Development and Application of a Methodology for Sediment Source Identification. II: Optimization Approach

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1 Development and Application of a Methodology for Sediment Source Identification. II: Optimization Approach Latif Kalin 1 ; Rao S. Govindaraju ; and Mohamed M. Hantush 1 Abstract: In this paper, we re-examine the question of identification of source areas using optimization methods for two Iowa watersheds. The question of nonuniqueness is addressed first, with the result that only eight sediment contributing regions are allowed to be active simultaneously as in the companion study. Tukey s procedure is utilized to examine if the differences in erodibility of the different regions are statistically significant. Results were presented in the form of maps to obtain a spatial picture of the potential of sediment generating areas. Comparisons with physical characteristics of the watersheds showed good agreement between erosion potential and properties associated with erosion. DOI: / ASCE :3 194 CE Database subject headings: Watersheds; Sediment transport; Optimization; Erosion; Iowa. Introduction Optimization techniques are used in a wide class of problems that look for solutions under a set of physical constraints. Numerous applications of optimization can be found in water resources management. Several state of the art optimization techniques have been successfully employed in identification of groundwater pollution sources. The response matrix approach Gorelick et al. 1983, random walk model Bagtzoglou et al. 199, maximum likelihood method Wagner 199, minimization of relative entropy Woodbury and Ulrych 1996, and genetic algorithms Aral and Guan 1996; Aral et al. 001 are some examples of algorithms that have been utilized in an optimization framework. One of the more recent applications of optimization techniques to the problem of sediment source identification over a long time scale is through the use of fingerprinting techniques. The most common method is based on a numerical mixing model in which the relative contributions of different sources are identified by using tracer properties of sediments measured at a watershed outlet and at different source groups within the watershed. The composite fingerprint is represented by a set of linear equations and relative contributions of individual sources are established by minimizing the sum of squared residuals Collins et al. 1998; Walling et al. 1999; Owens et al. 1999; Russell et al Ina similar way, Kelley and Nater 000 used the chemical mass balance CMB air quality receptor model to apportion sediments to different catchment sources. The CMB model uses statistical 1 U.S. EPA, National Risk Management Research Laboratory, Cincinnati, OH School of Civil Engineering, Purdue Univ., West Lafayette, IN govind@ecn.purdue.edu Note. Discussion open until October 1, 004. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on May 13, 003; approved on July 1, 003. This paper is part of the Journal of Hydrologic Engineering, Vol. 9, No. 3, May 1, 004. ASCE, ISSN /004/ /$ optimization by minimizing variance-weighted differences between calculated and measured receptor concentrations. In this paper sediment source areas are identified by utilizing the KINEROS model in an optimization framework. The study watersheds and the data used are the same as those of the companion paper Kalin et al The KINEROS model is formulated based on the assumption that flow is Hortonian. Previous applications of KINEROS to the study watersheds indicates that the runoff generating mechanism in this watershed is truly Hortonian Kalin et al Mathematical Formulation Features of the KINEROS model along with the appropriate equations can be found in Woolhiser et al Only a brief description of the sediment transport equations is provided here. The mass balance equation describing the sediment dynamics at any point along a surface flow path is given by t AC s x QC s e x,t q s x,t (1) where Q flow discharge; A flow cross-sectional area; C s volumetric sediment concentration; e rate of erosion of the soil bed; and q s rate of lateral sediment inflow for channels. The rate of erosion for overland planes e is partitioned into two parts: splash erosion rate (g s ) caused by the splash of rainfall on bare soil and hydraulic erosion rate (g h ) due to the interplay between shear force of water on the loose soil bed and the tendency of soil particles to settle under the force of gravity e c f k h rq c g C mx C s A () In Eq., c f and c g constants that can be estimated from universal soil loss equation Foster et al ; r rainfall rate; q 194 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004

