6

8

3

7

5

1

3

1

8

9

9

11

Table 4.9. Optimization procedures associated with highest estimation error for 2-parameter optimization problem

(Kj-Kf, x c

=

Parameter

Ki

K f

Ki

K f

P

0.25

0.50

Simplex SCE-UA Simplex SCE-UA and and and and

SLS SLS HMLE HMLE

2

1

6

2

2

8

6

11

4

1

3

2

10

9

7

6

Optimization Procedures Associated with Highest Estimation Error

(Ki-Kf-tc)

Ki (0.25) Kf (0.25) tau-c Ki (0.50) Kf(0.50) tau-c

(0.25) (0.50) jDSimplex+SLS

83

Optimization Procedures Associated with Highest Estimation Error

(Kf-tc)

100%

80%

60%

40%

20%

I

Kf (0.25)

m

tau-c (0.25) Kf (0.50) tau-c (0.50)

Figure 4.7. Optimization procedures associated with the highest estimation error for each parameter. Results from 3-parameter problem and 2-parameter

(Kric) optimization problems.

84

a.

"5

100%

60%

I

z 40%

(0

"S

20%

o

£

0%

Ki (0.25)

Optimization Procedures Associated with Highest Estimation Error

(Ki-Kf;

TC

= 0.502)

Kf (0.25)

Ki (0.50) Kf (0.50)

« 100%

£ 80%

60%

40%

20%

0%

Ki (0.25)

Optimization Procedures Associated with Highest Estimation Error

(Ki-Kf; tc = 0.0)

Kf (0.25) Ki (0.50)

Kf (0.50)

Figure 4.8. Optimization procedures associated with the highest estimation error for each parameter. Results from

2

-parameter problems (Ki-Kf) shown above for

T

c

=

0.502

and

T

C

= 0.0.

85

of Kf was improved by use of the SCE-UA algorithm in both error cases. For the correlated error case p=

0.25, the SLS criterion outperformed the HMLE estimator. For the case where p

= 0.50, the HMLE criterion provided better overall estimates for all of the parameters.

Gauging the success of the optimization procedures employed for the 3-parameter problem posed difficulties since it is known that sediment entrainment by raindrop impact has a behavior similar to that of entrainment by hydraulic shear, and that unique parameter identification may not be possible unless the value of Ki is determined separately (Lopes, 1987). However, few patterns of success were noted. The estimates of Kf were consistent with those of the 2-parameter problems, where the SCE-UA algorithm outperformed the Simplex, and the SLS criterion was the best estimator with lower levels of error. The behavior of z c with respect to the algorithm and objective function employed was less consistent; the SLS criterion provided better estimates in both error cases while the SCE-UA algorithm was most successful for p

= 0.25, while the

Simplex was more successful for p

= 0.50. The Simplex and HMLE procedure was a notable example of success since it was the most unsuccessful procedure in the error-free data case. It better estimated the value of Kj in both error cases and provided better overall estimates for all three parameters in Equation 3 for p

= 0.25.

86

EFFECT OF ALGORITHM & OBJECTIVE FUNCTION

ON FLOW-INDUCED EROSION EQUATIONS

The effect of the algorithm and objective function employed on each of the flowinduced erosion equations showed similar trends to those previously stated (see Figures

4.7 and 4.8). For example, for p=

0.25 in the K r x c

, and

K.i-K f

(x c

= 0.502) parameter optimization problems, the HMLE estimator was consistently associated with higher error than that of the SLS criterion. The reverse was true for p

= 0.50 case; that is, these same optimization problems were associated with higher error when employing the SLS criterion.

The 2-parameter optimization problem for K,-K f

(x c

= 0.0) did not show the same trends. In fact, for both p

= 0 25 and p

= 0.50 the HMLE criterion performed slightly better than the SLS estimator overall. This outcome supports the hypothesis that as the error quantity was increased, the HMLE criterion provided better estimates of the parameters by reducing the number of local optima on the response surface. In the case where the value of x c was fixed at zero, greater error in the estimate of Kj would be expected since the structure of the equation is exponential. Less error would be expected for the case where x c was fixed at a value of 0.502, since this expression of excess shear stress represented a process of decay.

For the K,-K f optimization problems, the Simplex algorithm performed slightly better overall for all three flow-induced equations. For the Krx c case, the SCE-UA performed slightly better than the Simplex in estimating

Kf.

However, the differences in

87

the highest error associated with each of the algorithms did not usually differ by more than unity, and thus were not considered to be significant differences in the abilities of the algorithms to estimate the K f for each of the flow-induced erosion equations.

OBJECTIVE FUNCTION

The selection of the objective function plays a major role in forming the shape of the response surface The more elliptical in shape the response surface is, the easier it is for the optimization method to search for the best parameter values The response surface configurations for the SLS criterion for all three equations are more elliptical than those of the HMLE criterion for the Kr-x c case. For this reason, it may be that when the error is low, the SLS criterion provided better parameter estimates than when the HMLE objective function was used Because the HMLE stabilizes a non-stationary variance, it would not necessarily follow that the HMLE criterion have a significant effect on a data set where correlated error was present However, as previously noted, the HMLE provided better parameter estimates at higher levels of correlated error.

No significant differences in the shapes of the response surface configurations were noted between the two estimators for the relationship between

Ki-Kf.

However, differences between the flow-induced equations did exist. Equations 1 and 3 demonstrated more than one "region of attraction" as well as several discontinuities in their response surface configurations. The contour plot for Equation 2, although

88

smoother and continuous, demonstrated a long, narrow valley that could cause difficulties in parameter identification.

The success of the HMLE criterion at a higher level of error suggests that the local minima were reduced as the degree of correlation in errors was increased. Results from parameter optimization for KrX c and Equation 2 is a notable example: for p = 0.50, the HMLE criterion resulted in better estimates of Kf, 8 out of 12 times and for t c

, 10 out of 12 times. Figure 4.9 shows that for Equation 2, the SLS response surface configuration is flatter and has more local minimum than that of the HMLE criterion. By contrast,

Figure 4.10 demonstrates that for p = 0.25, the SLS response surface is steeper than the

HMLE estimator.

BEST VALUE OF OBJECTIVE FUNCTION

AND AVOIDANCE OF LOCAL MINIMA

An examination of Tables A1-A36 demonstrates that in all of the parameter optimization problems posed when error was present, the SCE-UA algorithm consistently resulted in a smaller value of the objective function, except for those cases in which the comparison resulted in a tie. This observation suggests that local optima do indeed exist on the response surface. One explanation for the Simplex's lack of convergence to the global optima may be that it is located in a small crater-shaped region that lies in a relatively flat area on the response surface. Figures 4.11 and 4.12 demonstrate that the

HMLE

Criterion

SLS

Criterion

Figure 4.9. Comparison of response surfaces for HMLE and SLS criteria

(Equation 2, dry run, 15% slope, p = 0.50).

SLS

Criterion

HMLE

Criterion

Figure 4.10. Comparison of response surfaces for HMLE and SLS criteria

(Equation 2, wet run, 10% slope, p = 0.25).

Simplex

Surface

mww>

SCE-UA

Surface

Figure 4.11. Comparison of response surfaces for SCE-UA and Simplex algorithms.

(Equation 2, diy run, 10% slope, HMLE, p = 0.50).

91

Simplex

Surface

SCE-UA

Surface

Figure 4.12. Comparison of response surfaces for SCE-UA and Simplex algorithms.

(Equation 2, dry run, 10% slope, SLS, p = 0.25).

93

minimum lies in a relatively flat area for the Simplex algorithm, whereas the SCE-UA response surface is steeper.

However, contrary to the theoretical expectation, Simplex usually obtained better estimates for the parameters in the 2-parameter problems even though SCE-UA was better able to converge to the global minimum. One theory that may explain this result is that when error was introduced into the sediment concentration data, the error affected the values of the hydraulic parameters as well. Fixing the values of the hydraulic parameters may have impacted the analysis in such a way that the lowest value of the objective function did not necessarily correspond to the best estimates of the erosion parameters.

Another theory that might explain why the lowest value of the objective function did not correspond to the best estimates of the parameters is the use of the four-point implicit method to numerically solve the kinematic wave equations Although very conservative estimates of the change in time were used in this analysis, an analytical solution may perhaps lessen the uncertainty in the model processes when error is present.

In the 3-parameter problem posed, the success of the algorithms and objective functions was independent of the algorithms' abilities to locate the global minimum on the response surface. This is more than likely due to the similar behavior of the two entrainment terms (Lopes, 1987) as few patterns of success arose from this estimation problem. The success of the Simplex and HMLE procedure when error was present

(given its failure with the error-free data set) supports the result that an early termination of the Simplex algorithm was likely to result in better estimates of the parameters.

94

In the case of the error-free data, the SCE-UA algorithm generally did not succeed in finding a lower value of the objective function for the 2-parameter problems. This was a result of the SCE-UA algorithm terminating due to the population of points converging into a sufficiently small space such that any further search would not result in a significant improvement of the parameter estimates. This was indeed the case since very small differences in parameter estimates resulted between the use of the two algorithms. This again suggests that a small crater-shaped region exists in a relatively flat area of the response surfaces.

DATA SET VARIABILITY

All three parameters demonstrated different sensitivities to the three antecedent moisture conditions tested, without any effects due to the level of correlated error. Even though the estimation procedures were sensitive to the calibration data variability, no trends were noted in the selected objective function's ability to reduce this sensitivity.

According to averages in the percent error of estimation calculated for the three antecedent moisture conditions (see Tables 4.10-4.13 and Figure 4.13), the dry runs provided the best estimates for the parameter x c

, 4 out of 6 times, for the 3-parameter problem and, 5 out of 6 times, for the 2-parameter problems. This result may be related to the length and variability of the data which are crucial factors in the activation of a threshold parameter. Data from dry runs contained more information on varying soil

Table 4.10. Average estimation error for different initial moisture conditions

for 3-parameter optimization problem.

95

Parameter p Equation

Avg. %

Avg. %

Avg. %

No.

Error

Error

Error

Dry

Wet Very Wet

Ki

K

T

K

K f c f

Tc f

0.25

0.50

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

99.99

119.16

94.40

3.20

5.36

2.18

11.39

5.23

3.68

94.86

77.60

84.64

8.14

7.96

10.41

20.67

12.95

11.24

68.78

97.93

78.00

30.33

18.52

10.57

44.84

27.28

10.59

83.96

75.93

96.09

45.20

6.07

16.54

66.97

11.86

23.39

99.56

73.15

79.68

8.81

2.63

5.46

38.85

4.66

6.41

105.31

119.52

80.11

7.31

10.19

7.01

29.25

25.76

27.80

Table 4.11. Average estimation error for different initial moisture conditions

for 2-parameter optimization problem

( K r z c

)

96

Parameter p Equation Avg. % Avg. % Avg. %

No. Error Error Error

Dry Wet Very Wet

K f

0.25

T c

K x f c

0.50

1

2

3

1

2

3

1

2

3

1

2

3

3.10

4.32

3.17

6.17

6.67

2.40

7.31

11.45

11.35

18.52

9.33

11.46

22.97

10.35

12,88

33.12

18.34

13.20

44.34

6.45

16.20

63.64

12.78

25.26

6.02

2.41

2.41

38.21

3.60

3.78

5.16

7.41

9.31

30.49

21.31

29.38

Table 4.12. Average estimation error for different initial moisture conditions

for 2-parameter optimization problem (K.-Kj f

; To = 0.502

97

Parameter p Equation Avg. % Avg. % Avg. %

No. Error Error Error

Dry Wet Very Wet

Ki

K f

Ki

K f

0.25

0.50

1

2

3

1

2

3

1

2

3

1

2

3

91.93

133.70

63.08

54.03

122.77 104.40

3.55

1.56

1.50

31.42

1.48

1.82

110.68

51.29

95.59 114.29

55.87 118.05

3.31

4.04

2.33

1.94

0.59

2.11

109.99

48.34

76.80

2.82

2.42

3.43

123.00

56.12

104.55

5.75

1.48

37.38

Table 4.13. Average estimation error for different initial moisture conditions for 2-parameter optimization problem

(Kj-Kf; x c

= 0.0)

98

Parameter p Equation Avg. % Avg. % Avg. %

No. Error Error Error

Dry Wet Very Wet

Kj

K r

K

K f

0.25

0.50

1

2

3

1

2

3

1

2

3

1

2

3

108.20

124.01

101.55

1.60

9.05

2.03

100.04

78.64

145.72

2.53

1.05

1.47

76.11

149.73

96.83

1.26

1.26

18.67

97.12

38.44

103.91

3.33

1.60

18.98

104.09

77.93

61.84

3.73

2.10

101.55

77.63

64.65

134.79

5.98

3.12

3.25

99

moisture contents. Moreover, dry run simulations were twice the duration of the wet and very wet runs.

Wet antecedent moisture conditions provided the best estimates for the parameter

Ki, 11 out of 18 times for all four parameter optimization problems, followed by very wet runs providing the best estimates, 7 out of 18 times. The dry runs were not associated with the best estimates for the estimates ofKj. This result also may be related to the length of the data record. Given that entrainment by raindrop impact is similar to entrainment by hydraulic shear, the model cannot separate the two processes when optimizing for the appropriate parameters. The longer the simulation, the less clear the distinction becomes between the two processes of entrainment. The higher degree of success with the very wet simulations may be related to a higher rainfall intensity rate.

Effect of Initial Moisture Conditions on Parameter Estimates tau-c

Parameter

Figure 4.13. Number of best estimates associated with each initial moisture condition according to average percent estimation error for all four optimization procedures.

100

The effect of antecedent moisture was not as evident with respect to the parameter estimate of Kf. Each antecedent moisture condition provided the best estimates 8 out of

24 times for all of the parameter optimization problems posed. However, all estimation errors for Kf greater than 15% occurred in the wet runs. This result was not due to the effect of random error since all three flow-induced equations generated the same result, but due to the very low sediment concentration values associated with these runs. A higher flow rate would make it easier for the estimation procedure to determine the transport capacity of the flow.

EVALUATION OF FLOW-INDUCED EROSION EQUATIONS

A very high level of error was associated with Equation 1, which relates sediment entrainment to the stream power of the flow (Bagnold, 1966). Even when no error was present in the data, the error of estimation for the dry and wet runs was usually twice that of the other two erosion equations. This may have been due to an inability of Equation 1 to predict erosion under fully dynamic conditions.

In general, Equations 2 and 3 were associated with lowest estimation error for all three parameters In the Kric estimation problem, Equation 3 generated the best estimates of x c when p

= 0.25 (6.46% vs. 9.54% ) whereas Equation 2 performed best for the case where p = 0.50 (14.52% vs. 22.03%). Equation 2 was chosen for use in further studies because the results indicate that it would perform well regardless of the level of

101

error. By contrast, greater inaccuracies in the estimate of x c may exist unless the level of error is known to be low.

Another factor considered in the selection of the best flow-induced erosion equation was the shape of the response surface for K,-Kf Equation 2 was the only one of the three that was unaffected by discontinuities in the response surface. Moreover, the

Simplex and SLS procedure only demonstrated a 50% success rate for the error-free data case. These factors showed that Equation 2 was more robust than the other two equations.

OPTIMIZATION PROBLEM SELECTED FOR NATURAL DATA STUDIES

Four different parameter optimization problems were posed with the primary objective of studying the behavior of the erosion parameters with respect to the optimization algorithm and objective function employed. The inclusion of Ki in the optimization problem clearly resulted in a significant amount of uncertainty in the analysis.

For this reason, K, is often determined by measuring sediment yields from small plot studies. In very small areas, it is assumed that all of the erosion is induced by raindrop impact, since the area is too small to experience erosion by hydraulic shear. In the absence of plot studies, regression equations that relate soil properties to K; have been used to determine its value.

Once the value of K; has been estimated, it can be fixed so that the values of the parameters relating erosion by hydraulic shear can be determined. Other soil erosion

102

studies have also indicated this to be the best approach for estimating parameter values

(Lopes, 1987; Page, 1988).

The Simplex algorithm was selected for use in the rainfall simulator plot and watershed studies because it was more successful in estimating the true values of the parameters. The only relevant exception to this rule was in the K r t c optimization problem where Kf was better estimated by SCE-UA. Because

t

c is a threshold parameter and is more difficult to estimate than K f

, it was considered best to choose the algorithm that could better estimate the inactive parameter. Inaccurate estimates of

T

c would also have a greater impact on model predictions than any differences in the parameter estimates for

Kf generated by Simplex or SCE-UA.

The selection of the objective function for use in the natural data studies depended on the amount of error assumed to be present in the data. The synthetic data study clearly demonstrated that higher levels of error in the sediment concentration data would be better served by the HMLE estimator. However, it was assumed that the amount of correlated error present in the natural data would not warrant the use of the HMLE criterion.

Analysis of Plot Data

To test the selected flow-induced erosion equation, data from WEPP rainfall simulator plots located in the Western United States were used. A 2-parameter

103

optimization problem was posed by fixing the value of

Kj.

The value of

Ki had already been determined experimentally using small plot studies by the USDA-ARS WEPP team.

ESTIMATION OF HYDRAULIC PARAMETERS

Values of the hydraulic parameters and the Nash-Sutcliffe coefficient appear in

Tables 4.14 and 4.15 and their corresponding hydrographs are in Figures 4.14-4.23. With the exception of the dry run for plot 105, all of the simulations generated hydrographs fit the measured data well. This is important because the hydrology drives the erosion component of the model. If the hydraulic parameters are not well estimated, then it may not be possible to obtain good estimates of the erosion parameters.

TRUE VALUES, INITIAL VALUES AND PARAMETER BOUNDS

Because preliminary investigations of the rainfall simulator plots indicated that the value of critical shear stress (x c

) approached zero, and that in general, most values of

t

c for rangeland soils did not exceed a value of 6.0 (Foster et al., 1989), these values were designated the minimum and maximum values of the parameter for all simulations. The range of values for the parameter Kf changed from site to site, and were determined by trial and error (see Table 4.16).

Since the value of Kf is model dependent, it's true value was unknown. Therefore,

T c was the only parameter that could be compared to its true value as determined by the critical conditions for incipient motion to occur. After obtaining values of the Reynolds

Table 4.14. Optimized hydraulic parameters for the rainfall simulator plots.

Re Manning's n Ns Plot

No.

