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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, USDAARS
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 USDAARS for providing me with rainfall simulator plot data, Nick Mokhotu for providing Kendall watershed data, and
Carolyn Audilet of the BioSciences 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
FlowInduced 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
NONCONVEXITY 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 ERRORFREE DATA CASE:
THE TWOPARAMETER PROBLEM
THE ERRORFREE DATA CASE:
THE THREEPARAMETER 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 FLOWINDUCED 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 FLOWINDUCED 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 3parameter problem and 2parameter
(K rt
) optimization problems
83
Figure 4.8.
Optimization procedures associated with the highest estimation error for each parameter. Results from 2parameter problems (KjKf) 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 SCEUA and Simplex algorithms.
(Equation 2, dry run, 10% slope, HMLE, p = 0.50)
91
Figure 4.12. Comparison of response surfaces for SCEUA 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 2parameter problem
(KjKf)
Error statistics for 2parameter Problem (Krx c
)
56
Error Statistics for 3parameter problem
Optimization procedures associated with highest estimation error for 3parameter optimization problem
(KjKrx c
)
Optimization procedures associated with highest estimation error for 2parameter optimization problem
(Kf
x c
)
Optimization procedures associated with highest estimation error for 2parameter optimization problem (KjKf; t c
= 0.502)
79
81
81
82
Optimization procedures associated with highest estimation error for 2parameter optimization problem
(KjKf; x c
= 0.0)
Average estimation error for different initial moisture conditions for 3parameter optimization problem
Average estimation error for different initial moisture conditions for 2parameter optimization problem (Krt c
)
Average estimation error for different initial moisture conditions for 2parameter optimization problem
(KjKf; x c
= 0.502)
Average estimation error for different initial moisture conditions for 2parameter optimization problem
(KjKf; 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
129164
13
ABSTRACT
Two optimization algorithms and two objective functions were applied to determine erosion parameters for a physicallybased, eventoriented 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 errorfree 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 flowinduced 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, waterbased outdoor recreation, and fisheries.
Thus, the ability to predict soil erosion under current and alternate landuse conditions is important in managing land and water quality.
Physicallybased 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 physicallybased 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 rainfallrunoff 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 empiricallybased 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 physicallybased 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 physicallybased 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 flowinduced 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 steadystate rainfall intensities
3. Evaluate the capabilities of the selected flowinduced models to predict erosion under fully dynamic conditions.
4. Assess the ability of the selected flowinduced 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 flowinduced 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
Errorfree 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 (USDAARS) 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 flowinduced erosion. Two optimization algorithms, one based on a local search procedure (Simplex method) and one specifically designed to find a global minimum
(SCEUA), 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 flowinduced 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 flowinduced 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 rainfallrunoff 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 steadystate 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.
Physicallybased 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 rainfallrunoff 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 physicallybased erosion models. Based on physical laws and theoretical principles, physicallybased 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 physicallybased 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 steadystate assumptions. A kinematicwave 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 microwatershed. 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 physicallybased 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 transportcapacity 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 nonlinear 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
FlowInduced 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 welldefined drainage patterns did not exist. This implies that erosion caused by hydraulic shear may occur even in the absence of a welldefined 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.
Flowinduced 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 flowinduced detachment as a linear function of hydraulic shear, flume studies have shown this relationship to be nonlinear (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 flowinduced 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 excessshear type, and is based on the theoretical assumption that bedload 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 flowinduced 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 DarcyWeisbach 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.
Physicallybased 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 physicallybased 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 physicallybased models (Alberts et al.,
1989; Flanagan, 1991)
Qualitatively, texture can be used as index of erodibility. In general, finetextured soils are usually cohesive and difficult to detach. The small particles of fine textured soils are easy to transport unless the aggregates are large. Coarsetextured soils easily detach, but the large particles are difficult to transport. Mediumtextured 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 flowinduced 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 physicallybased 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 nonhomogenous 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 (nonhomogeneous 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 twostage method for nonlinear 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 twoparameter 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 thresholdtype 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, nonconvexity of the response surface, discontinuous derivatives and presence of multilocal optima.
THE SHUFFLED COMPLEX EVOLUTION 
UNIVERSITY OF ARIZONA (SCEUA) ALGORITHM
Duan et al. (1992) developed a new global optimization procedure called Shuffled
Complex Evolution  University of Arizona (SCEUA). The SCEUA 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 SCEUA 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 SCEUA methods were effective in finding the globally optimal parameters. However, the SCEUA method was found to be three times more efficient. The efficiency of the SCEUA method was confirmed for the Sacramento
42
Soil Moisture Accounting model (SACSMA) (Sorooshian et al., 1993). In this study, the
SCEUA 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 SCEUA procedure and the (local search) Simplex method utilizing a physicallybased 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 SCEUA 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 rainfallrunoff 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 twoparameter model, a long flatbottomed 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
Localtype 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
NONCONVEXITY 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 straightforward. 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 (SMANWSRFS). 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 lognormal 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 sitespecific 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 flowinduced 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 spatiallyvaried, unsteady and onedimensional 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 fourpoint 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 welldefined rills do not develop, the overland flow equations were also used to solve the sediment continuity equation for flowinduced detachment and transport in this study. The sediment continuity equation normally used for onedimensional 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 flowinduced 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 = flowinduced 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 raindropinduced entrainment equation was selected for use in this study.
Notwithstanding the successes of other raindropinduced 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 raindrop induced erosion on forest roads.
If hydraulic shear is considered to be negligible in raindropinduced 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 flowinduced 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 USDAARS 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 VJet
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 twentyfour 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
1.209E01 10.2
34 16.7 14.2 1.209E01 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
1.0175E04
1.0175E04
7.9060E05
8.1
7.5
8.6
66
102
7.8 28.7
15.1 36.0
63.5 Fine Sandy
Loam
48.9 Loam
7.9060E05 8.8
1.0810E04 11.2
105
120
15.1 36.0 48.9 Loam
44.2 33.4 22.4 Clay
1.0810E04 9.8
1.0643E05 11.2
121
Kendall
44.2 33.4 22.4 Clay
62.7 23.0 14.2 Sandy Clay
1.0643E05 11.6
8.1047e06 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 welldefined channel. Thus Kendall
Watershed was selected for this study because it can be modeled as a single plane.
Three rainfallrunoff events from the years 19751977 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 SCEUA 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
NashSutcliffe coefficient (r
2
) was used as a measure of goodnessoffit 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 raindropinduced 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 flowinduced 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
(xx 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 firstorder autoregressive model known as a Markov model (Lipschutz, 1968).
This model assumes that the additive errors are autocorrelated to a lagone by a simple linear relationship given by: e = pe,_
1
+ii t
(3.25) where s t
= additive errors at time /, p = the firstlag 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 errorfree 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 2.62.10) 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 SCEUA was designed to find the global minimum for error surfaces with multiple local optima Both methods required a range of reasonable parameter values.
A twoparameter optimization problem (Krx c
) was posed for the natural data sets, which included the rainfall simulator plots and Kendall Watershed rainfallrunoff events.
Only the most successful combination of flowinduced 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 SCEUA algorithm is based on an extension of the Simplex localsearch algorithm, the stopping criteria are the same However, the SCEUA 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 SCEUA 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 SCEUA methods, the objective function tolerances were set at 0.001. For the SCEUA method in the twoparameter 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 flowinduced 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 flowinduced 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 4.14.6 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 7188E04 5.00E05 4.00E04 2.25E04
t
2
/l°
5 m°
5
3.7365E03
1.00E03 1.00E02 4.25E03
T
2
/L
0005
M
0
'
5
3.9685E05 2.50E05 1.50E04 9.00E05
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
5.00E004 1 00E003
Kf
1.50E003 2.00E003
SLS Criterion
5.00
4.00o
a
3.00
2.00
—12000 
— 91.26 i26J»;
1.00
2.00E004 6 00E004 100E003
Kf
1 40E003
—r
1
•••'• i— c
—
1 80E003
Figure 4.1. K f
x c
Contour Plots for Equation 1.
