IMPROVED ESTIMATION OF SPLASH AND SHEET EROSION IN RANGELANDS:

IMPROVED ESTIMATION OF SPLASH AND SHEET EROSION IN RANGELANDS:

IMPROVED ESTIMATION OF SPLASH AND SHEET EROSION IN RANGELANDS:

DEVELOPMENT AND APPLICATION OF A NEW RELATIONSHIP AND NEW

APPROACHES FOR SENSITIVITY AND UNCERTAINTY ANALYSES by

Haiyan Wei

______________________________

A Dissertation Submitted to the Faculty of the

SCHOOL OF NATURAL RESOURCES

In Partial Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

WITH A MAJOR IN RENEWABLE NATURAL RESOURCES STUDIES

In the Graduate College

THE UNIVERSITY OF ARIZONA

2007

2

THE UNIVERSITY OF ARIZONA

GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Haiyan Wei entitled Improved estimation of splash and sheet erosion in rangelands: development and application of a new relationship and new approaches for uncertainty and sensitivity analyses and recommend that it be accepted as fulfilling the dissertation requirement for the

Degree of Doctor of Philosophy

____________________________________________________________Date: 07/13/07

Dr. David D. Breshears

____________________________________________________________Date: 07/13/07

Dr. Mark A. Nearing

____________________________________________________________Date: 07/13/07

Dr. D. Phillip Guertin

____________________________________________________________Date: 07/13/07

Dr. Jeffry J. Stone

Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

____________________________________________________________Date: 07/13/07

Dissertation Director: Dr. David D. Breshears

3

STATEMENT BY AUTHOR

This dissertation 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 dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted 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 instance, however, permission must be obtained from the author.

Signed: Haiyan Wei

4

ACKNOWLEDGEMENTS

First I would like to gratefully thank my research advisor, Dr. Mark A. Nearing, for his great insights, perspectives, patience and guidance throughout my doctoral work.

He encouraged me to grow as a modeler as well as an independent thinker. He taught me how to question thoughts and express ideas. He put in lots of hours on the numerous versions of the chapters and guided me through the publication process. His mentorship was paramount to me and is greatly appreciated.

I sincerely thank my academic advisor, Dr. David D. Breshears, for the generous time and commitment that he put into this dissertation. He carefully read through and commented on the revisions of this dissertation. He also encouraged me to develop synthetic thinking skills and presentation skills, as well as scientific writing skills. He is a great mentor that sets high standards for his students and guides them to meet those standards.

I am also very grateful for having an exceptional doctoral committee and wish to thank Dr. D. Phillip Guertin and Dr. Jeffry J. Stone for the lengthy and valuable discussions, the accessibility during busy semesters, and the practical support that helped me to understand my research area better and to write up this dissertation.

I extend many thanks to my colleagues and friends at the Agriculture Research

Service, as well as in Dr. Breshears’ research group. I owe a special note of gratitude to

Dr. Chris B. Zou, who has read through my draft and provided helpful, specific feedback.

My sincere thanks also go to Cecily Westphal, Dee Simons, Cheryl Craddock,

Anne Hartley, Mary White and Cheryl Fusco, for the administrative assistance that made it possible for me to complete my degree.

Finally, I deeply appreciate my parents for their belief in me. And I greatly value my friendship with Sophia and T. B. for listening to me and supporting me every step of the way.

Funding was provided by the United State Department of Agriculture, Agriculture

Research Service at Tucson, Arizona.

DEDICATION

To My Parents:

Naiang Wei, and Yuzhen Wu.

5

6

TABLE OF CONTENTS

ABSTRACT........................................................................................................................ 7

INTRODUCTION .............................................................................................................. 9

PRESENT STUDY........................................................................................................... 17

REFERENCES ................................................................................................................. 21

APPENDIX A: A NEW SPLASH AND SHEET EROSION EQUATION FOR

RANGELANDS ......................................................................................................... 25

APPENDIX B: A COMPREHENSIVE SENSITIVITY ANALYSIS FRAMEWORK

FOR MODEL EVALUATION AND IMPROVEMENT USING A CASE STUDY

OF THE RANGELAND HYDROLOGY AND EROSION MODEL ....................... 45

APPENDIX C: A DUAL-MONTE-CARLO APPROACH TO QUANTIFY

PREDICTIVE UNCERTAINTY USING A CASE STUDY OF THE RANGELAND

HYDROLOGY AND EROSION MODEL ................................................................ 78

APPENDIX D: SENSITIVITY ANALYSIS FORTRAN CODE ................................. 100

APPENDIX E: DUAL-MONTE-CARLO UNCERTAINTY ANALYSIS FORTRAN

CODE........................................................................................................................ 111

7

ABSTRACT

Soil erosion is a key issue in rangelands, but current approaches for predicting soil erosion are based on research in croplands and may not be appropriate for rangelands. An improved model is needed that accounts for the dominant erosion processes that operate in rangelands rather than croplands. In addition, effective application of such a model of rangeland erosion requires improved methods for assessing both model sensitivity and uncertainty if the model is to be applied confidently in natural resources management.

I developed a new equation for calculating the combined rate of splash and sheet erosion (D ss

, kg/m

2

) using existing rainfall-simulation data sets from the western United

States that is distinct from that for croplands: D ss

= K ss

I

1.052

q

0.592

, where K ss

is the splash and sheet erosion coefficient, I (m/s) is rainfall intensity, and q (mm/hr) is runoff rate.

This equation, which accounts for inter-relationship between I and q, was incorporated into a new model, the Rangeland Hydrology and Erosion Model (RHEM). This new model was better at predicting observed erosion rates than the commonly used, existing soil erosion model Water Erosion Prediction Project (WEPP).

New approaches for assessing model uncertainty and sensitivity were developed and applied to the model. The new approach for quantifying localized sensitivity indices, when combined with techniques such as correlation analysis and scatter plots, can be used effectively to compare the sensitivity of different inputs, locate sensitive regions in the parameter space, decompose the dependency of the model response on the input parameters, and identify nonlinear and incorrect relationships in the model. The approach for assessing model predictive uncertainty, called “Dual-Monte-Carlo” (DMC), uses two

8

Monte-Carlo sampling loops to not only calculate predictive uncertainty for one input parameter set, but also examine the predictive uncertainty as a function of model inputs across the full range of parameter space. Both approaches were applied to RHEM and yielded insights into model behavior.

Collectively, this research provides an important advance in developing improved predictions of erosion rates in rangelands and simultaneously provides new approaches for model sensitivity and uncertainty analyses that can be applied to other models and disciplines.

9

INTRODUCTION

Rangelands cover nearly 50% of the Earth’s land surface (Williams, 1968).

Measurements of rangeland soil loss indicate that although rangeland erosion rates are sometimes insignificant, this is certainly not so in many cases (Nearing et al., in press).

Rangeland soil loss has been regarded as an indicator of rangeland health (Pyke et al.,

2002). Rangeland soils cannot tolerate the same rates of erosion as most cultivated soils can because rangeland soils have shallower topsoil depth and slower rates of soil formation associated with dry climates (Rollins, 1982). Erosion models are widely used for calculating soil loss rates and assessing the effectiveness of conservation practices

(Federal Register, 1997, 2004a, 2004b). Current erosion prediction approaches are based on research in croplands. Rangelands differ from croplands in soils, plants, microtopography, management practices (Puigdefabregas, 2005; Bochet et al., 2002; Abrahams and Parsons, 1991a, 1991b), and in the way these surface conditions interact with hydrological and erosion processes (Greene et al., 1994; Spaeth, 1996; Puigdefabregas,

2005; Dunkerley, 2000; Parsons et al., 1992). Therefore, the current cropland-based erosion models may not be appropriate for rangelands.

An appropriate rangeland erosion model should be developed that is explicitly based on rangeland soils and is capable of capturing the dominant erosion processes that occur on rangelands. Splash and sheet erosion combined usually dominate the soil loss rates associated with water erosion on most undisturbed rangelands, which generally have adequate vegetation cover. Although interrill erosion in croplands includes the combined

10 processes of splash and sheet erosion, these combined erosion processes are based on measurements and represented in ways that are not necessarily applicable to rangelands:

1) most of the previous interrill equations were developed from either cropland fields or from laboratory agricultural soil pans, which are not representative of rangeland soils; and 2) interrill erosion was conceptualized and modeled for small size plots (ca. 1 m

2

), which are not large enough to encompass the relative high spatial heterogeneity of rangeland soils and surface conditions. In addition, interrill erosion is usually modeled as a function of rainfall intensity (I) and runoff rate (q) such that I and q considered to be independent of one another, but these factors are likely to be inter-related in both croplands and rangelands. In summary, a new splash and sheet erosion equation should be developed that addresses the issues of rangeland-specific soils, consideration of spatial scale, and inter-relationship between rainfall intensity and runoff rate.

As environmental models such as those used for erosion become more complex, the number of parameters increases and the overall parameter space increases greatly. As the parameter space increases, it becomes more challenging for modelers to ensure that the model does not have nonsensical responses to any set of possible input parameter values. Many numerical models utilize a large number of input parameters and complex interactions between inputs and algorithms within the model, which often results in problematic model responses to certain parameter input sets. Consequently, methods are needed that are able to survey model behaviors across the full range of parameter space and thus to detect model deficiencies and unreasonable responses. This is critical for

11 developing robust models related to erosion or other areas of environmental science, as well as for gaining a better understanding of associated model behavior.

In addition, the application of models of erosion and of other environmental processes for addressing natural resources management issues requires consideration of the uncertainty associated with model predictions. Information is needed that describes how the model output deviates from the ‘true prediction’. If the uncertainty associated with the model output (predictive uncertainty) can be quantified and propagated into model output, it can often provide useful information for management purposes. For example, although average annual soil loss values predicted by erosion models are often used as an indicator of erosion risk in the assessment of alternate conservation practices

(Federal Register, 1997, 2004a, 2004b), it can be difficult to justify decisions made on individual parcels of land based on this single erosion value unless there is also some estimate of the uncertainty associated with the prediction. Considering the uncertainty associated with a prediction is particularly important when evaluating conservation goals that are tied to specified thresholds values (e.g., a tolerance limit for amount of erosion loss over a period of concern). Despite the importance of model uncertainty, there are key limitations with current approaches for assessing that uncertainty, as detailed below.

Literature Review on Splash and Sheet Erosion Equations

Initial modeling of interrill erosion was based on the concept of Nichols and

Sexton (1932) that rainfall intensity is more important than rainfall amount in determining erosion. In 1981, based on experiments with a series of rainfall storms at

12 different intensities on a wide range of agriculture soils, Meyer (1981) found that the relationship between interrill soil erosion and rainfall intensity could be expressed as an exponential function:

E = a I b

(2) where, E is the soil loss rate, I is the rainfall intensity, a and b are coefficients. From

Meyer’s experiments, b ranged from 1.63-2.30 and was near 2 for most of the low-clay soils. A simplified version of Equation (2) was used in the older version of the WEPP model as the interrill erosion equation, with a substituted by K i

(interrill erodibility)

(Nearing et al.,1989), as presented in Equation (3):

E = K i

I

2

(3)

Kinnell (1991) proposed that the effect of runoff on sediment delivery should not be ignored, but rather that interrill erosion should combine the impacts of rainfall and runoff:

E = K i

I q (4)

Equation (4) was used in a revised version of the WEPP model (Foster et al.,

1995). Subsequently, additional interrill erosion equations were developed and evaluated

(Kinnell, 1993; Truman and Bradford, 1995; Zhang et al., 1995; Parsons and Stone,

2004), all of which had a form similar to Equation (5) with different coefficients e

1

and e

2

:

E = K i

I e1

q e2

(5)

However, a common problem with these previous interrill equations is that they were all developed based on the assumption that rainfall intensity and runoff rate are

13 independent of each other, allowing e

1

and e

2

to be optimized individually (Huang, 1995), yet this approach ignores recognized interactions between rainfall and runoff response.

In summary, a new splash and sheet erosion equation is needed for rangeland applications. The equation should be developed from rangeland soils and should account for the relationship between rainfall and runoff rates.

Literature Review on Sensitivity Analyses

There are many different techniques for conducting sensitivity analysis (SA; see reviews and comparisons by Helton, 1993; Campolongo and Saltelli, 1997; Saltelli and

Campolongo, 2000; Ionescu-Bujor and Cacuci, 2004). These various approaches range in scales of parameter space from local to global. Local SA evaluates the model response to local changes in input parameters. In this approach, deterministic or derivative-based indices and related summaries such as response surfaces and scatter plots are evaluated to assess sensitivity. Variations of this approach are also referred to as “One factor At a

Time (OAT)” or “regional” SA. If one wants information on the “overall” effect of an input factor on the model, the limitation of this derivative-based method is evident: it is not a representative “overall effect” sensitivity index for most numerical models that involve nonlinear relationships and strong interactions (Saltelli and Campolongo, 2000).

Local SA has been employed to build useful tools for screening model performance

(Morris, 1991; Ionescu-Bujor and Cacuci, 2004). Notably, the method of Morris is widely used to identify key parameters by comparing the “elementary effect” of different input parameters on model output (Saltelli and Campolongo, 2000; Francos et al., 2003).

14

Global SA calculates the “total effect” of a parameter on a model across the entire parameter space. Most of the global SA methods are variance-based, in which the global sensitivity index is estimated as a function of the contribution of each input factor to the total variance of the model output. The Fourier Amplitude Sensitivity Test (FAST) (as well as the extended FAST) is one of the variance-based global SA approaches that have been successfully applied in many different fields (Helton, 1993; Saltelli and

Campolongo, 2000; Crosetto and Tarantola, 2001). An alternative approach of global SA, called Multi-Objective Generalized Sensitivity Analysis (MOGSA), is based on an extension of local (or regional) sensitivity analysis (Hornberger and Spear, 1981; Liu et al., 2004).

In general, sensitivity analyses are useful for assessing model behavior, with local

SA providing insights into specific sets of input values and global SA providing a more holistic perspective. However, neither approach provides for a specific comparison of the relative sensitivity among input parameters to two or more specific input sets. For example, researchers may wish to know how the relative importance of input parameters at one site compares to that at another site, given the assumed and differential parameters for the sites. In addition, an approach is needed that allows a more comprehensive assessment of the parameter space to verify that unrealistic responses are not occurring under certain conditions.

15

Literature Review on Uncertainty Analysis

There are several different types of uncertainty analysis (UA) methods that are used to estimate the uncertainty associated with a model prediction. For example, measurement uncertainty analyses, which often involve repetitive measurements and the so-called “First-order” or “N th

-order” uncertainty estimation, were designed to determine measurement inaccuracies (Kline, 1985). These have been regarded as an effective tool for evaluating and calibrating instruments and minimizing instrumentation costs (ASME,

1983). Generalized Likelihood Uncertainty Estimation (GLUE) is a method that was developed to evaluate model performance by looking at how closely the predicted value matches the site-specific observation using an objective function or goodness of fit test

(Freer and Beven, 1996; Brazier et al., 2000; Brazier et al., 2001; Aronica et al., 2002).

GLUE is a useful tool for evaluating model performance for specific sites considering the model uncertainty from model structure and input parameters. Sampling-based uncertainty analysis is another method that can be used if one wants to know how a model responds to input over specified ranges (Birchall and James, 1994; Cacuci and

Ionescu-Bujor, 2004). This method usually first addresses the input uncertainty by assigning ranges of interest to each parameter, then randomly samples different combinations of input parameter sets and calculates the outputs to examine uncertainty by looking at the range and the distribution of the outputs.

In summary, UA is useful in assigning uncertainty to model output, but a key limitation of most existing approaches (excluding GLUE) is that they usually evaluate an aggregated uncertainty rather than evaluating the uncertainty specific to a given set of

16 input parameter values. And although the GLUE approach does focus on uncertainty for a given set of input values, that approach is limited to a single site and an associated set of site-specific observations. Lacking is a more comprehensive approach that allows simultaneous assessment of uncertainty for several different, specific sets of parameter values. As discussed above relative to uncertainty analysis, such an approach is needed to enable comparisons among two or more sets of conditions (such as different storm types or sites). More specifically, currently lacking is a method that can be used not only for model users to calculate the uncertainty intervals for their specific sites, but also for model developers to examine the model uncertainty across the full range of the parameter space.

17

PRESENT STUDY

The methods, results, and conclusions of this study are presented in the papers appended to this dissertation. The following is a summary of the methods and the most important findings in this document.

Summary of Methods

A new equation was developed for calculating the combined rate of splash and sheet erosion (D ss

) using existing rainfall-simulation data sets from rangelands in the western United States; these data are distinct from those obtained from croplands. This equation, which accounts for inter-relationship between I and q, was incorporated into a new model, the Rangeland Hydrology and Erosion Model (RHEM). Predictions from this model and the commonly used, existing soil erosion model Water Erosion Prediction

Project (WEPP) were compared to field observations.

A new approach for assessing model sensitivity was developed and applied to the model RHEM. A parameter space which combines the full ranges of all the input parameters and the new sensitivity equation was used to calculate the sensitivity indices for all the input parameters across the full parameter space. The results were saved into a sensitivity matrix which can be easily used to analyze various model behaviors, including nonsensical model errors. This method used a new sensitivity equation that quantifies the model response to change in different input parameters in a manner that allows comparison of the relative sensitivity among input parameters for two or more specific

18 input sets. The approach also allows a more comprehensive assessment of the parameter space to verify that unrealistic responses are not occurring under certain conditions.

A new approach for assessing model predictive uncertainty was also developed, called “Dual-Monte-Carlo” (DMC) uncertainty analysis. It first builds a parameter space that combines the full ranges of all the input parameters, and then calculates the model uncertainty intervals generated from input parameters for points sampled across the full parameter space. The first Monte-Carlo simulation was used to randomly sample points from the parameter space, and the second Monte-Carlo simulation was then used to calculate the model output distribution curve for each point selected from the first Monte-

Carlo simulation.

Summary of Results

A new splash and sheet erosion equation was developed based on the WEPP-

IRWET dataset, which is a large set of rangeland rainfall simulation experimental data collected across the western U.S.. The new equation for splash and sheet erosion D ss

) is process-based and considers the interaction between rainfall intensity I and runoff rate q:

D ss

= K ss

I

1.052

q

0.592

, where K ss is the splash and sheet erosion coefficient.

Predictions of soil loss from the new equation and those from the WEPP model both based on the same input information, when compared to field observations, highlight the ability of the new equation to improve predictions of soil erosion on rangelands.

The new SA approach was applied to model RHEM and the results yielded useful insights into model behavior, including interactions between model parameters: 1) local

19 sensitivity can vary greatly from site to site, highlighting the importance of the new approach; 2) the relative importance of varied substantially among the different input parameters; 3) the approach enabled identification of the sensitive ranges and relationships between input parameters; 4) the approach enabled decomposition of the dependency of model response on input parameter values; and 5) the approach effectively detected model errors. In summary, the new SA approach developed here can be used as an element of the iterative model development process whereby model response can be surveyed and problems identified in order to construct a robust model.

The new UA approach for evaluating model predictive uncertainty calculate using

Dual-Monte-Carlo simulation was applied to RHEM and was tested by comparing the variations of RHEM predicted soil loss from the DMC with that from the measured soil loss (Nearing et al., 1999). Both the predicted and measured data showed a strong relationship between the coefficient of variation of soil loss and the expected value of soil loss. A statistical test showed that there was no significant difference between those relationships for the predicted and measured data. An example application of the DMC results for RHEM to evaluate the erosion risk for different scenarios highlighted how the new approach can enable increased capability to assist in quantitatively-based decision making.