Antecedent

Moisture

31

34

56

59

63

Dry

Wet

Very Wet

Dry

Wet

Very Wet

Dry

Very Wet

Dry

Very Wet

Dry

Very Wet

66

Dry

Very Wet

102 Dry

Wet

Very Wet

105

120

121

Dry

Wet

Very Wet

Dry

Wet

Very Wet

Dry

Wet

Very Wet

30.3

36.2

55.8

40.2

53.6

51.6

47.0

7.0

15.2

42.9

42.0

57.9

53.5

53.0

35.1

52.5

57.6

55.0

36.2

38.3

58.3

56.2

59.1

48.7

58.8

49.4

2.90280e-02

2.81550e-02

3.7089e-02

6.32860e-02

2.70770e-02

2.70040e-02

3.84050e-02

4.00000e-02

3.82490e-02

4.74690e-02

3.21660e-02

4.90720e-02

4.07300e-02

4.07330e-02

2.81580e-02

5.99740e-02

5.99220e-02

5.12160e-02

3.70150e-02

5.99220e-02

5.89490e-02

5.77280e-02

5.52580e-02

3.73930e-02

5.9990 le-02

6.67600e-02

4.41380e-01

5.33810e-02

5.52800e-02

5.38480e-02

3.80480e-02

1.01360e-02

3.42550e-03

1 66690e-02

1.50470e-01

1.1394e+00

4.21110e-02

9.11000e-01

2.52970e-02

3.01810e-01

5.18280e-01

1.27120e-01

3.39010e-02

1.72610e-01

1.77970e-01

6.46050e-02

1.04820e+00

1.43670e-01

8.01780e-02

1.75689e+00

4.28810e-01

5.04840e-02

Ks

5.54130e-08

2.96770e-07

3.34560e-06

1.73860e-07

3.93120e-08

9.92950e-08

3.60950e-07

1.72180e-07

7.51070e-08

4.73970e-08

Objective

Function

240.22

403.34

410.53

848.78

429.16

421.81

174.16

303.92

81.84

1426.10

248.59

205.43

231.16

455.05

90.188

1358.2

690.77

3291.8

856.17

936.53

703.61

232.21

213.07

857.57

355.68

837.64

Nash-Sutcliffe

Coefficient

0.96

0.95

0.97

0.98

0.84

0.77

0.71

0.92

-0.39

0.74

0.92

0.83

0.95

0.97

0.66

0.94

0.92

0.88

0.93

0.98

0.98

0.78

0.99

0.65

0.96

0.92 o

Table 4.15. Parameter bounds and starting values for rainfall simulator plots

Plot Nos.

31,34

56,59,63,66

102, 105

120, 121

t c

[M/LT

2

]

Lower Bound Upper Bound Starting Value

0.0

0.0

0.0

0.0

6.0

6.0

6.0

6.0

1.150

0.100

0.875

0.875

Kr

[T

2

/L

05

M

05

]

Lower Bound Upper Bound

5.0e-02

1.0e-04

5.0e-02

5.0e-02

1.0e+00

2.5e-01

1.0e+00

2.0e-02

Starting Value

4.25e-01

1.25e-01

1.0e-01

1.0e-04

106 number from the overland flow at steady-state, the Shield's diagram was used to identify the value of x c

.

SEDIMENT GRAPHS FOR RAINFALL SIMULATOR PLOTS

In only

4

of the 24 simulations did the Nash-Sutcliffe coefficient indicate a good fit between the measured and the simulated sediment graphs (see Table 4.16 and Figures

4.14 - 4.23). It was in these cases that some of the largest errors in estimation occurred for the critical shear stress parameter. Conversely, when the error of estimation was low, the Nash-Sutcliffe coefficient indicated a poor fit between the simulated and measured data.

Plots 31 and 34 are a case in point, as the synthetic data were generated based on the soil properties at these sites. The only simulation (plot 31, Wet Run) to produce a sediment graph that matched the measured data (Nash-Sutcliffe Coefficient = 0.77) was also associated with the highest estimation error for x c for all six runs. Two simulations generated good estimates of t c

, (less than 11 % error), however, the results showed virtually no match between the measured and simulated sediment graphs.

Because WESP generated hydrographs that provided a good fit to the measured data, these anomalies may be linked to a structural problem in the erosion component of the WESP model. The analysis of the synthetic data identified a problem early on with the interactions of the deposition parameter occurring at very low slopes. WESP assumed that entrainment and deposition occurred simultaneously. If too much deposition

Table 4.16. Erosion parameter estimates for rainfall simulator plots.

Plot

No.

31

34

56

59

63

66

102

105

120

121

Antecedent

Moisture

I>ry

Wet

Very Wet

IDry

Wet

Very Wet

Dry

Very Wet

Dry

Very Wet

Dry

Very Wet

Dry

Very Wet

Dry

Wet

Very Wet

Dry .

Wet

Yei\ Wet

Dry

Wet

Very Wet

Dry

Wet

Very Wet

%

Slope

10.2

10

8.5

7.1

8.6

8.0

11

.2

9.8

11.2

11

.6

Fixed Value

K,

Parameter Estimates

a

Kr

285,000 1.6899F.-01

5.6370E-02

5.0001E-02

6.0261E-02

5.0171E-02

5.9516E-02 x t

4.8227E-0I

5.7948E-04

3.7531E-01

3.0237E-01

5.6329E-01

6 8658E-04

222.855

186.445

9.4761E-02

1.0115E-01

3.4933E-02

8.2727E-02

1.5110E-02

2.1566E-02

1.9131E-02

2.7595E-02

1.8051E-03

2.5757E-04

2.5573E-02

4.3905E-04

5.9127E-05

3.6865E-04

3.1051E-03

1.5909E-04

315,178

947,294

1.0976E-01

5.0007E-02

5.0027E-02

5.0063E-02

6.4687E-02

5.0036E-02

2.3611E-03

3.3085E-01

7.9432E-01

6.3875E-01

2.0695E-03

5.3836E-01

3.3447E-04 6.4087E-04

1.8990E-04 6.1951E-03

7.1875E-04 3.5684E<00

9.I398E-04

3.8742E-04

3.1719E-04

8.7357E-03

3.5665E-03

3.4180E-03 a

True value of Kr is unknown

b

Value of

T

C

5.4042E-01

5.4042E-01

5.4042E-01

4.7452E-01

5.2724E-01

5.2724E-01

5.6473E-01

5.8086E-01

6.4540E-01

5.8086E-01

5.0149E-01

5.0149E-01

4.7641E-01

4.3880E-01

5.6570E-01

6.3427E-01

6.5140E-01

6.8569E-01

6 1712E-01

6.I712E-0I

6.7510E-02

6.7510E-02

6.7510E-02

6.5820E-02

6.7510E-02

6.7510E-02

% Error of

True Value r.

(10.76015%)

(99.89277%)

(30.55216%)

(36.27877%)

(6.83749%)

(99.86978%)

(99.68036%)

(99.95566%)

(96.03765%)

(99.92441%)

(99.98821%)

(99.92649%)

(99.34823%)

(99.96374%)

(99.58262%)

(47.83767%)

(21.94044° b)

(6.84566%)

(99.66465%)

(12.76251%)

(99.05070%)

(90.82343%)

(5185.73545%)

(86.72789° o)

(94.71708%)

(94.93705%) b

True value of ic was determined using Shield's diagram

Value of

Obj. Ftn

1.8823E+03

3 1588E+02

9.6677E<02

2.2344E^03

5.3285E+03

2.2295E+03

6.7520E + 02

6.6359E < 02

3.9634E<02

2.5003E -02

8.1605E+01

1.1197E+01

1.0836E^02

6.8785E+02

4.7677E+02

8.5356E+02

6.7471E'02

3.5213Ei02

7.4228F.t02

3.8301F.f02

1.5280E'03

3.5753E^02

2.9100E<02

3.7593E+03

2.3602E+03

7.5927E*02

Nash-SutclifTe

Coefficient

4.50E-01

7.70E-01

1.60E-01

4.10E-01

-1.37E+00

-8.20F.-03

1.30E-01

7.30E-01

-6.50E-01

8.00E-01

3.90E-01

9.50F--01

3.10E-0!

5.90E-02

-3.80E-01

-7.20E-01

-2.60E-01

1.70E-01

2.90E-01

I.20E-01

-2.20E-01

1.30E-01

-3.10E+00

-1.02E + 00

-1.70E-01

2.20E-01

No. of

Iterations

71

156

100

100

101

93

92

91

60

77

48

59

59

63

45

39

20

74

55

39

64

57

65

67

62

64

Dry Run

60

•c 50

1

40

"

3 30 o:

1 20

* 10

1 ® j , * * — —

9

— ?

• i

1

0

10

20 30

Time (min)

40

Wet Run

30

20

10

50

0

60

50

40

50 -

E. 40 m

To 30 -

tr

I

%

20 c ce 10 -

T

30

- 25

CO

- 20 i

- 15

- 10

C o

O

- 5

•9 to

•1

0

40

10

20

Time (min)

Very Wet Run

0 5

Time (min)

—4— Observed Runoff •

—*— Observed Sed Cone •

Simulated Runoff

Simulated Sed Cone

Figure 4.14. Hydrographs and sediment graphs for plot number 31.

Dry Run

0>

ID

cr

30

*§ 20

c

20 40

Time (min)

60

Wet Run

10 15

Time (min)

20

Very Wet Run

0 5 10

Time (min)

•—Observed Runoff •

—A—Observed Sed Cone •

Simulated Runoff

Simulated Sed Cone

Figure 4.15. Hydrographs and sediment graphs for plot number 34.

Dry Run

30 -

2 5 -

20

£ 15 -

- - 2 1

0

10

20

Time (min)

30 40 50

90

T

7 5 -

60

15 45

5 3 0 -

Very Wet Run • •

0

5

10

Time (min)

-«—Observed Runoff

<4—Observed Sed Cone

15 20

• Simulated Runoff

• Simulated Sed Cone

- 50

- - 4 0

5

30 o

- 20 15

- 10

Figure 4.16. Hydrographs and sediment graphs for plot number 56.

60 t

Drv Run

20

Time (min)

30

Very Wet Run

10 15

Time (min)

-«—Observed Runoff

—Obserred Sed Cone

Simulated Runoff

Simulated Sed Cone

Figure 4.17. Hydrographs and sediment graphs for plot number 59.

Dry Run

60

40

1u 30

1 20

- 10

0

10

20

Time (min)

30 40

50

106 -r

Very Wet Run

10

"Time (min)

• Observed Runoff

-A—Observed Sed Cone

Simulated Runoff

Simulated Sed Cone

Figure 4.18. Hydrographs and sediment graphs for plot number 63.

Dry Run

— 40 -

20 30

Time (min)

- 4

40

.V.

50

Very Wet Run

» • #

105

90

75

4)

B

60

45

30

/

15

J

0

\£-

10

Time

(min)

- Observed Runoff

- Observed Sed Cone l

B

15

Simulated Runoff

Simulated Sed Cone

28

24

20

E

16 t

12

8 o

•s

w

4

20

0

Figure 4.19. Hydrographs and sediment graphs for plot number 66.

Dry Run

20 30

Time (min)

40

Wet Run

1 2 0

I

+ 15 »

10 15

Time (min)

20

Very Wet Run

Time (min)

-•— Observed Runoff

-A— Observed Sed Cone

Simulated Runoff

Simulated Sed Cone

10 o

Figure 4.20. Hydrographs and sediment graphs for plot number 102.

114

40

30

20

10

0

Dry Run

50

40

30

20

10

0

0

T

18

15 <o

10

20 30

Time (min)

40 50

60

Wet Run i

\

\ +/

1

<

J**

/

1—

i

i

10 15

20

Time (min)

II

1

•!

r\

1

*1

1

25

25 jj?

CJ>

J*

15

c.

10 O

«5 "S

5 03

30

0

Very Wet Run

- 30

10 u

• Observed Runoff

-A— Observed Sed Cone

Simulated Runoff

Simulated Sed Cone

Figure 4.21. Hydrographs and sediment graphs for plot number 105.

115

Dry Run

K 30

20 30

Time (min)

Wet Run

%

01 30

20

Time (min)

Very Wet Run

0)

to

a:

•B

C

Time (min)

• Observed Runoff R*. _ Simulated Runoff

-A—Observed Sed Cone Simulated Sed Cone

Figure 4.22. Hydrographs and sediment graphs for plot number 120.

Dry Run

- 30

10 15

Time (min)

20

Very Wet Run

105 x

90 -

75

60 -

- 40

- 30

0

5

10

15

20

25

Time (min)

-•— Observed Runoff

— Observed Sed Cone

• Simulated Runoff

• Simulated Sed Cone

Figure 4.23. Hydrographs and sediment graphs for plot number 121.

118

occurred in the simulated event, then simulated sediment concentrations would have been too low. This was in fact the case where the simulated sediment concentrations were severely underestimated on the rising limb of nearly all of the sediment graphs and overestimated on the falling limb.

PROCESSES OF DEPOSITION & ENTRAINMENT

To ascertain the effects that different values of the deposition parameter would have on parameter estimates, P was assigned a value of 0.25 and 0.75. Very small changes in the value of the objective function and estimates of

t

c were noted, while the estimates of Kf showed the greatest fluctuation. Since lower values of P reflect turbulent flow conditions, the transport capacity of the flow

(Kf) was increased. For higher values of P, estimates of Kf were decreased

This outcome indicates that the equation describing the downward sediment flux may be inappropriate for the WESP model However, there may be other confounding factors. For example, all of the successful sediment graphs estimated the critical shear stress parameter at a value close to zero. Such a low value for critical shear stress is not realistic. Sediment particles are entrained by flow whenever the magnitude of instantaneous fluid force acting on the sediment particle exceeds the resistance force for the particle to be moved. A greater force will be needed to initially detach a sediment particle from the soil matrix than one that will be required to re-entrain that same particle

119

once it has been deposited on the soil surface. Presently, WESP cannot account for any processes of re-entrainment, which may explain the very low estimates of the critical shear stress parameter. Moreover, the true effective particle diameter is unknown and small errors in its estimate may adversely effect the parameter estimation procedure.

ACTIVATION OF EROSION BY HYDRAULIC SHEAR

Equation 2 was selected as the best model of the three tested for describing flowinduced erosion. However, the synthetic data used in this analysis were also generated by the same flow-induced erosion model, and may not be appropriate for describing entrainment by hydraulic shear in broad-shallow overland flow areas under natural conditions. All of the equations tested in this analysis incorporated existing transport formulas that were developed based on experimental work in channels. Equation 2, for example, incorporated the bedload formula of Yalin (1963). Although it was assumed that the hydraulics of overland flow were different than those of a channel in this analysis, it may be that different equations and parameters are necessary for describing erosion by hydraulic shear.

It is also not known if entrainment by hydraulic shear actually occurred on the rainfall simulator plots studies. Bare soil conditions do not normally occur in rangeland environment, and had to be artificially created to conduct these plot studies (Simanton et al., 1985). Below ground biomass was left undisturbed, while all of the vegetation was clipped and rocks were removed from the soil surface Under these circumstances,

120

infiltration was enhanced, thereby reducing runoff and entrainment than what might have normally occurred under natural bare soil conditions. If erosion by hydraulic shear did not occur, then this would have also accounted for very low values in the estimates of

T

C

.

Analysis of Watershed Events

To test the effect of scale on the selected flow-induced erosion equation, data were used from a small experimental watershed located in the USDA-ARS Walnut Gulch

Experimental Watershed near Tombstone, Arizona. A 2-parameter optimization problem was posed by fixing the value of Ki Because the WESP model does not currently incorporate any adjustment factors for the erosion parameters when cover is present, the value of Ki was adjusted so that the detachment by raindrop impacted accounted for approximately 80% of the total sediment yield This value was arbitrarily selected in accordance with erosion studies that have shown that sediment entrainment by raindrop impact predominates in rangeland environments (Nearing et al., 1989).

PARAMETER ESTIMATES

Before optimizing for the erosion parameters, the values of the hydraulic parameters were determined for each event (see Table 4.17). Parameter bounds and starting values for the watershed events are shown in Table 4.18. The agreement between the simulated and the observed watershed responses for both runoff and sediment yield indicate that WESP satisfactorily described the sedimentation processes occurring in

121

Kendall Watershed (see Table 4.19). Estimates of the critical shear stress parameter confirmed this result in at least 2 of the 3 events studies. Even the highest estimation error

(138%) associated with

t

c accurately estimated the sediment yield for that event.

Moreover, this estimate was at least on the same order of magnitude as to its true value, a result that was contrary to the successful sediment graphs generated by the rainfall simulator events.

The difficulty of calibration was eased for the watershed events inasmuch as the sediment yield instead of sediment concentration values was used as a basis for comparing simulated to measured data. Typically with the rainfall simulator plots, simulated sediment concentrations were under predicted on the rising limb of the sediment graph, and over predicted on the falling limb due to the problems previously discussed. This imbalance may have been equilibrated by the comparison of a single value in the objective function.

However, WESP may be better able to describe the processes of sedimentation when they occur on a larger scale.

Table 4.17. Hydraulic parameters for selected Kendall Watershed events.

Event Date

Re

Manning's n Ns

Ks

13-Sep-75

28-Jul-76

5-Sep-76

24.2

18.6

16.8

4.46700e-02

2.50060e-02

4.63320e-02

1.36050e-01

1.05400e-01

5.23160e-03

1.06300e-06

165350e-06

2.94730e-06

Objective Nash-Sutcliffe

Function Coefficient

194.03

7.403

547.88

0.90

0.99

0.71

Table 4.18. Parameter bounds and starting values for selected Kendall Watershed events.

Event Date

13-Sep-75

28-Jul-76

S-Sep-76

Lower Bound

0.0

0.0

0.0

[M/LT

2

)

Upper Bound

6.0

6.0

6.0

Starting Value

1.150

0.100

0.875

Lower Bound

.00e-13

.00e-13

.00e-13

Kn

1T j

/L

05

M°-

5

]

Upper Bound

1 00e-06

1.00e-06

1.00e-06

Starting Value

1.00e-08

1.00e-08

1.00e-08

Table 4.19 Erosion parameter estimates for selected Kendall Watershed events.

Event Date

13-Sep-75

28-Jul-76

5-Sep-76

Fixed Value ofKi

105000

a

Kf

6.131 le-07

4.5648e-06

2.0037e-07

T

c

0.03856

0.03099

0.09504

b

Value of

0.04241

T

0.04113

0.03984 c

%

Error of

True Value of

(9.08041%)

b

True value of tt was determined using Shield's diagram

T

(24.64381%)

(138.55924%) c

Value of the No. of

Objective Function iterations

1.3805e-12

2.0354e-16

2.9381e-13

108

116

98

123

CONCLUSIONS AND FUTURE RESEARCH

Summary and Conclusions

The primary goal of this study was to determine the adequacy of the optimization procedures in identifying unique, optimal parameter values. In the first phase of this study, synthetic error-free data, as well as data contaminated with correlated and random error, provided the means for determining the effectiveness of the four optimization procedures evaluated Four different optimization problems were posed so that the behavior the erosion parameters could be fully studied. Using a fully-dynamic processbased approach, three sediment transport equations describing flow-induced erosion were compared.