2 rainfall excess; and C s volumetric sediment concentration. The first term in the right-hand side of the equation represents the splash erosion term and the last term represents hydraulic erosion. The term k(h) represents the reduction in splash erosion caused by the increasing depth h of water and is given by the empirical relation k h e c hh (3) where c h damping coefficient. It is clear from Eq. 3 that splash erosion is modeled as decreasing exponentially with increasing flow depth. Therefore, hydraulic erosion is the dominant sediment generating mechanism in large storms. C mx is the equilibrium concentration and KINEROS offers six options to estimate the transport capacity of the flow in channels or on a plane element. Throughout this study, the Bagnold/Kilinc Kilinc and Richardson 1973 formula is used for the estimation of C mx, which states k and zero otherwise. Note that f (p) i (t), i 1,...,NE can be computed from KINEROS for any rainfall event. Eq. 6 enables the computation of sedimentograph for a rainfall event as a linear combination of individual sedimentographs generated by applying unit erodibility to each element of the watershed. This requires knowledge of C (p) 0,i values, and of the function f (p) i (t). This is a forward problem. The backward problem constitutes the identification of sediment source areas of differentiation of elements according to their erodibilities. In most practical situations, for the purpose of sediment source identification, information available to the modeler is in the form of sedimentograph, precipitation data, and watershed characteristics in terms of land use and soil information. So, the problem becomes one of estimating C (p) 0,i values based on observed sediment discharges. The C (p) 0,i can be regarded as an index to assess the relative erodibility of element i during rainfall event p. The goal of this paper is to estimate the values of C (p) 0,i in an optimization framework. If the error between observed Q (p) o (t n ) and computed sediment discharge Q(t n ) at nth observation time t n is defined as C mx C 0 u c 1.67 h w u c (4) NE n p Q o p t n i 1 C p 0,i f p i t n (8) where w hs with w being specific weight of water; h flow depth; s slope; c Shields critical shear stress; u velocity of water; and C 0 scaling parameter. During large events, the critical shear stress is almost always exceeded. There are no definite guidelines for estimating C 0. In fact, derivation of Eq. 4 was based on relating sediment discharge to stream power under the assumed model of q s K c u m (5) As only large rainfall events are considered in this study, it is assumed that all sediment particles are flushed to the watershed outlet in a single event. Noting that the sediment transport Eq. 1 is linear, the sedimentograph is linearly dependent on C 0. The KINEROS requires the watershed to be subdivided into subwatersheds or elements. The superposition of the sedimentographs obtained by routing sediment generated from each element to the outlet was found to be practically identical to the sedimentograph generated over the entire watershed, again a consequence of linearity. As in most physically based distributed models, the study watershed may be partitioned into (NE) elements. Then for a large rainfall event p, the following relationship applies from linearity: NE Q p t i 1 C p 0,i f p i t (6) In this equation, Q (p) (t) represents the sedimentograph, the time distribution of sediment discharge, resulting from rainfall event p. C (p) 0,i value of the C 0 parameter for element i during rainfall event p and f (p) i (t) unit sedimentograph resulting from rainfall event p under the condition C 0,k i,k (7) where i,k kroneker delta function which is equal to one for i then the objective function to be minimized is N F n 1 n p (9) where N total number of data points in the observed sedimentograph. Since erodibilities cannot be negative, the constraint C (p) 0,i 0 has to be imposed during optimization. Model Application There is no easy way of measuring the degree of erodibility for different parts of a watershed. One possible way is by conducting expensive and time-consuming tracer experiments which are not available for the study watersheds. Therefore, the applicability of optimization was first tested by several scenarios with synthetic data from W- watershed see Kalin et al. 004 for watershed description. Measured data were used for rainfall. Some areas within W- watershed were assigned high priorities in terms of sediment production, and input parameters were selected accordingly. The sedimentographs obtained by running KINEROS were used as input to the optimization model to see how accurately it identifies areas with high potential for sediment generation. This was repeated with different parts of the watershed being assigned as high sediment producing areas. The high risk areas found from the proposed methodology were then compared to the initially assigned areas to evaluate model performance. After validation with synthetic data, the model was applied on watersheds W- and W-3 with observed rainfall data, runoff hydrographs, and sedimentographs. This procedure was repeated for several rainfall events and a statistical procedure was used to see if statistically significant differences in erodibilities among different elements could be obtained. Details on data and study watersheds are given in the accompanying paper Kalin et al. 004, therefore it is not repeated here. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004 / 195