Collectively, this research provides an important advance in developing improved predictions of erosion rates in rangelands and simultaneously provides new approaches for model sensitivity and uncertainty analyses that can be applied to other models and disciplines.

20

Dissertation Format

This dissertation has five appendix documents. Appendix A is a paper to be submitted on the new splash and sheet erosion equation. Appendix B is a paper on the new sensitivity analysis that was published in the Transactions of American Society of

Agricultural and Biological Engineers (Wei et al., 2007). Appendix C is a paper on the new DMC uncertainty analysis method and is currently in review for journal publication.

Appendix D is the Fortran code for the new sensitivity analysis approach. Appendix E is the Fortran code for the new DMC uncertainty analysis approach.

21

REFERENCES

Abrahams A., and A. Parsons. 1991a. Resistance to Overland Flow on Desert Pavement and Its Implications for Sediment Transport Modeling. Water resources research. 27

(8): 1827–1836.

Abrahams A., and A. Parsons. 1991b. Relation between infiltration and stone cover on a semiarid hillslope, southern Arizona. Journal of Hydrology.122 (1-4): 49-59.

Aronica G., D. Bates, and M. Horritt. 2002. Assessing the uncertainty in distributed model predictions using observed binary pattern information within GLUE.

Hydrological Processes. 16 (10): 2001-2016.

Birchall A., and A. James. 1994. Uncertainty analysis of the effective dose per unit exposure from radon progeny and implications for ICRP risk-weighting factors.

Radiation Protection Dosimetry. 53: 133-140.

Bochet E., J. Poesen, and J. Rubio. 2002. Influence of plant morphology on splash erosion. in a Mediterranean matorral. Zeitschrift Fur Geomorphologie. 46 (2): 223-

243.

Brazier R., K. J. Beven, J. Freer, and J. Rowan. 2000. Equifinality and uncertainty in physically based soil erosion models: application of the GLUE methodology to

WEPP- the Water Erosion Prediction Project - for sites in the UK and USA. Earth

Surface Process and Landforms. 25: 825-845.

Brazier R., K. Beven, S. Anthony, and J. Rowan. 2001. Implications of model uncertainty for the mapping of hillslope-scale soil erosion predictions. Earth Surface Processes and Landforms. 26 (12): 1333-1352.

Cacuci D., and M. Ionescu-Bujor 2004. A comparative review of sensitivity and uncertainty analysis of large-scale systems—II: Statistical methods. Nuclear Science and Engineering. 147:189-203.

Campolongo F., and A. Saltelli. 1997. Sensitivity analysis of an environmental model: an application of different analysis methods. Reliability Engineering and System Safety.

57: 49-69.

Crosetto M., and S. Tarantola. 2001. Uncertainty and sensitivity analysis: tools for GISbased model implantation. International Journal of Geographical Information Science.

15 (5), 415-437.

22

Dunkerley D. 2000. Assessing the influence of shrubs and their interspaces on enhancing infiltration in an arid Australian shrubland. Joural of Rangeland. 22 (1): 58-71.

Federal Register. 1997. USDA Commodity Credit Corporation, FSA. 7 CFR Parts 704 and 1410. Conservation Reserve Program. 62 (33): 7601-7635.

Federal Register. 2004a. USDA Commodity Credit Corporation, NRCS. 7 CFR Part 1469.

Conservation Security Program. 69 (118): 34502-34532.

Federal Register. 2004b. USDA Commodity Credit Corporation, NRCS. 7 CFR Part 491.

Farm and Ranch Lands Protection Program. 71 (144): 42567-42572.

Foster G. R., D. C. Flanagan, M. A. Nearing, L. J. Lane, L. M. Risse and S. C. Finkner.

1995. Chapter 11. Hillslope erosion component. In: Flanagan, D. C, and M. A.

Nearing. (editors) USDA-Water Erosion Prediction project: Hillslope profile and watershed model documentation. NSERL Report No. 10. USDA-ARS National Soil

Erosion Research Laboratory, West Lafayette, IN.

Francos A., F. J. Elorza, F. Bouraoui, G. Bidoglio, and L. Galbiati. 2003. Sensitivity analysis of distributed environmental simulation models: understanding the model behavior in hydrological studies at the catchment scale. Reliability Engineering and

System Safety 79, 205–218.

Freer J., and K. Beven. 1996. Bayesian estimation of uncertainty in runoff prediction and the value of data: An application of the GLUE approach. Water Resources Research.

32 (7): 2161-2173.

Greene R., P. Kinnell, and J. Wood. 1994. Role of plant cover and stock trampling on runoff and soil-erosion from semiarid weeded rangelands. Australian Journal of Soil

Research. 32 (5): 953-973.

Helton J. C. 1993. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering and System Safety.

42: 327-367.

Hornberger G. M., and R. C. Spear. 1981. An approach to the preliminary analysis of environmental systems. Journal of Environmental Management. 12 (1): 7-18.

Huang C. H. Empirical-analysis of slope and runoff for sediment delivery from interrill areas. 1995. Soil Science Society of America Journal 59 (4): 982-990.

Ionescu-Bujor M., and D. G. Cacuci. 2004. A comparative review of sensitivity and uncertainty analysis of large-scale systems—I: deterministic Methods. Nuclear

Science and Engineering. 147: 189-203.

23

Kinnell P. 1991. The effect of flow depth on sediment transport induced by raindrops impacting shallow flows. Transactions of American Society of Agricultural and

Biological Engineers 34 (1): 161-168.

Kinnell P. 1993. Runoff as a factor influencing experimental determined interrill erodibilities. Australian Journal of Soil Research. 31 (3): 333-342.

Kline S. J. 1985. The purpose of uncertainty analysis. Journal of Fluids Engineering. 107

(2): 153-160.

Liu Y., H. V. Gupta, S. Soroochian, and L. A. Bastidas. 2004. Exploring parameter sensitivities of the land surface using a locally coupled land-atmosphere model.

Journal of Geophysical Research. 109, D21101.

Meyer L. D. 1981. How rain intensity affects interrill erosion. Transactions of American

Society of Agricultural Engineers. 24 (6): 1472-1475.

Nearing M., G. Foster, L. Lane, and S. C. Finkner. 1989. A process-based soil erosion model for USDA-Water Erosion Prediction Project Technology. Transactions of

American Society of Agricultural Engineers. 32 (5): 1587-1593.

Nearing M. A., G. Govers, and D. Norton. 1999. Variability in soil erosion data from replicated plots. Soil Science of America Journal. 63: 1829-1835.

Nearing M. A., M. H. Nichols, J. J. Stone, K. G. Renard, and J. R. Simanton (in press)

Sediment yields from unit-source semi-arid watersheds at Walnut Gulch. Water

Resources Research.

Nichols M. L., and J. D. Sexton. 1932. A method of studying soil erosion. Agricultural

Engineering. 13:101-103.

Parsons A., A. Abrahams, and J. R. Simanton. 1992. Microtopography and soil-surface materials on semiarid piedmont hillslopes, southern Arizona. Journal of Arid

Environments. 22 (2):107-115.

Parsons A., and P. Stone. 2006. Effects of intra-storm variations in rainfall intensity on interrill runoff and erosion. Catena. 67 (1): 68-78.

Puigdefabregas J. 2005. The role of vegetation patterns in structuring runoff and sediment fluxes in drylands. Earth Surface Processes and Landforms. 30 (2): 133-147.

Pyke D. A., J. E. Herrick, P. Shaver, and M. Pellant. 2002. Rangeland health attributes and indicators for qualitative assessment. Journal of Range Management. 55 (6): 584-

597.

24

Rollins B. M. 1982. “T” factor on rangelands. Proceedings of the workshop on estimating erosion and sediment yield on rangelands. ARS (Western region), U.S.D.A., Oakland,

Calif.

Saltelli S. T., and F. Campolongo. 2000. Sensitivity analysis as an ingredient of modeling.

Statistical Science. 15 (4): 377-395.

Spaeth K. E., F. B. Pierson, M. A.Weltz, and J. B. Awang. 1996. Gradient analysis of infiltration and environmental variables as related to rangeland vegetation. American

Society of Agricultural Engineers. 39 (1): 67-77.

Truman C., and J. M. Bradford. 1995. Laboratory determination of interrill soil erodibility. Soil Science Society of America Journal. 59 (2): 519-526.

Wei H., M. A. Nearing, and J. J. Stone. 2007. A comprehensive sensitivity analysis framework for model evaluation and improvement using a case study of the

Rangeland Hydrology and Erosion Model. Transactions of American Society of

Agricultural and Biological Engineers. 50 (3): 945-953.

Williams R. E., B. E. Allred, R. M. Denio, and H. A. Paulsen. 1968. Conservation, development, and use of the world’s rangeland. Journal of Range Management. 21:

355-360.

Zhang X. C., M. A. Nearing, M. P. Miller, L. D. Norton, and L. T. West. 1998. Modeling interrill sediment delivery. Soil Science Society of America Journal. 62 (2): 438-444.

APPENDIX A:

A NEW SPLASH AND SHEET EROSION EQUATION FOR RANGELANDS

(To be submitted for journal publication)

25

26

ABSTRACT

Soil loss rates predicted from erosion models for rangelands have the potential to be important quantitative indicators for rangeland health and conservation practice effects.

Rainfall splash, overland sheet flow and concentrated flow are the driving forces of soil erosion processes at the hillslope scale. Although interrill erosion in croplands includes the combined processes of splash and sheet erosion, these combined erosion processes are based on measurements and represented in ways that are not necessarily applicable to rangelands: 1) most of the previous interrill equations were developed from either cropland fields or from laboratory agricultural soil pans, which are not representative of rangeland soils; and 2) interrill erosion was conceptualized and modeled for small size plots (ca. 1 m

2

), which are not large enough to encompass the relative high spatial heterogeneity of rangeland soils and surface conditions. In addition, interrill erosion is usually modeled as a function of rainfall intensity (I) and runoff rate (q) such that I and q considered to be independent of one another, but these factors are likely to be interrelated in both croplands and rangelands. Splash and sheet erosion combined usually dominate the soil loss rates associated with water erosion on most undisturbed rangelands, which generally have adequate vegetation cover. These important erosion process need to be addressed to develop an appropriate process-based rangeland erosion model. In this study we developed a new equation for calculating the combined rate of splash and sheet erosion (D ss

) using existing data sets on rainfall simulation from rangelands in western

U.S.; these data are distinct from those obtained from croplands. We propose the following equation: D ss

= K ss

I

1.052

q

0.592

, where K ss

denotes the splash and sheet erosion

27 coefficient, I is rainfall intensity, and q is runoff rate. This equation takes into account a key inter-relationship between I and q revealed in the data. Our proposed equation was better at predicting observed erosion rates than the predominantly used, existing soil erosion model: the Water Erosion Prediction Project (WEPP). The new equation should enable improved estimation of water erosion on rangelands in the western U.S. and perhaps on other rangelands of the world.

28

INTRODUCTION

Rangelands cover nearly 50% of the earth’s land surface (Williams, 1968). They are characteristically located in arid and semiarid climates where historically ranching was economically advantageous compared to farming. They tend to be characterized by shallow soil with low organic matter and poor structure with relatively sparse vegetation coverage (Wight and Lovely, 1982). Rangeland soils cannot tolerate the same rates of erosion as do most cultivated soils due to their shallower topsoil depth and the slow rates of soil formation that occur in dry climates.

Rates of soil erosion on rangelands are generally considered to be low, but reliable, published data from measurements are few. Measurements of sediment yields from 7 unit source (02.-5.4 ha) watersheds in the USDA-ARS Walnut Gulch

Experimental Watershed near Tombstone, AZ between 1995 and 2005 indicated a range of between 0.07 and 5.66 t ha

-1

yr

-1

.(Nearing et al., in press). Similar measurements by the USDA-ARS at the Univ. of Arizona Santa Rita Experimental have shown sediment yield values of between 0.06 and 4.21 t ha

-1

yr

-1

(Lane and Kidwell, 2003). Even fewer documented measurements on on-site erosion rates have been made under natural rainfall conditions in order to quantify hillslope soil loss rates. Estimates of soil loss within two small watersheds located within the Walnut Gulch Experimental Watershed gave estimates of mean erosion rates in eroding areas of 5.6 and 3.2 t ha

-1

yr

-1

(Ritchie et al.,

2005; Nearing et al., 2005). Maximum erosion rates within the watersheds were

29 calculated at greater than 10 t ha

-1

yr

-1

. These results in general do not support the idea that rangeland erosion rates are insignificant in all cases.

Soil erosion models are widely used tools for calculating soil erosion rates, indicating rangeland health, and assessing management practices. However, currently there is no erosion model available for rangeland applications. Empirical models such as

USLE and RUSLE were developed from cropland data and no rangeland plots were built to obtain the rangeland model parameters. The process-based erosion model, WEPP, has parameter estimation equations for rangelands, but the model process equations were originally developed from cropland soils.

Rangelands and croplands are different relative to erosion processes and surface factors that may affect erosion rates. For cropland situations, erosion tends to be dominated by a combination of rill and interrill erosion. Rills are relatively small, actively scouring flow channels that generate a significant, and often dominant, amount of erosion on a regular basis from cultivated agricultural fields (Meyer et al., 1975;

Dabney et al., 1993). Interrill areas are the relatively flat areas between the rills wherein soil loss is dominated by splash and thin sheet-flow erosion (Meyer, 1981). For rangelands the situation is quite different. Because the soils are un-tilled, they are typically consolidated, and hence significant rilling does not occur readily under most undisturbed situations. In most situations erosion in rangelands on the plot and hillslope scales is dominated by splash and sheet erosion. A reliable splash and sheet erosion equation is necessary for building a rangeland erosion model.

30

The objective of this paper was to develop a splash and sheet erosion equation specific for rangeland application, based on a large set of rangeland rainfall simulation data.

A Review of Previous Interrill Equations

Initial modeling of interrill erosion was based on the concept of Nichols and

Sexton (1932) that rainfall intensity is more important than rainfall amount in determining erosion. In 1981, based on experiments with a series of rainfall storms at different intensities on a wide range of agriculture soils, Meyer (1981) found that the relationship between interrill soil erosion and rainfall intensity could be expressed as an exponential function:

E = a I b

(1) where, E is the soil loss rate, I is the rainfall intensity, a and b are coefficients. From

Meyer’s experiments, b ranged from 1.63-2.30 and was near 2 for most of the low-clay soils. A simplified version of Equation (2) was used in the older version of the WEPP model as the interrill erosion equation, with a substituted by K i

(interrill erodibility)

(Nearing et al., 1989), as presented in Equation (3):

E = K i

I

2

(2)

Kinnell (1991) proposed that the effect of runoff on sediment delivery should not be ignored, but rather that interrill erosion should combine the impacts of rainfall and runoff:

E = K i

I q (3)

31

Equation (4) was used in a revised version of the WEPP model (Foster et al.,

1995). Subsequently, additional interrill erosion equations were developed and evaluated

(Kinnell, 1993; Truman and Bradford, 1995; Zhang et al., 1995; Parsons and Stone,

2004), all of which had a form similar to Equation (5) with different coefficients e

1

and e

2

:

E = K i

I e1

q e2

(4)

However, a common problem with these previous interrill equations is that they were all developed based on the assumption that rainfall intensity and runoff rate are independent of each other, allowing e

1

and e

2

to be optimized individually (Huang, 1995), yet this approach ignores recognized interactions between rainfall and runoff response.

Also, all the previous interrill erosion equations were developed from either cropland runoff plots or laboratory pans filled with agricultural soils. Rangeland soils are different from agricultural soils in that they are consolidated, untilled, shallower, and contain lower organic matter content. On croplands, erosion tends to be dominated by a combination of rill and interrill erosion with rills capable of generating a significant amount of erosion (Meyer et al., 1975; Dabney et al., 1993). While on rangelands, the surface water flow tends to be tortuous and often spreads as it moves across the hillslope.

The vegetation hummocks and complex slopes also tend to absorb the water and sediment in transit so that less water is available to form concentrated flow and significant rilling does not occur readily under most undisturbed situations. On most undisturbed rangelands, rainfall splash and sheet erosion dominants the erosion rates.

32

In addition, there is an issue with the representative plot size being used for modeling interrill erosion. Most interrill plots are by size of 0.6m*1.2m, which might be appropriate for agricultural land but not large enough for representing rangeland conditions. Rangeland surfaces are more complex with randomly distributed rock cover, plant residual, animal activity and various rangeland plants. The natural of high heterogeneity associated with rangeland surface require a large representative plot to measure and model rangeland splash and sheet erosion.

The objective of this paper is to develop a new process-based splash and sheet erosion equation, taking into account the interaction between rainfall and runoff responses based on a large set of large rangeland rainfall simulation plots. A comparison of soil loss predictions from the new equation with estimations from Water Erosion

Prediction Project (WEPP) model was also conducted to show the improved ability of the new equation in predicting soil erosion from rangelands.

METHODS AND DATA

The data we used for developing the new splash and sheet erosion equation was previously collected by the WEPP in 1987, 1988, and by the Interagency Rangeland

Water Erosion Team (IRWET) in 1990, 1991, and 1992. The IRWET project was coordinated closely with the WEPP model development, so that the experimental design and the data format were compatible with that of WEPP. The WEPP-IRWET rangeland dataset is a valuable erosion database that contains measurements of simulated rainfall,

33 soil, plant properties, runoff and sediment discharge on 204 plots from 49 rangeland sites distributed across 16 western states (Figure 1). The database covers a wide range of rangeland soil types (Table 1). For detailed information on WEPP and IRWET experiment design, the reader may refer to Flanagan and Nearing (1995), Nearing et al.

(1989), and Laflen et al. (1997).

Figure 1. Study site distributions.

#

19

20

21

15

16

17

18

11

12

13

14

22

23

24

25

6

7

4

5

1

2

3

8

9

10

38

39

40

41

35

36

37

42

43

44

45

26

27

28

29

30

31

32

33

34 site number of plots

3

4

6

2

2

2

2

2

2

2

2

6

6

6

6

2

2

2

2

2

2

2

2

2

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

H188

H287

H288

I187

J187

K187

K188

K288

H392

K287

B190

B290

C190

C190

E191

A187

A287

C187

D187

D188

D287

D288

E588

F187

H187

H292

I192

I292

J192

J292

K192

K292

E291

E391

F191

F291

F391

G191

G291

G391

H192

L193

L293

M193

M293

Table 1. Study site descriptions. state city soil name

NM

CA

CA

CA

ND

CA

NE

SD

SD

SD

NM

NE

TX

TX

KS

AZ

AZ

TX

OK

OK

OK

OK

OK

MT

SD

ND

ID

ID

ID

ID

AZ

AZ

CA

CA

UT

UT

KS

KS

CO

CO

CO

WY

WY

WY

ND

Tombstone

Tombstone

Sonora

Chickasha

Chickasha

Chickasha

Chickasha

Woodward

Sidney

Cottonwood

Cottonwood

Cottonwood

Cottonwood

Los Alamos

Cuba

Susanville

Susanville

Susanville

Killdeer

Susanville

Wahoo

Wahoo

Amarillo

Amarillo

Eureka

Eureka

Eureka

Akron

Akron

Akron

Newcastle

Newcastle

Newcastle

Killdeer

Killdeer

Buffalo

Buffalo

Blackfoot

Blackfoot

Prescott

Prescott

San L Obispo

San L Obispo

Cedar City

Cedar City

Stronghold

Forest

Purves

Grant

Grant

Grant

Grant

Woodward

Vida

Pierre

Pierre

Pierre

Pierre

Hackroy

Querencia

Jauriga

Jauriga

Jauriga

Jauriga

Burchard

Burchard

Olton

Olton

Martin

Martin

Martin

Stoneham

Stoneham

Stoneham

Kishona

Kishona

Kishona

Parshall

Parshall

Forkwood

Forkwood

Robin

Robin

Lonti

Lonti

Diablo

Diablo

Taylors Flat

Taylors Flat soil texture

Sandy loam

Sandy clay loam

Cobbly clay

Loam

Loam

Sandy loam

Sandy loam

Sandy loam

Loam

Clay

Clay

Clay

Clay

Sandy loam

Sandy loam

Sandy loam

Sandy loam

Sandy loam

Sandy loam

Loam

Loam

Loam

Loam

Silty clay loam

Silty clay loam

Silty clay

Loam

Fine sandy loam

Loam

Very fine sandy loam

Clay loam

Very fine sandy loam

Sandy loam

Fine sandy loam

Silt loam

Loam

Silt loam

Silt loam

Sandy loam

Sandy loam

Clay loam

Clay loam

Sandy loam

Sandy loam

34

35

RESULTS AND DISCUSSION

The new splash and sheet erosion equation takes the form of equation 6, with D ss as the splash and sheet erosion rate, and the K ss

as the splash and sheet erosion coefficient.