Based on the synthetic data analysis, the most successful optimization procedure and flow-induced erosion equation were selected for use in the second phase of the study.

Ten rainfall simulator events from four different areas in the Western United States were selected for analysis. Three rainfall-runoff events for a small watershed were also examined.

From the evaluation of the synthetic and natural data studies, the following conclusions can be made:

(1) The Simplex algorithm was more successful than the SCE-UA in estimating the parameters. This result was contrary to the theoretical expectation since the

SCE-UA algorithm achieved a 100% success rate in finding a lower value of the objective function. Although the outcome may be attributed to the fact

124

that the hydraulic parameters were fixed even after error was introduced into the analysis or to the use of the finite difference scheme, it was assumed that the behavior of the model was such that the global minimum on the response surface was located in areas that produced more extreme values of the parameters.

(2) The SLS criterion generated the best estimates of the parameter when the error level was low, whereas the HMLE estimator performed better when a higher level of correlated error was present in the data.

(3) Of the three flow-induced erosion equations studied, Equation 1, which was related to the stream power of the flow (Bagnold, 1966), was the only equation that was consistently associated with a high amount of estimation error. This may be due to an inability to predict erosion under fully dynamic conditions.

(4) Equation 2 was determined to be the best model describing flow-induced erosion. Although Equation 3 generated better estimates of the parameters at a lower level of error, the difference in performance between the two equations was small. Moreover, at a higher level of error, the performance of Equation 2 surpassed that of Equation 3.

(5) All four of the estimation procedures demonstrated the same sensitivities to calibration data variability. The parameter for critical shear stress (x c

) was better estimated for dry runs. Ki, which describes entrainment by raindrop

125

impact, was better estimated in the wet and very wet runs. The parameter related to the transport capacity of the flow, Kf, was unaffected by calibration data variability.

(6) The selected flow-induced erosion equation did not succeed in reproducing sediment graphs with physically, realistic parameter values for the rainfall simulator plots studied. This outcome may have been the result of an inappropriate equation used to describe deposition, an inactivation of the process of entrainment by hydraulic shear and/or the use of a flow-induced erosion equation that was developed from observations in channels and was not appropriate for describing erosion in broad-shallow overland flow areas.

(7) The agreement between the simulated and the observed hydrographs and sediment yields indicate that the WESP model is able to describe the sedimentation processes occurring in small watersheds However, because only one value for total sediment yield is used for comparison in the objective function, problems in under and over prediction at different points on the sediment graph are not at issue.

(8) Although the Simplex and HMLE optimization procedure was found to be more sensitive to very wet runs in the 3-parameter problem, it was determined that this sensitivity was more likely the result of a greater degree of parameter activation rather than a sensitivity to a variable rainfall rate.

126

Recommendations for Future Research

The verification of the processes in an erosion model is a critical step in developing a valid erosion prediction tool. In the WESP model, a detailed evaluation of the process of deposition is needed so that the model can accurately represent the physical system it simulates. In the synthetic data analysis, it became clear that the equation used to describe deposition was problematic, whereas in the natural data analysis, the effects of deposition were more obscured. This uncertainty must be resolved before confidence can be placed in the predictive capabilities of the WESP model.

Research into deriving flow-induced erosion equations is also needed. The equations studied in this analysis were developed from observations in channels and therefore their use outside the domain from which they were developed could lead to erroneous results. If the equation were applied to a channel, the parameter K f could be related to entrainment by way of shear stress acting at the fluid/soil interface, headcutting and sidewall sloughing. The latter two of these mechanisms are clearly not appropriate for broad-shallow overland flow. A need exists for the development and verification of universal, fundamentally derived equations for relating erosion by hydraulic shear in broad-shallow overland flow areas.

Research into the effects of using a numerical technique to solve the continuity equation for sediment transport is necessary to determine if the noise introduced is negatively impacting parameter identification. The reason for the global minimum to be consistently located in an area that corresponded to poorer estimates of the parameters

127

still remains unresolved. This result may have been related to the use of the implicit, fourpoint finite difference scheme, or even possibly to errors in the formulation of the basic equations that were discussed above.

To date, no erosion tests have been performed to relate the soil and cover properties to erodibility using statistical regression techniques for the WESP model. This approach can be problematic since the results are questionable for applications outside the range for which they were derived However, the current methodology requires that a large number of varied data sets be evaluated to decide how the parameters are affected by a wide variety of soil and cover characteristics.

The study presented herein was confined to an analysis of erosion in overland flow areas. A similar investigation into the methods of parameter identification in areas of channel flow is necessary to the development of the WESP model. Results of such a study might indicate where the problems of parameter identification exist in broad-shallow overland flow areas.

APPENDIX A

Tables

A1 - A36

Results of Parameter Optimization for Synthetic Data

Table A1. Results of K; -Kf

-T c parameter optimization problem with error-free data, Equation 1.

Q

P

0

Antecedent

Moisture

Dry

Wet

Very Wet

a

o-

Slope

10

15

10

15

10

15

Search

Algorithm

Simplex

SCE-UA

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCH-l'A

Simplex

SCF.-UA

Simplex

SCH-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HM1.F.

SLS

11MI.E

SI.S

IIMLE

SLS

HMI.E

SLS

HMLE

SLS

HMLE

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

* Values of Ki are multiplied by 10,000

28.517

28.518

44.378

28.531

28.483

28.483

28.481

28.477

28.403

28.397

57.573

28.405

69.022

28.538

74.563

28.502

b

Ki

28.575

28.500

65.817

28.588

28.635

28 618

36.003

28.637

Parameter Estimates (°'o of Error from True Values) c

Kf

(0.26316%)

(0.00000%)

(130.93684%)

(0.30877%)

(0.47368%)

(0.41404%)

(26.32632%)

(0.48070%)

1.7190F.-04

1.7194E-04

1.7349E-04

1.7192E-04

I.7186E-04

1.7187E-04

1.7195E-04

1.7187E-04

(0.01164°o)

(0.03491® o)

(0.93670%)

(0.02327^b)

(0.01164%)

(0.00582°b)

(0.04073%)

(0.00582%)

(0.340359b)

(0.36140%)

(102.01053%)

(0.33333%)

(142.18246%)

(0.13333%)

(161.62456%)

(0.00702%)

(0.05965%)

(0.06316%)

(55.71228%)

(0.10877%)

(0.05965%)

(0.05965%)

(0.06667%)

(0.08070%)

1.7189E-04

1.7189E-04

1.7361E-04

1.7191 E-04

1.7559E-04

1.7191E-04

1.7240E-04

1.7190E-04

1.7183E-04

1.7187E-04

2.5956E-04

1.7187E-04

1.7I88E-04

1.7188E-04

1.7188E-04

1.7188E-04

(0.00582%)

(0.00582%)

(1.00652%)

(0.01745° b)

(2.15848%)

(0.01745%)

(0.30254%)

(0.01164° o)

(0.029099 b)

(0.005829o)

(51.012339b)

(0.00582%)

(0.00000%)

(0.000009 b)

(0.000009b)

(0.00000° b)

.50199

.50200

.80276

.50200

.50199

.50200

.50200

.50200

.50183

50184

.56037

50194

.56785

.50230

.54353

.50213

d

tauc

.50225

.50232

.57411

.50235

.50202

.50203

.50968

.50204

(0.04980%)

(0.063759b)

(14.364549 b)

(0.06972%)

(0.00398%)

(0.00598° o

)

(1.52988%)

(0.00797° b)

(0.03386%)

(0.03187%)

(11.62749%)

(0.011959

0

)

(13.11753%)

(0.05976%)

(8.272919 b)

(0.02590%)

(0.001999o)

(0.00000%)

(59.91235%)

(0.000009b)

(0.00199%)

(0.000009' o

)

(0.0000094)

(0.00000%)

2.0331E-08

2.6321F.-08

6.5I69E-09

6.8692E-10

1.7373E-04

2.1479E-08

1.1334E-08

5.5804F.-10

2.2875E-08

2.2925E-08

3.2388E-04

7.6016F.-10

2.3046E-08

2.3018E-08

7.6715E-09

7.9847E-10

Value of

Obj. Ftn.

5.9059E-08

5.8872E-08

3.2616E-09

9.3542E-10

6.7305E-08

6.7219E-08

1.5863E-09

1.0825E-06

No.

Iterations

186

1554

162

1720

165

1393

128

1681

268

1536

114

1961

150

1411

231

1706

225

1303

98

1354

233

1406

217

1269

b

True value ofl.7188E-04

and

starting value of 2.25E-04

d

True value of 0.502 and starting value of 1.15

Table A2. Results of K; -Kf-x c parameter optimization problem with error-free data, Equation 2.

a

P

0

Antecedent

Moisture

Drv

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMI.E

SLS

HM1J-

SLS

HMLE

SLS

HMLE

SI.S

HMLE

SLS

HMLE

Correlation coefficient of sediment concentration; for p = 0, no random error

True value of 28.5 and starting value of 42.5

Values of Ki are multiplied by 10,000

28.531

28.539

28.534

28.531

28.500

28.497

28.561

28.503

28.450

28.536

61 108

28.406

28.541

28.527

45.423

28.504

b

Ki

28.560

28.536

14.235

28.563

29.055

28.618

39.562

28.723

Parameter Estimates (% of Error from True Values)

c

KI

(0.21053%)

(0.12632%)

(50.05263%)

(0.22105%)

(1.94737%)

(0.41404%)

(38.81404%)

(0.78246%)

3.7366E-03

3.7366E-03

3.7565E-03

3.7366E-03

3.7362E-03

3.7364E-03

3.7272E-03

3 7364E-03

(0.00268%)

(0.00268%)

(0.53526%)

(0.00268%)

(0.00803%)

(0.00268%)

(0.24890%)

(0.00268%)

(0.17544%)

(0.12632%)

(114.41404%)

(0.32982%)

(0.14386%)

(0.09474%)

(59.37895%)

(0.01404%)

3.7359E-03

3.7366E-03

3.7015E-03

3.7356E-03

3.7363E-03

3.7363E-03

3.7231E-03

3.7362E-03

(0.01606%)

(0.00268%)

(0.93671%)

(0.02409%)

(0.00535%)

(0.00535%)

(0.35862%)

(0.00803%)

(0.10877%)

(0.13684%)

(0.11930%)

(0.10877%)

(0.00000%)

(0.01053%)

(0.21404%)

(0.01053%)

3.7363E-03

3.7362E-03

3.7363E-03

3.7363E-03

3.7365E-03

3.7365E-03

3.7362E-03

3.7365E-03

(0.00535%)

(0.00803%)

(0.00535%)

(0.00535%)

(0.00000%)

(0.00000%)

(0.00803%)

(0.00000%)

.50198

.50198

.50198

.50199

.50199

.50199

.50196

.50199

.50189

.50203

.51029

.50181

.50197

.50196

.50459

.50194

d

tauc

.50204

.50203

.49892

.50204

.50210

.50202

.50353

.50206

(0.00797%)

(0.00598%)

(0.61355%)

(0.00797%)

(0.01992%)

(0.00398%)

(0.30478%)

(0.01195%)

(0.02191%)

(0.00598%)

(1.65139%)

(0.03785%)

(0.00598%)

(0.00797%)

(0.51594%)

(0.01195%;

(0.00398%)

(0.00398%)

(0.00398%)

(0.00199%)

(0.00199%)

(0.00199%)

(0.00797%)

(0,00199%)

Value of

Obj. Ftn.

4.7347E-08

4.7447E-08

2.3846E-09

7.1169E-10

6.7749E-08

6.0946E-08

2.0538E-09

9.2029E-10

1.5559E-08

4.7447E-08

9.6494E-09

4.3201E-10

2.0133E-08

2.0130E-08

3.9103E-09

4.4843E-10

1.9403E-08

1.9763E-08

6.4762F.-10

6.4908E-10

2.8262E-08

2.8308E-08

1.3702E-09

9.4509E-10

No.

Iterations

273

1472

172

1632

165

1515

191

1694

230

1335

216

1493

188

1461

154

1481

370

1472

244

1606

255

1652

195

1734

b ' True value of3.7365E-03 and starting value of 4.25E-03

d

True value of0.502 and starting value of 1.15

Table A3. Results of Kj -Kf-t c parameter optimization problem with error-free data, Equation 3.

a

P

0

Antecedent

Moisture

Dry

Wei

Very Wet

a

0

0

Slope

10

15

10

15

10

15

Seardi

Algorithm

Simplex

SCE-UA

Simplex

SCF.-IA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMI.E

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SI.S

HMLE

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

* Values of Ki are multiplied by 10,000

28.516

28.509

64.342

28.531

28.514

28.504

11.336

28.512

74.982

28.469

36.335

28.418

28.492

28.531

32.149

28.506

b

Ki

35.153

28.558

74.996

28.564

50.489

28.652

74.970

28.646

Parameter Estimates (% of Error from True Values) c

Kf

(23.34386%)

(0.20351%)

(163.143869b)

(0.22456%)

3.9630E-05

3.9684E-05

4.2010E-05

3.9686E-05

(0.13859%)

(0.00252%)

(5.85864%)

(0.00252%)

(77.15439%)

(0.53333%)

(163.05263%)

(0.51228%)

3.9561E-05

3.9683E-05

5.5614E-05

3.9684E-05

(0.31246%)

(0.00504%)

(40.13859%)

(0.00252%)

(163.09474%)

(0.10877%)

(27.49123%)

(0.28772%)

(0.02807%)

(0.10877%)

(12.80351%)

(0.02105%)

(0.05614%)

(0.03158%)

(125.76140%)

(0.10877%)

(0.04912%)

(0.01404%)

(60.22456%)

(0.04211%)

4.0136E-05

3.9679E-05

3.9598E-05

3.9677E-05

3.9684E-05

3.9684F.-05

3.9684E-05

3.9685E-05

3.9684E-05

3.9684F.-05

3.7152E-05

3.9682E-05

3.9685E-05

3.9686E-05

3.9469E-05

3.9685E-05

(1.13645%)

(0.01512%)

(0.21923%)

(0.02016%)

(0.00252%)

(0.00252%)

(0.00252%)

(0.00000%)

(0.00252%)

(0.00252%)

(6.38276%)

(0.00756%)

(0.00000%)

(0.00252%)

(0.54429%)

(0.00000%)

.50198

.50198

.47459

.50197

.50201

.50204

.49057

.50202

.52847

.50191

.50402

.50185

.50196

.50199

.50259

.50199

d

tauc

.50393

.50201

.55255

.50204

.50656

.50200

.77582

.50202

(0.38446%)

(0.00199%)

(10.06972%)

(0.00797%)

(0.90837%)

(0.00000%)

(54.54582%)

(0.00398%)

(5.27291%)

(0.01793%)

(0.40239%)

(0.02988%)

(0.00797%)

(0.00199%)

(0.11753%)

(0.00199%)

(0.00398%)

(0.00398%)

(5.46016%)

(0.00598%)

(0.00199%)

(0.00797%)

(2.27689%)

(0.00398%)

Value of

Obj. I tii.

3.3798E-06

5.5114E-08

1.4855E-05

8.2215E-10

2.2236E-05

6.2682E-08

1.2I97E-03

9.9707E-10

No.

Iterations

137

1329

168

1980

153

1549

161

1775

2.3237E-04

1.9111E-08

1.9291E-09

5.6308E-10

2.4085E-08

2.4316E-08

2.1964E-09

6.0122E-10

2.5454E-08

2.6064E-08

4.0288E-05

8.6428E-10

1.6427E-08

1.8361E-08

2.6470E-06

5.3448E-10

218

1260

99

1327

208

1246

255

1347

261

1569

346

2130

268

1415

220

1907

b

True value of 3.9685E-05 and starting value of 9.00E-05

d

Truevalueof0.502andstartingvaIueofl.lS

Table A4. Results of K;-KrT c parameter optimization problem for p = 0.25, Equation 1.

a

p

0.25

/Snteccdent

Moisture

Dry

Wet

Yen Wet

a

°0

Slope

10

15

10

15

10

15

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMIJi

SI.S

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMIJ£

SLS

HMLE

Conflation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42,5

* Values of Ki are multiplied by 10,000

45.608

74.978

71.419

74.995

10.001

10.003

10.002

10.003

50.659

37.793

51 190

61.271

36.781

29.947

42.200

74.999

b

Ki

60.937

10.002

58.039

10.004

47.796

74.977

45.179

75.060

Parameter Estimates (?b of Error from True Values) c

Kf

(113.81404%)

(64.90526%)

(103.64561%)

(64.89825%)

(67.70526%)

(163.07719%)

(58.52281%)

(163.36842%)

1.8566E-04

1.7941E-04

1.7899E-04

1.7515E-04

1.7J41E-04

1.7287E-04

1.7673E-04

1.7783E-04

(8.01722%)

(4.38096%)

(4.13661%)

(1.90249%)

(0.27345%)

(0.57598%)

(2.82174%)

(3.46172%)

(77.75088%)

(32.60702%)

(79.61404%)

(114 98596%)

(29.05614%)

(5.07719%)

(48.07018%)

(163.15439%)

2.5003E-04

2.3265E-04

2.4887E-04

2.4239E-04

1.8989E-04

1.9192E-04

2.0716F.-04

2.2927E-04

(60.02807%)

(163.08070%)

(150.59298%)

(163.14035%)

(64.90877%)

(64.90175%)

(64.90526%)

(64.90175%)

1.5670E-04

1.4814E-04

1.4907E-04

1.4759E-04

1.6375E-04

1.6320E-04

1.6264E-04

1.6278E-04

(45.46777%)

(35.35606%)

(44.79288° o)

(41.02281%)

(10.47824%)

(11.65930%)

(20.52595%)

(33.38957%)

(8.83174%)

(13.81196%)

(13.27089%)

(14.13195%)

(4.73004%)

(5.05003%)

(5,37584%)

(5.29439%)

.76312

.70999

.76182

.76293

.62453

.63025

.72402

.83997

.30055

.30002

.30222

.30001

.33326

.31964

.30002

.30002

d

tauc

.61311

.50537

.58397

.48659

.53373

.56058

.56263

.59672

(22.13347%)

(0.67131%)

(16.32869%)

(3.06972%)

(6.32072%)

(11.66932%)

(12.07769%)

(18.86853%)

(52.01594%)

(41.43227%)

(51.75697%)

(51.97809%)

(24.40837%)

(25.54781%)

(44.22709%)

(67.32470%)

(40.12948° o)

(40.23506%)

(39.79681%)

(40.23705%)

(33.613559b)

(36.32669%)

(40.23506%)

(40.23506%)

Value of

Obj. Pin.