HMLE Criterion
4.00
3.00
2.00
1.00
4.00E002
Kf
SLS Criterion
5.00
4.00
s
S2
3.00
2.00
1.00
125.00
1.00E002 3 DOE002 4.00E002
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 00E004
Kf
SLS Criterion
4.00E004
5.00'
4.00
3.00
$
2.00—
1.00
128.00
200E004
4 0QE004
Figure 4.3. Kf
x c
Contour Plots for Equation 3.
The response surface configurations for
Ki and
K f
(Figures
4.44.6) 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 ERRORFREE DATA CASE: THE TWOPARAMETER PROBLEM
For the errorfree 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 twoparameter optimization problem, the synthetic errorfree data study demonstrated a 100% success rate for both the Simplex and SCEUA algorithms and for both objective functions in finding the true parameter values for plots with slopes of 10% and 15% for all three flowinduced 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
5.00E004
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 00E002
Kf
3 00E002
SLS Criterion
250.00
177
D)
Q
Q
a
150.00
50.00
2.00E002
Kf
3 00E002
Figure 4.5. Ki  Kf Contour Plots for Equation 2.
72
HMLE Criterion
250.004
200.00j
5 150.00H
100.00J
50.00]
1.00E004 200E004
Kf
3 QOE004
SLS Criterion
250.00J
200.00{
5 150.00)
IOO.OO
H
50.00]
1.00E004
2.00E004 3 00E004
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 errorfree data for slopes of 5%, plots with these values were eliminated from the analysis.
THE ERRORFREE DATA CASE: THE THREEPARAMETER PROBLEM
For the threeparameter, errorfree synthetic data case, only the SCEUA 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 4.14.3) 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 threeparameter 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 A1A36. The following discussion is limited to the cases where error is present in the data, given the success of the errorfree data case for the twoparameter optimization problems. For the analysis of the threeparameter problem when error is present, the errorfree 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, SCEUA and SLS, Simplex and HMLE and SCEUA 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 errorfree data sets and the correlated error cases.
Equation 1 in the 2parameter 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 SCEUA and HMLE and the SCEUA 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 flowinduced 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 twoparameter cases are evaluated separately from that of threeparameters since there were estimation problems associated with two of the four procedures in the errorfree 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 A1A36).
Not only were large errors present in the estimates of Ki for all of the correlated error cases, but with the threeparameter, errorfree data set as well Excluding the errors in estimation for the errorfree, three parameter problem, the average percent error for K; at both levels of correlated error was 92.13% (93 .77% and 90.48% for KjKf 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
KjK f respectively) (see Tables 4.34.5).
Table 4.3. Error Statistics for 2parameter problem (KjKf).
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 4.34.6).
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 2parameter 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 3parameter 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 3parameter problem were consistent with those of the 2parameter 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
KiKf x c problems (see Tables 4.3  4.5). In general, the error associated with the estimates of Ki was slightly higher in the 2parameter case than in the 3parameter 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 3parameter problem is incorporated into all of the parameter estimates.
The average estimation error, however, can be misleading. The abilities of each of the flowinduced 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 twoparameter 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 3parameter optimization problem
(KiKrt c
).
Parameter p
0.25
Ki
K f
Tc
Ki
K f
Tc
0.50
Simplex SCEUA Simplex SCEUA 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 2parameter optimization problem (Kpx c
).
Parameter p
0.25 K f
Xc
K f
T c
0.50
Simplex SCEUA Simplex SCEUA 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 2parameter optimization problem (KiKf; x c
= 0.502).
Parameter p
0.25
Ki
K f
K;
K f
0.50
Simplex SCEUA Simplex SCEUA 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 2parameter optimization problem
(KjKf, x c
=
Parameter
Ki
K f
Ki
K f
P
0.25
0.50
Simplex SCEUA Simplex SCEUA 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
(KiKftc)
Ki (0.25) Kf (0.25) tauc Ki (0.50) Kf(0.50) tauc
(0.25) (0.50) jDSimplex+SLS
83
Optimization Procedures Associated with Highest Estimation Error
(Kftc)
100%
80%
60%
40%
20%
I
Kf (0.25)
m
tauc (0.25) Kf (0.50) tauc (0.50)
Figure 4.7. Optimization procedures associated with the highest estimation error for each parameter. Results from 3parameter problem and 2parameter
(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
(KiKf;
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
(KiKf; 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 (KiKf) shown above for
T
c
=
0.502
and
T
C
= 0.0.
85
of Kf was improved by use of the SCEUA 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 3parameter 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 2parameter problems, where the SCEUA 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 SCEUA 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 errorfree 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 FLOWINDUCED 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.iK 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 2parameter 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 flowinduced equations. For the Krx c case, the SCEUA 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 flowinduced 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 Krx 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 nonstationary 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
KiKf.
However, differences between the flowinduced 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 A1A36 demonstrates that in all of the parameter optimization problems posed when error was present, the SCEUA 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 cratershaped 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>
SCEUA
Surface
Figure 4.11. Comparison of response surfaces for SCEUA and Simplex algorithms.
(Equation 2, diy run, 10% slope, HMLE, p = 0.50).
91
Simplex
Surface
SCEUA
Surface
Figure 4.12. Comparison of response surfaces for SCEUA 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 SCEUA response surface is steeper.
However, contrary to the theoretical expectation, Simplex usually obtained better estimates for the parameters in the 2parameter problems even though SCEUA 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 fourpoint 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 3parameter 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 errorfree 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 errorfree data, the SCEUA algorithm generally did not succeed in finding a lower value of the objective function for the 2parameter problems. This was a result of the SCEUA 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 cratershaped 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 4.104.13 and Figure 4.13), the dry runs provided the best estimates for the parameter x c
, 4 out of 6 times, for the 3parameter problem and, 5 out of 6 times, for the 2parameter 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 3parameter 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 2parameter 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 2parameter 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 2parameter optimization problem
(KjKf; 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 tauc
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 flowinduced 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 FLOWINDUCED 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 flowinduced 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 errorfree 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 SCEUA. 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 SCEUA.
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 flowinduced erosion equation, data from WEPP rainfall simulator plots located in the Western United States were used. A 2parameter
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 USDAARS WEPP team.
ESTIMATION OF HYDRAULIC PARAMETERS
Values of the hydraulic parameters and the NashSutcliffe coefficient appear in
Tables 4.14 and 4.15 and their corresponding hydrographs are in Figures 4.144.23. 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
2.90280e02
2.81550e02
3.7089e02
6.32860e02
2.70770e02
2.70040e02
3.84050e02
4.00000e02
3.82490e02
4.74690e02
3.21660e02
4.90720e02
4.07300e02
4.07330e02
2.81580e02
5.99740e02
5.99220e02
5.12160e02
3.70150e02
5.99220e02
5.89490e02
5.77280e02
5.52580e02
3.73930e02
5.9990 le02
6.67600e02
4.41380e01
5.33810e02
5.52800e02
5.38480e02
3.80480e02
1.01360e02
3.42550e03
1 66690e02
1.50470e01
1.1394e+00
4.21110e02
9.11000e01
2.52970e02
3.01810e01
5.18280e01
1.27120e01
3.39010e02
1.72610e01
1.77970e01
6.46050e02
1.04820e+00
1.43670e01
8.01780e02
1.75689e+00
4.28810e01
5.04840e02
Ks
5.54130e08
2.96770e07
3.34560e06
1.73860e07
3.93120e08
9.92950e08
3.60950e07
1.72180e07
7.51070e08
4.73970e08
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
NashSutcliffe
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
5.0e02
1.0e04
5.0e02
5.0e02
1.0e+00
2.5e01
1.0e+00
2.0e02
Starting Value
4.25e01
1.25e01
1.0e01
1.0e04
106 number from the overland flow at steadystate, 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 NashSutcliffe 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 NashSutcliffe 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 (NashSutcliffe 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
5.6370E02
5.0001E02
6.0261E02
5.0171E02
5.9516E02 x t
4.8227E0I
5.7948E04
3.7531E01
3.0237E01
5.6329E01
6 8658E04
222.855
186.445
9.4761E02
1.0115E01
3.4933E02
8.2727E02
1.5110E02
2.1566E02
1.9131E02
2.7595E02
1.8051E03
2.5757E04
2.5573E02
4.3905E04
5.9127E05
3.6865E04
3.1051E03
1.5909E04
315,178
947,294
1.0976E01
5.0007E02
5.0027E02
5.0063E02
6.4687E02
5.0036E02
2.3611E03
3.3085E01
7.9432E01
6.3875E01
2.0695E03
5.3836E01
3.3447E04 6.4087E04
1.8990E04 6.1951E03
7.1875E04 3.5684E<00
9.I398E04
3.8742E04
3.1719E04
8.7357E03
3.5665E03
3.4180E03 a
True value of Kr is unknown
b
Value of
T
C
5.4042E01
5.4042E01
5.4042E01
4.7452E01
5.2724E01
5.2724E01
5.6473E01
5.8086E01
6.4540E01
5.8086E01
5.0149E01
5.0149E01
4.7641E01
4.3880E01
5.6570E01
6.3427E01
6.5140E01
6.8569E01
6 1712E01
6.I712E0I
6.7510E02
6.7510E02
6.7510E02
6.5820E02
6.7510E02
6.7510E02
% 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
NashSutclifTe
Coefficient
4.50E01
7.70E01
1.60E01
4.10E01
1.37E+00
8.20F.03
1.30E01
7.30E01
6.50E01
8.00E01
3.90E01
9.50F01
3.10E0!