D ss

= K ss

I e1

q e2

(6)

We used the very wet run measurements in the WEPP-IRWET database, because they were designed for multiple rainfall intensities. The very wet run started with rainfall intensity around 60 mm/hr, then went up to about 120 mm/hr, and then dropped down back to 60 mm/hr. Rainfall intensity changes occurred only after steady state runoff from the plots was reached. There were two replicated plots for each site in the WEPP dataset, and 6 replicated plots for each site in the IRWET dataset. Thus, there were either 6 (for

WEPP data) or 18 (for IRWET data) sets of steady state runoff discharge, sediment discharge (D ss

) and the rainfall intensities available for examining the relationships between I, q and D ss

.

Exponential relationships were found between I, q and D ss

. For site B190, as an example, the sediment discharge increased exponentially as the runoff rate increased, and also as rainfall intensity increased (Figure 2). The runoff rate is also an exponential function of rainfall intensity (Figure 3). The exponent coefficients in the three relationships varied with soil types. The statistics of the exponential coefficients in each relationship for all the sites were given in Table 2. The average r

2

of the three relationships are 0.62, 0.52 and 0.64, respectively. An average value of 1.731 of e

5

with average r

2

of 0.52 in the equation q = c

3

I e5 indicates that there was a strong relationship between rainfall intensity and runoff rate, which should not be ignored for developing the

36 interrill erosion equation. Also, in Equation (6), the ratio of e

1

to e

2

should be greater than

1, instead of equal to 1 as in Equation (4).

0.4

0.4

0.3

y = 0.0003x

1.5696

R

2

= 0.7294

0.3

y = 446513x

1.4097

R

2

= 0.5322

0.2

0.2

0.1

0.1

0

0

0 20 40 60 80 100

0 0.00001

0.00002

0.00003

0.00004

Runoff rate (mm/hr)

Rainfall intensity (m/s)

Figure 2 Exponential relationships between D ss

(sediment discharge) and Runoff rate (q), and between D ss

and I for site B190.

100

80 y = 201643x

0.7759

R

2

= 0.5446

60

40

20

0

0 0.00001

0.00002

0.00003

0.00004

Rainfall intensity (m/s)

Figure 3. Exponential relationships between q (runoff rate) and I (rainfall intensity) for site B190.

37

Table 2. Statistics of the three exponents in relationships between sediment discharge

(Dss), rainfall intensity (I), and runoff rate (q).

Average

Minimum

Maximum

Standard Deviation

Coefficient of Variation (%)

Average r

2 e

3

in D ss

= c

1 q e3

e

4

in D ss

= c

2

I e4

e

5

in q= c

3

I e5

1.152 2.162 1.731

0.261

2.521

0.393

0.720

6.033

1.036

0.219

6.213

1.010

34.09

0.62

47.94

0.52

58.38

0.64

Equation (7), (8), (9) were determined from Table 2 using the average value of the exponent coefficients. q = c

1

I

1.731

(7)

D ss

= c

2

I

D ss

= c

3

q

2.162

(8)

1.152

(9)

However, if Equation (7) is substituted into Equation (9) and then compared with

Equation (7), one would find that the three equations are nearly, but not fully internally consistent, i.e. 1.731*1.152 =1.994 ≠ 2.162. To make the equation set balanced, we introduced a constant x to reduce 2.162, and increase 1.731 and 1.152. The value of x was computed as 1.027 (which is close to 1) and the new balanced equations were obtained of the form:

D

D ss ss c c

1

2 q

= ×

I e

I q e

2.162 /1.027

= ×

I

= ×

= ×

I e q e

2.104

1.183

1.778

(10)

By combining the equations in (10) and solving Equation (6), while maintaining the ratio of e

1

/e

2

= 1.778, the final equation for rangeland splash and sheet erosion was obtained:

38

D ss

= k ss

×

I e

1 .

052

× q

0 .

592

The values of the splash and sheet erosion equation K ss

were determined from

(11)

Equation 11 for each plot and then averaged for the each site.

Our data showed that there was strong interaction between runoff rate and rainfall intensity (Figure 4). For 38 sites out of 49 sites, the r

2

of relationship q= c

3

I e5

was greater than 0.5. For 41 sites out of the 49 sites, the value of coefficient e

5

in equation q= c

3

I e5 was greater than 1. Physically, the rainfall intensity is one of the main factors that affects infiltration rate and thus affects the runoff generation rate. To develop an erosion equation that combines the effects of rainfall intensity and runoff, such interaction between the two factors cannot be ignored.

Figure 4. The exponential coefficient and r

2

of the relationship between runoff rate (q) and rainfall intensity (I).

39

VALIDATION

To evaluate the effectiveness of the new equation, we calculated the predicted soil loss rates for some of the large rainfall simulation plots using the obtained K ss

values, then compared to the predictions from the WEPP model using the optimized erodibility values for the same plot (Figure 5). The comparison showed that the new equation greatly improved the predictive ability. As the WEPP model user summary and documentation suggests (Flanagan and Nearing, 1995), the predicted soil loss from WEPP in Figure 5 combined the interrill erosion and rill erosion, with interrill erodibility obtained from small plot measurements, and rill erodibility factors optimized from large plot measurements.

0.03

Equation 11 y = 0.7436x + 0.0013

R

2

= 0.7586

WEPP interrill equation

Equation 11

0.02

0.01

WEPP equation y = 0.4863x + 0.0036

R

2

= 0.2251

0

0 0.01

0.02

Observ ed soil loss (kg/m

2

)

0.03

Figure 5. Comparison of soil loss prediction from the new equation and WEPP.

40

CONCLUSIONS

A new splash and sheet erosion equation ( D ss

= K ss

I

1.052

q

0.592

) was developed based on the WEPP-IRWET dataset, which is a large set of rangeland rainfall simulation experimental data collected across the western U.S.. The new equation is process-based and it considers the interaction between rainfall intensity I and runoff rate q . Such interaction cannot be ignored when developing splash and sheet erosion model since it is not only physically sound, but also statistically tested from the WEPP-IRWET dataset.

Due to the limitation of using small plots to represent the heterogeneity of rangeland surfaces, large plots were used to develop the new equation. We compared the predicted soil loss from the new equation with that from the WEPP model based on the same input information. Results indicated that the new equation improved the ability of predicting soil erosion on rangelands.

41

REFERENCES

Abrahams A., and A. Parsons. 1991a. Resistance to Overland Flow on Desert Pavement and Its Implications for Sediment Transport Modeling. Water Resources Research. 27

(8):1827–1836.

Abrahams A., and A. Parsons. 1991b. Relation between infiltration and stone cover on a semiarid hillslope, southern Arizona. Journal of Hydrology.122 (1-4):49-59.

Abrahams A., A. Parsons, and J. Wainwright. 1995. Effects of vegetation change on interrill runoff and erosion, Walnut-Gulch, southern Arizona. Geomorphology 13:37-

48.

Bochet E., J. Poesen, and J. Rubio. 2002. Influence of plant morphology on splash erosion. in a Mediterranean matorral. Zeitschrift Fur Geomorphologie. 46 (2):223-243.

Chartier M. P., and C. M. Rostagno. 2006. Soil erosion thresholds and alternative states in northeastern Patagonian rangelands. Rangeland Ecology Management. 59: 616–

624.

Dabney S., C. Murphree, and D. Meyer. 1993. Tillage, row spacing, and cultivation affect erosion from soybean cropland. Transactions of American Society of Agricultural

Engineers. 36 (1):87-94.

Dunkerley D. 2000. Assessing the influence of shrubs and their interspaces on enhancing infiltration in an arid Australian shrubland. Rangeland Journal. 22 (1):58-71.

Flanagan D. C, and M. A. Nearing. (editors) 1995. USDA-Water Erosion Prediction project: Hillslope profile and watershed model documentation. NSERL Report No. 10.

USDA-ARS National Soil Erosion Research Laboratory, West Lafayette, IN.

Foster G. R., D. C. Flanagan, M. A. Nearing, L. J. Lane, L. M. Risse and S. C. Finkner.

1995. Chapter 11. Hillslope erosion component. In: Flanagan, D.C, and M.A. Nearing.

(editors) USDA-Water Erosion Prediction project: Hillslope profile and watershed model documentation. NSERL Report No. 10. USDA-ARS National Soil Erosion

Research Laboratory, West Lafayette, IN.

Greene R., P. Kinnell, and J. Wood. 1994. Role of plant cover and stock trampling on runoff and soil-erosion from semiarid weeded rangelands. Australian Journal of Soil

Research. 32 (5): 953-973.

Huang C. H. Empirical-analysis of slope and runoff for sediment delivery from interrill areas. 1995. Soil Science Society of America Journal 59 (4): 982-990.

42

Kinnell P. 1991. The effect of flow depth on sediment transport induced by raindrops impacting shallow flows. Transactions of American Society of Agricultural Engineers.

34 (1):161-168.

Kinnell P. 1993. Runoff as a factor influencing experimental determined interrill erodibilities. Australian Journal of Soil Research. 31(3): 333-342.

Laflen J. M., W. J. Elliot, J. R. Simanton, C. S. Holzhey, and K. D. Kohl. 1991.WEPP –

Soil Erodibility Experiments for Rangeland and Cropland Soils. Journal of Soil and

Water Conservation. 46 (1): 39-44.

Laflen J. M., W. J. Elliot, D. C. Flanagan, C. R. Meyer, and M. A. Nearing. 1997. WEPP-

Predicting water erosion using a process-based model. Journal of Soil and Water

Conservation. 52 (2): 96-102.

Lane J. L., and M. R. Kidwell. 2003. Hydrology and Soil Erosion. pp. 92-100 In: M. P.

McClaran, P. F. Ffolliott, and C. B. Edminster. Santa Rita Experimental Range: 100

Years (1903-2003) of Accomplishments and Contributions. USDA-Forest Service

Proceedings RMRS-P-30. Rocky Mountain Research Station, Ft. Collins, CO (publ.)

Mergen D. E., M. Trlica, J. L. Smith, and W. H. Blackburn. 2001. Stratification of variability in runoff and sediment yield based on vegetation characteristics. Journal of the American Water Resources Association. 37 (3): 617-628.

Meyer L. D., G. R. Foster, and S. Nikolov. 1975. Effect of flow-rate and canopy on rill erosion. Transactions of American Society of Agricultural Engineers. 18 (5): 905-911.

Meyer L. D. 1981. How rain intensity affects interrill erosion. Transactions of American

Society of Agricultural Engineers. 24 (6): 1472-1475.

Nearing M. A. 2000. Evaluating soil erosion models using measured plot data:

Accounting for variability in the data. Earth Surface Processes and Landforms. 25:

1035-1043.

Nearing M. A., G. Foster, L. Lane, and S. C. Finkner. 1989. A process-based soil erosion model for USDA-Water Erosion Prediction Project Technology. Transactions of

American Society of Agricultural Engineers. 32 (5): 1587-1593.

Nearing M. A., G. Govers, and L. D. Norton. 1999. Variability in soil erosion data from replicated plots. Soil Science of America Journal. 63 (6): 1829-1835.

Nearing M. A., A. Kimoto, M. H. Nichols, and J. C. Ritchie. 2005. Spatial patterns of soil erosion and deposition in two small, semiarid watersheds, Journal of Geophysical

Research-Earth Surface. 110, F04020, doi: 10.1029/2005JF000290.

43

Nearing M. A., M. H. Nichols, J. J. Stone, K. G. Renard, and J. R. Simanton (in press)

Sediment yields from unit-source semi-arid watersheds at Walnut Gulch. Water

Resources Research.

Nichols M. L., and J. D. Sexton. 1932. A method of studying soil erosion. Agricultural

Engineering. 13: 101-103.

Parsons A., A. Abrahams, and J. R. Simanton. 1992. Microtopography and soil-surface materials on semiarid piedmont hillslopes, southern Arizona. Journal of Arid

Environments. 22 (2): 107-115.

Parsons A., A. Abrahams, and J. Wainwright. 1996. Response of interrill runoff and erosion rates to vegetation change in southern Arizona. Geomorphology. 14: 311-317.

Parsons A., and P. Stone. 2006. Effects of intra-storm variations in rainfall intensity on interrill runoff and erosion. Catena. 67 (1): 68-78.

Pellant M., P. Shaver, D. A. Pyke, and J. E. Herrick. 2005. Interpreting indicators of rangeland health, version 4. Technical Reference 1734-6. U.S. Department of the

Interior, Bureau of Land Management, National Science and Technology Center,

Denver CO. BLM/WO/ST-00/001+1734/REV05. 122 pp.

Pierson F. B., Carlson D. H., and Spaeth K. E. 2002. Impacts of wildfire on soil hydrological properties of steep sagebrush-steppe rangeland International Journal of

Wildland Fire. 11 (2): 145-151.

Pierson F. B., K. E. Spaeth, M. A.Weltz, and D. H. Carlson. 2002. Hydrologic response of diverse western rangelands. Journal of Range Management. 55: 558-570.

Puigdefabregas J. 2005. The role of vegetation patterns in structuring runoff and sediment fluxes in drylands. Earth Surface Processes and Landforms. 30 (2): 133-147.

Pyke D. A., J. E. Herrick, P. Shaver, and M. Pellant. 2002. Rangeland health attributes and indicators for qualitative assessment. Journal of Range Management. 55 (6): 584-

597.

Ritchie J. C., M. A. Nearing, M. H. Nichols, and C. A. Ritchie. 2005. Patterns of soil erosion and redeposition on Lucky Hills watershed, Walnut Gulch Experimental

Watershed, Arizona. Catena 61 (2-3): 122-130.

Robichaud P. R. 2000. Fire effects on infiltration rates after prescribed fire in Northern

Rocky Mountain forests. USA Journal of Hydrology. 231: 220-229 Sp. Iss. SI.

44

Rollins B. M. 1982. “T” factor on rangelands. Proceedings of the workshop on estimating erosion and sediment yield on rangelands. ARS (Western region), U.S.D.A., Oakland,

Calif.

Spaeth K. E., F. B. Pierson, M. A.Weltz, and J. B. Awang. 1996. Gradient analysis of infiltration and environmental variables as related to rangeland vegetation. American

Society of Agricultural Engineers. 39(1):67-77.

Tiscareño-Lopez M., V. L. Lopes, J. J. Stone, and L. J. Lane. 1993. Sensitivity analysis of the WEPP watershed model for rangeland applications. 1. Hillslope Processes.

Transactions of American Society of Agricultural Engineers. 36: 1659–1672.

Truman C., and J. M. Bradford. 1995. Laboratory determination of interrill soil erodibility. Soil Science Society of America Journal. 59 (2): 519-526.

Wight R. J., and C. J. Lovely. 1982. Application of soil loss tolerance concept to rangelands. Proceedings of the workshop on estimating erosion and sediment yield on rangelands. ARS (Western region), U.S.D.A., Oakland, Calif.

Williams R. E., B. E. Allred, R. M. Denio, and H. A. Paulsen. 1968. Conservation, development, and use of the world’s rangeland. Journal of Range Management. 21:

355-360.

Zhang X. C., M. A. Nearing, M. P. Miller, L. D. Norton, and L. T. West. 1998. Modeling interrill sediment delivery. Soil Science Society of America Journal. 62 (2): 438-444.

45

APPENDIX B:

A COMPREHENSIVE SENSITIVITY ANALYSIS FRAMEWORK FOR MODEL

EVALUATION AND IMPROVEMENT USING A CASE STUDY OF THE

RANGELAND HYDROLOGY AND EROSION MODEL

(Published in Transactions of American Society of Agricultural and Biological Engineers

2007 Vol. 50 (3): 945-953)

46

ABSTRACT

The complexity of numerical models and the large numbers of input factors result in complex interdependencies of sensitivities to input parameter values, and high risk of having problematic or nonsensical model responses in localized regions of the input parameter space. Sensitivity analysis (SA) is a useful tool for ascertaining model responses to input variables. One popular method is local SA, which calculates the localized model response of output to an input parameter. This article describes a comprehensive SA method to explore the parameter behavior globally by calculating localized sensitivity indices over the entire parameter space. This article further describes how to use this framework to identify model deficiencies and improve model function.

The method was applied to the Rangeland Hydrology and Erosion Model (RHEM) using soil erosion response as a case study. The results quantified the localized sensitivity, which varied and was interdependently related to the input parameter values. This article also shows that the localized sensitivity indices, combined with techniques such as correlation analysis and scatter plots, can be used effectively to compare the sensitivity of different inputs, locate sensitive regions in the parameter space, decompose the dependency of the model response on the input parameters, and identify nonlinear and incorrect relationships in the model. The method can be used as an element of the iterative modeling process whereby the model response can be surveyed and problems identified and corrected in order to construct a robust model.

Keywords: Hydrology, Local sensitivity, Morris’ screen method, RHEM, Soil erosion

47

INTRODUCTION

Many numerical models involve the utilization of a large number of input parameters, which often results in complex interactions between inputs and algorithms within the model. In all models, it is generally desirable to understand the relationships between output sensitivities and input parameter values, and how these relationships affect model predictions. This is important not only for gaining a better understanding of the model behavior, but also for detecting model deficiencies and unreasonable responses induced by the high level of model complexity and the high number of model input parameters.

Sensitivity analysis (SA) is a method widely used to ascertain the response of a simulation model to changes in its input parameters. In practice, SA is not only applied to examine the importance of input parameters but is also considered an important element of the model development process. SA helps to elucidate the impact of different model structures, prepare for model parameterization, and direct research priorities by focusing on the parameters that contribute the most to uncertainty to the model response (Saltelli and Campolongo, 2000; Breshears et al., 1992).

Many different SA techniques are available (see reviews and comparisons by

Helton, 1993; Campolongo and Saltelli, 1997; Saltelli and Campolongo, 2000; and

Ionescu-Bujor and Cacuci, 2004). Methods such as response surface, regional SA, scatter plots, differential analysis, and Monte Carlo analysis may give impressive results, but they are not widely applied due to the lack of quantitative sensitivity indices, or their

48 difficulty in application to complex models (e.g., differential analysis). The most well known SA methodology categories are the so-called local SA and global SA. Local SA, also termed the “one factor at a time” (OAT) or deterministic approach, is a derivativebased method. It aims to quantify the exact local response of output ( Y ) to a particular input factor ( x i

) at a selected point ( x

0

) within the full input parameter space for the model.