3 Table. Estimated C 0 Values for Selected Events for W- Watershed Scenario Fig. 1. Plane elements of W- for first configuration As a starting point, the simplest case corresponding to only three subareas was tried initially. There are only three overland flow planes elements in this case, numbered as 3, 4, and 5 Fig. 1, and consequently three unknown C 0 values. As discussed earlier, the functions f i (p) (t) for each of the three elements (i 3,4,5) were generated using the KINEROS model. All possible scenarios were considered. Table 1 summarizes the combination of C 0 values used in each scenario. The C 0 values in the table are selected randomly as no clear guidelines are given in the KINEROS manual for the range of C 0 values. Entries in the far left column are used to determine which elements are assigned erodibilities. For the entries in this left column, the first number indicates the number of sources and the numbers after the underscore sign are elements that are allowed to erode. The C 0 values assigned to each element are listed in the corresponding row. For example in scenario 35 two elements are contributing to sediment generation Elements 3 and 5 with both being assigned a C 0 value of 10. Sedimentographs were generated for each scenario using 13 rainfall events i.e., p 1,,...,13). The generated sedimentographs and f i (p) (t) functions served as inputs to the optimization model. Due to the small number of variables and constraints, and simplicity of the problem, it was decided to use MS/Excel Solver to estimate C 0 values. By minimizing the objective function 9, C 0 values were estimated. Results for four of the 13 events are shown in Table. These indicate that, estimated C 0 values are 5/30/ All /1/ All /8/ All /5/ All close to originally assigned values shown in Table 1. This helps to demonstrate that the methodology works for sediment source identification. Next, a configuration with eight elements labeled from 5 to 1 as in Fig. was considered for W-. A total of eight C 0 values need to be estimated in this case. For the 13 rainfall events, a very large number ( 8 1)*13 3,315 of simulations need to be carried out. A total of 3 scenarios were considered and are Table 1. Combination of C 0 Values used in Each Scenario for Low Resolution for W- Watershed Scenario Fig.. Plane elements of W- and W-3 with eight subwatersheds 196 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004

4 Table 3. Combination of C 0 Values used in Each Scenario for W- with Eight Possible Source Areas as Shown in Fig. Scenario shown in Table 3. This required 416 simulations for the 13 rainfall events. Among the 3 scenarios listed in Table 3, there are 8 scenarios with a single source, 8 scenarios with sources, 5 with 3 sources, 5 with 4 sources, 3 with 5 sources and one scenario for each of 6, 7, and 8 sources. As expected, with an increasing number of sources the identification of the right combination of sediment sources becomes increasingly uncertain. Therefore, more attention is paid to scenarios with fewer sources at this stage. The format and labeling scheme of Table 3 are the same as Table 1. For example, the scenario named in Table 3 corresponds to a case where 7 different parts of W- are eroding. These eroding elements are 5, 6, 7, 8, 10, 11, and 1, and the assigned C 0 values are 15.0, 1.5, 10.0, 7.5, 5.0,.5, and 1.0, respectively. Again the functions f (p) i (t) for each of the eight elements (i 5,6,...,1) and the sedimentographs for each scenario with 13 (p 1,,...,13) rainfall events were generated. The objective function 9 was minimized to solve for C (p) 0,i values. For brevity, only results are shown in Table 4 for the W- watershed for a single event details in Kalin 00. The last column in Table 4 is a statistic used to measure the model performance between sedimentographs computed using Eq. 6 and sedimentographs generated by running the watershed model with the C 0 values listed in Table 3. The statistic used is (1 R N ) where R N is the Nash- Sutcliffe efficiency measure. Generally, (1 R N ) values are very small, around 10 8, leading to the conclusion that the model performs quite well in terms of matching the sedimentographs, except for two events: 6/14/198 and 8/15/1977. The (1 R N ) values for these two events are around In an absolute sense, 10 5 is still a very small error but a relative comparison to (1 R N ) values of other events suggests that model performance was not as good for these two events. The performance of this methodology may depend on the rainfall pattern. To evaluate the accuracy of source identification with eight elements, R N values between optimized C 0 values and the original values were also computed for each scenario and rainfall event, and are summarized in Table 5. To distinguish the two R N estimates, the latter will be denoted by R NC measure of agreement between C 0 values, and the former will be denoted by R NS measure of agreement between sedimentographs. Careful investigation of results in Table 5 reveals that it is possible to obtain very different C 0 values than the originally assigned C 0 values after optimization, and yet have very high R NS values. This kind of nonuniqueness is encountered in many parameter estimation problems. For example, for the event 6/14/ JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004 / 197