5.90E02
3.80E01
7.20E01
2.60E01
1.70E01
2.90E01
I.20E01
2.20E01
1.30E01
3.10E+00
1.02E + 00
1.70E01
2.20E01
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 reentrain that same particle
119
once it has been deposited on the soil surface. Presently, WESP cannot account for any processes of reentrainment, 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 flowinduced erosion model, and may not be appropriate for describing entrainment by hydraulic shear in broadshallow 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 flowinduced erosion equation, data were used from a small experimental watershed located in the USDAARS Walnut Gulch
Experimental Watershed near Tombstone, Arizona. A 2parameter 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
13Sep75
28Jul76
5Sep76
24.2
18.6
16.8
4.46700e02
2.50060e02
4.63320e02
1.36050e01
1.05400e01
5.23160e03
1.06300e06
165350e06
2.94730e06
Objective NashSutcliffe
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
13Sep75
28Jul76
SSep76
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
.00e13
.00e13
.00e13
Kn
1T j
/L
05
M°
5
]
Upper Bound
1 00e06
1.00e06
1.00e06
Starting Value
1.00e08
1.00e08
1.00e08
Table 4.19 Erosion parameter estimates for selected Kendall Watershed events.
Event Date
13Sep75
28Jul76
5Sep76
Fixed Value ofKi
105000
a
Kf
6.131 le07
4.5648e06
2.0037e07
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
1.3805e12
2.0354e16
2.9381e13
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 errorfree 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 fullydynamic processbased approach, three sediment transport equations describing flowinduced erosion were compared.
Based on the synthetic data analysis, the most successful optimization procedure and flowinduced 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 rainfallrunoff 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 SCEUA in estimating the parameters. This result was contrary to the theoretical expectation since the
SCEUA 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 flowinduced 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 flowinduced 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 flowinduced 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 flowinduced erosion equation that was developed from observations in channels and was not appropriate for describing erosion in broadshallow 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 3parameter 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 flowinduced 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 broadshallow overland flow. A need exists for the development and verification of universal, fundamentally derived equations for relating erosion by hydraulic shear in broadshallow 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 broadshallow 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 errorfree data, Equation 1.
Q
P
0
Antecedent
Moisture
Dry
Wet
Very Wet
a
o
Slope
10
15
10
15
10
15
Search
Algorithm
Simplex
SCEUA
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCHl'A
Simplex
SCF.UA
Simplex
SCHl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
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
1.7194E04
1.7349E04
1.7192E04
I.7186E04
1.7187E04
1.7195E04
1.7187E04
(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%)
1.7189E04
1.7189E04
1.7361E04
1.7191 E04
1.7559E04
1.7191E04
1.7240E04
1.7190E04
1.7183E04
1.7187E04
2.5956E04
1.7187E04
1.7I88E04
1.7188E04
1.7188E04
1.7188E04
(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%)
2.0331E08
2.6321F.08
6.5I69E09
6.8692E10
1.7373E04
2.1479E08
1.1334E08
5.5804F.10
2.2875E08
2.2925E08
3.2388E04
7.6016F.10
2.3046E08
2.3018E08
7.6715E09
7.9847E10
Value of
Obj. Ftn.
5.9059E08
5.8872E08
3.2616E09
9.3542E10
6.7305E08
6.7219E08
1.5863E09
1.0825E06
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.7188E04
and
starting value of 2.25E04
d
True value of 0.502 and starting value of 1.15
Table A2. Results of K; Kfx c parameter optimization problem with errorfree data, Equation 2.
a
P
0
Antecedent
Moisture
Drv
Wet
Very Wet
%
Slope
10
15
10
15
10
15
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
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%)
3.7366E03
3.7366E03
3.7565E03
3.7366E03
3.7362E03
3.7364E03
3.7272E03
3 7364E03
(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%)
3.7359E03
3.7366E03
3.7015E03
3.7356E03
3.7363E03
3.7363E03
3.7231E03
3.7362E03
(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%)
3.7363E03
3.7362E03
3.7363E03
3.7363E03
3.7365E03
3.7365E03
3.7362E03
3.7365E03
(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.
4.7347E08
4.7447E08
2.3846E09
7.1169E10
6.7749E08
6.0946E08
2.0538E09
9.2029E10
1.5559E08
4.7447E08
9.6494E09
4.3201E10
2.0133E08
2.0130E08
3.9103E09
4.4843E10
1.9403E08
1.9763E08
6.4762F.10
6.4908E10
2.8262E08
2.8308E08
1.3702E09
9.4509E10
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 of3.7365E03 and starting value of 4.25E03
d
True value of0.502 and starting value of 1.15
Table A3. Results of Kj Kft c parameter optimization problem with errorfree 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
SCEUA
Simplex
SCF.IA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
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%)
3.9630E05
3.9684E05
4.2010E05
3.9686E05
(0.13859%)
(0.00252%)
(5.85864%)
(0.00252%)
(77.15439%)
(0.53333%)
(163.05263%)
(0.51228%)
3.9561E05
3.9683E05
5.5614E05
3.9684E05
(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%)
4.0136E05
3.9679E05
3.9598E05
3.9677E05
3.9684E05
3.9684F.05
3.9684E05
3.9685E05
3.9684E05
3.9684F.05
3.7152E05
3.9682E05
3.9685E05
3.9686E05
3.9469E05
3.9685E05
(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.
3.3798E06
5.5114E08
1.4855E05
8.2215E10
2.2236E05
6.2682E08
1.2I97E03
9.9707E10
No.
Iterations
137
1329
168
1980
153
1549
161
1775
2.3237E04
1.9111E08
1.9291E09
5.6308E10
2.4085E08
2.4316E08
2.1964E09
6.0122E10
2.5454E08
2.6064E08
4.0288E05
8.6428E10
1.6427E08
1.8361E08
2.6470E06
5.3448E10
218
1260
99
1327
208
1246
255
1347
261
1569
346
2130
268
1415
220
1907
b
True value of 3.9685E05 and starting value of 9.00E05
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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
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%)
1.8566E04
1.7941E04
1.7899E04
1.7515E04
1.7J41E04
1.7287E04
1.7673E04
1.7783E04
(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%)
2.5003E04
2.3265E04
2.4887E04
2.4239E04
1.8989E04
1.9192E04
2.0716F.04
2.2927E04
(60.02807%)
(163.08070%)
(150.59298%)
(163.14035%)
(64.90877%)
(64.90175%)
(64.90526%)
(64.90175%)
1.5670E04
1.4814E04
1.4907E04
1.4759E04
1.6375E04
1.6320E04
1.6264E04
1.6278E04
(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.