The most common form of a local sensitivity index is:

Y

∂ x i

 x

0

=

Y ( x

1

0

, ...

, x i

0

+ ∂

∂ x i x i

, ...

, x

I

0

)

Y ( x

0

)

(1) where i = (1, ..., I ), and I is the number of total input parameters. The local sensitivity index measures the partial derivatives of Y with respect to x i

at point x

0

.

Alternative forms of the technique measure the effect on

Y

of perturbing the x i

values by either a fixed amount (e.g., a fixed percent) or by some estimate of the standard deviation of the input (Saltelli and Campolongo, 2000).

The concept of local sensitivity analysis is simple, and it is effective if the localized sensitivity is of interest. However, if the modeler wants information on the overall effect of an input factor on the model, the limitation of this derivative-based method is evident: it is not a representative overall sensitivity index for most numerical models that involve nonlinear relationships and strong interactions (Saltelli and

Campolongo, 2000). For a non-linear model, the local sensitivity index is invalid if the chosen

∂ x i

is too large. This has been demonstrated by Breshears et al. (1992). When he conducted a sensitivity analysis based on different ranges of

∂ x i

, the sensitivity index varied; thus, the local sensitivity index was demonstrated to be a magnitude-based index.

49

For a model with strong interactions due to the dependency of one parameter on the others, the modeler must obtain various local sensitivity indices for a specific factor xi at different points

x

0

.

Global SA was so named because it considers and calculates the total effect of a parameter on a model across the entire parameter space. Most of the global SA methods are variance-based, i.e., the global sensitivity index is represented by the contribution of each input factor to the total variance of the model output. The Fourier amplitude sensitivity test (FAST, and the extended FAST) is one of the variance-based global SA approaches, and it has been successfully applied in many different fields (Helton, 1993;

Saltelli and Campolongo, 2000; Crosetto and Tarantola, 2001). An alternative global SA approach, called multi-objective generalized sensitivity analysis (MOGSA), is based on an extension of regional sensitivity analysis (Hornberger and Spear, 1981). MOGSA investigates the sensitivities of individual parameters by examining whether a prior distribution of the parameters separates under a specific behavioral classification via a K-

S probability index (Liu et al., 2004).

In contrast to global SA, local SA does not help to capture the overall effect of an input factor on a model output, but it is the only way to investigate the parameter sensitivity for specific input scenarios. This is especially important for complex models that involve non-linear effects or strong interactive relationships. For such complex models, the effect of a given parameter may be highly localized; hence, an overall sensitivity index will not be applicable to every case and will be misleading in many cases (i.e., regions of the input parameter space).

50

The localized sensitivity concept has been employed to build very useful tools, such as Morris’ screening method (Morris, 1991), the forward sensitivity analysis procedure (FSAP) of Ionescu-Bujor and Cacuci (2004), and the adjoint sensitivity analysis procedure (ASAP). The method of Morris is widely used to identify key parameters (Saltelli and Campolongo, 2000; Francos et al., 2003). It divides the range of each input parameter xi into p levels, using Latin hypercube randomly sampled points from the p × I parameter space ( I is the number of the parameters) and calculates the

“elementary effect” using equation 1, with

∂ x i

the predetermined multiple of 1/( p - 1). For each x i

, the elementary effects associated with each selected point will form a distribution.

The advantage of this method is that the estimates of the means and standard deviations of the distributions can be used as indicators of the importance of the input parameters. A large mean indicates an important overall influence on the output. A large standard deviation indicates that the influence is highly dependent on the values of the inputs, and that the effect is either non-linear or highly dependent on other factors.

Morris provided an effective framework for analyzing local sensitivity across the entire parameter space. However, the elementary effect of Morris uses a multiple of 1/( p - 1) as the change in the input factor (

∂ x i

), which does not always ensure that the elementary effect is representative of the exactly localized response of the output at point x

0

. This also limits application of this method beyond the screening of the input factors.

The objective of this article is to provide a new local sensitivity analysis framework that can be used effectively to show the interdependencies of sensitivity to multiple model inputs, and which can be used in the model development process to help

51 identify undesirable or illogical model responses. This study uses an algorithm similar to

Morris’ framework, but uses a different local sensitivity index to build a localized sensitivity matrix over the entire parameter space. The sensitivity matrix is further analyzed to make SA more effective as an aid in the model development process. In this article, we illustrate the use of this framework to: (1) examine the localized sensitivity to each parameter, and list and classify the importance of the input parameters; (2) locate the sensitive region for an input parameter; (3) decompose the dependency of model response on input parameter values to understand the parameter interactions, using correlations and regressions; and (4) generate scatter plots to survey the model response and reveal the nonlinear relationships, thresholds, and potential weaknesses or problems of the model structure.

This article, taking the erosion predictions in the Rangeland Hydrology and

Erosion Model (RHEM) as an example, not only highlights the local sensitivities, but also describes how to investigate the interactions between RHEM parameters and how to identify unusual RHEM behavior. Results from this study will be helpful in improving the understanding of the model behavior and parameter interactions in RHEM, and in improving the integrity of the model predictions.

METHODOLOGY

Sensitivity Equation

The local sensitivity index in this article is quantified by the following equation:

52

S i

( x

0

)

=

Y ( x

1

0

, ...

, x i

0

+ ∂ x i x i

0

, ...

Y (

∂ x x i

0

)

, x

I

0

)

Y ( x

0

)

(2)

S i

( x

0 where

)

is the sensitivity of output Y to input factor x i

at point x

0

( x

0

1

, ...

, x i

0

, ...

, x

0

I

)

;

S i

( x

0

) is a non-dimensional, localized index that represents the normalized response of the output to an increase in input value x i

. The absolute magnitude of

S i

( x

0

) indicates the degree of sensitivity of Y to x i

at point x

0

. A positive (or negative) S i

indicates a positive (or negative) relationship between Y and x i

, i.e., an increase in x i

will cause an increase in Y . The percentage of

∂ x i

/ x i

0

is expected to be small enough to ensure that S i

is representative at point x

0

. One of the merits of equation

2 is that if

∂ x i

/ x i

0

S i

remains a constant percentage, then the value of

( x

0

)

can be used to compare the sensitivity of the output to an input variable at its different magnitudes. It can also be used to compare the sensitivity of the output to different individual input factors, for example, the sensitivity of Y to x i

and x j

at point x

0

.

Procedure

Figure 1 is a flowchart of the methodology used in this article. It starts with selecting the input and output parameters to be analyzed. The ranges of each input parameter should then be given to build the parameter space of interest, which can encompass the full realistic range of all input parameters. Points x

0

were then randomly selected from the parameter space, and sensitivity indices were calculated for each

53 parameter at the selected point. Latin hypercube (LH) sampling (McKay et al., 1979) was used for random sampling of points x

0

. At each point selected by LH, the model was run

(1 +

I

) times. The first run was used to calculate and save the output value at point x

0

with no perturbation, and the next I runs were used to calculate new output values after increasing each parameter, one at a time, by a predetermined percentage

(

∂ x i

/ x i

0

)

. The local sensitivity index for each input parameter at this point was then calculated using equation 2 based on the (1 + I ) values of the output at this point.

A numerical model

Build an input parameter space of interest, and identify the targeting output parameter.

Sensitivity loops

1. Randomly select a point from the parameter space.

2. Calulate the sensitivity index for each parameter at that point.

3. Go back to 1 until a sufficient number of points is selected.

Sensitivity matrix

Sensitivity of output to each input parameter at each selected point

Sensitivity statistics

Relative importance and classification of parameters

Sensitive regions

Rank the sensitivities, and identify sensitive regions for parameters

Scatter plots

Survey the model behavior

Dependency

Correlation analysis, regression analysis

Identify sensitive regions Identify unusual model behavior

New parameter space

Model weakness

Modify the model

Figure 1. Flowchart of the sensitivity analysis described in this article.

54

In this study, the sampling procedure and local sensitivity index calculation were repeated 10,000 times, after which the parameter space was well covered and the points were well distributed. At the conclusion of the runs, a sensitivity index matrix had been constructed from the results, containing the values of each parameter at each selected point and the local sensitivity of the output to each input parameter at each point. The absolute values of the sensitivities were also generated for further analysis. A Fortran program was written to connect the model, the LH sampling, the sensitivity loops, and the building of the sensitivity matrix.

The sensitivity matrix was used for further analysis. It was first used to examine the localized sensitivity for each parameter over the whole parameter space. The estimated means and standard deviations of the distributions of each S i

were used to rank and classify the effects of the parameters. The sensitivity matrix was also used to identify if there were contiguous sensitive regions for particular parameters. The sensitivity index

( S i

) and the model output were plotted to examine the model response at different output values. Sensitivities for different input parameters ( S i

and S j

) were plotted to analyze the relationship between two parameters. Regression and correlation analysis were conducted to analyze the dependency of sensitivity ( S i

) on the input parameter values. Scatter plots of

S i

versus the value of the i th

input parameter were generated to identify unusual model responses and model weaknesses.

55

Latin Hypercube Sampling

Latin hypercube (LH) sampling (McKay et al., 1979; Stein, 1987) was used to select random points from the uniformly distributed parameter space. McKay et al. (1979) compared several sampling techniques, and they concluded that the LH method had a number of desirable advantages over the other techniques. LH first divides the range of each input variable into N strata of equal probability 1/ N , and then samples a value from each stratum randomly. The values of each input variable are combined at random to locate a point in the parameter space. One of the advantages of this method that makes it appropriate for this study is that LH ensures full coverage over the range of each variable so that all areas of the sample space will be represented by the selected input values.

RHEM Model and Input Parameter Space

The Rangeland Hydrology and Erosion Model (RHEM) was developed from the

Water Erosion Prediction Project (WEPP) model (Flanagan and Nearing, 1995; Nearing et al., 1989; Laflen et al., 1997). It predicts the hydrology, erosion, and deposition from single storms based on fundamentals of infiltration, surface runoff, hydraulics, and erosion mechanics on the hillslope scale. In this study, we selected 14 input parameters used in the hydrology and erosion components of RHEM for the sensitivity analysis. The amount of soil erosion from the hillslope, soil loss (kg/m

2

), was selected as the targeted output variable. The parameter space of interest for this study is the entire applicable space of the RHEM model. Thus, the full range of all reasonable values that might occur for each input parameter was used to build a 14-dimensional parameter space (Table 1).

56

The ranges were based on recommendations in the WEPP manual (Flanagan and

Livingston, 1995) and the WEPP database (Elliot et al., 1989; Simanton et al., 1991;

Laflen et al., 1991; Alberts et al., 1995).

Table 1. Parameters and parameter ranges used in this study.

Input

Parameter slp sln rain dur xip ki kr

τ c ke ns fr fe rsp psd

Lower

Bound

3

Upper

Bound Unit Description

30 % Slope

10

20

100

120 m Slope length mm Rainfall volume

0.5 2 h Rainfall duration

1 20 -- Rainfall peak intensity variable

1000 2

×

10

6

kg*s/m

4

Interrill erodibility

0.00001 0.004 s/m Rill erodibility

0.0001 7 N/m

2

Critical shear stress

0.8 40 mm/h

Effective hydraulic conductivity

0.00025 0.7

4.07 200

1.11 100

0.5

-7

5

-1 m Matric potential

-- Friction factor for runoff

-- Friction factor for erosion m Rill spacing

-- Particle size distribution

Table 1 lists the name, range, and description of each input parameter studied.

The range of each input parameter was required for the sensitivity study. The slope gradient ( slp ) and slope length ( sln ) were the two parameters that represent the slope condition. The rainfall parameters included storm total rainfall ( rain ), peak rainfall intensity divided by average intensity ( xip ), and storm duration ( dur ). Hydraulic parameters were the Green-Ampt Mein-Larsen hydraulic conductivity ( ke ) and effective matric potential ( ns ). Erosion parameters were the interrill erodibility coefficient ( ki ), the rill erodibility coefficient ( kr ), and critical shear stress (

τ c

). Friction parameters were

57 runoff friction ( fr ) and erosion friction ( fe ). Rill spacing on the hillslope was rsp . The particle size distribution parameter ( psd ) was used in the model to build a lognormal distribution curve, from which five pairs of particle size and fraction data were obtained and passed to the transport capacity and deposition calculations. Building a lognormal distribution curve requires the values of the mean and standard deviation; psd is the mean value, and it is always negative due to the logarithm transformation. A constant standard deviation of 2.163 was used for the distribution of all types of sediments, which was based on the WEPP database (Elliot et al., 1989; Simanton et al., 1991; Laflen et al., 1991;

Alberts et al., 1995).

The increment of each input parameter and the total number of samples were also required for the SA program. The increment was arbitrarily set at 5% in this study. A small value of the increment is preferred to make the sensitivity index representative of the exact localized effect, but the increment must be large enough to avoid rounding errors in the calculations. The total number of points should be determined by considering not only the number of the input parameters but the complexity of the model.

In this study, 10,000 points were used as a representative sampling of the full input parameter space.

RESULTS

Approximately 50% of the 10,000 events did not generate rainfall excess, which means that runoff and erosion from these events was zero, and approximately 20% of the

58 events yielded runoff less than 5 mm, which was considered to be too small to be of interest in terms of output. As a result, only the 3180 events (of the 10,000 total events) that generated runoff greater than 5 mm were saved in the sensitivity matrix for further analysis. The rainfall excess regime is mathematically complicated and is controlled in the model by the relationships between the rainfall and infiltration processes, which is why only a portion of the combinations of the input parameters yielded runoff events of interest. This result is important because it is probably true for many numerical models that only a portion of the entire parameter space yields relevant results, and it is always interesting to know what this proportion is and where it is located in the entire parameter space.

To reveal the location of the parameter space of interest, we compared the histograms of each parameter in the entire parameter space with that in the saved sensitivity matrix. The distributions of all parameters in the entire parameter space fell into uniform distributions according to the LH sampling method used in this study.

However, the results showed that the distribution of some parameters changed after screening out the non-significant events. For example, Figure 2 shows two distributions of the input parameter rain. Figure 2a shows the distribution of rain in the parameter space of 10,000 events, and Figure 2b shows the distribution of rain in the parameter space of the 3180 events that produced runoff greater than 5 mm. It can be seen that many events with small rain values were removed. In addition to rain, we found that some events with high ke and ns values were also removed. This indicates that these three parameters control the amount of runoff.

59

600

500

400

300

200

100

0

20 40 60 rain value (mm)

(a)

80 100 120

600

500

400

300

200

100

0

20 40 60 rain value (mm)

(b)

80 100 120

Figure 2. Distributions of parameter rain (a) in the whole parameter space and (b) in the parameter space of interest.

Localized Sensitivity

Absolute local sensitivity can be used to compare the relative importance of the input parameters. Each row of the sensitivity matrix generates a ranking of parameter importance based on the rank of the absolute sensitivity values at each point. However, the importance of a parameter varied from point to point. For example, at point 23, the

60 ranking of the parameters was: rain , dur , ke , ns , xip , ki , sln , psd , slp , fe , fr , rsp , kr , and

τ c

.

At point 30, the ranking was: psd , rain , dur , xip, ke , ki , ns , slp , fe , sln , fr , rsp , kr, and

τ c

.

Figure 3 lists the four most important parameters based on the count of events. For 98.0% of the 3180 events, the total rainfall depth ( rain ) was the most important parameter, but psd , dur , xip , slp ,

τ c

, and ke also showed up as top-ranked parameters, and these parameters accounted for the remaining 2.0% of the events. Storm duration ( dur ) was the second most sensitive parameter for 68.5% of the total events, and slp , rain , xip , ki , kr , ke , fe , and psd accounted for the rest (31.5%) of the events. The third and fourth ranked parameters were more widely distributed among the input parameters (Figure 3). The results show that for a complex model in which the input parameters interact with each other, the sensitivity for input parameters may vary greatly from point to point in the parameter space.

61

100

80

60

40

20

0 the first important parameter slp sln rain dur xip ki kr τc ke ns fr fe rsp psd

100

80

60

40

20

0 the 2 nd

important parameter slp sln rain dur xip ki kr τc ke ns fr fe rsp psd

100

80

60

40

20

0 the 3 rd

important parameter slp sln rain dur xip ki kr τc ke ns fr fe rsp psd

100

80

60

40

20

0 the 4 th

important parameter slp sln rain dur xip ki kr τc ke ns fr fe rsp psd

Figure 3. Distribution of the four top-ranked input parameters based on the count of events. The top graph shows that rain was the most sensitive parameter for 97.99% of the events, with psd, dur, xip, slp,

τ c, and ke the most sensitive parameters for the remaining

2.01% of the events.

Classification of Input Parameter Effects

The distribution of the sensitivities ( S i

) for each input parameter can be generated from the sensitivity matrix, and the characteristics of the distributions can be used to address and classify the effects of the input parameters on the model output (Morris, 1991;

Saltelli and Campolongo, 2000). The mean of S i

describes the overall effect of the

62 parameter on model response, and the standard deviation of S i

, which indicates the spread tendency of S i

, describes the interaction or nonlinear effect of the parameter.

Figure 4 shows the distribution of each

S i

based on absolute values. The parameters are listed by the ranking of their mean sensitivities for the overall effect: rain , dur , xip , ke , ns , ki , psd , slp , sln , fe , fr ,

τ c

, kr , and rsp . Overall, the rainfall parameters

( rain , dur , and xip ) were the most sensitive variables from RHEM. The next important group of variables included the hydrologic parameters ke and ns .

10

5

0 rain dur xip ke ns ki psd slp sln fe fr tc kr rsp

Input parameter

Figure 4. Statistics of absolute sensitivities for each input parameter. The sensitivity scale

(y-axis) only shows values less than 10. The input parameters (x-axis) are ranked in order of mean sensitivity, which is represented by the small square symbols within the boxplots. Each S i

is represented as a boxplot. The box height indicates the 25th and 75th percentiles, the “x” symbols indicate the 5th and 95th percentiles, and the “-“symbols indicate the minimum and maximum values. The input parameters are described in table

1.

Using Figure 4, we can also investigate the effect of an input parameter over the entire parameter space. For example, the parameter rain had the highest mean value of sensitivity (5.13), which means that given a small increase in rainfall (5% in this study),

63 the soil loss output will change by 5.13 times 5% on average. The maximum sensitivity index of rain was 23.48, which indicates that, in this extreme case, a small change in rainfall will induce a change in calculated soil loss of 23 times the percentage change in the rainfall amount. The minimum sensitivity of rain was 0.16, which indicates that a change in rain will induce a change in soil loss in all cases tested. The standard deviation of S rain

was 2.96, which was the largest value on the list. This indicated that the sensitivity to rain varied greatly from case to case, and the effect of rain on soil loss was highly interactive with other parameters.

Tiscareno-Lopez et al. (1994) conducted a sensitivity analysis on a similar soil erosion model (WEPP) on the USDA-ARS Walnut Gulch Experimental Watershed located near Tombstone, Arizona. The results from his study indicated that, on that watershed, rainfall amount was the most sensitive parameter, followed by ke . From our results, rain was the most important parameter for 98.0% of events, followed by slp

, psd

, ki,

τ c

, or ke depending on the combination of input values. From Figures 3 and 4, we can see that RHEM is a complex model, whereby the localized sensitivities vary greatly from site to site.

Estimates of the sensitivity distributions can also be used to compare and classify the input parameters. A plot of the mean and standard deviation of each S i

is given in

Figure 5. This figure has often been used to classify the effects of parameters as a preliminary analysis (Morris, 1991; Saltelli and Campolongo, 2000; Francos et al., 2003).