5 Table 4. Estimated C 0 Values for Event of 5/30/198 for W- Watershed for Different Synthetic Source Generating Regions Scenario R N E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E and scenario 1 11, the optimization routine found C 0 values of 6.59 for Element 7, 1.74 for Element 8, 1.94 for element 10, and.00 for Element 1, while the correct identification could have been C 0 10 assigned to Element 11 and zero to other elements. The value of (1 R NS ) is 3.11E 6 for this case implying a very good prediction. If the C 0 values assigned in the original scenario were to be used, the (1 R NS ) value would be 1.6E 3 which leads to R NS Although there is a significant difference in C 0 values, both result in very high R NS values implying that nonuniqueness can be a severe limitation. The results are graphically illustrated in Fig. 3. The sedimentograph obtained by running the watershed model with the C 0 s assigned in the scenario goal 11-hollow circles and the sedimentograph calculated from Eq. 6 computed 11-dashed line are shown on the left axes and f (p) 11 (t) is shown on the right axes. As can be seen, it is very difficult to distinguish one graph from another except for some very small differences in the rising limbs. In other words, both sets of C 0 values resulted in sedimentographs close in shape to the target sedimentograph. In contrast to the very high R NS, R NC is 0.554, a rather low value. At first glance, this may appear to be contradictory. However, it should be noted that the correlation between R NS value of R NC and R NC is not double sided. A high always results in high R NS. But, a high R NS does not necessarily imply a high R NC. There could be some solutions that may result in larger R NS, though giving small or even negative R NC. As with all numerical models, some numerical errors are present that one must contend with. This error gets compounded to some degree when superposing the results using several f (p) i (t) functions. Due to these small differences, the optimization routine can find different combinations than the one defined in the scenario depending on the rainfall pattern. The problem of nonuniqueness associated with identification of appropriate C 0 values is expected to become worse with increasing watershed resolution and more elements because of the corresponding increase in degrees of freedom. To demonstrate this further, R NC values are averaged over 13 rainfall events as a function of the number of sources and plotted against the number of sources in Fig. 4 a. Also in Fig. 4 a shown is the plot of standard deviations of average R NC values against number of sources. The coefficient of variations COV are shown in Fig. 4 b for completeness. From the general trend in these figures, it 198 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004

6 Table 5. Summary of Nash-Sutcliffe statistics, R NC between C 0 Values Obtained by Optimization and Originally Assigned Values for W- Watershed for Different Rainfall Events Scenario 5/30/198 6/1/1980 6/13/1983 6/14/198 6/15/198 6/18/1980 7/8/1981 8/1/1981 8/15/1977 8/6/1981 8/9/1975 9/5/1980 9/17/198 JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004 /

7 Fig. 3. Sedimentographs for scenario 1 11 and rainfall event 6/14/198 is apparent that increasing the number of sources decreases the mean R CN values, but increases the standard deviations and COVs, making the procedure of identifying sediment sources by optimization less reliable. This implies that increasing the number of sources leads to larger uncertainty in source identification. This conclusion is in agreement with the results of Javitz et al They showed during the testing of a chemical mass balance CMB model to identify the source of air pollution that five to seven sources are usually linearly independent of each other. They also stated that if the number of sources exceeds 7, the ability of the model to adequately differentiate the sources becomes uncertain. Consequently, further subdivision of the watershed was not considered. Applications to W- and W-3 with Observed Sediment Data Simulations carried out with artificial scenarios provided the range of applicability of optimization methods in sediment source and strength identification. Following this, observed sediment discharge data were used. Simulations were carried out for both the W- and W-3 watersheds. A total of 14 events for W- and 11 events for W-3 were utilized. It should be observed that there is no independent method of verification. Furthermore, contrary to the artificial scenarios, the number of data points