4.7627E03
4.4972E03
7.1661E05
7.0619E05
9.9575E02
9.9374E02
1.5306F.03
I.4992E03
No.
Iterations
91
1388
108
1603
72
1408
62
1458
3.8926E03
3.8126E03
1.1590E04
1.0592E04
3.5515E02
3.5482E02
1.1088E03
1.0505E03
4.8371E01
4.8118E01
1.5834E02
1.5818E02
1.6943E+00
1.6942E+00
5.5249E02
5.5244E02
86
1200
84
1112
219
1223
213
1287
59
1289
59
1707
109
1220
106
1491
b
True value of 1.7188E04 and starting value of 2.25E04
d
True value of0.502 and starting value of 1.15
Table A5. Results of KjKrT c parameter optimization problem for p = 0.25, Equation 2.
a
p
0.25
Antecedent
Moisture
Drv
Vet
Vers' Wet
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
°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%)
3.4363E03
3.3964E03
3.3736E03
3.3006E03
3.7037E03
3.6908E03
3.6907E03
3.6985E03
(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%)
3.8553E03
3.8081E03
5.4826E03
4.0727E03
4.4796E03
4.4796E03
5.0611E03
4.1902E03
3.6448E03
3.9508E03
3.7700E03
3.9809E03
3.5576E03
3.7312E03
3.7244E03
3.7290E03
(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.
1.0394E01
1 0347E01
1.5771E03
1.5164E03
2.0055E01
2.0049E01
3.2153E03
3.2125E03
2.1080E02
2.0206E02
1.0374E03
3.3806E04
7.6516E02
7.6150E02
2.6174E03
1.7600E03
6.6409E0I
6.5411E01
2.1922E02
2.1644E02
1.7104E+00
1.6546E+00
5.5101E02
5.5096E02
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 3.7365e03 and starting value of4.25E03
d
True value of 0.502 and starting value of 1.15
Table A6. Results of KiKric parameter optimization problem for p = 0.25, Equation 3.
a
p
0.25
Antecedent
Moisture
Drv
Wet
Very Wet
Simplex
SCEUA
Simplex*
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCKUA
Simplex
SCEUA
Simplex
SCKUA
Simplex
SCEUA
%
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 enor
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%)
4.1063E05
4.0594E05
4.1287E05
4.0580E05
(3.47234%)
(2.29054%)
(4.03679%)
(2.25526%)
73.601
40.223
39.608
74.996
(158.24912%)
(41.13333%)
(38,97544%)
(163.14386%)
4.0532E05
4.0230E05
4.0101E05
4.0013E05
(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%)
4.7698E05
4.7589E05
4.6577E05
4.8566E05
3.8821F.05
3.9097E05
3.9262E05
3.8314E05
3.9368E05
3.6758E05
3.7409E05
3.6758E05
4.0147E05
4.1630E05
4.3333E05
4.2518E05
(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%)
2.9156E02
2.9151E02
7.8505EO4
7.2810E04
1.0137E01
1.0098E01
3.2838E03
3.2105E03
5.4046E01
5.3129E01
1.5363E02
1.5356E02
1.5987E+00
1.5719E+00
5.1405E02
5.0967E02
Value of
Obj. Ftn.
7.9436E02
7.8138E02
1.2600E03
1.2403E03
1.8504E01
1.8481E01
2.9492E03
2.935IE03
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 3.9685E05 and starting value of 9.00E05
d True value of0.502 and starting value of 1.15
Table A7. Results of KjKfT 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
SCKl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
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%)
1.6179E04
1.6685E04
1.6450E04
1.6385E04
1.5342E04
(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%)
1.5221E04
1.4967E04
1.5086E04
(10.74005%)
(11.44403%)
(12.92181%)
(12.22946%)
(55.42807%)
(40.94386%)
(86.32281%)
(37.85965%)
(62.45263%)
(163.12281%)
(62.39298%)
(163.15088%)
2.1841E04
2.1106E04
2.1689E04
1 5625E04
2.7685E04
3.0821E04
2.7651E04
3.0I15E04
(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%)
1.8867E04
1.9231E04
1.6765E04
1.7539E04
1.5794E04
1.5744E04
1.5924E04
1.5739E04
(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.
6.2797E03
6.2419E03
9.7054E05
9.6933E05
1.9481E01
1.3373E01
2.1424E03
2.I455E03
No.
Iterations
104
1357
84
1437
86
1387
94
1354
2.2242E03
2.2161E03
7.2536E03
3.3113E05
3.0411F.02
3.0821E04
9.7577E04
9.2587E04
3.2846E01
3.2768E01
9.3066E03
9.2673E03
2.3084E+00
2.3072Ef00
7.7294E02
7.6895E02
93
1529
83
1971
52
1519
60
1569
63
1156
74
1304
83
1192
108
1183
b
True value of 1.7188E04 and starting value of 2.25E04
d
TruevalueofO. 502 and starting value of 1.15
Table A8. Results of KjKrtc 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
SCEUA
Simplex
SCKl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
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%)
3.3446E03
3.3502E03
3.3280E03
3.3494E03
3.8384E03
3.8403E03
4.2688E03
3.8058E03
(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%)
3.6158E03
3.5362E03
3.6822E03
3.6173E03
3.4363E03
3.3587E03
3.4258E03
3.4043E03
3.3675E03
3.3301E03
3.5798E03
3.6173E03
3.1849E03
3.1845E03
3.3909E03
3.1903E03
(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.
6.3853E02
6.3434E02
1.0105E03
1.0068E03
2.5694E01
2.5648E01
4.2773E03
4.1112E03
4.7657E02
4.7355E02
1.4739E03
1.4525E03
8.0968E02
8.0036E02
1.6654E03
1.6151E03
4.7238E01
4.7215E01
1.5423E02
1.4525E03
1.2809E+00
1.2809E + 00
4.2692E02
4.2691E02
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 3.7365E03 and starting value of 4.25E03
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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
%
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%)
4.5I60E05
4.5004E05
4.5160E05
4.5072E05
(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%)
4.2973E05
4.3I26E05
4.2252F.05
4.1781E05
(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%)
3.0956E05
3.0445E05
3.0285E05
3.0409E05
3.7196E05
3.7256E05
3.5078E05
3.3341E05
4.1786E05
4.1099E05
3.9500E05
3.9330E05
3.5095E05
3.5521E05
3.4498E05
3.5429E05
(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.
6.0723E02
5.9338E02
9.6386E04
9.4023E04
1.8502E01
1.8489E01
2.9236E03
2.920 5E03
No.
Iterations
86
1394
86
1564
142
1488
143
1477
4.8774E02
4.7281E02
1.5490E03
1.5252E03
6.4836E02
6.4745E02
1.6669E03
6.5782E04
4.8326E01
4.8301E01
1.5843E02
1.5834E02
1.2555E+00
1.2452E+00
4.0979E02
4.0209E02
67
1119
71
1321
113
1119
85
1215
170
1491
164
1976
139
1555
181
1896
b True value of 3.9685E05 and starting value of 9.00E05 d True value of0.502 and starting value of 1.15
Table AlO. Results of Kfx c parameter optimization problem with errorfree 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SI.S
HMLE
SLS
HMLE
b
Kf
1.7190E04
1.7191E04
1.7191E04
1.7190E04
1.7187E04
I.7I87E04
1.7186E04
1.7186E04
1.7192E04
1.7190E04
1.7193E04
1.7192E04
1.7191E04
1.7190E04
1.7193E04
1.7190E04
1.7188E04
1.7188E04
1.7I88E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
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.
5.9222E08
5.9184E08
1.4089E01
9.4199E10
6.8309E08
6.8481E08
I.4354E01
1.1015E09
No.
Iterations
96
634
79
542
86
544
78
601
2.7545E08
2.7556E08
1.3935E01
8.8865E10
2.1481E08
2.7556E08
I.2872E0I
6.0297E10
114
642
100
585
118
642
140
569
2.3099E08
2.3326E08
1.3544E01
7.7244E10
2.3099E08
2.6522E08
1.4021E01
8.I667EI0
92
535
85
563
92
528
74
618
b
True value of 1.7188E04 and starting value of 2.25E04
* Values of Ki are multiplied by 10,000
U>
00
Table All. Results of Kfx c parameter optimization problem with errorfree 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCF.UA
Simplex
SCElIA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
1IMI.F.