From Figure 5, the input parameters can be generally classified into three groups: rain is alone in the first group with both the highest overall effect and the highest interaction or

64 nonlinear effect; dur , ke , xip , ns , and psd are in the second group with median effects; and the rest of the parameters are in the third group, which contains the least important parameters.

3.5

3.0

2.5

2.0

1.5

1.0

psd ns ke xip

0.5

0.0

0

τ c sln fr fe slp ki krrsp

1 2 dur

3

Mean of S i

4 rain

5 6

Figure 5. Plot of the estimated means and standard deviations of the distribution of Si.

This plot helps to classify the effect of each input parameter (Morris, 1991).

Identification of Sensitive Regions

Identification of the sensitive regions could be very useful for both the model developer and model user. Ranking the sensitivity matrix by the sensitivity (

S i

) to a parameter and then examining the continuity of the input parameter values is one way to locate the sensitive regions for this parameter. The distribution of these regions is dependent on the model complexity, the number of the input parameters, and the size of the parameter space. For a simple model with few input parameters, and especially when the sensitive regions tend to be tightly packed and well connected with a strong gradient within the parameter space (i.e., highly distinct from the non-sensitive regions), the

65 sensitive regions may be easier to identify. For a complex model, such identification may be difficult, but the process may be made easier if the non-related parameters are removed from the matrix and or if the parameter space is redefined to a smaller subset of the full input parameter space. The framework of this method is simple and allows reanalysis of the sensitivities based on the smaller, newly defined parameter space of interest (Figure 1).

For example, the overall effect of critical shear stress (

τ c

) is small, but based on the ranked sensitivity matrix by

S

τ c

, we found that the model could be very sensitive to

τ c for some extreme events. Furthermore, in the ranked matrix, we found that the sensitivity to

τ c

tended to be higher when ki was small. This is a reasonable relationship because a low ki means that interrill contributions to erosion will be insignificant, and that rill erosion will be far more important. Since

τ c affects rill erosion, it is reasonable that it is a very sensitive parameter for these situations. Figure 6 shows the relationship between

S

τ c and ki . The value of

S

τ c was not always high for all cases of small ki because

S

τ c

also depended on the values of other parameters (particularly kr and

τ c

itself).

66

-3

-4

-5

-6

-7

-8

0

-1

-2

1

0

500000 1000000 1500000 ki values (kg*s/m

4

)

2000000 2500000

Figure 6. Plot of

S

τ c

vs. ki values showing that

τ c

may be very sensitive when ki is relatively small.

Model Response Related to Output Values

A plot of the sensitivity index ( S i

) versus the output values ( soil loss ) revealed the interesting fact that as the soil loss levels decrease, the sensitivities to the input parameters tend to increase. Figure 7 is an example of S rain

plotted against the soil loss values. The sensitivity indices for all the input values showed a similar relationship with soil loss values. This relationship indicates that there is more uncertainty involved for small soil loss events, which is consistent with the results of Nearing (2000), who showed that model predictions of erosion compared to field measurements show less relative error for larger magnitudes of measured erosion.

67

20

15

10

5

30

25

0

0 2 4 soil loss (kg/m

2

)

6 8

Figure 7. Sensitivity of soil loss to rain (S rain

) vs. magnitude of soil loss.

Relationship between Sensitivities for Different Input Parameters

Figure 8 is a plot of S ki

versus S kr that shows the relationship between the two parameters: the sensitivity for ki can be large only when the sensitivity for kr is low, and vice versa. This relationship makes sense because soil loss in RHEM is controlled by both interrill erosion and rill erosion, which are associated with ki and kr , respectively.

The total soil loss is the summation of the soil loss from the rill area and the interrill area.

If the interrill erosion accounts for the larger proportion of the total soil loss, then S ki

will be higher than S kr

. On the other hand, if the rill erosion accounts for the larger proportion, then the total soil loss will be more sensitive to kr than to ki . In addition to kr , rill erosion is also controlled by

τ c

, and the contribution to erosion by rilling only occurs when the flow shear exceeds the critical hydraulic shear stress (

τ c

). In fact, 2,483 of the 3,180 events (78.08%) did not generate rill erosion, and S kr

and

S

τ c

were both zero for these events. This explains why the overall sensitivity to ki (Figure 2 and Figure 3) was greater

68 than to kr and

τ c

, and why Figure 8 shows many points located at S ki

= 1. Furthermore, the sensitivity of soil loss to ki also depends on other parameters, such as psd and fr (see next section), and S ki is not always 1 when S kr and

S

τ c

are 0.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.2

0.4

0.6

0.8

Sensitivity of soil loss to ki

1 1.2

Figure 8. Plot of sensitivities of S ki

vs. S kr

.

Dependencies of Sensitivity Indices on Input Parameter Values

Regression analysis and correlation analysis were used to understand the dependence of the sensitivity for an input parameter on the parameter values (Table 2).

The coefficient of determination (r

2

) of the regression describes the percentage of the variance of the sensitivity index that can be explained by the magnitudes of the input parameters. The correlation coefficients of S i

and the values of each input parameter reveal the dependence of S i

on each input parameter. For example, it can be seen from

Table 2 that approximately 50% of the variance of S fr

, S sln

, and S ke

can be explained by the magnitudes of the entire input parameter set. The correlation matrix in Table 2 helps to further decompose this dependency. As can be seen, S fr

is dependent on parameters kr

69 and ki ; thus, the sensitivity of runoff friction ( fr ) is related to the magnitude of the erosion parameters.

Table 2. Dependencies of sensitivity indices on the input parameter values.

Input Sensitivity Indices fr fe rsp psd ki kr

τ c ke

Ns slp sln rain dur xip

Parameters

S fr

S sln

S ke

S ki

S rain

S ns

S xip

S fe

S slp

S dur

S r

2

of the regression of S i on input parameter values kr

S psd

S tc

S rsp

0.52 0.51 0.48 0.44 0.43 0.40 0.40 0.38 0.37 0.22 0.22 0.12 0.07 0.02

Correlation coefficient of

S i

to each input parameter

-0.17 0.25 0.08 -0.17 -0.13 0.06 0.05 0.15 0.01 0.06 0.08 0.35 -0.05 -0.01

-0.22 -0.31 -0.19 -0.16 0.16 -0.11 -0.12 -0.19 0.28 0.06 0.02 -0.07 -0.01 0.04

0.15 0.09 0.34 0.10 0.10 0.25 -0.07 0.04 -0.12 -0.04 -0.01 0.04 0.02 -0.03

-0.12 0.04 -0.28 -0.06 -0.18 -0.14 0.15 0.01 0.03 -0.01 -0.02 0.00 -0.05 0.01

0.16 -0.10 0.32 0.09 0.09 0.18 0.11 -0.06 -0.04 0.23 0.00 -0.01 0.03 0.00

0.33 -0.11 -0.00 0.21 0.09 0.00 0.02 -0.05 -0.22 -0.09 -0.08 -0.41 0.10 -0.02

-0.37 0.07 -0.02 -0.19 0.00 0.01 -0.00 -0.00 0.14 0.04 0.05 -0.00 -0.10 0.07

0.17 -0.02 0.03 0.16 -0.01 -0.02 0.00 -0.03 -0.09 -0.03 -0.05 -0.00 -0.02 -0.03

-0.06 -0.07 -0.44 -0.04 -0.10 -0.20 0.10 -0.02 0.04 0.01 -0.01 -0.03 -0.02 0.02

-0.05 -0.07 -0.24 -0.04 -0.10 -0.36 0.09 -0.01 0.04 0.03 -0.01 -0.03 -0.01 0.02

-0.12 -0.35 -0.20 -0.10 0.13 -0.13 -0.13 -0.21 0.26 0.05 -0.00 -0.07 0.00 0.03

0.27 -0.10 0.01 0.25 0.08 -0.00 0.02 -0.06 -0.19 -0.10 -0.09 -0.37 0.06 -0.02

-0.04 -0.04 -0.02 -0.10 0.00 0.02 0.00 0.04 0.02 0.03 0.03 -0.03 0.06 -0.04

0.22 0.03 0.02 0.17 -0.02 0.01 0.01 0.02 -0.16 -0.06 0.01 -0.09 0.03 -0.01

The coefficients in Table 2 reveal many insights into the relationships between the parameters. For example, S ns

and S ke are dependent on the values of rain , dur , and xip .

This relationship reflects the fact that runoff generation in RHEM is controlled by both the rainfall regime (associated with rainfall parameters rain , dur , and xip ) and the infiltration regime (associated with hydrologic parameters ke and ns ).

70

Table 2 also shows that there is negative correlation between S ke

and ke , which indicates that the response of soil loss to ke is dependent on the magnitude of ke itself.

The negative correlation coefficient indicates that the higher the ke

value, the greater the sensitivity for parameter ke . This relationship makes sense because a high ke value is often associated with a small amount of runoff and soil loss, and the sensitivity of soil loss to input parameters increases as the soil loss value decreases (as shown in Figure 7).

Scatter Plots to Identify Characteristics of Model Behavior

Scatter plots of the sensitivity index ( S i

) at each point over the values of the i th parameter at this point can help the modeler survey the model response and identify nonlinear relationships, thresholds, and potential model problems. Figure 9 is a scatter plot of S psd

over the corresponding psd values. The parameter psd is important because it is the only parameter that accounts for particle size distribution in this study. Figure 9 is revealing because it shows an unexpected and undesirable model response around the psd value of -3.2. The same sensitivity procedures, focused on a narrow region of psd (-3.0, -

3.4) for a closer look, confirmed the inconsistent model behavior. For example, for sediment with a psd of -3.31, a 5% increase in psd could induce a 70% increase in soil loss, which was much more sensitive than simulations with psd values outside this region.

This is not a reasonable model response for this variable. A careful examination suggested that the cause of this problem is that the transport capacity was calculated based on different sediment particle sizes.

71

-8 -7 -6 -5 -4 psd

-3 -2

6

4

2

-1

0

0

14

12

10

8

Figure 9. Scatter plot of sensitivity of soil loss to psd (Spsd) vs. psd, revealing a discontinuous model response when psd is close to -3.2.

Another model problem was found by looking at the plot of S xip over xip (Figure

10). According to the modeler’s understanding of the erosion process and model structure, an increase in xip should only induce increases in soil loss; thus, S xip

should always be positive. However, Figure 10 shows there are some situations where S xip

is negative, and they show up at various magnitudes of xip . A closer look at those points with negative

S xip

values showed the general tendency that xip was more sensitive as the value of xip increased.

72

10

8

6

4

2

0

-2

0

-4

-6

-8

-10

2 4 6 8 10 xip

Figure 10. Scatter plot of sensitivity of soil loss to xip (S xip

) vs. xip values.

The cause of this problem was found by careful examination of one of the problematic events. In RHEM, the rainfall parameters ( rain , xip , and dur ) are read from an input file and transformed into a double-exponential hyetograph to simulate the timestep rainfall process. The parameter rain controls the total area under the hyetograph, dur controls the duration of the hyetograph, and xip controls the peak of the hyetograph. The hydrograph is calculated on a fixed time step based on the hyetograph and on the infiltration calculations. Water becomes available for runoff generation only when the rainfall rate at a time step exceeds the infiltration rate at the time step. Thus, the shape of the hyetograph and the infiltration curve control the duration and amount of runoff, and with the roughness, slope length, and gradient, they control the peak rate of runoff and consequent soil loss value.

The erosion model uses a double-exponential transformation approximation for creating the synthetic hyetograph from rainfall input values. Because of the way in which

73 this done, there are a small number of combinations of the three rainfall parameters such that when the modeler increases xip and maintains constant values of rain and dur , the model increases the rainfall intensity peak but also reduces the rainfall amount at the beginning of the hyetograph. This small difference in the shape of the hyetograph can make a large difference in the hydrograph in a limited number of cases (Figure 10). Less runoff can be generated after the increase in xip , and the modeler can thus obtain a negative S xip

for these events. This also explains why the higher the xip value, the more sensitive xip

becomes for points with negative

S xip

, because the higher the xip

value, the more distortion occurs at the beginning of the hyetograph. Figure 11 is an example of how increasing xip may induce a decrease in runoff. The total runoff amounts from the two events are 17.94 mm ( xip = 3.94) and 14.81 mm ( xip = 4.13), respectively.

150 xip = 3.94

xip = 4.13

100

50

0

30 40 50

Time (min)

60 70

Figure 11. Hydrographs from rainfall events with the same rain (41.82 mm) and dur

(47.83 min) but with different xip values. Due to the double-exponential transformation in the model, an increase in xip sometime causes an increase in rainfall peak and a decrease in rainfall at the beginning of the hyetograph. This distortion of the hyetograph may cause an observed decrease of runoff and soil loss corresponding with the increase in xip, explaining the unexpected negative S xip

values in Figure 10.

74

The three examples above describe how scatter plots of local sensitivity indices over the local parameter values may help to identify nonlinear relationships, thresholds, and model problems. Scatter plots such as Figures 9 and 10 are also very useful when the modeler is attempting to fix model problems, as shown in the flowchart (Figure 1).

CONCLUSION

A sensitivity analysis method based on the concept of local sensitivity and Latin hypercube sampling was conducted, using the soil erosion component in RHEM as a case study. The local sensitivity indices of soil loss to 14 input parameters of RHEM at 10,000 points from the full parameter space were obtained and used to build a sensitivity matrix.

The sensitivity matrix was analyzed by correlation analysis and scatter plots to draw useful insights into model response and interactions between model parameters: (1) the results highlighted the importance of local sensitivity, which varies from site to site for a complex model such as RHEM; (2) these analyses showed the relative importance of different input parameters; (3) the results also showed the ability of the method to identify the sensitive range and relationships between input parameters; (4) the method was used to decompose the dependency of model response on input parameter values; (5) the method effectively detected model errors. The method described in this article can be used as an element of the iterative model development process whereby model response can be surveyed and problems identified and corrected in order to construct a robust model.

75

REFERENCES

Alberts E. E., M. A. Nearing, M. A. Weltz, L. M. Risse, F. B. Pierson, X. C. Zhang, J. M.

Laflen, and J. R. Simanton. 1995. Chapter 7: Soil component. In USDA Water

Erosion Prediction Project: Hillslope Profile and Watershed Model Documentation. D.

C. Flanagan and M. A. Nearing, eds. NSERL Report No. 10. West Lafayette, Ind.:

USDA-ARS National Soil Erosion Research Laboratory.

Breshears D. D., T. B. Kirchner, and F. W. Whicker. 1992. Contaminant transport through agroecosystems: Assessing relative importance of environmental, physiological, and management factors. Ecological Applications 2( 3): 285-297.

Campolongo F., and A. Saltelli. 1997. Sensitivity analysis of an environmental model:

An application of different analysis methods. Reliability Eng. System Safety 57 (1):

49-69.

Crosetto M., and S. Tarantola. 2001. Uncertainty and sensitivity analysis: Tools for GISbased model implantation. International Journal of Geographical Information Science.

15 (5): 415-437.

Elliot W. J., A. M. Liebenow, J. M. Laflen, and K. D. Kohl. 1989. A compendium of soil erodibility data from WEPP cropland soil field erodibility experiments 1987 & 88.

NSERL Report No. 3. West Lafayette, Ind.: USDA-ARS National Soil Erosion

Research Laboratory.

Flanagan D. C., and S. J. Livingston, eds. 1995. USDA Water Erosion Prediction Project:

User summary. NSERL Report No. 11. West Lafayette, Ind.: USDA-ARS National

Soil Erosion Research Laboratory.

Flanagan D. C, and M. A. Nearing. 1995. USDA Water Erosion Prediction Project:

Hillslope Profile and Watershed Model Documentation. NSERL Report No. 10. West

Lafayette, Ind.: USDA-ARS National Soil Erosion Research Laboratory.

Francos A., F. J. Elorza, F. Bouraoui, G. Bidoglio, and L. Galbiati. 2003. Sensitivity analysis of distributed environmental simulation models: Understanding the model behavior in hydrological studies at the catchment scale. Reliability Engineering

System Safety. 79 (2): 205-218.

Helton J. C. 1993. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Engineering System Safety. 42

(2-3): 327-367.

76

Hornberger G. M., and R. C. Spear. 1981. An approach to the preliminary analysis of environmental systems. Journal of Environmental Management. 12 (1): 7-18.

Ionescu-Bujor M., and D. G. Cacuci. 2004. A comparative review of sensitivity and uncertainty analysis of large-scale systems: I. Deterministic methods. Nuclear Sci.

Eng. 147 (3): 189-203.

Laflen J. M., W. J. Elliot, R. Simanton, S. Holzhey, and K. D. Kohl. 1991. WEPP soil erodibility experiments for rangeland and cropland soils. Journal of Soil and Water

Conservation. 46 (1): 39-44.

Laflen J. M., W. J. Elliot, D. C. Flanagan, C. R. Meyer, and M. A. Nearing. 1997. WEPP:

Predicting water erosion using a process-based model. Journal of Soil and Water

Conservation. 52 (2): 96-102.

Liu Y., H. V. Gupta, S. Soroochian, and L. A. Bastidas. 2004. Exploring parameter sensitivities of the land surface using a locally coupled land-atmosphere model.

Journal of Geophysical Research. 109: D21101.

McKay M. D., W. J. Conover, and R. J. Beckman. 1979. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.

Technometrics. 21 (2): 239-245.

Morris M. D. 1991. Factorial sampling plans for preliminary computational experiments.

Technometrics. 33 (2): 161-174.

Nearing M. A. 2000. Evaluating soil erosion models using measured plot data:

Accounting for variability in the data. Earth Surface Processes and Landforms. 25 (9):

1035-1043.

Nearing M. A., G. R. Foster, L. J. Lane, and S. C. Finkner. 1989. A process-based soil erosion model for USDA Water Erosion Prediction Project technology. Transactions of American Society of Agricultural Engineers. 32 (5): 1587-1593.

Saltelli S. T., and F. Campolongo. 2000. Sensitivity analysis as an ingredient of modeling.

Statistical Science. 15 (4): 377-395.

Simanton J. R., M. A. Weltz, and H. D. Larson. 1991. Rangeland experiments to parameterize the Water Erosion Prediction Project model: Vegetation canopy cover effects. Journal of Range Management. 44 (3): 276-281.

Stein M. 1987. Large sample properties of simulations using Latin hypercube sampling.

Technometrics. 29 (2): 143-151.

77

Tiscareno-Lopez M., V. L. Lopes, J. J. Stone, and L. J. Lane. 1994. Sensitivity analysis of the WEPP watershed model for rangeland: 1. Hillslope processes. Transactions of

American Society of Agricultural Engineers. 37 (1): 151-158.