4.7627E-03

4.4972E-03

7.1661E-05

7.0619E-05

9.9575E-02

9.9374E-02

1.5306F.-03

I.4992E-03

No.

Iterations

91

1388

108

1603

72

1408

62

1458

3.8926E-03

3.8126E-03

1.1590E-04

1.0592E-04

3.5515E-02

3.5482E-02

1.1088E-03

1.0505E-03

4.8371E-01

4.8118E-01

1.5834E-02

1.5818E-02

1.6943E+00

1.6942E+00

5.5249E-02

5.5244E-02

86

1200

84

1112

219

1223

213

1287

59

1289

59

1707

109

1220

106

1491

b

True value of 1.7188E-04 and starting value of 2.25E-04

d

True value of0.502 and starting value of 1.15

Table A5. Results of Kj-KrT c parameter optimization problem for p = 0.25, Equation 2.

a

p

0.25

Antecedent

Moisture

Drv

Vet

Vers' Wet

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

°-o

Slope

10

15

10

15

10

15

Objective

Function

SI.S

HMLE

SLS

HMLE

SI.S

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

54.938

10.002

47.453

10.086

57.506

10.001

10.039

10.001

69.897

10.000

74.996

46.711

74.590

45.125

10.553

10.485

b

Ki

35.497

74.999

27.869

74.994

72.722

74.988

62.357

74.988

Parameter Estimates (% of Error from True Values) c

Kf

(24.55088%)

(163.15439%)

(2.21404%)

(163.13684%)

(155.16491%)

(163.11579%)

(118.79649%)

(163.11579%)

3.4363E-03

3.3964E-03

3.3736E-03

3.3006E-03

3.7037E-03

3.6908E-03

3.6907E-03

3.6985E-03

(8.03426%)

(9.10210%)

(9.71230%)

(11.66600%)

(0.87783%)

(1.22307%)

(1.22575%)

(1.01699%)

(145.25263%)

(64.91228%)

(163.14386%)

(63.89825%)

(161.71930%)

(58.33333%)

(62.97193%)

(63.21053%)

(92.76491%)

(64.90526%)

(66.50175%)

(64.61053%)

(101.77544%)

(64.90877%)

(64.77544%)

(64.90877%)

3.8553E-03

3.8081E-03

5.4826E-03

4.0727E-03

4.4796E-03

4.4796E-03

5.0611E-03

4.1902E-03

3.6448E-03

3.9508E-03

3.7700E-03

3.9809E-03

3.5576E-03

3.7312E-03

3.7244E-03

3.7290E-03

(3.17945%)

(1.91623%)

(46.73090° i.)

(8.99773%)

(19.88760%)

(19.88760%)

(35.45029%)

(12.14238%)

(2.45417%)

(5.73531%)

(0.89656%)

(6.54088%)

(4.78790%)

(0.14184%)

(0.32383%)

(0.20072%)

Correlation coefficient of sediment concentration; for p = 0, no random error

True value of 28.5 and starting value of 42.5

Values of Ki are multiplied by 10,000

d

tauc

.45495

.46583

.44273

.45167

.50037

.49801

.49338

.49921

.55796

.52588

.73766

.57978

.69323

.66232

.74669

.60827

.49106

.52373

.51987

.53416

.50869

.53455

.53323

.53590

(9.37251%)

(7.20518%)

(11.80677%)

(10.02590%)

(0.32470%)

(0.79482%)

(1.71713%)

(0.55578%)

(11.14741%)

(4.75697%)

(46.94422%)

(15.49402%)

(38.09363%)

(31.93625%)

(48.74303%)

(21.16932%)

(2.17928%)

(4.32869%)

(3.55976%)

(6.40637%)

(1.33267%)

(6.48406%)

(6.22112%)

(6.75299%)

Value of

Obj. Ftn.

1.0394E-01

1 0347E-01

1.5771E-03

1.5164E-03

2.0055E-01

2.0049E-01

3.2153E-03

3.2125E-03

2.1080E-02

2.0206E-02

1.0374E-03

3.3806E-04

7.6516E-02

7.6150E-02

2.6174E-03

1.7600E-03

6.6409E-0I

6.5411E-01

2.1922E-02

2.1644E-02

1.7104E+00

1.6546E+00

5.5101E-02

5.5096E-02

No.

Iterations

137

1348

143

1586

101

1516

104

1512

75

1268

72

1258

83

1308

138

1369

171

1618

153

2029

104

1380

125

2276

b

True value of 3.7365e-03 and starting value of4.25E-03

d

True value of 0.502 and starting value of 1.15

Table A6. Results of Ki-Kric parameter optimization problem for p = 0.25, Equation 3.

a

p

0.25

Antecedent

Moisture

Drv

Wet

Very Wet

Simplex

SCE-UA

Simplex*

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCK-UA

Simplex

SCE-UA

Simplex

SCK-UA

Simplex

SCE-UA

%

Slope

10

15

10

15

10

15

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SI.S

HMIJ£

SLS

HMLE

SLS

HMLE

Correlation coefficient of sediment concentration; for p = 0,no random en-or

True value of 28.5 and starting value of 42.5

Values of Ki are multiplied by 10,000

24.160

18.774

14.743

10.009

61.138

74.999

34.524

74.885

35.337

74.999

54.903

66.811

39.021

10.012

12.381

10.005

b

Ki

31.092

74.998

23.292

74.999

Parameter Estimates (% of Error from True Values)

c

Kf

(9.09474%)

(163.15088%)

(18.27368%)

(163.15439%)

4.1063E-05

4.0594E-05

4.1287E-05

4.0580E-05

(3.47234%)

(2.29054%)

(4.03679%)

(2.25526%)

73.601

40.223

39.608

74.996

(158.24912%)

(41.13333%)

(38,97544%)

(163.14386%)

4.0532E-05

4.0230E-05

4.0101E-05

4.0013E-05

(2.13431%)

(1.37331%)

(1.04826%)

(0.82651%)

(15.22807%)

(34.12632%)

(48.27018%)

(64.88070%)

(114.51930%)

(163.15439%)

(21.13684%)

(162.75439%)

(23.98947%)

(163.15439%)

(92.64211%)

(134.42456%)

(36.91579%)

(64.87018%)

(56.55789%)

(64.89474%)

4.7698E-05

4.7589E-05

4.6577E-05

4.8566E-05

3.8821F.-05

3.9097E-05

3.9262E-05

3.8314E-05

3.9368E-05

3.6758E-05

3.7409E-05

3.6758E-05

4.0147E-05

4.1630E-05

4.3333E-05

4.2518E-05

(20.19151%)

(19.91685%)

(17.36676%)

(22.37873%)

(2.17715%)

(1.48167%)

(1.06589%)

(3.45471%)

(0.79879° o)

(7.37558%)

(5.73516%)

(7.37558%)

(1.16417%)

(4.90110%)

(9.19239%)

(7.13872%)

.60646

.60307

.59533

.59140

.50383

.51510

.50565

.52038

.51416

.48835

.47179

.46069

.49691

.52394

.57851

.55854

d

tauc

.51556

.52733

.51558

.52716

.53039

.51416

.51133

.52225

(2.70120%)

(5.04582%)

(2.70518%)

(5.01195%)

(5.65538%)

(2.42231%)

(1.85857%)

(4.03386%)

(20 80876%)

(20.13347%)

(18.59163 %)

(17.80876%)

(0.36454%)

(2.60956%)

(0.72709%)

(3.66135%)

(2.42231%)

(2.71912%)

(6.01793%)

(8.22908%)

(1.01394%)

(4.37052%)

(15.24104%)

(11.26295%)

2.9156E-02

2.9151E-02

7.8505E-O4

7.2810E-04

1.0137E-01

1.0098E-01

3.2838E-03

3.2105E-03

5.4046E-01

5.3129E-01

1.5363E-02

1.5356E-02

1.5987E+00

1.5719E+00

5.1405E-02

5.0967E-02

Value of

Obj. Ftn.

7.9436E-02

7.8138E-02

1.2600E-03

1.2403E-03

1.8504E-01

1.8481E-01

2.9492E-03

2.935IE-03

No.

Iterations

81

1467

94

1404

116

1355

130

1336

63

1218

69

1286

68

1237

116

1281

149

1353

136

1912

123

1432

139

1677

b

True value of 3.9685E-05 and starting value of 9.00E-05

d True value of0.502 and starting value of 1.15

Table A7. Results of Kj-Kf-T c parameter optimization problem for p = 0.50, Equation 1.

a

p

0.50

Antecedent

Moisture

Do-

Wel

Very Wet

%

Slope

10

15

10

15

10

15

Seardi

Algorithm

Simplex

SCK-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

IIMI.F.

SLS

HMLE

SLS

HMLE

SLS

HMLE

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

* Values of Ki are multiplied by 10.000

16.344

10.003

35.566

10.005

74.939

74.995

72.959

74.999

44.297

40.169

53.102

17.710

46.299

74.990

46.282

74.998

b

Ki

36.745

63.052

42.576

35.562

Parameter Estimates (% of Error from True Values) c

Kf

(28.92982%)

(121.23509%)

(49.38947%)

(24.77895%)

1.6179E-04

1.6685E-04

1.6450E-04

1.6385E-04

1.5342E-04

(5.87037%)

(2.92646°o)

(4.29369%)

(4.67186%)

41.434

75.000

74.921

74.997

(45.38246%)

(163.15789%)

(162.88070%)

(163.14737%)

1.5221E-04

1.4967E-04

1.5086E-04

(10.74005%)

(11.44403%)

(12.92181%)

(12.22946%)

(55.42807%)

(40.94386%)

(86.32281%)

(37.85965%)

(62.45263%)

(163.12281%)

(62.39298%)

(163.15088%)

2.1841E-04

2.1106E-04

2.1689E-04

1 5625E-04

2.7685E-04

3.0821E-04

2.7651E-04

3.0I15E-04

(27.07121%)

(22.79497%)

(26.18687%)

(9.09355%)

(61.07168%)

(79.31697%)

(60.87387%)

(75.20945%)

(42.65263%)

(64.90175%)

(24.79298%)

(64.89474%)

(162.94386%)

(163.14035%)

(155.99649%)

(163.15439%)

1.8867E-04

1.9231E-04

1.6765E-04

1.7539E-04

1.5794E-04

1.5744E-04

1.5924E-04

1.5739E-04

(9.76844%)

(11.88620%)

(2.46102%)

(2.04212%)

(8.11031%)

(8.40121%)

(7.35397%)

(8.43030%)

d

Lauc

.51077

.55765

.51166

.49498

.30160

.33260

.30503

.31943

.70960

.68467

.71786

.44403

.96980

1.05510

.54230

.55606

.34081

.34261

.30992

.30006

.33820

.30009

(1.74701%)

(11.08566%)

(1.92430%)

(1.39841%)

(39.92032%)

(33.74502%)

(39.23705%)

(36.36853%)

(41.35458%)

(36.38845%)

(43.00000%)

(11.54781%)

(93.18725%)

(110.17928%)

.96824 (92.87649%)

1.04020 (107.21116%)

(8.02789%)

(10.76892%)

(32.10956%)

(31.75100%)

(38.26295%)

(40.22709%)

(32.62948%)

(40.22112%)

Value of

Obj. Pin.

6.2797E-03

6.2419E-03

9.7054E-05

9.6933E-05

1.9481E-01

1.3373E-01

2.1424E-03

2.I455E-03

No.

Iterations

104

1357

84

1437

86

1387

94

1354

2.2242E-03

2.2161E-03

7.2536E-03

3.3113E-05

3.0411F.-02

3.0821E-04

9.7577E-04

9.2587E-04

3.2846E-01

3.2768E-01

9.3066E-03

9.2673E-03

2.3084E+00

2.3072E-f00

7.7294E-02

7.6895E-02

93

1529

83

1971

52

1519

60

1569

63

1156

74

1304

83

1192

108

1183

b

True value of 1.7188E-04 and starting value of 2.25E-04

d

TruevalueofO. 502 and starting value of 1.15

Table A8. Results of Kj-Kr-tc parameter optimization problem for p = 0.50, Equation 2.

a

P

0

•Antecedent

Moisture

Dry

Wet

Verv Wet

a

%

Slope

10

15

10

15

10

15

Search

• Ugonthm

Simplex

SCE-UA

Simplex

SCK-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SIS

HMLE

SLS

HMLK

SLS

HMLE

SLS

HMLE

Correlation coefficient of sediment concentration; for p - 0, no random error

c

True value of28.5 and startingvalue of42.5

* Values of Ki are multiplied by 10,000

56.776

62.240

21.058

11.437

74.999

74.999

74.994

74.993

45.021

10.000

44.595

11.437

72.806

10.046

52.210

10.004

h

Ki

36.908

10.002

31.694

10.000

45.408

74.994

10.087

74.998

Parameter Estimates (°o of Error from True Values)

c

Kf

(29.50175°b)

(64.90526° o)

(I 1.20702%)

(64.91228%)

(59.32632%)

(163.13684%)

(64.60702%)

(163.15088%)

3.3446E-03

3.3502E-03

3.3280E-03

3.3494E-03

3.8384E-03

3.8403E-03

4.2688E-03

3.8058E-03

(10.48842%)

(10.33855%)

(10.93269%)

(10.35996%)

(2.72715%)

(2.77800%)

(14.24595%)

(1.854689b)

(57.96842%)

(64.91228%)

(56.47368%)

(59.87018%)

(155.45965%)

(64.75088%)

(83.19298%)

(64.89825%)

(99.21404%)

(118.38596%)

(26.11228%)

(59.87018%)

(163 15439%)

(163 15439%)

(163.13684%)

(163.13333%)

3.6158E-03

3.5362E-03

3.6822E-03

3.6173E-03

3.4363E-03

3.3587E-03

3.4258E-03

3.4043E-03

3.3675E-03

3.3301E-03

3.5798E-03

3.6173E-03

3.1849E-03

3.1845E-03

3.3909E-03

3.1903E-03

(3.23030%)

(5.36063%)

(1.45323%)

(3.19015%)

(8.03426%)

(10.11107%)

(8.31527%)

(8.89067%)

(9.87555%)

(10.87649%)

(4.19376° b)

(3.19015%)

(14.76248%)

(14.77318%)

(9.24930%)

(14.61796°. b)

.41126

.40890

.43650

.46929

.31173

.31147

.31717

.31503

.47828

.44970

.49336

.46928

.43091

.38628

.42601

.40582

d

tauc

.45871

.44389

.44927

.44378

.5581 1

.56736

.62982

.56052

(8.62351%)

(11.57570%)

(10.50398%)

(11.59761%)

(11.17729%)

(13.01992%)

(25.46215%)

(11.65737%)

(4.72510%)

(10.41833%)

(1.72112%)

(6.51793%)

(14.16135%)

(23.05179%)

(15.13745%)

(19.15936%)

(18.07570%)

(18.54582%)

(13.04781%)

(6.51594%)

(37.90239%)

(37.95418%)

(36.81873%)

(37.24502%)

Value of

Obj. Ftn.

6.3853E-02

6.3434E-02

1.0105E-03

1.0068E-03

2.5694E-01

2.5648E-01

4.2773E-03

4.1112E-03

4.7657E-02

4.7355E-02

1.4739E-03

1.4525E-03

8.0968E-02

8.0036E-02

1.6654E-03

1.6151E-03

4.7238E-01

4.7215E-01

1.5423E-02

1.4525E-03

1.2809E+00

1.2809E + 00

4.2692E-02

4.2691E-02

No.

Iterations

127

1479

148

1479

117

1408

121

1544

127

1400

196

1977

130

1419

165

1832

88

1309

83

1377

244

1244

216

1232

b

True value of 3.7365E-03 and starting value of 4.25E-03

d

Truevalueof0.502andsUitingvaIueofl.15

Table A9. Results of K;-Krt c parameter optimization problem for p = 0.50, Equation 3.

a

p

0.25

Antecedent

Moisture

[>>•

Wet

Very Wet

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

%

Slope

10

15

10

15

10

15

Objective

Function

SLS

HMLF.

SI.S

HMLF.

SLS

HMLE

SLS

HMLF

SLS

HMLE

SLS

HMLE

Correlation coefficient of sediment concentration; for p = 0, no random error

True value of 28.5 and starting value of 42.5

Values of Ki are multiplied by 10,000

35.053

74.985

51.563

74.999

63.033

74.986

38.351

34.107

36.040

47.750

69.993

74.999

32 355

10.133

55.641

10.005

b

Ki

25.534

74.993

25.534

74.998

Parameter Estimates (% of EiTor from True Values) e

Kf

(10.40702%)

(163.13333%)

(10.40702%)

(163.15088%)

4.5I60E-05

4.5004E-05

4.5160E-05

4.5072E-05

(13.79614%)

(13.40305%)

(13.79614° o)

(13.57440%)

16.426

10.000

11.403

74.886

(42.36491%)

(64.91228%)

(59.98947%)

(162.75789%)

4.2973E-05

4.3I26E-05

4.2252F.-05

4.1781E-05

(8.28525%)

(8.67078%)

(6.46844%)

(5.28159%)

(22.99298° o)

(163.10526%)

(80.92281%)

(163.15439%)

(121.16842%)

(163.10877%)

(34.56491%)

(19.67368%)

(26.45614%)

(67.54386%)

(145.58947%)

(163.15439%)

(13.52632%)

(64.44561%)

(95.23158%)

(64.89474%)

3.0956E-05

3.0445E-05

3.0285E-05

3.0409E-05

3.7196E-05

3.7256E-05

3.5078E-05

3.3341E-05

4.1786E-05

4.1099E-05

3.9500E-05

3.9330E-05

3.5095E-05

3.5521E-05

3.4498E-05

3.5429E-05

(21.99572%)

(23.28336%)

(23.68653%)

(23.37407%)

(6.27189%)

(6.12070° b)

(11.60892° o)

(15.98589%)

(5.29419%)

(3.56306%)

(0.46617° b)

(0.89454%)

(11.56608%)

(10.49263° b)

(13.07043%)

(10.72446? o)

d

tauc

.56664

.58289

.56664

.58379

.54620

.54622

.53260

.54249

.33811

.34677

.33131

.34601

.46856

.47486

.40717

.36372

.58684

.58399

.57358

.57425

.30167

.30000

.30028

.30014

(12.87649%)

(16.11355%)

(12.87649%)

(16.29283%)

(8.80478%)

(8.80876%)

(6.09562%)

(8.06574%)

(32.64741%)

(30.92231%)

(34.001999o)

(31.07371%)

(6.66135%)

(5.40637%)

(18.89044%)

(27.54582%)

(16.90040%)

(16.33267%)

(14.25896%)

(14.39243%)

(39.90637%)

(40.23904%)

(40.18327%)

(40.21116%)

Value of

Obj. I'til.