SLS
HMLE
SLS
HMLE
SLS
HMLE
h
Kf
3.7367E03
3.7370E03
3.7367E03
3.7369E03
3.7364E03
3.7364E03
3.7364E03
3.7364E03
3.7359E03
3.7362E03
3.7361E03
3.7359E03
3.7365E03
3.7363E03
3.7362E03
3.7364E03
3.7365E03
3.7366E03
3.7365E03
3.7365E03
3.7365E03
3.7363E03
3.7365E03
3.7365E03
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%)
1.6010E08
1.6409E08
1.2686E01
5.1506E10
2.0885E08
2.0157EQ8
1.2720E01
6.4399E10
2.0885E08
2.6814E08
1.3327E01
7.3187E10
2.9058E08
2.1989E07
1.4101 E01
1.1814E09
Value of
Obj. Ftn.
4.7698E08
6.2210E08
1.3419E01
7.6014E10
6.3I05E08
6.2210E08
1.4180E01
1.0104E09
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 3.7365E03 and starting value of 4.25E03
* Values of Ki are multiplied by 10,000
U>
V©
Table A12. Results of Kfx c parameter optimization problem with errorfree 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
SCEl'A
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Seardi
•Algorithm
Simplex
SCEl'A
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SI.S
HMI.E
SLS
HMLE
SLS
HMLE
b
Kf
3.9687E05
3.9685E05
3.9686E05
3.9686E05
3.9684E05
3.9684E05
3.9683E05
3.9682E05
3.9680E05
3.9681E05
3.9681E05
3.9680E05
3.9684E05
3 9684E05
3.9684E05
3.9684E05
3.9685E05
3.9684E05
3.9684E05
3.9686E05
3.9686E05
3.9686E05
3.9686E05
3.9686E05
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.
5.4527E08
5.5368E08
1.3792E01
8.4665E10
6.3439E08
6.4531E08
1.4218E01
1.0774E09
No.
Iterations
121
579
123
563
126
582
148
555
1.9352E08
1.9477E08
1.3030E01
6.2143E10
2.4184E08
2.4269E08
1.3046E01
6.6268E10
130
527
143
721
133
590
159
686
2.6225E08
2.9370E08
1.3873E01
5.1408E09
1.7655E08
1.9244E08
1.2934E01
7.1185E09
87
448
91
434
93
502
86
538
b
True value of 3.9685E05 and starting value of 9.00E05
* Values of Ki are multiplied by 10,000
O
Table A13. Results of Kfi 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
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Seardi
Algorithm
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
1.8167E04
1 8101E04
1.7064E04
1.7652E04
1.7391E04
1.7154E04
1.8613E04
1.7310E04
2.2462E04
2.2455E04
2.2950E04
2.2444E04
1.9250E04
1.9181E04
1.9999E04
2.0I70E04
1.6248E04
1.6I63E04
1.6251E04
1.6111E04
1.6133E04
1.6131E04
1.6095E04
1.6090E04
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%)
3.8232E03
3.8224E03
6.2403E01
1.2101E04
3.5487E02
3.5482E02
8.2172E01
1.1195E03
4.8662E01
4.8582E01
1.1287E+00
1.6012E02
1.7073E
4
00
1.7073E+00
1.3482E+00
5.5809E02
Value of
Obj. Ftn.
4.5690E03
4.5620E03
5.7323E01
7.0922E05
9.9754E02
9.9709E02
8.8676E01
1.5454E03
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 .7188E04 and starting value of 2.25E04
• Values of Ki are multiplied by 10,000
Table A14. Results of Kft 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMIJi
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.4678E03
3 8383E03
3.3171E03
3.3648E03
3.7191E03
3.7082E03
3.8113E03
3.7260E03
3.8053E03
3.8231E03
3.9652E03
3.9703E03
4.5319E03
4.4746E03
4.2132E03
4.2039E03
3.8298E03
3.8305E03
3.8363E03
3.8623E03
3.6614E03
3.6614E03
3.6565E03
3.6594E03
a
Correlation coefficient ofsediment concentration; for p0, 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%)
2.0394E02
2.0387E02
7.1091E01
3.1789E04
7.6247E02
7.6181E02
8.6366E01
1.7681E03
6.5759E01
6.5757E01
1.1742E+00
2.1755E02
1.6738E+00
1.6738E+00
1.3479E+00
5.5695E02
Value of
Obj. Ftn.
1.0406E01
1.0403E01
8.5087E01
1.5748E03
No.
Iterations
2.0151E01
2.0150E01
9.1560E01
3.2158E03
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 3.7365E03 and starting value of 4.25E03
* Values of Ki are multiplied by 10,000
K>
Table A15. Results of Kft 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
SCEL'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Seardi
.Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HN1LE
SLS
HMJE
SLS
I1MLE
SLS
HMLE
SLS
HMI.E
SLS iRni
b
Kf
4.0797E05
4.1024E05
4.1480E05
4.0888E05
4.0212E05
4.0275E05
4.0095E05
3.6594E05
4.7908E05
4.7753E05
5.4167E05
4.7421E05
3.8936E05
3.9076E05
4.0185E05
3.9173E05
3.9624E05
3.9983E05
3.8928E05
3.9152E05
4.0848E05
4.0874E05
4.1358E05
4.1647E05
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.
7.9542E02
7.9503E02
8.3058E01
1.2574E03
1.8483E01
1.8482E01
9.0694E0]
5.5695E02
No.
Iterations
71
616
72
582
91
674
90
508
2.9160E02
2.9I62E02
8.1624E01
7.9923E04
1.0240E01
1.0239E01
9.1820E01
3.2925E03
5.4274E01
5.4225E01
1.1216E+00
1.5401E02
1.5875E+00
1.5875E+00
1.5401E02
5.1658E02
47
445
48
540
49
488
540
565
89
678
57
644
95
713
96
735
b
Truevalueof3.9685E05andstartingvalueof9.00E05
* Values ofKi are multiplied by 10,000
U>
Table A16. Results of Kft 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMI.E
SI.S
HMI.E
SLS
HMLF,
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
1.6193E04
1.6333E04
1.6313E04
1.6290E04
1.5631E04
1.5469E04
1.8613E04
1.5461E04
2.0599E04
2.0552E04
2.0196E04
1.9049F.04
2.9280E04
2.9419E04
2.9502E04
2.9879E04
1.8613E04
1.8681E04
I.7115E04
1.6986E04
I.6214E04
1.6220E04
1.6205E04
1.6216E04
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.
6.3024E03
6.2963E04
5.9972E01
9.7285E05
No.
Iterations
50
593
41
551
61 1.3541E0!
1.3530E01
9.0675E01
2.1792E03
540
34
627
2.2380E03
2.2378E07
5.7251EOI
7.0034E05
3.0528E02
3.0512E02
8.1319E01
9.8180E04
54
521
46
548
39
555
32
657
3.2960E01
3.2950E01
1.0238E+00
9.2912E03
2.4453E+00
2.4453E+00
1.4130E+00
8.1509E02
67
507
45
497
99
440
105
431
b
Truevalueof 1.7I88E04andstartingvalueof2.25E04
* Values of Ki are multiplied by 10,000
Table A17. Results of Kfx 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMI.E
SLS
HMLE
SLS
HMLE
h
Kf
4.4276E03
4.4237E03
4.6484E03
4.4768E03
3.884 IF.03
3.8684E03
3.7488E03
3.8357E03
3.5491E03
3.5478E03
3.6750E03
3.6493E03
3.3608E03
3.3560E03
3.4114E03
3.4132E03
3.5722E03
3.5616E03
3.5568E03
3.5290E03
3.3584E03
3.3584E03
3.3733E03
3.3684E03
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%)
4.7485E02
4.7461E02
8.4498E01
1.4593E03
8.0255E02
8.0170E02
8.5521E01
1.6351E03
4.7407E01
4.7402E01
1.1216E+00
1.5390E02
1.3112E+00
1.3112E+00
1.2993E+00
4.3577E02
Value of
Obj. Ftn.