APPENDIX C:

A DUAL-MONTE-CARLO APPROACH TO QUANTIFY PREDICTIVE

UNCERTAINTY USING A CASE STUDY OF THE RANGELAND

HYDROLOGY AND EROSION MODEL

(In review for journal publication)

78

79

ABSTRACT

Natural resources models serve as important tools to support decision making by simulating and predicting natural processes. All model predictions have uncertainty associated with them. The quantification of model predictive uncertainty, particularly when expressed as the confidence interval around a model prediction value, may serve as important supplementary information for assisting the decision making process. In this paper, we describe a new method called “Dual-Monte-Carlo” (DMC) to calculate model predictive uncertainty based on input parameter uncertainty. DMC uses two Monte-Carlo sampling loops to not only calculate predictive uncertainty for one input parameter set, but also examine the predictive uncertainty as a function of model inputs across the full range of parameter space. We illustrated the application of DMC to the process-based, rainfall event-driven Rangeland Hydrology and Erosion Model (RHEM). The results demonstrated that DMC effectively generated model predictive uncertainty from input parameter uncertainty and provided useful information for decision making. To evaluate the uncertainty results for RHEM, we compared the calculated uncertainty intervals with the natural variation associated with measured erosion plot data. The calculated dataset showed a similar relationship between the variation of soil loss and the magnitude of soil loss to that observed dataset. We also found that the uncertainty intervals were strongly related to specific model input and output parameter values, yielding regression relationships (r

2

>0.97) that enable the accurate estimation of the uncertainty interval for any point in the input parameter space without the need to run the Monte-Carlo

simulations each time the model is used. Soil loss predictions and their associated uncertainty intervals for three example storms and three site conditions were used to illustrate how DMC can be a useful tool for directing decision making.

Key Words: model predictive uncertainty, soil erosion, RHEM, decision making

80

81

INTRODUCTION

The environmental indicators predicted by natural resources models are important for assisting decision making. However, just like one often wants to know how a measured length closes to the true length of an objective, a universal problem of applying models is how the model output deviates from the “true prediction”. If the uncertainty associated with the model output (predictive uncertainty) can be quantified and propagated into model output, it may provide useful information for management purposes. For example, average annual soil loss values predicted by erosion models have served as a single indicator to help assess erosion risk and to choose conservation practices (Federal Register, 1997, 2004a, 2004b). However, a single soil loss value alone may not provide adequate information about the erosion state, and it can be difficult to justify decisions made on individual parcels of land based on this single erosion value. It is also difficult to know to what level of confidence in a decision addresses the desired conservation goal, or more importantly, to what level it may cross certain thresholds that may require a different design of conservation practice. Knowing the level of uncertainty associated with the impact of a specific conservation plan may allow one to quantify the risk of failure of that practice as applied to a particular situation.

Model predictive uncertainty comes from multiple sources. The input parameter set that users provide is usually the representation of the average condition of a study site

(for example, the average slope of a hillslope element). However, assigning a value to that representative variable inherently involves a certain degree of uncertainty, which will

82 directly affect a level of uncertainty on the model prediction. Model predictive uncertainty also comes from model structure and internal parameter uncertainty (Chaves and Nearing, 1991). Structural uncertainty is associated with the inadequacy and the incompleteness of the model to represent the natural process. Internal parameter uncertainty refers to the coefficients set as constant values inside the model, as well as limitations with model equation structures. The variation of input parameter uncertainty at different sites and the complicated interaction between input parameters make the predictive uncertainty both site-specific and implicit.

There are different types of uncertainty analysis methods available for different purposes. For example, measurement uncertainty analyses, which often involve repetitive measurements and the so-called “First-order” or “N th

-order” uncertainty estimation, were designed to determine the measurement inaccuracies (Kline, 1985). These have been regarded as an effective tool for evaluating and calibrating instruments and minimizing instrumentation costs (ASME, 1983). Generalized Likelihood Uncertainty Estimation

(GLUE) is a method that was developed to evaluate model performance by looking at how the predicted value closes to the site-specific observation using an objective function or goodness of fit test (Freer and Beven, 1996; Brazier et al., 2000; Brazier et al., 2001;

Aronica et al., 2002). GLUE is a useful tool for evaluating model performance for specific sites considering the model uncertainty from model structure and input parameters. Sampling-based uncertainty analysis is another method that can be used if one wants to know how a model responds to input over specified ranges (Birchall and

James, 1994; Cacuci and Ionescu-Bujor, 2004). This method usually first addresses the

83 input uncertainty by assigning ranges of interest to each parameter, then randomly samples different combinations of input parameter sets and calculates the outputs to examine uncertainty by looking at the range and the distribution of the outputs. However, currently there is a lack of method that can be used not only for model users to examine the uncertainty intervals for their specific sites, but also for model developers to examine the model uncertainty across the full range of the parameter space.

The objective of this paper was to develop a “Dual-Monte-Carlo uncertainty analysis (DMC)” method to calculate predictive uncertainty and to apply it to the

Rangeland Hydrology and Erosion Model (RHEM) (Wei et al., 2007) as an example. The methodology of DMC is similar to the sampling-based uncertainty analysis mentioned above, but in our case we use two Monte-Carlo sampling loops for assessing the entire ranges of all the input parameters. This is done not only for purposes of calculating the model uncertainty for specific sites and conditions, but also to examine the uncertainty intervals as a function of model inputs across the full range of the parameter space. We evaluated soil loss predictions and their associated uncertainty intervals for three example storms and three management practices to illustrate how DMC can be a useful tool for directing decision making.

METHODS

Delineate the input parameter space ( I )

First Monte-Carlo simulation

Randomly sample a point x

0 from the space ( I )

Run model to obtain the output value y ( x

0

)

Second Monte-Carlo simulation

1. Assign uncertainty to each input parameter;

2. Define a local space ( i ) around point x

0 based on local uncertainty of inputs;

3. Randomly sample points from the local space ( i ) and calculate the corresponding output values. All the output values from the local space will form a distribution;

4. Calculate confidence intervals from the output distribution.

Uncertainty interval at point x

0

Uncertainty intervals for all the input parameter sets

Figure 1. A flowchart of the methodology outlined in this paper.

The DMC approach starts with delineating an input parameter space (Figure 1).

The input parameter space (denoted as I ) was built by overlaying the full range of each selected input parameter. Then the first Monte-Carlo simulation was conducted to

84

85 randomly select an individual point, x

0

(e.g. one input parameter set), from the parameter space I using the Latin Hypercube (LH) sampling method (McKay et al., 1979). The model was executed and the output value y(x

0

)

was computed and saved. The second

Monte-Carlo simulation was conducted to calculate the uncertainty interval at x

0

, a process which required four steps: 1) assign the uncertainty to each input parameter at point x

0

by providing distribution information to all the input parameters; 2) build a local input space (denoted as i ) around point x

0

by combining the uncertainty of all the input parameters; 3) conduct the Monte-Carlo simulation to randomly sample points from the local space i , and calculate the corresponding output value for each, and 4) determine the uncertainty interval on certain confidence level from the distribution of all the output values from the local space i . After finishing the second Monte-Carlo simulation and saving the results, the initial Monte-Carlo simulation would be conducted again and the process repeated until sufficient points, x

0

, are selected for the space

I

to be well covered.

A uniform distribution was used for the first Monte-Carlo simulation to ensure parameter space I to be well covered and sampled. The distribution type of the second Monte-Carlo simulation depends on the nature of the input parameters and it could vary for different parameters.

RHEM and Model Inputs for Running DMC

RHEM is a newly conceptualized model which was adapted from relevant portions of the WEPP (Water Erosion Prediction Project) model (Flanagan and Nearing,

1995; Nearing et al., 1989; Laflen et al., 1997) and modified to specifically address

86 rangeland conditions. It predicts the soil loss for rangelands based on the simulation of hydrology and erosion processes. A previous study was conducted to assess the sensitivity of the predicted erosion to the model input variables (Wei et al., 2007). RHEM provides a good case study for this paper because soil loss rates on natural rangelands are generally low compared to other agricultural environments, and low erosion rates have been shown to be associated with high variability (Nearing et al., 1999). Quantitative estimations of uncertainty on the RHEM soil loss predictions will increase the ability of

RHEM to direct decision making on choosing appropriate conservation practices.

To run the DMC, the output and input parameters of interest need to be determined, and the ranges and the uncertainty information for each input parameter must be assigned. In this study, the amount of soil erosion from the hillslope, soiloss (kg/m

2

) was the targeted output parameter. Twelve required input parameters were selected for

RHEM and the full range of parameter values for each input variable was assigned (Table

1). The sources of the ranges came from recommendations for rangeland applications in the WEPP model documentation (Flanagan and Nearing, 1995), and WEPP rainfall simulation experimental databases (Elliot et al., 1989; Simanton et al., 1991; Laflen et al.,

1991; Alberts et al., 1995).

87

Table 1. Parameters and parameter ranges used in this study. slp slplen rain dur xip k ss k r

τ c ke fr fe psd

Parameter

Lower bound

Upper bound

3

10

20

0.5

1

30

100

120

2

20

50 4600

0.00004 0.003

0.0001

0.8

4.07

1.11

-7

5.71

40

200

100

-1

Unit

% m mm hr

-

- s/m

N/m

2 mm/hr

-

-

-

Description

Slope

Slope length

Rainfall volume

Rainfall duration

Rainfall intensity variable

Splash and sheet erosion coefficient

Concentrated flow erosion coefficient

Critical shear stress

Effective hydraulic conductivity

Friction factor for runoff

Friction factor for erosion

Particle size distribution

Among the 12 input variables, slp , and sln are the two parameters that represent the slope condition; rain , xip , and dur are the rainfall parameters; ke is the hydraulic conductivity; k ss

, k r

, and

τ c are erodibility factors; and fr , and fe are friction factors. The particle size distribution factor, psd , was used in the model to build a lognormal distribution curve, from which five pairs of particle size and fraction data are obtained and passed to the transport capacity and deposition calculations ( psd represents the mean value of the lognormal distribution). Based on the WEPP rangeland database, a standard deviation of 2.16 was used for the distribution of all types of sediments.

88

Table 2. Input parameter uncertainty and the range of parameter values used to obtain the information on uncertainty.

Input parameter k ss k r

(s/m)

τ c

(N/m

2

) ke (mm/hr) psd fr fe

Min* Max*

80 457

0.000048 0.0013

0.20 2.23

1.14 36.94

-6.56

20.73

9.16

-1.61

193.78

52.92

Standard deviation

0.8539

Coefficient of variation (%)

30.13

30.73

-43.11log(ke)+1.38

-7.81**

12.23

14.55

* Database size used to calculate input parameter uncertainty.

** The variation of psd was assumed equal to that of the primary sediment distribution.

The inputs on storm and slope condition were considered as the driving force and given as constants; hence no uncertainties were assigned to these parameters. The uncertainties of the remaining seven parameters were addressed using either standard deviation or coefficient of variation (CV) (Table 2). From the WEPP database, we compiled repetitive measurements on the input parameters to determine if the standard deviation or coefficient of variation (CV) could be used to describe the uncertainty. Since the range of the values of individual parameters in the database used to characterize input distributions was similar to the domain of the input parameter space I (see Table 1 and

Table 2), we believe that input uncertainty in Table 2 can be applied to the entire parameter space I .

The uncertainty of an input parameter may be dependent on the magnitude of the parameter. In our case, we found there was an exponential relationship between hydraulic conductivity k e

and the coefficient of variation of k e

(r

2

= 0.68) (Figure 2).

89

3

2.5

2

1.5

1

0.5

0

0 y = -0.4311Ln(x) + 1.3823

R

2

= 0.675

5 10 15

Ke(mm/hr)

20 25

Figure 2. An example of how input parameter uncertainty was determined.

Sampling Methods for the Monte-Carlo Simulations

Because of the different purposes of the two Monte-Carlo samplings, we used different sampling methods in each case. Latin Hypercube sampling (LH) was used for the first Monte-Carlo to select random points from the uniformly distributed parameter space I . McKay et al. (1979) compared several sampling techniques, and the LH method was concluded to have a number of desirable properties over others. One of the advantages of this method that makes it appropriate for this study is that LH ensures the full coverage over the range of each variable so that all areas of the sample space will be represented by the selected input values. In our case, 10,000 points were sampled from space I . The purpose of the second Monte-Carlo simulation was to randomly sample

90 input parameter values and build a local space i based on the characteristics of the input distributions (for a given normal distribution, standard deviation or coefficient of variation). We used an inverse normal distribution function to generate the input parameter value for any given probability (0-1), with the probability based on random numbers generated from a random function. We sampled 1,000 points to build the space i for each point selected from space I .

RESULTS AND DISCUSSION

For the RHEM model, the magnitude of the expected soil loss value and uncertainty intervals based on a 95% confidence level at each point x

0

were highly related to the model-estimated soil loss y ( x

0

) such that uncertainty increased with magnitude of soil loss (Figure 3) The expected soil loss value was computed as the mean of the 1000 values from the output distribution around each point x

0

. The relationship between the expected soil loss value and the estimated value is shown in the middle line of the graph.

The slope of 1.02 indicates that the expected values were very close to but on average slightly greater than the predicted values. This means that for the model RHEM, the output distributions were not significantly skewed (either positively or negatively).

Different results in this regard might be expected from different models. If the two values

(expected and predicted) were significantly different, it would mean the output distributions were highly skewed, in which case such model predictive uncertainty analysis would become even more important. The other two lines in Figure 3 are the lower and upper limits of 95% confidence intervals vs. the estimated soil loss value,

91 which shows that the uncertainties were highly dependent on the magnitude of the soil loss value.

25 y = 1.636x + 0.0452

R

2

= 0.946

20 lower bound of uncertainty interval upper bound of uncertainty interval expected value

15

10 y = 1.0161x + 0.0068

R

2

= 0.9963

y = 0.5106x - 0.0079

R

2

= 0.9745

5

0

0 5 10

Predicted soil loss (kg/m

2

)

15

Figure 3. Expected prediction and uncertainty intervals vs. the predicted value.

The magnitude of the model predictive uncertainty depends on not only the input parameter values and the associated uncertainties with them, but also the model structure, the sensitivity of model output to the input parameters, and interactions between the model parameters. To assess whether our results of model uncertainty were realistic, we compared our overall results to the measured data from erosion plots. Erosion datasets for such a comparison is rare. Here we use data from Nearing et al. (1999) to examine the natural and measurement variability of soil loss. Nearing et al. (1999) collected measured soil loss data from replicated plot pairs for 2061 storms, 797 annual erosion

92 measurements, and 53 multi-year, which represented thirteen different site locations, each with different soil types. He calculated the mean value ( M m

) and coefficient of variation

(

CV m

) from the replicated soil loss measurements, and found a linear relationship between the logarithm of CV m

and the logarithm of M m

(r

2

= 0.78).

100

10

Dashed line: log(CV)= -0.355 log (M) - 0.637

r

2

= 0.51

Solid line: log(CV)= -0.306 log (M) - 0.442

(Nearing et al. 1999)

1

0.1

0.001

0.010

0.100

1.000

Mean Soil Loss (kg/m

2

)

10.000

Figure 4. Coefficient of variation (CV) of soil loss from the output distribution at a point vs. the expected predicted soil loss value. The corresponding relationship developed from measured data (Nearing, 1999) is also showed on this figure.

A comparative type of relationship with our predicted data from RHEM was obtained using the coefficient of variation ( CV p

) and the expected value ( M p

) obtained from the output distributions. The variance between replicates is related to the magnitude of the soil loss, with higher variation associated with smaller events. This was true for both the measured data and our model predictions (Figure 4). To describe the relationship,

Nearing et al. (1999) developed a regression line: Log ( CV m

) = -0.306log ( M m

) - 0.442 (r

2

93

= 0.78). Our corresponding relationship for predicted variation was: Log ( CV p

) = -0.355 log ( M p

) - 0.637 (r

2

= 0.51). It is encouraging to see that the line derived from the measured data of Nearing et al. (1999) did fall within our predicted data, and that the two lines were quite close to each other. A test of the equality of the slope and intercept of the two regression lines was conducted and the results indicated that there is no statistical significant difference between the two lines (p<0.001).

The high correlations between the predicted soil loss value (r

2

of 0.95 and 0.97 on

Figure 4) and uncertainty intervals indicate that the relationship can be used to quantify the model uncertainty. Furthermore, we found that the model uncertainty intervals were also highly related to the output parameter, runoff depth and some input parameter values such as splash and sheet erosion coefficient, rainfall amount, and saturated hydraulic conductivity. We developed two regression functions that even had greater r

2

values for the upper and lower uncertainty intervals at 95% confidence level. upper interval (r

2

= 0.97)

= 0.01+1.59

soiloss + 0.01( soiloss * rain ) - 0.02( soiloss * k e

) -14.89( soiloss * runoff ) lower interval (r

2

= 0.99)

= -0.001+0.00002( soiloss * k ss

) + 0.003( soiloss * k e

) + 2.35( soiloss * runoff ) + 0.28

soiloss

The advantage of these equations is that they allow the model developers to provide to the user an estimate of the confidence range for a given model output without the need to run a full Monte-Carlo simulation around the user's point of interest ( x

0

) each time the model is used. We used the stepwise multiple variable linear regression in SAS software and selected the equations with the highest r

2

to generate the two functions, the variables

94 in the equations were selected from two output parameters (runoff depth in meters, runoff and soiloss ), all the input parameter and the also the product of input parameters.

Comparison showed that the predicted uncertainty intervals from the equations are very close to those calculated from DMC (Figure 5).

8

6

1:1 line

20

15

1:1 line

4

2

10

5

0

0 2 4 6

Lower Uncertainty Interval (kg/m

2

)

8

0

0 5 10 15

Upper Uncertainty Interval (kg/m

2

)

20

Figure 5. Predicted uncertainty intervals from the regression equations vs. uncertainty intervals calculated from DMC.

APPLICATION

To show how the predictive uncertainty can be used to assist decision relative to natural resources management, we applied our results to a 0.18 ha shrub watershed located in the USDA-ARS Walnut Gulch Experimental Watershed in southeastern

Arizona. We calculated the soil loss from RHEM and the associated uncertainty from the equations derived in the previous section for 3 different sizes of storms and 3 different site conditions: 1-year, 25-year, and 100-year rainfall with different durations were chosen with rainfall amounts of 17.78 mm, 51.31 mm, and 76.71 mm, respectively (Table

95

3). The values of hydraulic conductivity and soil erodibility coefficients for the current condition were obtained from the rangeland WEPP rainfall simulation dataset, and we arbitrarily decreased k e

and

τ c

, and increased k ss

and k r

by different amounts to simulate the different site conditions (Table 4), since poor condition could be expected to relate to low hydraulic conductivity due to the soil compaction, and to higher soil erodibility due to less vegetation coverage.

Table 3. Storm input parameters.

Frequency (year)

Duration (hr)

Rainfall (mm)

Rainfall input

1

0.5

25

1

100

3

17.78 51.31 76.71

Table 4. Hydrological and erosion parameters for different conditions.

Parameter k e

(mm/hr) k ss k r

(s/m)

τ c

(N/m

2

)

Current condition

28.7

435

0

0

Moderately degraded

Severely degraded

28.7*0.5 = 14.35 28.7*0.05 =1.44

435 * 3 = 1305

0.0001

1.20

435 * 6 = 2610

0.001

0.5

The magnitude of the uncertainty intervals varied with the sizes of storms and site conditions (Figure 6). To evaluate the risk of the two scenarios, we referred to Rollins’

(1982) general estimation on the “T” factor (soil erosion tolerance rate) for rangeland as a reference level, which is 1 ton/acre/year. Soil loss tolerance T is defined as the predetermined value of soil loss below which there will be no effect of soil loss in the

96 fertility and the productivity of soil in an economic sense. The upper bounds of the uncertainty interval for the moderately degraded and severely degraded conditions from the 25-year and 100-year rainfall events were quite close to the yearly ‘T’ value (Figure

6). The absolute uncertainty intervals increased as the size of the storm increased and as the degree of assumed land degradation increased.

1.2

1.0

0.8

0.6

T: 1 ton/acre/year c

0.4

b

0.2

c b

0.0

1 a b a a

3

(17.78mm) (51.31mm) (76.71mm) c

Figure 6. The predictive uncertainty for 3 different storms and 3 different site conditions.