in observed sedimentographs was not always abundant. The number of data points varies from event to event depending on the data collection efforts. For W-, this number varies from 13 to 37, and from 14 to 40 for W-3. The more the number of data points the better the model performance is expected to be. Again, by minimizing the summation of least square errors, C 0 values were estimated by the optimization routine and are shown for several events over W- and W-3 in Tables 6 and 7, respectively. The numbers are the C 0 values estimated by the optimization procedure followed by shaded values showing relative percentages of erodibilities C 0 of each element for each rainfall event. All the elements were assigned zero erodibility as the initial condition in the optimization. It was observed that the selected initial C 0 values do not have an impact on the optimization results. The average R NS for W- is 0.81, and 0.8 for W-3. Model performance should not be based on sedimentographs but evaluated in terms of consistency between C 0 values estimated from different rainfall events for each element. As mentioned earlier, C 0 values are expected to vary with rainfall pattern. Further, seasonal changes might have significant effect on soil erodibility, especially in agricultural areas. Observed and predicted sedimentographs for W- and W-3 from Eq. 6 for three events are shown in Figs. 5 and 6 for illustration. Observed and predicted hydrographs along with the rainfall histograms are also shown in the figures. Results and Discussion The first impression from Tables 6 and 7 is that the estimated C 0 values for each element show quite a large variation between rainfall events. To have a robust estimate of C 0 values, it Fig. 4. a Mean and standard deviation std of R NC plotted against number of sources values plotted against number of sources. b Coefficient of variation COV of R NC values 00 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004

8 Table 6. Estimated C 0 Values for W- Event R NS Number of data 05/30/ % /1/ % /13/ % /14/ % /15/ % /18/ % /30/ % /08/ % /01/ % /15/ % /6/ % /9/ % /05/ % /17/ % Mean Mean % Table 7. Estimated C 0 Values for W-3 Event R N Number of data 05/19/ % /5/ % /8/ % /04/ % /07/ % /13/ % /7/ % /03/ % /18/ % /01/ % /1/ % Mean Mean % JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004 / 01

9 Fig. 5. Observed hollow circles and predicted solid line sediment discharges top for three sample events from W-. Bottom graphs show hydrographs and rainfall histograms is first necessary to have a good agreement between the observed and computed runoff hydrograph, as the sedimentograph is a function of the hydrographs. This is vital, since for a uniformly eroding watershed the sedimentograph generally follows the same pattern as the hydrograph for a given rainfall event. The errors in hydrographs carry over and confound source identification results as well. For instance, if the computed hydrograph is shifted toward the origin when compared to the observed hydrograph, then the optimization scheme assigns less weight to the elements closer to the outlet to move the sedimentograph away from the origin. On the contrary, if the computed hydrograph is shifted away from the origin with respect to the observed hydrograph, Fig. 6. Observed hollow circles and predicted solid line sediment discharges top for three sample events from W-3. Bottom graphs show hydrographs and rainfall histograms 0 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004

10 then the optimization routine tends to assign more weights to elements closer to the outlet to bring the sedimentograph closer to the origin. Hence, one should be very careful in interpreting the results. Results for W- As mentioned earlier, although R NS values are sufficiently large, estimated C 0 values are not consistent between events. Some events need further attention. The computed hydrograph of the event 09/17/198 was shifted away from the origin compared to observed hydrograph see Kalin 00 and this effect is carried over to the sedimentograph. As anticipated, the optimization routine has assigned all the weight to Element 5, since it gives a faster response than all the other elements. Intuitively, the optimization routine is expected to assign some weight to Element 6 owing to the fact that they are both located at the mouth of the watershed. However, Element 6 is about 63% larger in area than Element 5 and this results in more dispersed sedimentograph for Element 6, since the average travel distance is longer. Consequently, although the starting time of the sedimentographs generated from Elements 5 and 6 are the same, sedimentograph base times differ and is larger for Element 6. The time lag for this event is around 5 min and this is reflected in the R NS value, which is Element 5 might really be the sole contributing element. However, considering the small R NS, it is clear that any estimate with this event is susceptible to error. Similar anomalies are present with the events 06/18/1980 and 08/6/1981, but in these