6.3662E02
No.
Iterations
6.3634E02
8.1897E01
9.7505E03
49
547
29
587
2.5719E01
2.5711E01
9.3817E01
3.9360E03
47
585
38
625
55
56]
49
631
57
580
88
689
50
542
44
546
105
547
100
535
b
True value of 3.7365E03 and starting value of 4.25E03
* Values of Ki are multiplied by 10,000
Table A18. Results of Kft 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
SCEUA
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Sinqjlex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEUA
Objective
Function
SLS
HMLE
SI.S
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
4.5501F.05
4.5350E05
4.5303E05
4.5353E05
4.3359E05
4.3021E05
4.3427E05
4.2205E05
3.0474E05
3.0264E05
3.0979E05
3.0348E05
3.7238E05
3.7227E05
3.4558E05
3.4951E05
4.2758E05
4.2452E05
4.2386E05
4.2723E05
3.5147E05
3.5100E05
3.5446E05
3.5069E05
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.
6.0609E02
6.0605E02
8.1064E01
9.6199E04
No.
Iterations
77
519
71
586
1.8522E01
1.8495E01
5.0675E01
2.9258E03
86
601
74
635
4.8998E02
4.8985E02
8.5059E01
1.5703E03
6.5275E02
6.5206E02
8.5943E01
1.7044E03
97
653
98
660
76
652
95
650
4.8377E01
4.8343E01
1.1279E+00
1.5910E02
1.2531E+00
1.2518E+00
1.2879E+00
4.0380E02
50
542
53
507
87
397
52
421
b
True value of3.9685E05 and starting value of 9.00E05
* Values of Ki are multiplied by 10,000
Table A19. Results of K;Kf (x c
= 0.502) parameter optimization problem with errorfree 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
SCEl'A
Simplex
SCEUA
Simplex
SCF.UA
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
1.7189E04
1.7189E04
1.7189E04
1.7189E04
1.7187E04
1.7186E04
1.7186E04
1.7186E04
1.7191E04
1.7191E04
1.7192F.04
1.7192E04
1.7189E04
1.7189E04
1.7188E04
1.7188E04
1.7187E04
1.7187E04
1.7187E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
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.
5.9781E08
5.9640E08
9.5534E10
9.4637F.10
6.7285E08
6.7266E08
1.0783E09
1.0766E09
No.
Iterations
76
523
88
548
93
603
87
546
2.6467E08
2.6482E08
7.2034E10
7.0814E10
2.3121E08
2.2957E08
6.5906E10
6.6260E10
92
483
92
601
122
614
118
602
2.2908E08
2.2868E08
7.5964E10
7.6111E10
2.3086E08
2.3396E08
7.6718E10
7.6630E10
91
822
92
505
111
522
110
417
b
True value of 1.7188E04 and starting value of2.25E04
• ValuesofKi are multiplied by 10,000
Table A20. Results of KiKf (x c
= 0.502) parameter optimization problem with errorfree data, Equation 2.
Parameter Estimates (% of Error from True Values)
a
P
0
Antecedent
Moisture
Dry
Wet
Very Wet
9o
Slope
10
15
10
15
10
15
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Seardi
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCElIA
Simplex
SCEUA
Objective
Function
SLS
HMIJ
S1.S
HMLE
SI.S
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.7366E03
3.7365F.03
3.7364E03
3.7364E03
3.7363E03
3.7364E03
3.7363E03
3.7363E03
3.7364E03
3.7364E03
3.7362E03
3.7362E03
3.7364E03
3.7364F.03
3.7365E03
3.7365E03
3.7364E03
3.7364E03
3.7364E03
3.7364E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
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.
4.8187E08
4.7982E08
7.3938E10
7.3957E10
6.1381E08
6.1282E08
9.5570E10
9.5564E10
No.
Iterations
99
838
98
532
77
634
131
636
1.7935E08
1.7906E08
5.4925E10
5.4967E10
2.0533E08
2.0506E08
6.30I5EI0
6.2898E10
2.0042E08
2.0002E08
6.6372E10
6.6189E10
2.8579E08
2.8531E08
9.4745E10
9.5404E10
84
550
100
854
95
667
85
597
101
598
118
525
97
593
119
653
b
True value of 3.7365E03 and starting value of 4.25E03
* Values of Ki are multiplied by 10,000
00
Table A21. Results of K;K f
(x c
= 0.502) parameter optimization problem with errorfree 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
SCEUA
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEl'A
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEl'A
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.9686E05
3.9685F.05
3.9684E05
3.9684E05
3.9683E05
3.9683E05
3.9683E05
3.9683E05
3.9683E05
3.9683E05
3.9683E05
3.9682E05
3.9685E05
3.9685E05
3.9685E05
3.9685E05
3.9685E05
3.9685E05
3.9685E05
3.9685E05
3.9684E05
3.9685E05
3.9685E05
3.9685E05
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.
5.4899E08
5.4823E08
8.5125E10
8.5498E10
6.2729E08
6.2719E08
9.9533E10
9.9492E10
No.
Iterations
112
595
89
609
125
632
99
592
2.0513E08
2.0539E08
6.4503E10
6.45I9E10
2.4453E08
2.4355E08
5.7563E10
5.8939E10
2.6143E08
2.6071E08
8.4940E10
8.4709E10
1.7629E08
1.7547E08
5.8003E10
5.8237E10
112
652
111
644
115
622
112
612
111
526
123
566
107
593
90
627
b
True value of 3.9685E05 and starting value of 9.00E05
*
Values of Ki are multiplied by 10,000
Table A22. Results of KjKf (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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCF.UA
Simplex
SCF.UA
Simplex
SCF.UA
Simplex
SCEUA
Objective
Function
SLS
HMIJ
SLS
HM1.E
SI.S
HMLE
SI.S
HMLE
SLS
HMLE
SLS
HMLE
h
Kf
1.7809E04
1.7862E04
1.7781E04
1.7765E04
1.6590E04
1.6845E04
1.6497E04
1.6400E04
1.8I49E04
1.8175E04
1.8149E04
1.8181E04
1.7556E04
1.7571E04
1.7556E04
1.7569E04
1.6370E04
1.6167F.04
1.6213E04
1.6139E04
1.7187E04
I.7185E04
1.7194E04
1.7194E04
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%)
4.1292E03
4.1248E03
1.3287E04
1.3232E04
3.6136E02
3.6121E02
I.1658E03
1.1653E03
5.1474E01
5.1357E01
1.6614E02
1.6605E02
1.7406E+00
1.7406E+00
5.7334E02
5.7334E02
Value of
Obj. Fin.
4.5071E03
4.4975E03
7.0809E05
7.0693E05
No. iterations
40
467
40
560
9.9731E02
9.9644E+00
1.5270E03
1.5261E03
38
501
36
566
58
440
47
472
43
533
43
507
34
478
43
432
134
455
106
499
b
True value of 1.7188E04 and starting value of 2.25E04
* Values of Ki are multiplied by 10,000
IS)
O
Table A23. Results of KjKf (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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.6951E03
3.6210E03
3.6135E03
3.6115E03
3.7338E03
3.7082E03
3.7440E03
3.7106E03
3.6232E03
3.6576E03
3.6527E03
3.6563E03
3.7580E03
3.7605E03
3.7515E03
3.7633E03
3.8225E03
3.8623E03
3.8093E03
3.8275E03
3.6523E03
3.6524E03
3.6470E03
3.6470E03
a
Conelation 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.
1.0528E01
1.0443E01
1.5589E03
1.5587E03
2.0104E01
2.0051E01
3.2I82E03
3.2125E03
No.