The abscissa is divided into three sections for 3 storm sizes (17.78mm, 51.31mm, and

76.71mm). Site conditions are indicated by letters a, b and c, where a is the current condition, b is the moderate degraded condition and c is the degraded condition. For each event, the predicted soil loss, upper and lower intervals are presented by ‘

’, ‘-’and

‘o’ respectively. The soil loss tolerance value (1 ton/acre/year; the horizontal dashed line) is also given in the figure as a reference to evaluate the erosional risk of each soil loss event.

97

CONCLUSIONS

We developed a Dual-Monte-Carlo (DMC) approach that can be used to calculate the predictive uncertainty for any model input parameter set and to simultaneously examine the uncertainty interval as a function of the output and input parameter values.

We used the Rangeland Hydrology and Erosion Model (RHEM) as an example to describe the framework and the application of DMC. The variations of RHEM predicted soil loss from the DMC approach was compared to measured soil loss variations published previously by Nearing et al. (1999). Both the predicted and measured data showed a strong relationship between the coefficient of variation of soil loss and the expected value of soil loss. A statistical test showed that there was no significant difference between those relationships for the predicted and measured data. An example of applying the results from DMC for RHEM to evaluate the erosion risk for different scenarios demonstrated that, with the capability of quantifying the predictive uncertainty from an input dataset, a model possesses an increased capability to assist in quantitatively-based decision making.

98

REFERENCES

Alberts E., M. A. Nearing, M. A. Weltz, L. M. Risse, F. B. Pierson, X. C. Zhang, J. M.

Laflen, and J. R. Simanton. 1995. Chapter 7: Soil component. In USDA Water

Erosion Prediction Project: Hillslope Profile and Watershed Model Documentation. D.

C. Flanagan and M. A. Nearing, eds. NSERL Report No. 10.West Lafayette, Ind.:

USDA-ARS National Soil Erosion Research Laboratory.

Aronica G., D. Bates, and M. Horritt. 2002. Assessing the uncertainty in distributed model predictions using observed binary pattern information within GLUE.

Hydrological Processes. 16 (10): 2001-2016.

Birchall A., and A. James. 1994. Uncertainty analysis of the effective dose per unit exposure from radon progeny and implications for ICRP risk-weighting factors.

Radiation Protection Dosimetry. 53: 133-140.

Brazier R., Beven K. J., Freer, and J. Rowan. 2000. Equifinality and uncertainty in physically based soil erosion models: application of the GLUE methodology to

WEPP- the Water Erosion Prediction Project - for sites in the UK and USA. Earth

Surface Process and Landforms. 25: 825-845.

Brazier R., K. Beven, S. Anthony, and J. Rowan. 2001. Implications of model uncertainty for the mapping of hillslope-scale soil erosion predictions. Earth Surface Processes and Landforms. 26 (12): 1333-1352.

Cacuci D., and M. Ionescu-Bujor. 2004. A comparative review of sensitivity and uncertainty analysis of large-scale systems—II: Statistical methods. Nuclear Science and Engineering. 147: 189-203.

Chaves H., and M. A. Nearing. 1991. Uncertainty analysis of the WEPP soil-erosion model. Transactions of American Society of Agricultural Engineers. 34 (6): 2437-

2445.

Elliot W. J., A. M. Liebenow, J. M. Laflen, and K. D. Kohl. 1989. A compendium of soil erodibility data from WEPP cropland soil field erodibility experiments 1987 & 88.

NSERL Report No. 3. West Lafayette, Ind.: USDA-ARS National Soil Erosion

Research Laboratory.

Federal Register. 1997. USDA Commodity Credit Corporation, FSA. 7 CFR Parts 704 and 1410. Conservation Reserve Program. 62 (33): 7601-7635.

Federal Register. 2004a. USDA Commodity Credit Corporation, NRCS. 7 CFR Part 1469.

Conservation Security Program. 69 (118): 34502-34532.

99

Federal Register. 2004b. USDA Commodity Credit Corporation, NRCS. 7 CFR Part 491.

Farm and Ranch Lands Protection Program. 71 (144): 42567-42572.

Flanagan D. C., and M. A. Nearing. 1995. USDA-Water Erosion Prediction project:

Hillslope profile and watershed model documentation. NSERL Report No. 10.

USDA-ARS National Soil Erosion Research Laboratory, West Lafayette, IN 47097-

1196.

Freer J., and K. Beven. 1996 Bayesian estimation of uncertainty in runoff prediction and the value of data: An application of the GLUE approach. Water Resources Research.

32 (7): 2161-2173.

Kline S. J. 1985. The purpose of uncertainty analysis. Journal of Fluids Engineering. 107

(2): 153-160.

Laflen J. M., W. J. Elliot, D. C. Flanagan, C. R. Meyer, and M. A. Nearing. 1997. WEPP-

Predicting water erosion using a process-based model. Journal of Soil and Water

Conservation. 52(2): 96-102.

Laflen, J. M., W. J. Elliot, R. Simanton, S. Holzhey, and K. D. Kohl. 1991. WEPP soil erodibility experiments for rangeland and cropland soils. Journal of Soil and Water

Conservation. 46 (1): 39-44.

McKay M. D., W. J. Conover, and R. J. Beckman. 1979. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.

Technometrics. 21: 239-245.

Nearing M. A., G. Govers, and D. Norton. 1999. Variability in soil erosion data from replicated plots. Soil Science of America Journal. 63: 1829-1835.

Nearing M. A., G. R. Foster, L. J. Lane, and S. C. Finkner. 1989. A process-based soil erosion model for USDA- Water Erosion Prediction Project technology. Transactions of American Society of Agricultural Engineering. 32: 1587-1593.

Simanton J. R., M. A. Weltz, and H. D. Larson. 1991. Rangeland experiments to parameterize the Water Erosion Prediction Project model: Vegetation canopy cover effects. Journal of Range Management. 44 (3): 276281.

ASME. 1983. The 1983 ASME/WAM symposium on uncertainty analysis- Theory and practice of experimental control. Journal of Fluids Engineering. 107 (2): 153-182.

Wei H., M. A. Nearing, and J. J. Stone. 2007. A Comprehensive Sensitivity Analysis

Framework for Model Evaluation and Improvement Using a Case Study of the

Rangeland Hydrology and Erosion Model. Transactions of American Society of

Agricultural and Biological Engineers. 50 (3): 945-953.

APPENDIX D:

SENSITIVITY ANALYSIS FORTRAN CODE

100

101 program main c purpose: 1) Combine the full range of each input parameter c 2) Randomly sample points from the parameter space c 3) Calculate the sensitivity indices for each parameter at each point c

implicit none integer mxplan, nm parameter (mxplan=75) real avsole,rf real avgslp(mxplan),slplen(mxplan) real rain(mxplan),stmdur real frcrff(mxplan),frcern(mxplan) real ks(mxplan),ns(mxplan),ki(mxplan),kr(mxplan),shcrit(mxplan)

real xip(mxplan),rspace(mxplan),media,xmxint(mxplan) real st(14) integer i character*51 filen

character*51 filensa character*65 ostrng real slplw,slpup,slplenlw,slplenup,rainlw,rainup,durlw,

1 durup,kilw,kiup,krlw,krup,cshlw,cshup,kelw,keup,

1 frlw,frup,felw,feup,nslw,nsup,xiplw,xipup,rspacelw,rspaceup,

1 increm,medialw, mediaup real xstmdr, xrain,te,effdrr(mxplan) real peakro(0:mxplan) real p1, p2 integer nsamp double precision seed real random real avesloss1,avesloss2, slosssvty, rf1, rf2, rfsvty real te1,effdrr1,te2,effdrr2 integer nintv real sslplw, sslpup, sslplenlw,sslplenup real srainlw, srainup, sdurlw, sdurup real skilw, skiup, skrlw, skrup real scshlw, scshup, skelw, skeup real sfrlw, sfrup, sfelw, sfeup real snslw, snsup, sxiplw, sxipup real srspacelw,srspaceup,nsedialw,nsediaup real avgslp1,slplen1 real xrain1,xstmdr1 real frcrff1,frcern1 real ks1,ns1,ki1,kr1,shcrit1

real xip1,rspace1,media1 c********************************************************************** c OPEN OUTPUT FILES c (UNIT = 32, STATUS=3) c (UNIT = 101, ..., 114 , status = unknown)

102 c********************************************************************** ostrng = 'Enter file name for single event output -->'

call open(32,3,1,ostrng,filen) write (32,1200) open (unit=101,file='svty_avgslp.txt',status= 'unknown') open (unit=102,file='svty_slplen.txt',status= 'unknown') open (unit=103,file='svty_rain.txt',status= 'unknown') open (unit=104,file='svty_stmdur.txt',status= 'unknown') open (unit=105,file='svty_ip.txt',status= 'unknown') open (unit=106,file='svty_ki.txt',status= 'unknown') open (unit=107,file='svty_kr.txt',status= 'unknown') open (unit=108,file='svty_shcrit.txt',status= 'unknown') open (unit=109,file='svty_ke.txt',status= 'unknown') open (unit=110,file='svty_ns.txt',status= 'unknown') open (unit=111,file='svty_fr.txt',status= 'unknown') open (unit=112,file='svty_fe.txt',status= 'unknown') open (unit=113,file='svty_rspace.txt',status= 'unknown') open (unit=114,file='svty_psd.txt',status= 'unknown') open (unit=115,file='svty_all.txt',status= 'unknown') c************************************************************** c Read all the bounds, and increment, and nsamp c************************************************************** ostrng = 'Enter lower and upper bounds on Slope--->' read (5,*) slplw, slpup ostrng = 'Enter lower and upper bounds on Slope Length--->' read (5,*) slplenlw, slplenup ostrng = 'Enter lower and upper bounds on rainfall volume--->' read (5,*) rainlw, rainup ostrng = 'Enter lower and upper bounds on rainfall duration--->' read (5,*) durlw,durup ostrng = 'Enter lower and upper bounds on ip--->' read (5,*) xiplw,xipup ostrng = 'Enter upper and lower bounds on Ki--->' read (5,*) kilw, kiup ostrng = 'Enter upper and lower bounds on Kr--->' read (5,*) krlw, krup

ostrng = 'Enter upper and lower bounds on Critical shear--->' read (5,*) cshlw, cshup ostrng = 'Enter upper and lower bounds on ke--->' read (5,*) kelw, keup ostrng = 'Enter lower and upper bounds on Ns --->' read (5,*) nslw, nsup ostrng = 'Enter lower and upper bounds on ff for runoff--->' read (5,*) frlw, frup ostrng = 'Enter lower and upper bounds on ff for erosion--->' read (5,*) felw, feup ostrng = 'Enter lower and upper bounds on rill spacing --->' read (5,*) rspacelw, rspaceup ostrng = 'Enter lower and upper bounds for PSD --->' read (5,*) medialw, mediaup

ostrng = 'Enter the numebr of levels--->' read (5,*) nsamp ostrng = 'Enter the local increment by percentage--->'

read (5,*) increm ostrng = 'Enter the seed for random number generator--->' read (5,*) seed increm = increm/100 c c write the inputs to output c write(101,*) slplw,slpup write(101,*) slplenlw,slplenup write(101,*) rainlw,rainup write(101,*) durlw,durup write(101,*) xiplw,xipup write(101,*) kilw,kiup write(101,*) krlw,krup write(101,*) cshlw,cshup write(101,*) kelw,keup write(101,*) nslw,nsup write(101,*) frlw,frup write(101,*) felw,feup write(101,*) rspacelw, rspaceup write(101,*) medialw,mediaup c*************************************************************** c Latin hypercube sampling and Sensitivity loops c*************************************************************** nm = 0 do 8000 i = 1, nsamp c Slope nintv = random(seed) * nsamp c calculate the lower and upper bound for the subintervals sslplw = (slpup - slplw) / nsamp * nintv + slplw sslpup = sslplw + (slpup - slplw)/nsamp c Randomly pick a number between the subinterval

avgslp(1) = random(seed)* (sslpup - sslplw) + sslplw c Slplen nintv = random(seed) * nsamp sslplenlw = (slplenup - slplenlw) / nsamp * nintv + slplenlw sslplenup = sslplenlw + (slplenup - slplenlw)/nsamp

slplen(1) = random(seed)* (sslplenup-sslplenlw)+sslplenlw c rainfall nintv = random(seed) * nsamp srainlw = (rainup - rainlw) / nsamp * nintv + rainlw srainup = srainlw + (rainup - rainlw) / nsamp

xrain = random(seed) * (srainup - srainlw) + srainlw c Rainfall duration nintv = random(seed) * nsamp sdurlw = (durup - durlw) / nsamp * nintv + durlw sdurup = sdurlw + (durup - durlw) / nsamp

xstmdr = random(seed)* (sdurup-sdurlw)+ sdurlw c xip

1111 nintv = random(seed) * nsamp

103

104 sxiplw = (xipup - xiplw) / nsamp * nintv + xiplw sxipup = sxiplw + (xipup - xiplw) / nsamp

xip(1) = random(seed)* (sxipup-sxiplw) + sxiplw

if ( (xip(1)*xrain/xstmdr) .gt. 254) goto 1111 c Ki nintv = random(seed) * nsamp skilw = (kiup - kilw) / nsamp * nintv + kilw skiup = skilw + (kiup - kilw) / nsamp

ki(1) = random(seed)* (skiup-skilw) + skilw c Kr nintv = random(seed) * nsamp skrlw = (krup - krlw) / nsamp * nintv + krlw skrup = skrlw + (krup - krlw) / nsamp

kr(1) = random(seed)* (skrup-skrlw) + skrlw c Shcrit nintv = random(seed) * nsamp scshlw = (cshup - cshlw) / nsamp * nintv + cshlw scshup = scshlw + (cshup - cshlw) / nsamp

shcrit(1) = random(seed)* (scshup-scshlw) + scshlw c Ke nintv = random(seed) * nsamp skelw = (keup - kelw) / nsamp * nintv + kelw skeup = skelw + (keup - kelw) / nsamp

ks(1) = random(seed)* (skeup-skelw) + skelw c nintv nintv = random(seed) * nsamp snslw = (nsup - nslw) / nsamp * nintv + nslw snsup = snslw + (nsup - nslw) / nsamp

ns(1) = random(seed)* (snsup-snslw) + snslw c runoff friction factor nintv = random(seed) * nsamp sfrlw = (frup - frlw) / nsamp * nintv + frlw sfrup = sfrlw + (frup - frlw) / nsamp

frcrff(1) = random(seed)* (sfrup-sfrlw) + sfrlw c erosion friction factor nintv = random(seed) * nsamp sfelw = (feup - felw) / nsamp * nintv + felw sfeup = sfelw + (feup - felw) / nsamp

frcern(1) = random(seed)* (sfeup-sfelw) + sfelw c PSD (media) nintv = random(seed) * nsamp nsedialw = (mediaup - medialw) / nsamp * nintv + medialw nsediaup = nsedialw + (mediaup - medialw) / nsamp

media = random(seed)* (nsediaup-nsedialw) + nsedialw c Rill space nintv = random(seed) * nsamp srspacelw = (rspaceup - rspacelw) / nsamp * nintv + rspacelw srspaceup = srspacelw + (rspaceup - rspacelw) / nsamp

rspace(1) = random(seed)* (srspaceup-srspacelw) + srspacelw c*********Calculate soil loss for base parameter set***************** call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro) avesloss1 = avsole/slplen(1)

105 rf1 = rf

if (rf1 .ge.0.005) then c**********************Sensitivity calculation********************** c #1 average slope avgslp1 = avgslp(1) avgslp(1) = avgslp(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(1) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

avgslp(1) = avgslp1 if( avesloss1 .ne. 0)

1 write(101,222)avgslp(1),slplen(1),xrain,xstmdr,xip(1),

1 ki(1),kr(1),shcrit(1),ks(1),ns(1),

1 frcern(1),frcrff(1), rspace(1),media,rf1,avesloss1 c********************************************************************* c #2 slope length slplen1 = slplen(1) slplen(1) = slplen(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(2) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

slplen(1) = slplen1 c********************************************************************** c #3 rainfall volume xrain1 = xrain xrain = xrain * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(3) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

xrain = xrain1 c********************************************************************** c #4 rainfall duration xstmdr1 = xstmdr xstmdr = xstmdr * (1+increm)

106 call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(4) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

xstmdr = xstmdr1 c********************************************************************** c #5 ip (max rainfall intensity / average intensity) xip1 = xip(1) xip(1) = xip(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(5) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

xip(1) = xip1 c********************************************************************** c #6 KI ki1 = ki(1) ki(1) = ki(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(6) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

ki(1) = ki1 c********************************************************************** c #7 Kr kr1 = kr(1) kr(1) = kr(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(7) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

kr(1) = kr(1) / (1+increm) c********************************************************************** c #8 critical shear stress

107 shcrit1 = shcrit(1) shcrit(1) = shcrit(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(8) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

shcrit(1) = shcrit1 c********************************************************************** c #9 KE ks1 = ks(1) ks(1) = ks(1) * (1+increm) te1 = te effdrr1 = effdrr(1) p1 = peakro(1)*3.6e6 call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(9) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

ks(1) = ks1 te2 = te effdrr2 = effdrr(1) p2 = peakro(1) *3.6e6 c********************************************************************** c #10 NS ns1 = ns(1) ns(1) = ns(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(10) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

ns(1) = ns1 c********************************************************************** c #11 runoff friction factor frcrff1 = frcrff(1) frcrff(1) = frcrff(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

108

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(11) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

frcrff(1) = frcrff1 c********************************************************************** c #12 erosion friction factor frcern1 = frcern(1) frcern(1) = frcern(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(12) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

frcern(1) = frcern1 c********************************************************************** c #13 rill sapcing rspace1 = rspace(1) rspace(1) = rspace(1) * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(13) = slosssvty

rfsvty = (rf2- rf1)/rf1/increm

rspace(1) = rspace1 c********************************************************************** c #14 PSD media1 = media media = media * (1+increm) call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,

1 frcern,frcrff,ks,ns,ki,kr,shcrit,xip,rspace,media,xmxint,

1 te,effdrr,peakro)

avesloss2 = avsole/slplen(1)

rf2 = rf

slosssvty = (avesloss2 - avesloss1)/avesloss1/increm st(14) = slosssvty

109

rfsvty = (rf2- rf1)/rf1/increm

media = media1 c********************************************************************** if ( avesloss1 .ne. 0) then write (115,333)st(1),st(2),st(3),st(4),

1 st(5),st(6),st(7),st(8),st(9),st(10),st(11),st(12),st(13),st(14)

nm = nm+1 endif c*********************************************************************

endif

8000 continue write(114, *) nm

close (unit=32) close (unit=101) close (unit=102) close (unit=103) close (unit=104) close (unit=105) close (unit=106) close (unit=107) close (unit=108) close (unit=109) close (unit=110) close (unit=111) close (unit=112) close (unit=113) close (unit=114)

write (6,9000)

222 format(f12.8,x,f8.4, x, f8.4, x, f8.4,x, f8.4, x,

1 f12.4,x,f10.6,x, f8.4, x, f8.4,x, f8.4, x,

1 f8.4,x, f8.4, x, f8.4, x, f8.4,x, f10.6, x,

1 f8.4,x, f8.4, x, f8.4)

555 format(f12.8,x,f12.8, x, f12.8, x, f12.8,x, f12.8, x,

1 f16.8,x,f12.8,x, f12.8, x, f12.8,x, f12.8, x,

1 f12.8,x, f12.8, x, f12.8, x, f12.8,x, f12.8, x,

1 f12.8,x, f12.8, x, f12.8)

333 format(f8.4,x,f8.4,x,f8.4,x,f8.4,x,f8.4,x,f8.4,x,f8.4,x,

1 f8.4,x,f8.4,x,f8.4,x,f8.4,x,f8.4,x,f8.4,x,f8.4)

1000 format(/,'SENSITIVITY ANALYSIS RESULTS:',/,

1'----------------------------',/)

9000 format (/,' HILLSLOPE SUCCESSFUL SIMULATION ',/)

1200 format(//10x,'Single storm rangeland erosion model output',/)

1300 format(/,'Parameterization flags',/)

3000 format ('Parameters',/,'-----------')

4000 format ('# parameter lower bound

1 upper bound unit ')

4100 format (2x,'slp',3x,'slplen',2x,'rain',4x,'dur',8x, 'xip',4x,

1 'ki',9x,'kr',4x,'schrit',4x,'ke',9x,'Ns',8x,'fr',4x,'fe',6x,

1'rspace',4x,'PSD',4x,

1'roff1',4x,'roof2',6x,'satvroof',6x,'absroof',2x,

1'sloss1',2x, 'sloss2',4x,'stsloss', 4x,'satvsl'/)

110

end

C----------------------------------------------------------------------

C FUNCTION RANDOM – free source code from the WEB.