events the computed hydrographs were shifted toward the origin compared to observed hydrographs. As a result, the estimated C 0 values are zero for all the elements except for Element 10, which is the farthest element from the outlet of watershed W-. Again as a result of this shifting, the R NS values are very low 0.48 for 06/18/1980 and 0.56 for 08/6/1981. Even if the above-mentioned three events are discarded, a strong and clear conclusion is not forthcoming. Some general trends may however be elucidated. One observation is that Elements 8 and 11 almost never contribute to erosion according to this procedure. The total contribution of these elements is just 0.5%. In general, Elements 5 and 10 seem to have the highest erodibilities. Element 5 has an average 3.1% relative erodibility and Element 10 has an average 30.8% relative erodibility. They account for 6.9% of the whole watershed in terms of erosion potential. Results for W-3 Although average R NS is about the same as for W-, results for this watershed show greater scatter. The lowest R NS is 0.6 corresponding to event 08/1/1986. The elements dominating the erosion process are 10 and 1 see Fig. for element configurations, which have average erodibilities of 30.7 and 6.6%, respectively, and account for 57.3% of the whole watershed. Element 11 never contributes to erosion. Again as mentioned earlier for W-, the true erosion potentials of different areas within W-3 are not known and therefore results cannot be verified independently. As in the companion paper Kalin et al. 004 Tukey s procedure was implemented to compare the erosion potentials of different areas in a statistical fashion. Test results are shown on Tables 8, 9, and 10 for W-, and Tables 11, 1, and 13 for W-3. Table 8. Tukey s Test over W-: Summary Statistics Groups Count Sum Average Variance , , Table 9. Tukey s Test over W-: Analysis of Variance Results Source of Variation Sum of squares Degree of freedom Mean square F p-value Between groups 14, Within groups 6, Total 77, Table 10. Tukey s Test over W-: w Values for Various Levels of Q w Table 11. Tukey s Test over W-3: Summary Statistics Groups Count Sum Average Variance , ,700 Table 1. Tukey s Test over W-3: Analysis of Variance Results Source of variation Sum of squares Degree of freedom Mean square F p-value Between groups 9, , Within groups 6, Total 7,0 87 Table 13. Tukey s Test over W-3: w Values for Various Levels of Q w JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004 / 03

11 Table 14. Grouping of Elements for W- According to Tukey s Procedure with Varying Levels Element Mean A A A A A A A B A A A B A B A B A B A B A B A B A B A B B B B C B B C Table 15. Grouping of Elements for W-3 According to Tukey s Procedure with Varying Levels Element Mean A A A A A B A B A A B A B A A B A B A A B A B A A B A B A A B A B A B B Tables 8 and 11 show summary statistics, and Tables 9 and 1 present results of the ANOVA test for W- and W-3, respectively. The p-values are 0.00 for W- and for W-3. In other words, relative contributions of different elements to erosion are not the same for confidence levels smaller than 99.8% for W- and 88.1% for W-3. The sample variances of the C 0 values between observations are much higher for W-3 than the sample variances of W- which lead to a smaller confidence interval for W-3. Tables 10 and 13 summarize w values for different levels for W- and W-3. Elements, with mean C 0 values differing by more than w, are said to be statistically significant in terms of their differences. Tables 14 and 15 show the grouping of elements with this procedure for various levels for W- and W-3, respectively. In case of W- for 0.10, Elements 10 and 5 have significantly higher erodibilities than Elements 11 and 8. If a level of 0.0 is used, in addition to the previous conclusion, it can be claimed that Element 9 also has a significantly lower contribution to the erosion process than Elements 10 and 5. Finally, when 0.30, it is seen that Element 5 has a considerably higher erodibility than all the other elements except Element 10. A similar conclusion can be drawn for W-3. For 0.10, the differences are not statistically significant. This in fact supports the ANOVA results, where the p-value was 0.119, meaning any smaller than this value would lead to the conclusion of equal means. Using 0.0 and 0.30 reveals the same conclusions, where only Elements 10 and 11 significantly differ from each other with Element 10 having higher erodibility potential. Except for Elements 10 and 11, the differences in strength for the remaining elements are statistically insignificant. In Figs. 7 a and 8 a high and low erosion potential areas of W- and W-3, respectively, are shown based on C 0 values. In both figures, darker colors portray highly erodible areas. In general, results are consistent with the results from the companion paper Kalin et al. 004 for both watersheds, shown in Figs. 7 b and 8 b. Although the ranking of elements according to their erodibilities are not exactly the same, the general trends are