Iterations
37
462
57
407
35
507
33
528
2.0644E02
2.0393E02
6.5504E04
6.5445E04
8.8096E02
8.8030E02
2.8562E03
2.8366E03
6.5806E01
6.5718E01
2.183IE02
2.1825E02
1.6627E+00
1.6627E f 00
5.5366E02
5.5366E02
32
543
31
486
114
454
109
459
34
442
50
475
62
441
35
519
b
True value of 3.7365E03 and starting value of 4.25E03
* Values of Ki are multiplied by 10,000
Table A24. Results of KjKf (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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMIE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
KI
3.8954F.05
3.8751E05
3.9046E05
3.8763E05
4.0073E05
4.0109E05
3.9178E05
3.9481E05
4.0037E05
4.0048E05
3.9963E05
4.0074E05
3.8645E05
3.8537E05
3.8645E05
3.8514E05
3.7892E05
3.7225E05
3.8I37E05
3.9637E05
4.0962E05
4.1035E05
4.0850E05
4.0934E05
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.
7.8892E02
7.8757E02
I.2500E03
1.2482E03
No.
Iterations
42
483
42
498
1.8493E01
1.8489E01
2.9480E03
2.9481E03
50
504
44
531
3.2584E02
3.2576E02
1.0201E03
1.0472E03
1.0129E01
1.0105E01
3.2328E03
3.2174E03
5.3468E01
5.3224E01
I.5547E02
1.5426E02
1.5784E+00
1.5759E+00
5.1789E02
5.1709E02
43
407
45
507
47
498
51
428
60
468
53
461
45
386
45
431
b
Truevaiueof3.9685E05andstartingvaIueof9.00E05
* Values of Ki are multiplied by 10,000
Table A25. Results of KjKf(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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
1.5339E04
1.6156E04
1.6518E04
I.6477E04
I.7I6IE04
1.7108E04
1.7097E04
1.7106E04
1.7332E04
1.7367E04
1.6945E04
1.6933E04
1.7634E04
1.7663E04
1.7615E04
1.7690E04
1.8463E04
1.8696E04
1.8551E04
1.8771E04
1.6626E04
1.6654E04
1.6647E04
1.6650E04
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.
6.3364E03
6.2615E03
3.6963E05
9.6949E05
No.
Iterations
38
519
4)
516
I.3662E0I
1.3628E01
2.1938E03
2.1934E03
38
424
48
489
2.4740E03
2.4730E03
6.6505E05
6.6261E05
5.4202E02
5.4085E02
1.7478E03
1.7411E03
3.3064E01
2.9910E01
9.5511E03
9.5473E03
2.3888E+00
2.3872E+00
7.9543E02
7.9533E02
35
479
40
542
61
467
39
452
44
467
46
669
46
462
38
525
b
True value of 1.7188E04 and starting value of 2.25E04
* 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
h
Kf
3.6017E03
3.5633E03
3 6017E03
3.5304E03
3.6232E03
3.6409E03
3.5803E03
3.5438E03
3.7329E03
3.7497E03
3.7115E03
3.7059E03
3.7354E03
3.7043E03
3.7001E03
3.7023E03
3.8228E03
3 8514E03
3.8228E03
3.8517E03
3.7447E03
3.7445E03
3.7494E03
3.7489E03
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%)
4.8016E02
4.7985E02
1.4777E03
1.4771E03
8.4566E02
8.2850E02
2.6136E03
2.6116E03
4.9532E0I
4.9170E01
1.6321E02
1.6225E02
1.4637E+00
1.4637E+00
4.7748E02
4.7748E02
Value of
Obj. Ftn.
6.5904E02
6.5875E02
1.0425E03
1.0415E03
No.
Iterations
2.6089E01
2.6076E01
4.1734E03
4.1700E03
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
Truevalueof3.7365E03andstaitingvalueof4.25E«03
* Values of Ki are multiplied by 10,000
Table A27. Results of KiKf (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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Sinqilex
SCEUA
Objective
Function
SLS
HMI.E
SLS
HK1LE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
4.0399E05
4.0479E05
4.0342E05
4.0504E05
4.0841E0 5
4.0883E05
4.0668E05
4.0766E05
3.8748E05
3.8648E05
3.8885E05
3.8669E05
3.8578E05
3.9377E05
3.8927E05
3.8928E05
3.6902E05
3.6819E05
3.7304E05
3.6824E05
4.0026E05
4.0322E05
4.1101E05
4.1236E05
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.
6.5937E02
6.5448E02
1.0347E03
1.0334E03
1.8767E01
1.8755E01
2.9552E03
2.9536E03
No.
Iterations
46
493
52
510
51
483
50
551
5.4626E02
5.3711E02
1.7216E03
1.7065E03
6.5304E02
6.5034E02
1.9585E03
1.9385E03
38
517
43
471
42
453
66
490
5.0894E0!
5.0852E01
1.6856E02
1.6655E02
1.6888E+00
I.6736E+00
4.9625E02
4.9437E01
66
429
46
452
43
396
47
477
b
True value of 3.9685E05 and starting value of 9.00E05
* Values ofKi are multiplied by 10,000
Table A28. Results of KjKf (t c
= 0.0) parameter optimization problem with errorfree 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Sinqriex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCElIA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
1.7190E04
1.7189E04
1.7189E04
1.7189E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
1.7187E04
1.7187E04
1.7189E04
1.7192E04
1.7189E04
1.7189E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
1.7188E04
1.7I88E04
1.7188E04
1.7188E04
1.7188E04
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.
4.9920E08
4.9906E08
7.8055F.10
7.7614E10
5.7422E08
5.7284E08
9.1509E10
9.1660E10
No.
Iterations
88
755
86
530
102
623
115
536
3.0779E08
3.0769E08
7.5819E10
4.3590E13
2.6375E08
2.6302E08
8.2139E10
8.2920E10
83
530
101
631
93
664
109
579
2.6539E08
2.64I7E08
8.8433E10
8.8102E10
2.7889E08
2.7925E08
8.6064E10
8.6085E10
93
23
95
683
117
628
113
763
b
True value of 1.7188E04 and starting value of 2.25E04
* Values of Ki are multiplied by 10,000
Table A29. Results of K;Kf ( t c
= 0.0) parameter optimization problem with errorfree 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
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objedive
Function
SLS
HMLE
SLS
HMIF.
SLS
HV1LE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.7366E03
3.7366E03
3.7366E03
3.7367E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
3.7366E03
3.7366E03
3.7364E03
3.7364E03
3.7364E03
3.7366E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
3.7365E03
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 3.7365E03 and starting value of 4.25E03
* Values of Ki are multiplied by 10,000
Value of
Obj. Ftn.
5.0414E06
5.0528E08
7.9930E10
7.992 IE10
6.2739E08
6.2779E08
I.0131E09
1.0127E09
No. herations
108
595
118
550
104
730
93
579
1.8605E08
1.8633E08
5.8503E10
5.7750E10
2.2040E08
2.2033E08
6.5821E10
5.7750E10
2.5993E08
2.5903E08
8.6620E10
8.6934E10
1.7698E08
6.2779E08
5.9013E10
5.8427E10
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 errorfree 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.9687E05
3.9687E05
3.9687E05
3.9687E05
3.9681E05
3.9682E05
3.9675E05
3.9687E05
3.9686E05
3.9686E05
3.9686E05
3.9686E05
3.9681E05
3.9681E05
3.9686E05
3.9681E05
3.9681E05
3.9681E05
3.9681E05
3.9681E05
3.9681E05
3.9681E05
3.9681E05
3.9681E05
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.
4.7344E08
4.7416E08
7.46I8E10
7.4826E10
6.8755E08
6.8649E08
9.4408E08
7.4826E10
No.
Iterations
150
655
161
529
109
694
160
529
3.1767E08
3.1775E08
7.4219E10
7.6982E10
2.4021E08
2.3916E08
7.5292E10
7.5117E10
127
794
130
644
141
649
117
616
2.2677E08
2.2710E08
7.5292E10
7.5117E10
2.4975E08
2.4995E08
8.3130E10
8.3181E10
125
836
144
531
127
572
114
626
b
True value of 3.9685E05 and starting value of 9.00E05
* 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
1.7490E04
1.7503E04
1.7463E04
1.7467E04
1.6957E04
1.6907E04
1.6972E04
1.6892E04
1.7396E04
1.7368E04
1.7374E04
1.7401E04
1.7028E04
1.7370E04
1.7018E04
1.6755E04
1.6168E04
1.5935F.04
1.6303E04
1.5934E04
1.7369E04
1.7359E04
1.7371E04
1.7374E04
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.