CFUNCTION RANDOM. PRODUCES A PSEUDORANDOM REAL ON THE OPEN INTERVAL

C (0.,1.).

C DIX (IN DOUBLE PRECISION) MUST BE INITIALIZED TO A WHOLE NUMBER

C BETWEEN 1.D0 AND 2147483646.D0 BEFORE THE FIRST CALL TO RANDOM

C AND NOT CHANGED BETWEEN SUCCESSIVE CALLS TO RANDOM.

C BASED ON L. SCHRAGE, ACM TRANS. ON MATH. SOFTWARE 5, 132 (1979).

C----------------------------------------------------------------------

-

FUNCTION RANDOM(DIX)

C PORTABLE RANDOM NUMBER GENERATOR

C USING THE RECURSION

C DIX = DIX*A MOD P

C

DOUBLE PRECISION A,P,DIX,B15,B16,XHI,XALO,LEFTLO,FHI,K

C

C 7**5, 2**15, 2**16, 2**31-1

DATA A/16807.D0/,B15/32768.D0/,B16/65536.D0/,P/2147483647.D0/

C

C GET 15 HI ORDER BITS OF DIX

XHI = DIX / B16

XHI = XHI - DMOD(XHI,1.D0)

C GET 16 LO BITS IF DIX AND FORM LO PRODUCT

XALO=(DIX-XHI*B16)*A

C GET 15 HI ORDER BITS OF LO PRODUCT

LEFTLO = XALO/B16

LEFTLO = LEFTLO - DMOD(LEFTLO,1.D0)

C FORM THE 31 HIGHEST BITS OF FULL PRODUCT

FHI = XHI*A + LEFTLO

C GET OVERFLO PAST 31ST BIT OF FULL PRODUCT

K = FHI/B15

K = K - DMOD(K,1.D0)

C ASSEMBLE ALL THE PARTS AND PRESUBTRACT P

C THE PARENTHESES ARE ESSENTIAL

DIX = (((XALO-LEFTLO*B16) - P) + (FHI-K*B15)*B16) + K

C ADD P BACK IN IF NECESSARY

IF (DIX .LT. 0.D0) DIX = DIX + P

C MULTIPLY BY 1/(2**31-1)

RANDOM=DIX*4.656612875D-10

RETURN

END

APPENDIX E:

DUAL-MONTE-CARLO UNCERTAINTY ANALYSIS FORTRAN CODE

111

112 program main c Purpose: to assign the confidence interval to soil loss values. c Procedures: c 1) First randomly sample a point from the parameter space. c 2) Assign probability distribution function to each parameter. c 3) Assign CV value for each parameter. c 4) For each point, randomly sample points around the base value c using the assigned distribution function and deviation. c 5) Calculate the soil loss values for the local space. c 6) Get the confidence intervals (90%, 80%, and 70%).

implicit none integer mxplan,mxsamp parameter (mxplan=75) parameter (mxsamp=1000000) real avsole,rf,avsolem,rfm real avgslp(mxplan),slplen(mxplan) real rain(mxplan),stmdur real frcern(mxplan),frcrff(mxplan) real ks(mxplan),sm(mxplan),ki(mxplan),kr(mxplan),shcrit(mxplan)

real rspace(mxplan),media, xip(mxplan) integer i,j,k,l,m character*51 filen

character*51 filensa

character*65 ostrng real slpm,slplenm,rainm,durm,kim,krm,cshm,

1 kem,fem,frm,nsm,rrm,rspacem,mediam,xipm real slpcv,slplencv,raincv,durcv,kicv,krcv,cshstd,

1 kecv,fecv,frcv,nscv,xipcv,rspacecv,mediacv real xstmdr, xrain real slplw,slpup,slplenlw,slplenup,rainlw,rainup,durlw,

1 durup,kilw,kiup,krlw,krup,cshlw,cshup,kelw,keup,

1 felw,feup,frlw,frup,nslw,nsup,rrlw,rrup,rspacelw,rspaceup,

1 medialw, mediaup,xiplw,xipup integer nintv real sslplw, sslpup, sslplenlw,sslplenup real srainlw, srainup, sdurlw, sdurup real sxiplw, sxipup real skilw, skiup, skrlw, skrup real scshlw, scshup, skelw, skeup real sfelw, sfeup, sfrlw, sfrup real snslw, snsup real srspacelw,srspaceup,nsedialw,nsediaup integer nsamp,msamp,m5,m10,m15,m85,m90,m95 double precision seed integer inf real dinvnorm,random,PPND7 real avesloss(mxsamp), runoff(mxsamp),ranksloss(mxsamp) real sumsl(mxsamp), avesl(mxsamp), std, stdsl(mxsamp), cv(mxsamp)

113 real small,locat c********************************************************************** c OPEN OUTPUT FILES, including the output from rem, and sensitivity for c each parameter c (UNIT = 32, STATUS=3) c (UNIT = 101, ..., 114 , status = unknown) c**********************************************************************

ostrng = 'Enter file name for single event output -->'

call open(32,3,1,ostrng,filen)

ostrng = 'Enter file name for uncertt output -->'

call open(101,3,1,ostrng,filen) ostrng = 'Enter file name for parconfid output -->'

call open(102,3,1,ostrng,filen) ostrng = 'Enter file name for rank output -->'

call open(103,3,1,ostrng,filen) c************************************************************** c Read all the bounds, and nsamp c************************************************************** ostrng = 'Enter mean and cv of Slope--->' read (5,*) slplw, slpup ostrng = 'Enter mean and cv of Slope Length--->' read (5,*) slplenlw, slplenup ostrng = 'Enter mean and cv of rainfall volume--->' read (5,*) rainlw, rainup ostrng = 'Enter mean and cv ofrainfall duration--->' read (5,*) durlw, durup ostrng = 'Enter mean and cv of xip--->' read (5,*) xiplw, xipup ostrng = 'Enter mean and cv of Ki--->' read (5,*) kilw, kiup ostrng = 'Enter mean and cv of Kr--->' read (5,*) krlw, krup

ostrng = 'Enter mean and cv of Critical shear--->' read (5,*) cshlw,cshup ostrng = 'Enter mean and cv of ke--->' read (5,*) kelw,keup ostrng = 'Enter mean and cv of Ns --->' read (5,*) nslw,nsup ostrng = 'Enter mean and cv of fr--->' read (5,*) frlw,frup ostrng = 'Enter mean and cv of fe--->' read (5,*) felw,feup ostrng = 'Enter mean and cv of rill spacing --->' read (5,*) rspacelw,rspaceup ostrng = 'Enter lower and cvper bounds for PSD --->' read (5,*) medialw,mediaup

ostrng = 'Enter the numebr of total samples--->' read (5,*) nsamp ostrng = 'Enter the number of simulations inside one loop--->' read (5,*) msamp seed = 65 c*************************************************************** c Latin hypercube sampling loops (number of points: nsamp)

c*************************************************************** write(102,222) do 100 i = 1, nsamp c-------------------------------------------- c Use Latin Hypercube sampling method, sample in the range of input c (1) Slope c divide the range into subintervals, which is sample numbers - 1 c identify which subinterval will be used, Randomly. c nintv = random(seed) * nsamp c calculate the lower and upper bound for the subintervals sslplw = (slpup - slplw) / nsamp * nintv + slplw sslpup = sslplw + (slpup - slplw)/nsamp c Randomly pick a number between the subinterval

slpm = random(seed)* (sslpup - sslplw) + sslplw c c (2) slplen nintv = random(seed) * nsamp sslplenlw = (slplenup - slplenlw) / nsamp * nintv + slplenlw sslplenup = sslplenlw + (slplenup - slplenlw)/nsamp

slplenm = random(seed)* (sslplenup-sslplenlw)+sslplenlw c (3) rainfall nintv = random(seed) * nsamp srainlw = (rainup - rainlw) / nsamp * nintv + rainlw srainup = srainlw + (rainup - rainlw) / nsamp

rainm = random(seed) * (srainup - srainlw) + srainlw c (4) Rainfall duration nintv = random(seed) * nsamp sdurlw = (durup - durlw) / nsamp * nintv + durlw sdurup = sdurlw + (durup - durlw) / nsamp

durm = random(seed)* (sdurup-sdurlw)+ sdurlw c (5) ip

1111 nintv = random(seed) * nsamp sxiplw = (xipup - xiplw) / nsamp * nintv + xiplw sxipup = sxiplw + (xipup - xiplw) / nsamp

xipm = random(seed)* (sxipup-sxiplw) + sxiplw

if ( (xipm*rainm/durm) .gt. 254) goto 1111 c (6) Ki nintv = random(seed) * nsamp skilw = (kiup - kilw) / nsamp * nintv + kilw skiup = skilw + (kiup - kilw) / nsamp

kim = random(seed)* (skiup-skilw) + skilw c (7) Kr nintv = random(seed) * nsamp skrlw = (krup - krlw) / nsamp * nintv + krlw skrup = skrlw + (krup - krlw) / nsamp

krm = random(seed)* (skrup-skrlw) + skrlw c (8) Shcrit nintv = random(seed) * nsamp scshlw = (cshup - cshlw) / nsamp * nintv + cshlw scshup = scshlw + (cshup - cshlw) / nsamp

114

cshm = random(seed)* (scshup-scshlw) + scshlw c (9) Ke nintv = random(seed) * nsamp skelw = (keup - kelw) / nsamp * nintv + kelw skeup = skelw + (keup - kelw) / nsamp

kem = random(seed)* (skeup-skelw) + skelw c (10) ns nintv = random(seed) * nsamp snslw = (nsup - nslw) / nsamp * nintv + nslw snsup = snslw + (nsup - nslw) / nsamp

nsm = random(seed)* (snsup-snslw) + snslw c (11) fr nintv = random(seed) * nsamp sfrlw = (frup - frlw) / nsamp * nintv + frlw sfrup = sfrlw + (frup - frlw) / nsamp

frm = random(seed)* (sfrup-sfrlw) + sfrlw c (12) fe nintv = random(seed) * nsamp sfelw = (feup - felw) / nsamp * nintv + felw sfeup = sfelw + (feup - felw) / nsamp

fem = random(seed)* (sfeup-sfelw) + sfelw c (13) PSD (media) nintv = random(seed) * nsamp nsedialw = (mediaup - medialw) / nsamp * nintv + medialw nsediaup = nsedialw + (mediaup - medialw) / nsamp

mediam = random(seed)* (nsediaup-nsedialw) + nsedialw c (14) Rill space nintv = random(seed) * nsamp srspacelw = (rspaceup - rspacelw) / nsamp * nintv + rspacelw srspaceup = srspacelw + (rspaceup - rspacelw) / nsamp

rspacem = random(seed)* (srspaceup-srspacelw) + srspacelw c*************************************************************** c Assign the uncertainty to each parameter c using CV or STD describe the uncertainty c data calculated from Wepp database c***************************************************************

slpcv = 0

slplencv = 0

raincv = 0

durcv = 0

xipcv = 0

kicv = 0.30134

krcv = 0.3073

cshstd = 0.8539

kecv = -0.2312*log(kem)+0.9023

frcv = 0.12225

fecv = 0.14554

nscv = 0

rspacecv = 0

mediacv = -0.078068 c c*************************************************************** c Call RHEM to calculate the soil loss and runoff for the base c parameter set

115

116 c*************************************************************** call rhem(avsole,rf,slpm,slplenm,durm,rainm,xipm,

1 frm,fem,kem,nsm,kim,krm,cshm,rspacem,mediam)

avsolem =avsole/slplenm

rfm = rf c do 200 j = 1, msamp

avgslp(1) = PPND7(random(seed),inf)* slpcv *slpm + slpm

slplen(1) = PPND7(random(seed),inf)* slplencv *slplenm + slplenm

xrain = PPND7(random(seed),inf)* raincv * rainm + rainm

xstmdr = PPND7(random(seed),inf)* durcv * durm + durm xip = PPND7(random(seed),inf)* xipcv * xipm + xipm

ki(1) = PPND7(random(seed),inf)* kicv * kim+ kim

kr(1) = PPND7(random(seed),inf)* krcv * krm + krm shcrit(1) = PPND7(random(seed),inf)* cshstd + cshm

ks(1) = PPND7(random(seed),inf)* kecv * kem + kem

sm(1) = PPND7(random(seed),inf)* nscv * nsm + nsm

frcrff(1) = PPND7(random(seed),inf)* frcv * frm + frm

frcern(1) = PPND7(random(seed),inf)* fecv * fem + fem

media = PPND7(random(seed),inf)* mediacv * mediam + mediam

rspace(1) = PPND7(random(seed),inf)* rspacecv * rspacem+ rspacem c c Call RHEM and write to the output c call rhem(avsole,rf,avgslp,slplen,xstmdr,xrain,xip,frcrff,frcern,

1 ks,sm,ki,kr,shcrit,rspace,media) avesloss(j) = avsole/slplen(1) runoff(j) = rf

200 continue c DO loop 190 and 180 sort the soil loss values ascendingly.

do 190 k = 1, msamp

small = avesloss(k)

locat = k

do 180 l = k+1, msamp if (avesloss(l) .lt. small) then

small = avesloss(l)

locat = l

endif

180 continue

avesloss(locat) = avesloss(k)

avesloss(k) = small

190 continue c Calculate the mean sumsl(i) = avesloss(1) do 300 k = 2, msamp sumsl(i) = sumsl(i) + avesloss(k)

300 continue

avesl(i) = sumsl(i)/msamp

117 c Calculate standard deviation stdsl(i) = 0

do 400 k = 1, msamp std= (avesloss(k) - avesl(i))**2 stdsl(i) = stdsl(i) + std

400 continue

stdsl(i) = (stdsl(i)/(msamp-1))**0.5 c Given the ranked soil loss values, find the confidence intervals. m5 = msamp * 0.05 m10 = msamp * 0.10 m15 = msamp * 0.15

m85 = msamp * 0.85 m90 = msamp * 0.90 m95 = msamp * 0.95

1 write(102,111) slpm,slplenm,rainm,durm,xipm,kim,krm,cshm,kem,nsm,

1 fem,frm,mediam,

1 rfm,avsolem,

1 avesloss(m5), avesloss(m95),

1 avesl(i),stdsl(i)

100 continue

close (unit=32) close (unit=101) close (unit=102)

write (6,9000)

111 format(f5.3,x,f6.3,x, f7.3,x, f6.3,x, f6.3,x, f12.3,x, f8.4,x,

1 f8.4,x,f8.4,x, f8.4,x,

1 f8.4,x, f8.4,x, f8.4,x,

1 f8.4,x, x,f8.3,x,

1 f10.5,x, f10.5, x, f10.5,x, f10.5)

333 format(f6.4,x, f6.2,x, f6.2,x,f6.2,x, f12.2,x, f8.4,x,f12.4,x,

1 f12.4, x,f8.4,x, f8.2,x, f8.2,x,f8.4,x, f8.4,x, f8.2,x,

1 f8.6,f8.4)

222 format(/,x,'slp',x,'slplen',2x,'rain',4x,'dur',6x,'xip',4x,

1 'ki',8x,'kr',4x,'schrit',4x,'ke',6x,'Ns',6x,'fr',6x,'fe',4x,

1 'PSD',x,'runoff',x,'sloss',2x,'5% interval',5x,'95%

1 interval',/)

444 format(f10.5,x,f10.5)

9000 format (/,' HILLSLOPE SUCCESSFUL SIMULATION ',/)

1200 format(//10x,'Single storm rangeland erosion model output',/)

1300 format(/,'Parameterization flags',/)

3000 format ('Parameters',/,'-----------')

4000 format ('# parameter lower bound

1 upper bound cv ')

4100 format (/,2x,'slp',3x,'slplen',2x,'rain',4x,'dur',8x, 'xip',5x,

1 'ki',9x,'kr',4x,'schrit',4x,'ke',9x,'Ns',8x,'fi',4x,'fr',6x,

1'rspace',4x,'PSD',x,'runoff(m)',x,'sloss(kg/m2)')

end

C----------------------------------------------------------------------

C FUNCTION RANDOM – free source code from the WEB.

C FUNCTION RANDOM. PRODUCES A PSEUDORANDOM REAL ON THE OPEN INTERVAL

C (0.,1.).

118

C DIX (IN DOUBLE PRECISION) MUST BE INITIALIZED TO A WHOLE NUMBER

C BETWEEN 1.D0 AND 2147483646.D0 BEFORE THE FIRST CALL TO RANDOM

C AND NOT CHANGED BETWEEN SUCCESSIVE CALLS TO RANDOM.

C BASED ON L. SCHRAGE, ACM TRANS. ON MATH. SOFTWARE 5, 132 (1979).

C----------------------------------------------------------------------

FUNCTION RANDOM(DIX)

C

C PORTABLE RANDOM NUMBER GENERATOR

C USING THE RECURSION

C DIX = DIX*A MOD P

C

DOUBLE PRECISION A,P,DIX,B15,B16,XHI,XALO,LEFTLO,FHI,K

C

C 7**5, 2**15, 2**16, 2**31-1

DATA A/16807.D0/,B15/32768.D0/,B16/65536.D0/,P/2147483647.D0/

C

C GET 15 HI ORDER BITS OF DIX

XHI = DIX / B16

XHI = XHI - DMOD(XHI,1.D0)

C GET 16 LO BITS IF DIX AND FORM LO PRODUCT

XALO=(DIX-XHI*B16)*A

C GET 15 HI ORDER BITS OF LO PRODUCT

LEFTLO = XALO/B16

LEFTLO = LEFTLO - DMOD(LEFTLO,1.D0)

C FORM THE 31 HIGHEST BITS OF FULL PRODUCT

FHI = XHI*A + LEFTLO

C GET OVERFLO PAST 31ST BIT OF FULL PRODUCT

K = FHI/B15

K = K - DMOD(K,1.D0)

C ASSEMBLE ALL THE PARTS AND PRESUBTRACT P

C THE PARENTHESES ARE ESSENTIAL

DIX = (((XALO-LEFTLO*B16) - P) + (FHI-K*B15)*B16) + K

C ADD P BACK IN IF NECESSARY

IF (DIX .LT. 0.D0) DIX = DIX + P

C MULTIPLY BY 1/(2**31-1)

RANDOM=DIX*4.656612875D-10

RETURN

END

Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project