similar. In other words, high and low erosion potential areas show similarities in both methods. It should be noted that significant statistical differences between relative erodibilities of the elements are only produced at rather high levels. On the other hand, though an value of 0.0 or 0.30 is rarely used in most hydrologic applications, it would appear that statistically significant results are not likely to be available given the nature of the data. To complement the analysis, erosion potentials of different parts of W- and W-3 watersheds were also evaluated using the universal soil loss equation USLE Wischmeier and Smith 1978 Fig. 7. Erosion potential map of W- based on a C 0, b C k Kalin et al / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004

12 Fig. 8. Erosion potential map of W-3 based on a C 0, b C k Kalin et al. 004 A R K LS C P (10) where A average annual soil loss in tons per acre per year; R rainfall and runoff factor for a given location, K soil erodibility index; L slope length factor; S slope steepness factor; C cover and management factor; and P conservation or support practice factor. The R, C, and P factors are the same for all elements of W- and W-3. Hence, erosion potentials were assessed based on only K and LS factors. K is equal to 0.38 for silt loam and 0.3 for silty clay loam, the basic two soil types in W- and W-3. LS is given by Wischmeier and Smith 1978 LS slope slope NN slope length.1 (11) with slope in % and slope length in meters. NN slope dependent parameter as shown in Table 16. KLS factors and their relative values, KLS were computed for all eight elements of W- and W-3. Based on these values, erosion potential maps were generated for both watersheds and are shown in Fig. 9. In W-, Element 10 has the lowest erodibility based on the USLE equation, but was judged as the highest erosion potential by both methods presented in this paper and the companion. Since, both of these methods rely on optimization, it seems that Element 10 has received the highest weight due to shifting of observed hydrographs away from the origin with respect to computed hydrographs. The remaining elements have comparable erosion potentials. In particular, Element 5 has the highest erosion potential in all three methods if Element 10 is ignored. Results in W-3 are more consistent. Elements 10 and 1 were among the high erosion risk areas in all three methods. The remaining elements show good agreement as well. between observed and computed sediment discharges, it was hoped to identify relative erodibilities of individual elements of the study watersheds through the C 0 coefficient which appears in the Kilinc/Bagnold sediment transport model. Results obtained through assigning artificial C 0 values to elements of W- with real rainfall data demonstrated the range of applicability of this methodology. Outcomes of the model revealed that sediment source areas could be reasonably identified as long as the number of sediment generating areas is less than eight. When the number of sources exceeds eight, the uncertainty of source identification increases due to the nonuniqueness problem. In other words, the number of source combinations resulting in similar sedimentographs under the same conditions increases. The plot of number of sources against average R NC, which is the Nash-Sutcliffe statistic between originally assigned and estimated C 0 values, supports this argument. The average standard deviations of R NC val- ues increase with an increasing number of sources, leading to the conclusion that reliability decreases with increasing number of sources. Next, the same procedure was performed with observed sediment data for both W- and W-3. Results were not very promising for either watershed, for several reasons. The first, and perhaps the most important one, is that small measurement errors are reflected in the optimization scheme as large differences in C 0 estimates showing the sensitivity of the optimization routine to measurement errors. It is believed that this procedure may work better with larger watersheds with a limited number of eroding elements, where the sedimentograph timings between elements show significant differences. In large watersheds, the durations of sedimentographs are long enough to better compensate for data errors. Another reason for having inconsistent C 0 values is the possible offset between observed and computed hydrographs in some events, which eventually affects sedimentographs. When the computed hydrograph is shifted toward the origin with respect to the observed hydrograph, the optimization routine tends to assign high weights to the more remote elements in the watershed, Summary and Conclusions An optimization routine has been utilized in this paper as an alternative method for identification of sediment source areas. Observed sediment discharge data along with rainfall served as the main input. By minimizing the sum of the square errors Table 16. NN Values in LS Factor of Universal Soil Loss Equation Slope 1% %1 slope 3% 3% slope 5% slope 5% JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004 / 05