6.0723E-02

5.9338E-02

9.6386E-04

9.4023E-04

1.8502E-01

1.8489E-01

2.9236E-03

2.920 5E-03

No.

Iterations

86

1394

86

1564

142

1488

143

1477

4.8774E-02

4.7281E-02

1.5490E-03

1.5252E-03

6.4836E-02

6.4745E-02

1.6669E-03

6.5782E-04

4.8326E-01

4.8301E-01

1.5843E-02

1.5834E-02

1.2555E+00

1.2452E+00

4.0979E-02

4.0209E-02

67

1119

71

1321

113

1119

85

1215

170

1491

164

1976

139

1555

181

1896

b True value of 3.9685E-05 and starting value of 9.00E-05 d True value of0.502 and starting value of 1.15

Table AlO. Results of Kf-x c parameter optimization problem with error-free data, Equation 1.

Parameter Estimates (% of Error from True Values)

a

P

0

Antecedent

Moisture

Dry

Wet

>

Very Wet

°0

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SI.S

HMLE

SLS

HMLE

b

Kf

1.7190E-04

1.7191E-04

1.7191E-04

1.7190E-04

1.7187E-04

I.7I87E-04

1.7186E-04

1.7186E-04

1.7192E-04

1.7190E-04

1.7193E-04

1.7192E-04

1.7191E-04

1.7190E-04

1.7193E-04

1.7190E-04

1.7188E-04

1.7188E-04

1.7I88E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

a

Correlation coefficient of sediment concentration; for p - 0, no random error c True value of0.502 and starting value of 1.15

(0.01164%)

(0.01745%)

(0.01745%)

(0.01164%)

(0.00582%)

(0.00582%)

(0.01164%)

(0.01164%)

(0.02327%)

(0.01164%)

(0.02909° b)

(0.02327%)

(0.01745%)

(0.0U64«?b)

(0.02909%)

(0.01164%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

c

tauc

.50212

.50214

.50213

.50210

.50190

.50190

.50190

.50185

.50200

.50200

.50198

.50199

.50199

.50203

.50203

.50199

.50218

.50217

.50221

.50219

.50223

.50212

.50213

.50219

(0.02390%)

(0.02789%)

(0.02590%)

(0.01992%)

(0.01992° i>)

(0.01992%)

(0.01992%)

(0.02988%)

(0.03586%)

(0.03386%)

(0.04183%)

(0.03785%)

(0.04582%)

(0.02390%)

(0.02590%)

(0.03785%)

(0.00000%)

(0.00000%)

(0.00398%)

(0.00199%)

(0.00199%)

(0.00598%)

(0.00598%)

(0.00199%)

Value of

Obj. Ftn.

5.9222E-08

5.9184E-08

1.4089E-01

9.4199E-10

6.8309E-08

6.8481E-08

I.4354E-01

1.1015E-09

No.

Iterations

96

634

79

542

86

544

78

601

2.7545E-08

2.7556E-08

1.3935E-01

8.8865E-10

2.1481E-08

2.7556E-08

I.2872E-0I

6.0297E-10

114

642

100

585

118

642

140

569

2.3099E-08

2.3326E-08

1.3544E-01

7.7244E-10

2.3099E-08

2.6522E-08

1.4021E-01

8.I667E-I0

92

535

85

563

92

528

74

618

b

True value of 1.7188E-04 and starting value of 2.25E-04

* Values of Ki are multiplied by 10,000

U>

00

Table All. Results of Kf-x c parameter optimization problem with error-free data, Equation 2.

Parameter Estimates (°/o of Error from True Values)

a

P

0

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCF.-UA

Simplex

SCE-lIA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

1IMI.F.

SLS

HMLE

SLS

HMLE

SLS

HMLE

h

Kf

3.7367E-03

3.7370E-03

3.7367E-03

3.7369E-03

3.7364E-03

3.7364E-03

3.7364E-03

3.7364E-03

3.7359E-03

3.7362E-03

3.7361E-03

3.7359E-03

3.7365E-03

3.7363E-03

3.7362E-03

3.7364E-03

3.7365E-03

3.7366E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7363E-03

3.7365E-03

3.7365E-03

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

Tnievalueof0.502andstartingvalueof 1.15

(0.00535%)

(0.01338%)

(0.00535%)

(0.01071%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.01606%)

(0.00803%)

(0.01071%)

(0.01606%)

(0.00000%)

(0.00535%)

(0.00803%)

(0.00268'! i>)

(0.00000%)

(0.00268%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00535%)

(0.00000%)

(0.00000%)

c

taue

.50202

.50199

.50203

.50206

.50199

.50199

.50198

.50198

.50200

.50203

.50202

.50201

.50198

.50187

.50199

.50199

.50191

.50195

.50193

.50191

.50200

.50195

.50193

.50199

(0.00398%)

(0.00199%)

(0.00598%)

(0.01195%)

(0.00199%)

(0.00199%)

(0.00398%)

(0.00398%)

(0.01793%)

(0.00996%)

(0.01394%)

(0.01793%)

(0.00000%)

(0.00996%)

(0.01394%)

(0.00199%)

(0.00000%)

(0.00598%)

(0.00398%)

(0.00199%)

(0.00398%)

(0.02590%)

(0.00199%)

(0.00199%)

1.6010E-08

1.6409E-08

1.2686E-01

5.1506E-10

2.0885E-08

2.0157E-Q8

1.2720E-01

6.4399E-10

2.0885E-08

2.6814E-08

1.3327E-01

7.3187E-10

2.9058E-08

2.1989E-07

1.4101 E-01

1.1814E-09

Value of

Obj. Ftn.

4.7698E-08

6.2210E-08

1.3419E-01

7.6014E-10

6.3I05E-08

6.2210E-08

1.4180E-01

1.0104E-09

No.

Iterations

86

629

84

504

88

629

91

549

85

441

82

498

82

385

98

443

91

514

99

595

85

575

92

510

b

True value of 3.7365E-03 and starting value of 4.25E-03

* Values of Ki are multiplied by 10,000

U>

Table A12. Results of Kf-x c parameter optimization problem with error-free data, Equation 3.

Parameter Estimates (% of Error from True Values)

a

P

0

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Seardi

•Algorithm

Simplex

SCE-l'A

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SI.S

HMI.E

SLS

HMLE

SLS

HMLE

b

Kf

3.9687E-05

3.9685E-05

3.9686E-05

3.9686E-05

3.9684E-05

3.9684E-05

3.9683E-05

3.9682E-05

3.9680E-05

3.9681E-05

3.9681E-05

3.9680E-05

3.9684E-05

3 9684E-05

3.9684E-05

3.9684E-05

3.9685E-05

3.9684E-05

3.9684E-05

3.9686E-05

3.9686E-05

3.9686E-05

3.9686E-05

3.9686E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of0.502 and starting value of 1.15

(0.00504%)

(0.00000%)

(0.00252%)

(0 .00252%)

(0.00252%)

(0.00252%)

(0.00504%)

(0.00756%)

(0.01260%)

(0.01008%)

(0.01008%)

(0.01260%)

(0.00252%)

(0.00252° i>)

(0.00252%)

(0.00252%)

(0.00000%)

(0.00252%)

(0.00252%)

(0.00252%)

(0.00252%)

(0.00252%)

(0.00252%)

(0.00126%)

.50199

.50198

.50199

.50198

.50203

.50204

.50203

.50201

.50194

.50195

.50195

.50194

.50198

.50197

.50196

.50197

c

tauc

.50203

.50200

.50202

.50202

.50197

.50197

.50196

.50192

(0.00598%)

(0.00000%)

(0.00398%)

(0.00398%)

(0.00598%)

(0.00598%)

(0.00797%)

(0.01594%)

(0.01195%)

(0.00996%)

(0.00996%)

(0.01195%)

(0.00398%)

(0.00598%)

(0.00797%)

(0.00598%)

(0.00199%)

(0.00398%)

(0.00199%)

(0.00398%)

(0.00598%)

(0.00797%)

(0.00598%)

(0.00199%)

Value of

Obj. Pin.

5.4527E-08

5.5368E-08

1.3792E-01

8.4665E-10

6.3439E-08

6.4531E-08

1.4218E-01

1.0774E-09

No.

Iterations

121

579

123

563

126

582

148

555

1.9352E-08

1.9477E-08

1.3030E-01

6.2143E-10

2.4184E-08

2.4269E-08

1.3046E-01

6.6268E-10

130

527

143

721

133

590

159

686

2.6225E-08

2.9370E-08

1.3873E-01

5.1408E-09

1.7655E-08

1.9244E-08

1.2934E-01

7.1185E-09

87

448

91

434

93

502

86

538

b

True value of 3.9685E-05 and starting value of 9.00E-05

* Values of Ki are multiplied by 10,000

O

Table A13. Results of Kf-i c parameter optimization problem for p = 0.25, Equation 1.

Parameter Estimates (% of Error from True Values)

a

p

0.25

Antecedent

Moisture

Dry

Wet

Very Wet

°'o

Slope

10

15

10

15

10

15

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Seardi

Algorithm

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

1.8167E-04

1 8101E-04

1.7064E-04

1.7652E-04

1.7391E-04

1.7154E-04

1.8613E-04

1.7310E-04

2.2462E-04

2.2455E-04

2.2950E-04

2.2444E-04

1.9250E-04

1.9181E-04

1.9999E-04

2.0I70E-04

1.6248E-04

1.6I63E-04

1.6251E-04

1.6111E-04

1.6133E-04

1.6131E-04

1.6095E-04

1.6090E-04

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True valueofO. 502 and starting value of 1.15

(5.69583%)

(5.31185%)

(0.72143%)

(2.69956%)

(1.18106%)

(0.19781%)

(8.29067%)

(0.70980%)

(30.68420%)

(30.64347%)

(33.52339%)

(30.57947%)

(11.99849%)

(11.59530°,o)

(16.35443%)

(17.34931%)

(5.46893%)

(5.96346%)

(5,45148%)

(6.26600%)

(6.13800%)

(6.14964%)

(6.35909%)

(6.38818%)

c

tauc

.54394

.54314

.54025

.52483

.52378

.50768

.55826

.52175

.30958

.30000

.33300

.30001

.31966

.31928

.30001

.30002

.67820

.67757

.69084

.67731

.63201

.62840

.67677

.68499

(8 35458%)

(8.19522%)

(7.61952%)

(4.54781%)

(4.33865%)

(1.13147%)

(11.20717%)

(3.93426%)

(35.09960%)

(34.97410%)

(37.61753%)

(34.92231%)

(25.89841%)

(25.17928%)

(34.81474%)

(36.45219%)

(38.33068%)

(40.23904%)

(33.66534%)

(40.23705%)

(36.32271%)

(36.39841%)

(40.23705%)

(40.23506%)

3.8232E-03

3.8224E-03

6.2403E-01

1.2101E-04

3.5487E-02

3.5482E-02

8.2172E-01

1.1195E-03

4.8662E-01

4.8582E-01

1.1287E+00

1.6012E-02

1.7073E

4

00

1.7073E+00

1.3482E+00

5.5809E-02

Value of

Obj. Ftn.

4.5690E-03

4.5620E-03

5.7323E-01

7.0922E-05

9.9754E-02

9.9709E-02

8.8676E-01

1.5454E-03

No.

Iterations

80

578

65

593

39

720

34

505

58

417

48

475

118

474

112

449

57

595

50

729

57

602

36

562

b

True value ofl .7188E-04 and starting value of 2.25E-04

• Values of Ki are multiplied by 10,000

Table A14. Results of Kf-t c parameter optimization problem for p = 0.25, Equation 2.

Parameter Estimates (% of Error from True Values)

a

p

0.25

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMIJi

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.4678E-03

3 8383E-03

3.3171E-03

3.3648E-03

3.7191E-03

3.7082E-03

3.8113E-03

3.7260E-03

3.8053E-03

3.8231E-03

3.9652E-03

3.9703E-03

4.5319E-03

4.4746E-03

4.2132E-03

4.2039E-03

3.8298E-03

3.8305E-03

3.8363E-03

3.8623E-03

3.6614E-03

3.6614E-03

3.6565E-03

3.6594E-03

a

Correlation coefficient ofsediment concentration; for p-0, no random error

c

True value of 0.502 and starting value of 1.15

(7.19122%)

(2.72447%)

(11.22441%)

(9.94781%)

(0.46568%)

(0.75739%)

(2.00187%)

(0.28101%)

(1.84130%)

(2.31768%)

(6.12070%)

(6.25719%)

(21.28730%)

(19.75378%)

(12.75793%)

(12.50903%)

(2.49699%)

(2.51572%)

(2.67095%)

(3.36679%)

(2.00990%)

(2.00990%)

(2.14104%)

(2.06343%)

.51257

.51216

.51391

.52361

.52415

.52407

.52542

.52488

.53361

.53600

.56325

.56337

.66655

.65632

.61720

.61646

c

tauc

.45753

.45290

.43406

.44187

.48945

.48677

50891

.49032

(8.85857%)

(9.78088%)

(13.53386%)

(11.97809%)

(2.50000%)

(3.03386%)

(1.37649%)

(2.32669%)

(6.29681%)

(6.77291%)

(12.20120%)

(12.22510%)

(32.77888%)

(30.74104%)

(22.94821%)

(22.80080%)

(2.10558%)

(2.02390%)

(2.37251%)

(4.30478%)

(4.41235%)

(4.39641%)

(4.66534%)

(4.55777%)

2.0394E-02

2.0387E-02

7.1091E-01

3.1789E-04

7.6247E-02

7.6181E-02

8.6366E-01

1.7681E-03

6.5759E-01

6.5757E-01

1.1742E+00

2.1755E-02

1.6738E+00

1.6738E+00

1.3479E+00

5.5695E-02

Value of

Obj. Ftn.

1.0406E-01

1.0403E-01

8.5087E-01

1.5748E-03

No.

Iterations

2.0151E-01

2.0150E-01

9.1560E-01

3.2158E-03

49

550

45

484

57

550

48

603

54

622

81

641

43

559

58

686

47

527

38

549

85

551

38

508

b

Tme value of 3.7365E-03 and starting value of 4.25E-03

* Values of Ki are multiplied by 10,000

K>

Table A15. Results of Kf-t c parameter optimization problem for p = 0.25, Equation 3.

Parameter Estimates (% of Error from True Values)

a p

0.25

.Antecedent

Moisture

Drv

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-L'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Seardi

.Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HN1LE

SLS

HMJ-E

SLS

I1MLE

SLS

HMLE

SLS

HMI.E

SLS iRni

b

Kf

4.0797E-05

4.1024E-05

4.1480E-05

4.0888E-05

4.0212E-05

4.0275E-05

4.0095E-05

3.6594E-05

4.7908E-05

4.7753E-05

5.4167E-05

4.7421E-05

3.8936E-05

3.9076E-05

4.0185E-05

3.9173E-05

3.9624E-05

3.9983E-05

3.8928E-05

3.9152E-05

4.0848E-05

4.0874E-05

4.1358E-05

4.1647E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

Tnie value of0.502 and starting value of 1.15

(2.80207%

(3.37407%

(4.52312%

(3.03137%

(1.32796%

(1.48671%

(1.03314%

(7.78884° b

(20.72068%

(20.33010%

(36.49238%

(19.49351%

(1.88736%

(1.53458%

(1.25992%

(1.29016%

(0.15371%

(0.75091%

(1.90752%

(1.34308%

(2.93058%

(2.99609%

(4.21570° b

(4.94393%

.51261

.52099

.48441

.48957

.51281

.51337

.52989

.54429

.61025

.60855

.67701

.60765

49480

.49824

.52451

.50067

c

tauc

.51097

.51448

.51953

.51254

.51024

.51133

.50819

.52488

(1.78685%)

(2.48606%)

(3.49203%)

(2.09960%)

(1.64143%)

(1.85857%)

(1.23307%)

(4.55777°

o)

(21.56375%)

(21.22510%)

(34.86255%)

(21.04582%)

(1.43426%)

(0.74900%)

(4.48406%)

(0.26494%)

(2.11355%)

(3.78287%)

(3.50398%)

(2.47610%)

(2.15339%)

(2.26494%)

(5.55578%)

(8.42430%)

Value of

Obj. Rn.

7.9542E-02

7.9503E-02

8.3058E-01

1.2574E-03

1.8483E-01

1.8482E-01

9.0694E-0]

5.5695E-02

No.

Iterations

71

616

72

582

91

674

90

508

2.9160E-02

2.9I62E-02

8.1624E-01

7.9923E-04

1.0240E-01

1.0239E-01

9.1820E-01

3.2925E-03

5.4274E-01

5.4225E-01

1.1216E+00

1.5401E-02

1.5875E+00

1.5875E+00

1.5401E-02

5.1658E-02

47

445

48

540

49

488

540

565

89

678

57

644

95

713

96

735

b

Truevalueof3.9685E-05andstartingvalueof9.00E-05

* Values ofKi are multiplied by 10,000

U>

Table A16. Results of Kf-t c parameter optimization problem for p = 0.50, Equation 1.