6.0561E03
6.0492E03
9.0575E05
9.0664E05
1.0677E01
1.0673E01
1.6887E03
1.6876E03
No.
Iterations
46
442
57
446
39
510
35
510
7.0902E03
7.0679E03
2.1399E04
2.1378E04
4.5031E02
4.4585E02
1.4369E03
1.4358E09
4.6901E01
4.6708E01
1.5502E02
1.5398E02
2.7874E+00
2.7873E+00
9.2654E02
9.2654E02
35
432
38
469
104
437
100
422
55
475
54
481
30
472
33
544
b
True value of 1.7188E04 and starting value of 2.25E04
* 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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEl'A
Simplex
SCEl'A
Simplex
SCEl'A
Objective
Function
SLS
HMLE
SLS
IIMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.7589E03
4.9798E03
3.7589E03
4.9801E03
3.7039E03
3.6855E03
3.6976E03
3.6833F.03
3.6539E03
3.6504E03
3.6875E03
3.6507E03
3.7186E03
3.7150E03
3.7219E03
3.7165E03
3.7095E03
3.6775E03
3.6739E03
3.6544E03
3.6292E03
3.6288E03
3.6488E03
3.6418E03
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%)
8.1763E02
8.1708E02
2.6591E03
2.6219E03
3.0739E01
3.0703E01
9.5810E03
9.5714E03
9.4521E01
9.4404E01
3.1328E02
3.1262E02
2.1079E+00
2.1079E+00
6.0593E02
6.0594E02
Value of
Obj. Pin.
2.2805E01
2.2773E01
3.6199E03
3.6142E03
4.7400E01
4.7361E01
7.4675E03
7.4659E03
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 3.7365K03 and starting value of 4.25E03
» 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
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCF.UA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Simplex
SCEllA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
11 MI J
SLS
HMLE
SLS
7
.
HMLE
b
Kf
3.9143E05
3.8994E05
3.9069E05
3.8944E05
4.0553E05
4.0660E05
4.0668E05
4.0718E05
3.9970E05
3.9668E05
3.9900E05
3.9709E05
2.5000E05
2.5001E05
2.5000E05
2.5001E05
4.0859E05
4.0524E05
4.1708E05
4.1024E05
3.8554E05
3.8555E05
3.8580E05
3.8688E05
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.
2.6258E01
2.6244E01
4.1093E03
4.1400E03
7.1027E01
7.0945E01
1.1229E02
1.1226E02
No.
Iterations
42
472
44
493
48
481
50
507
1.5650E01
1.5484E01
4.8211E03
4.8027E03
1.4230E01
1.3862E01
4.1192E03
3.9974E03
1.2074E+00
1.2047E+00
3.9658E02
3.9657E02
1.8356E+00
1.8356E+00
6.0839E02
6.0588E02
46
492
55
591
112
554
130
572
37
451
41
529
85
391
85
397
b
True value of 3.9685E05 and starting value of 9.00E05
* Values of Ki are multiplied by 10,000
Table A34. Results ofKiKf(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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCEUA
Sinqilex
SCElIA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMI.F.
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
1.7670E04
1.7649E04
1.7593E04
1.7588E04
1.6817E04
1.6681E04
1.6842E04
1.668IE04
1.7653E04
1.7790E04
1.7653E04
1.8702E04
1.6813E04
1.6776E04
1.6815E04
1.6810E04
1.5523E04
1.5957E04
1.5379E04
1.5198E04
1.7568E04
1.7566E04
1.7568E04
1.7583E04
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.
7.9130E03
7.9040E03
1.1467E04
1.1466E04
No.
Iterations
50
486
49
432
1.3561E01
I.3531E01
2.1862E03
2 1825E03
38
491
99
497
7.9657E03
7.7596E03
1.8704E04
1.8702E04
6.9699E02
6.9643E02
2.0395E03
2.0387E03
3.2048E01
3.2012E01
9.6950E03
9.6756E03
1.2889E+00
1.2889E+00
4.2962E02
4.2958E02
34
469
37
487
123
498
123
525
39
568
48
561
39
487
53
527
b
True value of 1.7188E04 and starting value of 2.25E04
• Values ofKi are multiplied by 10,000
Table A35. Results of KjKf (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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Search
Algorithm
Simplex
SCF.UA
Simplex
SCEl'A
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
b
Kf
3.7300E03
3.6936E03
3.7300E03
3.7030E03
3.6745E03
3.6880E03
3.6753E03
3.6851E03
3.7929E03
3.8099F.03
3.7854E03
3.8069E03
3.6738E03
3.6849E03
3.6731E03
3.6843E03
3.9669E03
3.9574E03
3.9484E03
3.9548E03
3.7501E03
3.7500E03
3.7488E03
3.7491E03
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.
2.1134E01
2.1117E01
3.2693E03
3.2691E03
5.4142E01
5.4098E01
8.1919E03
8.1659E03
No.
Iterations
32
457
32
361
38
434
39
555
1.0876E01
1.0804E01
3.4619E03
3.4311E03
1.5185E01
1.5151E01
4.8853E03
4.8650E03
33
518
38
504
40
511
35
465
1.0311E+00
1.0298E+00
3.4258E02
3.4128E02
9.3284E01
9.3285E01
3.0920E02
3.0940E02
67
441
55
463
114
397
95
443
b
True value of 3.7365E03 and starting value of 4.25E03
* Values of Ki are multiplied by 10,000
Table A36. Results of KjKf (
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
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Simplex
SCEUA
Objective
Function
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMLE
SLS
HMJii
SLS
HMLE
b
Kf
3.8820E05
3.8702E05
3 8776E05
3.8650E05
3.9513E05
3.9431E05
3.9513E05
3.9403E05
3.9192E05
3.9257E05
3.9488E05
3.9281E05
2.5000E05
2.5000E05
2.5000E05
2.5000E05
3.7892E05
3.7769E05
3.7934E05
3.8243E05
3.9018E05
3.9013E05
3.8579E05
3.8699E05
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%)
7.0308E02
7.0157E02
2.0821E03
2.0739E03
2.4882E01
2.4377E01
7.2479E03
7.1411E03
4.2353E01
4.2338E01
1.3682E02
1.3642E02
1.2892E+00
1.2893E+00
4.2346E02
4.2221E02
Value of
Obj. Ftn.
2.5991E01
2.5988E0I
4.0110E03
4.0096E03
6.7175E01
6.7143E01
1.0148E02
1.0132E02
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
Tnievalueof3.9685E05andstartingva!ueof9.00E05
• Values of Ki are multiplied by 10,000
165
LITERATURE CITED
Aitken, A.P. 1973. Assessing systematic errors in rainfallrunoff models. Journal of
Hydrology. 20(2): 121136.
Alberts, E E., J.M. Laflen, W.J., Rawls, J R. Simanton and M.A Nearing. 1989. WEPP
Profile Documentation. National Soil Erosion Research Laboratory. Report No. 2
USD A. 6.16.15.
Alonso, C.V., W.H. Neibling, G.R. Foster 1981. Estimating sediment transport capacity in watershed modeling. Transactions of the American Society of Agricultural
Engineers. 24: 12111226.
Bagnold, R.A 1966. Sediment discharge and stream power. United States Geological
Survey Circular 421. Washington D. C
Bagnold, R.A. 1980. An empirical correlation of bedload transport rates in flumes and natural rivers. Proceedings of the Royal Society Series A372, 453473.
Barnett, A.P. and J.S. Rogers. 1966. Soil physical properties related to runoff and erosion from artificial rainfall. Transactions of the American Society of Agricultural
Engineers. 9:123125, 128.
Bennet, H.H. 1939. Soil Conservation. McGrawHill Book Company, Inc. New York,
NY.
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