13 Fig. 9. Erosion potential map of W- and W-3 based on KLS to compensate for the error in the hydrograph. Similarly, a shift away from the origin results in a higher weight for the element closest to the watershed outlet. Results revealed that on the average, Elements 5 and 10 have the highest erodibility potential and Elements 8 and 11 have the lowest potential for W-. In fact, Elements 8 and 11 did not contribute to erosion at all. Elements 10 and 1 are the elements most susceptible to erosion in W-3 with Element 11 never contributing to erosion. These results were also supported using Tukey s procedure with various significance levels. From simulations carried out with artificial data, it was found that the number of source areas for this procedure to have a reasonable chance at identifying sources should not exceed eight. Violation of this constraint may lead to an improper estimation of hydrographs and sedimentographs. If more elements are needed, then they can be grouped in such a way that elements falling under the same category could be assigned the same erodibility or C 0 value. This can easily be done within the optimization framework by enforcing more constraints. Acknowledgments This research was funded by EPA, Award No. R It has not been subjected to Agency review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred. The writers gratefully acknowledge this support. Special thanks to Larry Kramer, USDA-ARS-NSTL- DLRS, Council Bluffs, IA, for supplying all the data sets. References Aral, M. M., and Guan, J Genetic algorithms in search of groundwater pollution sources. Advances in groundwater pollution control and remediation, M. M. Aral, ed., Vol. 9, Kluwer, Dodrecht, The Netherlands, Aral, M. M., Guan, J., and Maslia, L. L Identification of contaminant source location and release history in aquifers. J. Hydrologic Eng., 6 3, Bagtzoglou, A. C., Dougherty, D. E., and Tompson, A. F. B Application of particle methods to reliable identification of groundwater pollution sources. Water Resour. Manage., 6, Collins, A. L., Walling, D. E., and Leeks, G. J. L Use of composite fingerprints to determine the provenance of the contemporary of the suspended sediment load transported by rivers. Earth Surf. Processes Landforms, 3, Foster, G. R., Smith, R. E., Knisel, W. G., and Hakonson, T. E Modeling the effectiveness of on-site sediment controls, Paper 83-09, presented at the summer 1983 meeting, Bozeman, Mont., American Society of Agricultural Engineers, Saint Joseph, Mich. Gorelick, S. M., Evans, B., and Ramson, I Identifying sources of groundwater pollution: An optimization approach. Water Resour. Res., 19 3, Javitz, H. S., Watson, J. G., and Robinson, N. F Performance of the chemical balance method model with simulated land-scale aerosols. Atmos. Environ.,, Kalin, L. 00. Sediment source area identification over watersheds: Influence of spatial scale and sediment travel times. PhD dissertation, Purdue Univ., W. Lafayette, Ind. Kalin, L., Govindaraju, R. S., and Hantush, M. M Effect of geomorphologic resolution on runoff hydrograph and sedimentograph. J. Hydrol., 76, Kalin, L., Govindaraju, R. S., and Hantush, M. M Development and application of a methodology for sediment source identification. I: Modified unit sedimentograph approach. J. Hydrologic Eng., 9 3, Kelley, D. W., and Nater, E. A Source apportionment of bed sediments to watersheds in an Upper Mississippi basin using a chemical mass balance method. Catena, 41, Kilinc, M., and Richardson, E. V Mechanics of soil erosion from overland flow generated by simulated rainfall. Hydrology Paper 63, Colorado State Univ., Fort Collins, Colo. Owens, P. N., Walling, D. E., and Leeks, G. J. L Use of floodplain cores to investigate recent historical changes in overbank sedimentation rates and sediment sources in the catchment of the River Ouse, Yorkshire, UK. Catena, 36, Russell, M. A., Walling, D. E., and Hodgkinson, R. A Suspended sediment sources in two small lowland agricultural catchments in the UK. J. Hydrol., 5, 1 4. Wagner, B. J Simultaneous parameter estimation and contaminant source characterization for coupled groundwater flow and contaminant transport modeling. J. Hydrol., 135, Walling, D. E., Owens, P. N., and Leeks, G. J. L Fingerprinting 06 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004

14 suspended sediment sources in the catchment of River Ouse Yorkshire, UK. Hydrolog. Process., 13, Wischmeir, W. H., and Smith, D. D Predicting rainfall erosion losses a guide to conservation planning. U.S. Dept. of Agriculture, AH-537, Washington, D.C. Woodbury, A. D., and Ulrych, T. J Minimum relative entropy inversion: The release history of a groundwater contaminant. Water Resour. Res., 3 9, Woolhiser, D. A., Smith, R. E., and Goodrich, D. C KINEROS A kinematic runoff and erosion model: Documentation and user manual. U.S. Dept. of Agriculture, Agricultural Research Service, ARS-77, Tucson, Ariz. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 004 / 07