Parameter Estimates (% of Error from True Values)

a

p

0.50

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMI.E

SI.S

HMI.E

SLS

HMLF,

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

1.6193E-04

1.6333E-04

1.6313E-04

1.6290E-04

1.5631E-04

1.5469E-04

1.8613E-04

1.5461E-04

2.0599E-04

2.0552E-04

2.0196E-04

1.9049F.-04

2.9280E-04

2.9419E-04

2.9502E-04

2.9879E-04

1.8613E-04

1.8681E-04

I.7115E-04

1.6986E-04

I.6214E-04

1.6220E-04

1.6205E-04

1.6216E-04

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value ofO. 502 and starting value of 1.15

(5.78892%)

(4.97440%)

(5.09076%)

(5.22458%)

(9.05865%)

(10.00116%)

(8.29067%)

(10.04771%)

(19.84524%)

(19.57179%)

(17.50058%)

(10.82732%)

(70.35141%)

(71.16011%)

(71.64301%)

(73.83640%)

(8.29067%)

(8.68629%)

(0.42471%)

(1.17524%)

(5.66674%)

(5.63184%)

(5.71911%)

(5.65511%)

c

tauc

.47280

.47784

.48045

.47747

.31814

.30000

.55826

.30003

.65375

.65215

.64259

.60297

.99852

1.00245

1.00640

1.01295

.55258

.56168

.35622

.34167

.30000

.30004

.30002

.30000

(5.81673%)

(4.81275%)

(4.29283%)

(4.88645%)

(36.62550%)

(40.23904%)

(11.20717%)

(40.23307%)

(30.22908%)

(29.91036%)

(28.00598%)

(20.11355%)

(98.90837%)

(99.69124%)

(100.47809%)

(101.78287%)

(10.07570%)

(11.88845%)

(29.03984%)

(31.93825%)

(40.23904%)

(40.23108%)

(40.23506%)

(40.23904%)

Value of

Obj. Ftn.

6.3024E-03

6.2963E-04

5.9972E-01

9.7285E-05

No.

Iterations

50

593

41

551

61 1.3541E-0!

1.3530E-01

9.0675E-01

2.1792E-03

540

34

627

2.2380E-03

2.2378E-07

5.7251E-OI

7.0034E-05

3.0528E-02

3.0512E-02

8.1319E-01

9.8180E-04

54

521

46

548

39

555

32

657

3.2960E-01

3.2950E-01

1.0238E+00

9.2912E-03

2.4453E+00

2.4453E+00

1.4130E+00

8.1509E-02

67

507

45

497

99

440

105

431

b

Truevalueof 1.7I88E-04andstartingvalueof2.25E-04

* Values of Ki are multiplied by 10,000

Table A17. Results of Kf-x c parameter optimization problem for p = 0.50, Equation 2.

a

p

0.50

Parameter Estimates (% of Error from True Values)

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMI.E

SLS

HMLE

SLS

HMLE

h

Kf

4.4276E-03

4.4237E-03

4.6484E-03

4.4768E-03

3.884 IF.-03

3.8684E-03

3.7488E-03

3.8357E-03

3.5491E-03

3.5478E-03

3.6750E-03

3.6493E-03

3.3608E-03

3.3560E-03

3.4114E-03

3.4132E-03

3.5722E-03

3.5616E-03

3.5568E-03

3.5290E-03

3.3584E-03

3.3584E-03

3.3733E-03

3.3684E-03

Correlation coefficient of sediment concentration; for p = 0, no random error

True value of 0.502 and starting value of 1.15

(18.49592%)

(18.39154°;,)

(24.40519%)

(19.81266%)

(3.95022%)

(3.53004°o)

(0.32919%)

(2.65489%)

(5.01539%)

(5.05018%)

(1.64593%)

(2.33373%)

(10.05486%)

(10.18333%)

(8.70066%)

(8.65248%)

(4.39716%)

(4.68085%)

(4.80931%)

(5.55466%)

(10.11910%)

(10.11910%)

(9.72033%)

(9.85147%)

.44111

.43818

.43612

.42810

.34927

.34924

.35805

.35508

.46085

.45985

.48394

.48191

.39494

.39233

.41439

.41459

c

tauc

.44637

.44650

.48516

.45492

.56052

.55807

.53752

.55163

(11.08167%)

(11.05578%)

(3.35458%)

(9.37849%)

(11.65737%)

(11.16932%)

(7.07570%)

(9.88645%)

(8.19721%)

(8.39641%)

(3.59761%)

(4.00199%)

(21.32669%)

(21.84661%)

(17.45219%)

(17.41235%)

(12.12948%)

(12.71315%)

(13.12351%)

(14.72112%)

(30.42430%)

(30.43028%)

(28.67530%)

(29.26693%)

4.7485E-02

4.7461E-02

8.4498E-01

1.4593E-03

8.0255E-02

8.0170E-02

8.5521E-01

1.6351E-03

4.7407E-01

4.7402E-01

1.1216E+00

1.5390E-02

1.3112E+00

1.3112E+00

1.2993E+00

4.3577E-02

Value of

Obj. Ftn.

6.3662E-02

No.

Iterations

6.3634E-02

8.1897E-01

9.7505E-03

49

547

29

587

2.5719E-01

2.5711E-01

9.3817E-01

3.9360E-03

47

585

38

625

55

56]

49

631

57

580

88

689

50

542

44

546

105

547

100

535

b

True value of 3.7365E-03 and starting value of 4.25E-03

* Values of Ki are multiplied by 10,000

Table A18. Results of Kf-t c parameter optimization problem for p = 0.50, Equation 3.

Parameter Estimates (% of Error from True Values)

a

p

0.50

Antecedent

Moisture

Dry

Wet

Very Wet

°0

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Sinqjlex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SI.S

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

4.5501F.-05

4.5350E-05

4.5303E-05

4.5353E-05

4.3359E-05

4.3021E-05

4.3427E-05

4.2205E-05

3.0474E-05

3.0264E-05

3.0979E-05

3.0348E-05

3.7238E-05

3.7227E-05

3.4558E-05

3.4951E-05

4.2758E-05

4.2452E-05

4.2386E-05

4.2723E-05

3.5147E-05

3.5100E-05

3.5446E-05

3.5069E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

Traevalueof0.502andstartingvalueof 1.15

(14.65541°, o)

(14.27491%)

(14.15648%)

(14.28247- i)

(9.25791%)

(8.40620%)

(9.42926%)

(6.35001%)

(23.21028%)

(23.73945%)

(21.93776%)

(23.52778%)

(6.16606%)

(6.19378%)

(12.91924%)

(11.92894%)

(7.74348%)

(6.97241%)

(6.80610%)

(7.65529%)

(11.43505%)

(11.55348%)

(10.68162° i)

(11.63160%)

.60111

.59451

.60248

.60350

.30162

.30004

.32022

.30007

.32436

.31893

.33485

.32056

.45643

.45723

.38978

.39948

c

tauc

.57114

.56939

.56866

.56943

.55486

.54987

.55684

.53600

(13.77291%)

(13.42430%)

(13.27888%)

(13.43227%)

(10.52988%)

(9.53586%)

(10.92430%)

(6.77291%)

(35.386459b)

(36.46813%)

(33.29681%)

(36.14343%)

(9.07769%)

(8.91833%)

(22.35458%)

(20.42231%)

(19.74303%)

(18.42829%)

(20.01594%)

(20.21912%)

(39.91633%)

(40.23108%)

(36.21116%)

(40.22510%)

Value of

Obj. Ftn.

6.0609E-02

6.0605E-02

8.1064E-01

9.6199E-04

No.

Iterations

77

519

71

586

1.8522E-01

1.8495E-01

5.0675E-01

2.9258E-03

86

601

74

635

4.8998E-02

4.8985E-02

8.5059E-01

1.5703E-03

6.5275E-02

6.5206E-02

8.5943E-01

1.7044E-03

97

653

98

660

76

652

95

650

4.8377E-01

4.8343E-01

1.1279E+00

1.5910E-02

1.2531E+00

1.2518E+00

1.2879E+00

4.0380E-02

50

542

53

507

87

397

52

421

b

True value of3.9685E-05 and starting value of 9.00E-05

* Values of Ki are multiplied by 10,000

Table A19. Results of K;-Kf (x c

= 0.502) parameter optimization problem with error-free data, Equation 1.

Parameter Estimates (% of Error from True Values)

a

P

0

Antecedent

Moisture

Dr>'

Wet

Very Wet

°o

Slope

10

15

10

15

10

15

Search

Algorithm

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCF.-UA

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

1.7189E-04

1.7189E-04

1.7189E-04

1.7189E-04

1.7187E-04

1.7186E-04

1.7186E-04

1.7186E-04

1.7191E-04

1.7191E-04

1.7192F.-04

1.7192E-04

1.7189E-04

1.7189E-04

1.7188E-04

1.7188E-04

1.7187E-04

1.7187E-04

1.7187E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and staitingvalue of 42.5

(0.00582°o

(0.00582%

(0.00582%

(0.00582%

(0.00582%

(0.01164%

(0.01164%

(0.01164%

(0.01745%

(0.01745%

(0.02327%

(0.02327%

(0.00582%

(0.00582%

(0.00000%

(0.00000%

(0.00582%

(0.00582%

(0.00582%

(0.00000%

(0.00000%

(0.00000%

(0.00000° o

(0.00000%

28.515

28.515

28.520

28.511

28.483

28.483

28.482

28.480

28.425

28.425

28.406

28.409

28.420

28.441

28.503

28.499 c

Ki*

28.454

28.471

28.443

28.473

28.607

28.612

28.620

28.624

(0.16140%)

(0.10175%)

(0.20000%)

(0.09474%)

(0.37544%)

(0.39298%)

(0.42105%)

(0.43509%)

(0.26316%)

(0.26316%)

(0.32982%)

(0.31930%)

(0.28070%)

(0.20702%)

(0.01053%)

(0.00351%)

(0.05263%)

(0.05263%)

(0 07018%)

(0.03860%)

(0.05965%)

(0.05965%)

(0.06316%)

(0.07018%)

Value of

Obj. Ftn.

5.9781E-08

5.9640E-08

9.5534E-10

9.4637F.-10

6.7285E-08

6.7266E-08

1.0783E-09

1.0766E-09

No.

Iterations

76

523

88

548

93

603

87

546

2.6467E-08

2.6482E-08

7.2034E-10

7.0814E-10

2.3121E-08

2.2957E-08

6.5906E-10

6.6260E-10

92

483

92

601

122

614

118

602

2.2908E-08

2.2868E-08

7.5964E-10

7.6111E-10

2.3086E-08

2.3396E-08

7.6718E-10

7.6630E-10

91

822

92

505

111

522

110

417

b

True value of 1.7188E-04 and starting value of2.25E-04

• ValuesofKi are multiplied by 10,000

Table A20. Results of Ki-Kf (x c

= 0.502) parameter optimization problem with error-free data, Equation 2.

Parameter Estimates (% of Error from True Values)

a

P

0

Antecedent

Moisture

Dry

Wet

Very Wet

9-o

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Seardi

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-lIA

Simplex

SCE-UA

Objective

Function

SLS

HMIJ-

S1.S

HMLE

SI.S

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.7366E-03

3.7365F.-03

3.7364E-03

3.7364E-03

3.7363E-03

3.7364E-03

3.7363E-03

3.7363E-03

3.7364E-03

3.7364E-03

3.7362E-03

3.7362E-03

3.7364E-03

3.7364F.-03

3.7365E-03

3.7365E-03

3.7364E-03

3.7364E-03

3.7364E-03

3.7364E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of28.5 and starting value of 42.5

(0.00268°o)

(0.00000%)

(0.00268%)

(0.00268%)

(0.00535%)

(0.00268%)

(0.00535%)

(0.00535%)

(0.00268%)

(0.00268%)

(0.00803%)

(0.00803%)

(0.00268%)

(0.00268%)

(0.00000%)

(0.00000%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

28.513

28.521

28.518

28 514

28.493

28.495

28.493

28,494

28.552

28.543

28.607

28.605

28.571

28.574

28.508

28.508

c

Ki*

28.475

28.491

28.529

28.533

28.606

28.588

28.639

28.641

(0.08772%)

(0.03158%)

(0.10175%)

(0.11579%)

(0.37193%)

(0.30877%)

(0.48772%)

(0.49474%)

(0.18246%)

(0.15088%)

(0.37544%)

(0.36842%)

(0.24912%)

(0.25965%)

(0.02807%)

(0.02807%)

(0.04561%)

(0.07368%)

(0.06316%)

(0.04912%)

(0.02456%)

(0.01754%)

(0.02456%)

(0.02105%)

Value of

Obj. Ptn.

4.8187E-08

4.7982E-08

7.3938E-10

7.3957E-10

6.1381E-08

6.1282E-08

9.5570E-10

9.5564E-10

No.

Iterations

99

838

98

532

77

634

131

636

1.7935E-08

1.7906E-08

5.4925E-10

5.4967E-10

2.0533E-08

2.0506E-08

6.30I5E-I0

6.2898E-10

2.0042E-08

2.0002E-08

6.6372E-10

6.6189E-10

2.8579E-08

2.8531E-08

9.4745E-10

9.5404E-10

84

550

100

854

95

667

85

597

101

598

118

525

97

593

119

653

b

True value of 3.7365E-03 and starting value of 4.25E-03

* Values of Ki are multiplied by 10,000

00

Table A21. Results of K;-K f

(x c

= 0.502) parameter optimization problem with error-free data, Equation 3.

Parameter Estimates (°i> of Error from Tnie Values)

a

P

0

/Antecedent

Moisture

Dry

Wei

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-l'A

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-l'A

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.9686E-05

3.9685F.-05

3.9684E-05

3.9684E-05

3.9683E-05

3.9683E-05

3.9683E-05

3.9683E-05

3.9683E-05

3.9683E-05

3.9683E-05

3.9682E-05

3.9685E-05

3.9685E-05

3.9685E-05

3.9685E-05

3.9685E-05

3.9685E-05

3.9685E-05

3.9685E-05

3.9684E-05

3.9685E-05

3.9685E-05

3.9685E-05

a

Correlation coefficient of sediment concentration; for p 0, no random error

c

True value of 28.5 and starting value of 42.5

28.495

28.498

28.500

28.498

28.519

28.524

28.521

28.523

28.532

28.552

28.552

28.573

28.527

28.539

28.504

28.504

c

Ki*

28.472

28.477

28.538

28.540

28.638

28.628

28.640

28.629

(0.00252%)

(0.00000%)

(0.00252%)

(0.00252%)

(0.00504%)

(0.00504%)

(0.00504%)

(0.00504%)

(0.00504%)

(0.00504° b)

(0.00504° o)

(0.00756° b)

(0.00000° o)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000° b)

(0.00252%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.09825%)

(0.08070%)

(0.13333%)

(0.14035%)

(0.48421%)

(0.44912%)

(0.49123%)

(0.45263%)

(0.11228%)

(0.18246%)

(0.18246° b)

(0.25614%)

(0.09474° o)

(0.13684%)

(0.01404%)

(0.01404%)

(0.01754%)

(0.00702%)

(0.00000%)

(0.00702%)

(0.06667%)

(0.08421%)

(0.07368%)

(0.08070%)

Value of

Obj. Ptn.

5.4899E-08

5.4823E-08

8.5125E-10

8.5498E-10

6.2729E-08

6.2719E-08

9.9533E-10

9.9492E-10

No.

Iterations

112

595

89

609

125

632

99

592

2.0513E-08

2.0539E-08

6.4503E-10

6.45I9E-10

2.4453E-08

2.4355E-08

5.7563E-10

5.8939E-10

2.6143E-08

2.6071E-08

8.4940E-10

8.4709E-10

1.7629E-08

1.7547E-08

5.8003E-10

5.8237E-10

112

652

111

644

115

622

112

612

111

526

123

566

107

593

90

627

b

True value of 3.9685E-05 and starting value of 9.00E-05

*

Values of Ki are multiplied by 10,000

Table A22. Results of Kj-Kf (t

c

= 0.502) parameter optimization problem for p = 0.25, Equation 1.

Parameter Estimates (% of Error from True Values)

a

p

0.25

Antecedent

Moisture

Dry

Wet

Very Wet

°b

Slope

10

15

10

15

10

15

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCF.-UA

Simplex

SCF.-UA

Simplex

SCF.-UA

Simplex

SCE-UA

Objective

Function

SLS

HMIJ-

SLS

HM1.E

SI.S

HMLE

SI.S

HMLE

SLS

HMLE

SLS

HMLE

h

Kf

1.7809E-04

1.7862E-04

1.7781E-04

1.7765E-04

1.6590E-04

1.6845E-04

1.6497E-04

1.6400E-04

1.8I49E-04

1.8175E-04

1.8149E-04

1.8181E-04

1.7556E-04

1.7571E-04

1.7556E-04

1.7569E-04

1.6370E-04

1.6167F.-04

1.6213E-04

1.6139E-04

1.7187E-04

I.7185E-04

1.7194E-04

1.7194E-04

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

(3.61299%

(3.92134%

(3.45008%

(3.35699%

(3.47917%

(1.99558® •

(4.02025%

(4.58401%

(5.59111%

(5.74238%

(5.59111%

(5.77729%

(2.14103%

(2.22830%

(2.14103%

(2.21666%

(4.75913%

(5.94019%

(5.67256%

(6.10310%

(0.00582%

(0.01745%

(0.03491%

(0.03491%

10.823

10.000

10.823

10.003

11.251

10.002

11.251

10.000

67.508

74.996

73.278

74.995

10.000

10.000

10.001

10.001

c

Ki*

10.103

10.002

12.244

12.585

65.442

46.275

67.826

74.977

(64.55088%)

(64.90526%)

(57.03860%)

(55.84211%)

(129.62105%)

(62.36842%)

(137.98596%)

(163.07719%)

(62.02456%)

(64.91228%)

(62.02456%)

(64.90175%)

(60.52281%)

(64.90526%)

(60.52281%)

(64.91228%)

(136.87018%)

(163.14386%)

(157.11579%)

(163.14035%)

(64.91228%)

(64.91158%)

(64.90877%)

(64.90877%)

4.1292E-03

4.1248E-03

1.3287E-04

1.3232E-04

3.6136E-02

3.6121E-02

I.1658E-03

1.1653E-03

5.1474E-01

5.1357E-01

1.6614E-02

1.6605E-02

1.7406E+00

1.7406E+00

5.7334E-02

5.7334E-02

Value of

Obj. Fin.

4.5071E-03

4.4975E-03

7.0809E-05

7.0693E-05

No. iterations

40

467

40

560

9.9731E-02

9.9644E+00

1.5270E-03

1.5261E-03

38

501

36

566

58

440

47

472

43

533

43

507

34

478

43

432

134

455

106

499

b

True value of 1.7188E-04 and starting value of 2.25E-04

* Values of Ki are multiplied by 10,000

IS)

O

Table A23. Results of Kj-Kf (x

c

= 0.502) parameter optimization problem for p = 0.25, Equation 2.

Parameter Estimates (% of Error from True Values)

a

p

0.25

Antecedent

Moisture

Dry

Wei

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.6951E-03

3.6210E-03

3.6135E-03

3.6115E-03

3.7338E-03

3.7082E-03

3.7440E-03

3.7106E-03

3.6232E-03

3.6576E-03

3.6527E-03

3.6563E-03

3.7580E-03

3.7605E-03

3.7515E-03

3.7633E-03

3.8225E-03

3.8623E-03

3.8093E-03

3.8275E-03

3.6523E-03

3.6524E-03

3.6470E-03

3.6470E-03

a

Con-elation coefficient ofsediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

(1.10799%)

(3.09113%)

(3.29185%)

(3.34538%)

(0.07226%)

(0.75739%)

(0.20072%)

(0.69316%)

(3.03225%)

(2.11160%)

(2.24274%)

(2.14639%)

(0.57540%)

(0.64231%)

(0.40145%)

(0.71725%)

(2.30162%)

(3.36679%)

(1.94835%)

(2.43543%)

(2.25345%)

(2.25077%)

(2.39529%)

(2.39529%)

23.885

12.204

22.828

18.865

10.003

10.003

10.001

10.003

20.063

10.001

10.614

10.000

11.434

10.003

22.685

10.003

c

Ki*

49.633

74.993

74.979

74.999

55.076

74.997

53.181

74.998

(74.15088%)

(163.13333%)

(163.08421%)

(163.15439%)

(93.24912%)

(163.14737%)

(86.60000%)

(163.15088%)

(29.60351%)

(64.90877%)

(62.75789%)

(64.91228%)

(59.88070%)

(64.90175%)

(20.40351%)

(64.90175%)

(16.19298%)

(57.17895%)

(19.90175%)

(33.80702%)

(64.90175%)

(64.90175%)

(64.90877%)

(64.90175%)

Value of

Obj. Ftn.

1.0528E-01

1.0443E-01

1.5589E-03

1.5587E-03

2.0104E-01

2.0051E-01

3.2I82E-03

3.2125E-03

No.

Iterations

37

462

57

407

35

507

33

528

2.0644E-02

2.0393E-02

6.5504E-04

6.5445E-04

8.8096E-02

8.8030E-02

2.8562E-03

2.8366E-03

6.5806E-01

6.5718E-01

2.183IE-02

2.1825E-02

1.6627E+00

1.6627E f 00

5.5366E-02

5.5366E-02

32

543

31

486

114

454

109

459

34

442

50

475

62

441

35

519

b

True value of 3.7365E-03 and starting value of 4.25E-03

* Values of Ki are multiplied by 10,000

Table A24. Results of Kj-Kf (x

c

= 0.502) parameter optimization problem for p = 0.25, Equation 3.

Parameter Estimates (

0/ o of Error from True Values)

a

p

0.25

Antecedent

Moisture

Dry-

Wet

Very Wet

°0

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMI-E

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

KI

3.8954F.-05

3.8751E-05

3.9046E-05

3.8763E-05

4.0073E-05

4.0109E-05

3.9178E-05

3.9481E-05

4.0037E-05

4.0048E-05

3.9963E-05

4.0074E-05

3.8645E-05

3.8537E-05

3.8645E-05

3.8514E-05

3.7892E-05

3.7225E-05

3.8I37E-05

3.9637E-05

4.0962E-05

4.1035E-05

4.0850E-05

4.0934E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

(1.84201%)

(2.35353%)

(1.61018%)

(2.32330%)

(0.97770%)

(1.06841%)

(1.27756%)

(0.51405%)

(0.88699%)

(0.91470%)

(0.70052%)

(0.98022%)

(2.62064%)

(2.89278%)

(2.62064%)

(2.95074%)

(4.51808%)

(6.19882%)

(3.90072%)

(0.12095%)

(3.21784%)

(3.40179%)

(2.93562%)

(3.14728%)

58.718

74.999

59.820

26.834

14.060

10.001

14.520

10.020

10.214

10.004

13.163

10.009

65.731

74.959

65.731

74.996

c

Ki*

66.690

74.985

67.035

74.998

14.537

10.410

66.445

68.704

(134.00000%)

(163.10526%)

(135.21053%)

(163.15088%)

(48.99298%)

(63.47368%)

(133.14035%)

(141.06667%)

(64.16140%)

(64.89825%)

(53.81404%)

(64.88070%)

(130.63509%)

(163.01404%)

(130.63509%)

(163.14386%)

(106.02807%)

(163.15439%)

(109.89474%)

(5.84561%)

(50.66667%)

(64.90877%)

(49.05263%)

(64.84211%)

Value of

Obj. Ftn.

7.8892E-02

7.8757E-02

I.2500E-03

1.2482E-03

No.

Iterations

42

483

42

498

1.8493E-01

1.8489E-01

2.9480E-03

2.9481E-03

50

504

44

531

3.2584E-02

3.2576E-02

1.0201E-03

1.0472E-03

1.0129E-01

1.0105E-01

3.2328E-03

3.2174E-03

5.3468E-01

5.3224E-01

I.5547E-02

1.5426E-02

1.5784E+00

1.5759E+00

5.1789E-02

5.1709E-02

43

407

45

507

47

498

51

428

60

468

53

461

45

386

45

431

b

Truevaiueof3.9685E-05andstartingvaIueof9.00E-05

* Values of Ki are multiplied by 10,000

Table A25. Results of Kj-Kf(x c

= 0.502) parameter optimization problem for p = 0.50, Equation 1.

Parameter Estimates (% of Error from True Values)

a

p

0.50

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

1.5339E-04

1.6156E-04

1.6518E-04

I.6477E-04

I.7I6IE-04

1.7108E-04

1.7097E-04

1.7106E-04

1.7332E-04

1.7367E-04

1.6945E-04

1.6933E-04

1.7634E-04

1.7663E-04

1.7615E-04

1.7690E-04

1.8463E-04

1.8696E-04

1.8551E-04

1.8771E-04

1.6626E-04

1.6654E-04

1.6647E-04

1.6650E-04

a

Correlation coefficient of sediment concentration; for p = 0, no random error c True value of 28.5 and starting value of 42.5

(10.75751%)

(6.004191b)

(3.89807°o)

(4.13661%)

(0.15709%)

(0.46544%)

(0.52944%)

(0.47708%)

(0.83779%)

(1.04142%)

(1.41378%)

(1.48359%)

(2.59483%)

(2.76356%)

(2.48429%)

(2.92064%)

(7.41797%)

(8.77356%)

(7.92995%)

(9.20991%)

(3.26972%)

(3.10682%)

(3.14754%)

(3.13009%)

c

Ki*

66.097

45.474

36.467

37.035

70.512

74.998

74.778

74.998

11.125

10.018

19.174

19.446

11.828

10.001

12.668

16.799

10.004

17.238

10.007

74.697

74.999

74.999

74.999

74.999

(131.91930%)

(59.55789%)

(27.95439%)

(29.94737%)

(147.41053%)

(163.15088%)

(162.37895%)

(163.15088%)

(60.96491%)

(64.84912%)

(32.72281%)

(31.76842%)

(58.49825%)

(64.90877%)

(55.55088%)

(41.05614%)

(64.89825%)

(39.51579%)

(64.88772%)

(162.09474%)

(163.15439%)

(163.15439%)

(163.15439%)

(163.15439%)

Value of

Obj. Rn.

6.3364E-03

6.2615E-03

3.6963E-05

9.6949E-05

No.

Iterations

38

519

4)

516

I.3662E-0I

1.3628E-01

2.1938E-03

2.1934E-03

38

424

48

489

2.4740E-03

2.4730E-03

6.6505E-05

6.6261E-05

5.4202E-02

5.4085E-02

1.7478E-03

1.7411E-03

3.3064E-01

2.9910E-01

9.5511E-03

9.5473E-03

2.3888E+00

2.3872E+00

7.9543E-02

7.9533E-02

35

479

40

542

61

467

39

452

44

467

46

669

46

462

38

525

b

True value of 1.7188E-04 and starting value of 2.25E-04

* Values of Ki are multiplied by 10,000

Table A26. Results of K;-Kf ( t c

= 0.502) parameter optimization problem for p = 0.50, Equation 2.

Parameter Estimates (% of Error from True Values)

a

p

0.50

.Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

h

Kf

3.6017E-03

3.5633E-03

3 6017E-03

3.5304E-03

3.6232E-03

3.6409E-03

3.5803E-03

3.5438E-03

3.7329E-03

3.7497E-03

3.7115E-03

3.7059E-03

3.7354E-03

3.7043E-03

3.7001E-03

3.7023E-03

3.8228E-03

3 8514E-03

3.8228E-03

3.8517E-03

3.7447E-03

3.7445E-03

3.7494E-03

3.7489E-03

a

Correlation coefficient ofsediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

(3.60765%)

(4.63535%)

(3.60765%)

(5.51586%)

(3.03225%)

(2.55854%)

(4.18038%)

(5.15723%)

(0.09635%)

(0.35327%)

(0.66908%)

(0.81895%)

(0.02944%)

(0.86177%)

(0.97417%)

(0.91530%)

(2.30965%)

(3.07507%)

(2.30965%)

(3.08310%)

(0.21946%)

(0.21410%)

(0.34524%)

(0.33186%)

52.005

49.165

54.000

54.691

54.248

74.992

74.944

74.983

20.014

10.004

20.014

10.004

10.007

10.009

10.002

10.000

c

Ki*

49.166

62.892

49.160

74.984

20.063

10.038

51.375

74.473

(72.51228%)

(120.67368%)

(72.49123%)

(163.10175%)

(29.60351%)

(64.77895%)

(80.26316%)

(161.30877%)

(82.47368%)

(72.50877%)

(89.47368%)

(91.89825%)

(90.34386%)

(163.12982%)

(162.96140%)

(163.09825%)

(29.77544%)

(64.89825%)

(29.77544%)

(64.89825%)

(64.88772%)

(64.88070%)

(64.90526%)

(64.91228%)

4.8016E-02

4.7985E-02

1.4777E-03

1.4771E-03

8.4566E-02

8.2850E-02

2.6136E-03

2.6116E-03

4.9532E-0I

4.9170E-01

1.6321E-02

1.6225E-02

1.4637E+00

1.4637E+00

4.7748E-02

4.7748E-02

Value of

Obj. Ftn.

6.5904E-02

6.5875E-02

1.0425E-03

1.0415E-03

No.

Iterations

2.6089E-01

2.6076E-01

4.1734E-03

4.1700E-03

32

466

31

471

35

454

32

406

34

468

34

463

109

451

103

439

30

380

37

571

37

460

61

444

b

Truevalueof3.7365E-03andstaitingvalueof4.25E«03

* Values of Ki are multiplied by 10,000

Table A27. Results of Ki-Kf (x

c

= 0.502) parameter optimization problem for p = 0.50, Equation 3.

Parameter Estimates (° 'o of Error from True Values)

a

p

0.50

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Sinqilex

SCE-UA

Objective

Function

SLS

HMI.E

SLS

HK1LE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

4.0399E-05

4.0479E-05

4.0342E-05

4.0504E-05

4.0841E-0 5

4.0883E-05

4.0668E-05

4.0766E-05

3.8748E-05

3.8648E-05

3.8885E-05

3.8669E-05

3.8578E-05

3.9377E-05

3.8927E-05

3.8928E-05

3.6902E-05

3.6819E-05

3.7304E-05

3.6824E-05

4.0026E-05

4.0322E-05

4.1101E-05

4.1236E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

(1.79917%)

(2.00076%)

(1.65554%)

(2.06375%)

(2.91294%)

(3.01877%)

(2.47701%)

(2.72395%)

(2.36109%)

(2.61308%)

(2.01588%)

(2.56016%)

(2.78947%)

(0.77611%)

(1.91004%)

(1.90752%)

(7.01273%)

(7.22187%)

(5.99975%)

(7.20927%)

(0.85927%)

(1.60514%)

(3.56810%)

(3.90828%)

74.337

74.991

64.646

74.998

17.046

10.002

13.475

10.001

66.523

74.996

66.360

74.999

65.996

74.998

36.620

36.666

c

Ki*

15.293

10.003

14.442

10.002

14.037

10.006

16.830

10.010

(46.34035%)

(64 90175%)

(49.32632%)

(64.90526%)

(50.74737%)

(64.89123%)

(40.94737%)

(64.87719%)

(133.41404%)

(163.14386%)

(132.84211%)

(163.15439%)

(131.56491%)

(163.15088%)

(28.49123%)

(28.65263%)

(160.83158%)

(163.12632%)

(126.82807%)

(163.15088%)

(40.18947%)

(64.90526%)

(52.71930%)

(64.90877%)

Value of

Obj. Ftn.

6.5937E-02

6.5448E-02

1.0347E-03

1.0334E-03

1.8767E-01

1.8755E-01

2.9552E-03

2.9536E-03

No.

Iterations

46

493

52

510

51

483

50

551

5.4626E-02

5.3711E-02

1.7216E-03

1.7065E-03

6.5304E-02

6.5034E-02

1.9585E-03

1.9385E-03

38

517

43

471

42

453

66

490

5.0894E-0!

5.0852E-01

1.6856E-02

1.6655E-02

1.6888E+00

I.6736E+00

4.9625E-02

4.9437E-01

66

429

46

452

43

396

47

477

b

True value of 3.9685E-05 and starting value of 9.00E-05

* Values ofKi are multiplied by 10,000

Table A28. Results of Kj-Kf (t c

= 0.0) parameter optimization problem with error-free data, Equation 1.

Parameter Estimates (% of Error from True Values)

a

P

0

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Sinqriex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-lIA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

1.7190E-04

1.7189E-04

1.7189E-04

1.7189E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7187E-04

1.7187E-04

1.7189E-04

1.7192E-04

1.7189E-04

1.7189E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7188E-04

1.7I88E-04

1.7188E-04

1.7188E-04

1.7188E-04

a

Correlation coefficient of sediment concentration; for p - 0, no random error

c

True value of 28.5 and starting value of 42.5

(0.01164%)

(0.00756%)

(0.00582%)

(0.00582%)

(0.00000%)

(0.00000» o)

(0.00000%)

(0.00000%)

(0.00582%)

(0.00582%)

(0.00582%)

(0.02327%)

(0.00582%)

(0.00582%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

28.499

28.502

28.497

28.504

28.484

28.484

28.489

28.488

28.516

28.510

28.462

28.445

28.444

28.447

28 513

28.449

c

Ki*

28.451

28.457

28.494

28.485

28 520

28.501

28.503

28.500

(0.17193%)

(0.15088%)

(0.02105%)

(0.05263%)

(0.07018%)

(0.00351%)

(0.01053%)

(0.00000%)

(0.05614%)

(0.03509%)

(0.13333%)

(0.19474%)

(0.19649° o)

(0.18596%)

(0.04561%)

(0.17895%)

(0.00351%)

(0.00702%)

(0.01053%)

(0.01404%)

(0.05614%)

(0.05614%)

(0.03860%)

(0.04211%)

Value of

Obj. Ftn.

4.9920E-08

4.9906E-08

7.8055F.-10

7.7614E-10

5.7422E-08

5.7284E-08

9.1509E-10

9.1660E-10

No.

Iterations

88

755

86

530

102

623

115

536

3.0779E-08

3.0769E-08

7.5819E-10

4.3590E-13

2.6375E-08

2.6302E-08

8.2139E-10

8.2920E-10

83

530

101

631

93

664

109

579

2.6539E-08

2.64I7E-08

8.8433E-10

8.8102E-10

2.7889E-08

2.7925E-08

8.6064E-10

8.6085E-10

93

23

95

683

117

628

113

763

b

True value of 1.7188E-04 and starting value of 2.25E-04

* Values of Ki are multiplied by 10,000

Table A29. Results of K;-Kf ( t c

= 0.0) parameter optimization problem with error-free data, Equation 2.

Parameter Estimates (°o of Error from True Values)

a

P

0

Antecedent

Moisture

Dry

Wet

Very Wet

O-

Slope

10

15

10

15

10

15

Search

Algorithm

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objedive

Function

SLS

HMLE

SLS

HMI-F.

SLS

HV1LE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.7366E-03

3.7366E-03

3.7366E-03

3.7367E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7366E-03

3.7366E-03

3.7364E-03

3.7364E-03

3.7364E-03

3.7366E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

3.7365E-03

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

(0.00268%)

(0.00268%)

(0.00268%)

(0.00535%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.00268%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

(0.00000%)

28.502

28.500

28.502

28.501

28.499

28.531

28.498

28.498

28.487

28.486

28.441

28.439

28.590

28.600

28.630

28.439

c

Ki*

28.445

28.460

28.432

28.395

28.464

28.531

28.527

28.480

(0.19298%)

(0.14035%)

(0.23860%)

(0.36842%)

(0.12632%)

(0.10877%)

(0.09474%)

(0.07018%)

(0.04561%)

(0.04912%)

(0.20702%)

(0.21404%)

(0.31579%)

(0.35088%)

(0.45614%)

(0.21404%)

(0.00702%)

(0.00000%)

(0.00702%)

(0.00351%)

(0.00351%)

(0.10877%)

(0.00702%)

(0.00702%)

b

True value of 3.7365E-03 and starting value of 4.25E-03

* Values of Ki are multiplied by 10,000

Value of

Obj. Ftn.

5.0414E-06

5.0528E-08

7.9930E-10

7.992 IE-10

6.2739E-08

6.2779E-08

I.0131E-09

1.0127E-09

No. herations

108

595

118

550

104

730

93

579

1.8605E-08

1.8633E-08

5.8503E-10

5.7750E-10

2.2040E-08

2.2033E-08

6.5821E-10

5.7750E-10

2.5993E-08

2.5903E-08

8.6620E-10

8.6934E-10

1.7698E-08

6.2779E-08

5.9013E-10

5.8427E-10

103

629

98

563

97

730

87

731

100

708

123

638

115

879

94

638

Table A30. Results of K;-Kf (x c

= 0.0) parameter optimization problem with error-free data, Equation 3.

Parameter Estimates (% of Error from True Values)

a

P

0

Antecedent

Moisture

Do-

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.9687E-05

3.9687E-05

3.9687E-05

3.9687E-05

3.9681E-05

3.9682E-05

3.9675E-05

3.9687E-05

3.9686E-05

3.9686E-05

3.9686E-05

3.9686E-05

3.9681E-05

3.9681E-05

3.9686E-05

3.9681E-05

3.9681E-05

3.9681E-05

3.9681E-05

3.9681E-05

3.9681E-05

3.9681E-05

3.9681E-05

3.9681E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error c Tnievalueof28.5andstaitingvaiueof42.5

(0.00504%)

(0.00504%)

(0.00504%)

(0.00504%)

(0.01008%)

(0,00756%)

(0.02520%)

(0.00504%)

(0.00252%)

(0.00252%)

(0.00252%)

(0.00252%)

(0.01008%)

(0.01008%)

(0.00252%)

(0.01008%)

(0.01008® o)

(0.01008%)

(0.01008%)

(0.01008%)

(0.01008%)

(0.01008%)

(0.01008%)

(0.01008%)

28.504

28.503

28.505

28.502

28.489

28.487

28.487

28.490

28.435

28.451

28.416

28.415

28.474

28.519

28.436

28.515 c

Ki*

28.376

28.376

28.342

28.354

28.929

28.364

29.604

28.354

(0.43509%)

(0.43509%)

(0.55439%)

(0.51228%)

(1.50526%)

(0.47719%)

(3.87368%)

(0.51228%)

(0.22807%)

(0.17193%)

(0.29474%)

(0.29825%)

(0.09123%)

(0.06667%)

(0.22456%)

(0.05263%)

(0.01404%)

(0.01053%)

(0.01754%)

(0.00702%)

(0.03860%)

(0.04561%)

(0.04561%)

(0.03509%)

Value of

Obj. Ftn.

4.7344E-08

4.7416E-08

7.46I8E-10

7.4826E-10

6.8755E-08

6.8649E-08

9.4408E-08

7.4826E-10

No.

Iterations

150

655

161

529

109

694

160

529

3.1767E-08

3.1775E-08

7.4219E-10

7.6982E-10

2.4021E-08

2.3916E-08

7.5292E-10

7.5117E-10

127

794

130

644

141

649

117

616

2.2677E-08

2.2710E-08

7.5292E-10

7.5117E-10

2.4975E-08

2.4995E-08

8.3130E-10

8.3181E-10

125

836

144

531

127

572

114

626

b

True value of 3.9685E-05 and starting value of 9.00E-05

* Values of Ki are multiplied by 10,000 l/k

00

Table A31. Results of K;-K f

(x c

= 0.0) parameter optimization problem for p = 0.25, Equation 1.

Parameter Estimates (% of Error from True Values)

a

p

0.25

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

1.7490E-04

1.7503E-04

1.7463E-04

1.7467E-04

1.6957E-04

1.6907E-04

1.6972E-04

1.6892E-04

1.7396E-04

1.7368E-04

1.7374E-04

1.7401E-04

1.7028E-04

1.7370E-04

1.7018E-04

1.6755E-04

1.6168E-04

1.5935F.-04

1.6303E-04

1.5934E-04

1.7369E-04

1.7359E-04

1.7371E-04

1.7374E-04

a

Correlation coefficient of sediment concentration; for p = 0, no random error c True value of 28.5 and starting value of 42.5

65.024

74.993

62.333

74.995

10.000

10.018

10.001

10.001

10.324

10.001

10.446

10.001

44.803

10.018

47.898

74.607

c

Ki*

10.859

10.003

10.182

10.002

70.160

74.959

67.650

74.993

(1.75704%)

(1.83267%)

(1.59995%)

(1.62323%)

(1.34396%)

(1.63486"! b)

(1.25669%)

(1.72213%)

(1.21015%)

(1.04724%)

(1.08215%)

(1.23924%)

(0.93088%)

(1.05888%)

(0.98906%)

(2.51920%)

(5.93437%)

(7.28997%)

(5.14894%)

(7.29579%)

(1.05306%)

(0.99488%)

(1.06470%)

(1.08040%)

(61.89825%)

(64.90175%)

(64.27368%)

(64.90526%)

(146.17544%)

(163.01404%)

(137.36842%)

(163.13333%)

(63.77544%)

(64.90877%)

(63.3 473 7° o)

(64.90877%)

(57.20351%)

(64.84912%)

(68.06316%)

(161.77895%)

(128.15439%)

(163.13333%)

(118.71228%)

(163.14035%)

(64.91228%)

(64.84912%)

(64.90877%)

(64.90877%)

Value of

Obj. Ftn.

6.0561E-03

6.0492E-03

9.0575E-05

9.0664E-05

1.0677E-01

1.0673E-01

1.6887E-03

1.6876E-03

No.

Iterations

46

442

57

446

39

510

35

510

7.0902E-03

7.0679E-03

2.1399E-04

2.1378E-04

4.5031E-02

4.4585E-02

1.4369E-03

1.4358E-09

4.6901E-01

4.6708E-01

1.5502E-02

1.5398E-02

2.7874E+00

2.7873E+00

9.2654E-02

9.2654E-02

35

432

38

469

104

437

100

422

55

475

54

481

30

472

33

544

b

True value of 1.7188E-04 and starting value of 2.25E-04

* Values of Ki are multiplied by 10,000

Table A32. Results of K;-Kf (x c

= 0.0) parameter optimization problem for p = 0.25, Equation 2.

Parameter Estimates (% of Error from True Values)

a

p

0.25

.Antecedent

Moisture

Dry

Wet

Very Wet

O -

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-l'A

Simplex

SCE-l'A

Simplex

SCE-l'A

Objective

Function

SLS

HMLE

SLS

IIMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.7589E-03

4.9798E-03

3.7589E-03

4.9801E-03

3.7039E-03

3.6855E-03

3.6976E-03

3.6833F.-03

3.6539E-03

3.6504E-03

3.6875E-03

3.6507E-03

3.7186E-03

3.7150E-03

3.7219E-03

3.7165E-03

3.7095E-03

3.6775E-03

3.6739E-03

3.6544E-03

3.6292E-03

3.6288E-03

3.6488E-03

3.6418E-03

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

(0.59949%)

(33.27445%)

(0.59949%)

(33.28248%)

(0.87247%)

(1.36491%)

(1.04108%)

(1.42379%)

(2.21062%)

(2 30430%)

(1.31139%)

(2.29734%)

(0.47906%)

(0.57540%)

(0.39074%)

(0.53526%)

(0.72260%)

(1.57902%)

(1.67536%)

(2.19724%)

(2.87167%)

(2.88238%)

(2.34712%)

(2.53446%)

c

Ki*

51.326

74.945

51.326

74.808

52.403

74.963

56.202

74.777

74.794

74.995

50.493

74.930

72.503

74.995

71 661

75.000

46.694

56.493

50.995

63.494

10000

10.002

10.001

10.006

(80.09123%)

(162.96491%)

(80.09123%)

(162.48421%)

(83.87018%)

(163.02807%)

(97.20000%)

(162.37544%)

(162.43509%)

(163.14035%)

(77.16842%)

(162.91228%)

(154.39649%)

(163.14035%)

(151.44211%)

(163.15789%)

(63.83860%)

(98.22105%)

(78.92982%)

(122.78596%)

(64.91228%)

(64.90526%)

(64.90877%)

(64.89123%)

8.1763E-02

8.1708E-02

2.6591E-03

2.6219E-03

3.0739E-01

3.0703E-01

9.5810E-03

9.5714E-03

9.4521E-01

9.4404E-01

3.1328E-02

3.1262E-02

2.1079E+00

2.1079E+00

6.0593E-02

6.0594E-02

Value of

Obj. Pin.

2.2805E-01

2.2773E-01

3.6199E-03

3.6142E-03

4.7400E-01

4.7361E-01

7.4675E-03

7.4659E-03

No.

Iterations

30

515

31

578

33

479

33

502

32

485

31

585

121

427

107

397

59

527

37

440

56

470

54

451

b

True value of 3.7365K-03 and starting value of 4.25E-03

» Values of Ki are multiplied by 10,000

Table A33. Results of K;-Kf ( t c

= 0.0) parameter optimization problem for p = 0.25, Equation 3.

a

p

0.25

Antecedent

Moisture

Dry

Wet

Verv Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCF.-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Simplex

SCE-llA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

11 MI J

SLS

HMLE

SLS

7

.

HMLE

b

Kf

3.9143E-05

3.8994E-05

3.9069E-05

3.8944E-05

4.0553E-05

4.0660E-05

4.0668E-05

4.0718E-05

3.9970E-05

3.9668E-05

3.9900E-05

3.9709E-05

2.5000E-05

2.5001E-05

2.5000E-05

2.5001E-05

4.0859E-05

4.0524E-05

4.1708E-05

4.1024E-05

3.8554E-05

3.8555E-05

3.8580E-05

3.8688E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

Parameter Estimates (% of Error from True Values)

(1.36576%)

(1.74121%)

(1.55222%)

(1.86720%)

(2.18722%)

(2.45685%)

(2.47701%)

(2.60300%)

(0.71816%)

(0.04284°o)

(0.54177%)

(0.06048° b)

(37.00391%)

(37.00139

^ 0 )

(37.00391%)

(37.00139%)

(2.95830%)

(2.11415%)

(5.09739%)

(3.37407%)

(2.84994%)

(2.84742%)

(2.78443%)

(2.51228° b)

15.003

30.092

15.384

17.038

55.969

55.976

55.021

48.339

60.888

74993

59.797

74.976

14.950

10.028

14.950

10.016

c

Ki*

67.149

74.983

67.759

74.969

16.388

10.016

16.830

10.090

(135.61053%)

(163.09825%)

(137.75088%)

(163.04912%)

(42.49825%)

(64.85614%)

(40.94737%)

(64.59649%)

(113.64211%)

(163.13333%)

(109.81404%)

(163.07368%)

(47.54386%)

(64.81404%)

(47.54386%)

(64.85614%)

(47.35789%)

(5.58596%)

(46.02105%)

(40.21754%)

(96.38246%)

(96.40702%)

(93.05614%)

(69.61053%)

Value of

Obj. Pin.

2.6258E-01

2.6244E-01

4.1093E-03

4.1400E-03

7.1027E-01

7.0945E-01

1.1229E-02

1.1226E-02

No.

Iterations

42

472

44

493

48

481

50

507

1.5650E-01

1.5484E-01

4.8211E-03

4.8027E-03

1.4230E-01

1.3862E-01

4.1192E-03

3.9974E-03

1.2074E+00

1.2047E+00

3.9658E-02

3.9657E-02

1.8356E+00

1.8356E+00

6.0839E-02

6.0588E-02

46

492

55

591

112

554

130

572

37

451

41

529

85

391

85

397

b

True value of 3.9685E-05 and starting value of 9.00E-05

* Values of Ki are multiplied by 10,000

Table A34. Results ofKi-Kf(x

c

= 0.0) parameter optimization problem for p = 0.50, Equation 1.

Parameter Estimates (% of Error from True Values)

a

p

0.50

Antecedent

Moisture

Dry

Wet

Very Wet

0,

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCE-UA

Sinqilex

SCE-lIA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMI.F.

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

1.7670E-04

1.7649E-04

1.7593E-04

1.7588E-04

1.6817E-04

1.6681E-04

1.6842E-04

1.668IE-04

1.7653E-04

1.7790E-04

1.7653E-04

1.8702E-04

1.6813E-04

1.6776E-04

1.6815E-04

1.6810E-04

1.5523E-04

1.5957E-04

1.5379E-04

1.5198E-04

1.7568E-04

1.7566E-04

1.7568E-04

1.7583E-04

a

Correlation coefficient of sediment concentration; for p = 0, no random error c True value of 28.5 and starting value of 42.5

(2.804280 0

(2.68210%

(2.35630%

(2.32721%

(2.15848%

(2.94973%

(2.01303%

(2.94973%

(2.70538%

(3.50244%

(2.70538%

(8.80847%

(2.18175%

(2.39702%

(2.17012°'o

(2.19921%

(9.68699%

(7.16197%

(10.52478%

(11.57785%

(2.21084%

(2.19921%

(2.21084%

(2.29811%

66.690

49.592

67.515

74.985

36.834

37.044

36.834

35.497

17.081

12.911

14.223

14.162

72.494

74.991

65.149

67.166

c

Ki»

10.304

10.003

10.036

10.003

59.888

74.985

58.556

74.998

(63.84561%)

(64.90175%)

(64.78596%)

(64.90175%)

(110.13333%)

(163.10526%)

(105 45965%)

(163.15088%)

(40.06667%)

(54.69825%)

(50.09474%)

(50.30877%)

(154.36491%)

(163.12632%)

(128.59298%)

(135.67018%)

(134.00000%)

(74.00702%)

(136.89474%)

(163.10526%)

(29.24211%)

(29.97895%)

(29.24211%)

(24.55088%)

Value of

Obj. Ftn.

7.9130E-03

7.9040E-03

1.1467E-04

1.1466E-04

No.

Iterations

50

486

49

432

1.3561E-01

I.3531E-01

2.1862E-03

2 1825E-03

38

491

99

497

7.9657E-03

7.7596E-03

1.8704E-04

1.8702E-04

6.9699E-02

6.9643E-02

2.0395E-03

2.0387E-03

3.2048E-01

3.2012E-01

9.6950E-03

9.6756E-03

1.2889E+00

1.2889E+00

4.2962E-02

4.2958E-02

34

469

37

487

123

498

123

525

39

568

48

561

39

487

53

527

b

True value of 1.7188E-04 and starting value of 2.25E-04

• Values ofKi are multiplied by 10,000

Table A35. Results of Kj-Kf (x

c

= 0.0) parameter optimization problem for p = 0.50, Equation 2.

Parameter Estimates (% of Error from True Values)

a

p

0.50

Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Search

Algorithm

Simplex

SCF.-UA

Simplex

SCE-l'A

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

b

Kf

3.7300E-03

3.6936E-03

3.7300E-03

3.7030E-03

3.6745E-03

3.6880E-03

3.6753E-03

3.6851E-03

3.7929E-03

3.8099F.-03

3.7854E-03

3.8069E-03

3.6738E-03

3.6849E-03

3.6731E-03

3.6843E-03

3.9669E-03

3.9574E-03

3.9484E-03

3.9548E-03

3.7501E-03

3.7500E-03

3.7488E-03

3.7491E-03

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of28.5 and starting value of 42.5

(0.17396%)

(1.14813%)

(0.17396%)

(0.89656%)

(1.65931%)

(1.29801%)

(1.63790%)

(1.37562%)

(1.50943%)

(1.96441%)

(1.30871%)

(1.88412%)

(1.67804%)

(1.38097%)

(1.69678%)

(1.39703%)

(6.16620%)

(5.91195%)

(5.67108%)

(5.842379o)

(0.36398%)

(0.36130%)

(0.32919%)

(0.33721%)

10.121

10.003

10.459

10.004

10.000

10.005

10.000

10.003

27.955

10.013

25.333

10.000

23.378

10.023

23.647

10.015

c

Ki*

49.968

74.986

49.968

70.375

23.108

10.005

22.838

10.063

(75.32632%)

(163.10877%)

(75.32632%)

(146.92982%)

(18.91930%)

(64.89474%)

(19.86667%)

(64.69123%)

(1.91228%)

(64.86667%)

(11.11228%)

(64.91228%)

(17.97193%)

(64.83158%)

(17.02807%)

(64.85965%)

(64.48772%)

(64.90175%)

(63.30175%)

(64.89825%)

(64.91228%)

(64.89474%)

(64.91228%)

(64.90175%)

Value of

Obj. Ftn.

2.1134E-01

2.1117E-01

3.2693E-03

3.2691E-03

5.4142E-01

5.4098E-01

8.1919E-03

8.1659E-03

No.

Iterations

32

457

32

361

38

434

39

555

1.0876E-01

1.0804E-01

3.4619E-03

3.4311E-03

1.5185E-01

1.5151E-01

4.8853E-03

4.8650E-03

33

518

38

504

40

511

35

465

1.0311E+00

1.0298E+00

3.4258E-02

3.4128E-02

9.3284E-01

9.3285E-01

3.0920E-02

3.0940E-02

67

441

55

463

114

397

95

443

b

True value of 3.7365E-03 and starting value of 4.25E-03

* Values of Ki are multiplied by 10,000

Table A36. Results of Kj-Kf (

t c

= 0.0) parameter optimization problem for p = 0.50, Equation 3.

a

p

0.50

.Antecedent

Moisture

Dry

Wet

Very Wet

%

Slope

10

15

10

15

10

15

Search

.Algorithm

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Simplex

SCE-UA

Objective

Function

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMLE

SLS

HMJii

SLS

HMLE

b

Kf

3.8820E-05

3.8702E-05

3 8776E-05

3.8650E-05

3.9513E-05

3.9431E-05

3.9513E-05

3.9403E-05

3.9192E-05

3.9257E-05

3.9488E-05

3.9281E-05

2.5000E-05

2.5000E-05

2.5000E-05

2.5000E-05

3.7892E-05

3.7769E-05

3.7934E-05

3.8243E-05

3.9018E-05

3.9013E-05

3.8579E-05

3.8699E-05

a

Correlation coefficient of sediment concentration; for p = 0, no random error

c

True value of 28.5 and starting value of 42.5

Parameter Estimates (% of Error from True Values)

(2.17966%)

(2.47701%)

(2.29054%)

(2.60804° b)

(0.43341%)

(0.64004°o)

(0.43341 °o)

(0.71060%)

(1.24228%)

(1 07849%)

(0.49641%)

(1.01802%)

(37.00391%)

(37.00391%)

(37.00391%)

(37.00391%)

(4.51808%)

(4.82802%)

(4.41225%)

(3.63361%)

(1.68074%)

(1.69334%)

(2.78695%)

(2.48457%)

c

Ki»

65.802

74.931

65.430

74.984

64.561

74.989

64.561

74.983

74.750

74.996

62.083

74.994

14.950

10.002

14.950

10.011

59.086

63.494

61.318

51.443

75.000

74.993

75.000

74.999

(130.88421%)

(162.91579%)

(129.57895° b)

(163.10175%)

(126.52982%)

(163.11930%)

(126.52982%)

(163.09825%)

(162.28070%)

(163.14386%)

(117.83509%)

(163.13684%)

(47.54386%)

(64.90526%)

(47.54386%)

(64.87368%)

(107.31930%)

(122.78596%)

(115.15088%)

(80.50175%)

(163.15789%)

(163.13333%)

(163.15789%)

(163.15439%)

7.0308E-02

7.0157E-02

2.0821E-03

2.0739E-03

2.4882E-01

2.4377E-01

7.2479E-03

7.1411E-03

4.2353E-01

4.2338E-01

1.3682E-02

1.3642E-02

1.2892E+00

1.2893E+00

4.2346E-02

4.2221E-02

Value of

Obj. Ftn.

2.5991E-01

2.5988E-0I

4.0110E-03

4.0096E-03

6.7175E-01

6.7143E-01

1.0148E-02

1.0132E-02

No.

Iterations

44

486

42

472

43

489

43

419

65

424

43

490

85

392

80

420

46

512

45

617

111

468

125

484

b

Tnievalueof3.9685E-05andstartingva!ueof9.00E-05

• Values of Ki are multiplied by 10,000

165

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