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Published online 1 September 2009
Published in Agron J 101:1068-1079 (2009)
DOI: 10.2134/agronj2008.0207x
© 2009 American Society of Agronomy
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Erosion Index Derived from Terrain Attributes using Logistic Regression and Neural Networks

A. C. Pikea, T. G. Muellerb,*, A. Schörgendorferc, S. A. Shearerd and A. D. Karathanasisb

a Photo Science, Lexington, KY 40503-3302
b Dep. of Plant and Soil Sci., Univ. of Kentucky, Lexington, KY 40546-0091
c Dep. of Statistics, Univ. of Kentucky, Lexington KY, 40506-0027
d Biosystems and Agricultural Eng., Univ. of Kentucky, Lexington, KY40546-0276

* Corresponding author (mueller{at}uky.edu).


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Maps identifying areas prone to channel erosion within agricultural fields could be useful for conservation planners. The objective of this study was to test an approach for creating such maps with logistic regression and neural networks. Survey grade elevation measurements were obtained from on a Central Kentucky farm. The elevation measurements were used to create 4 by 4-m digital elevation models (DEMs) from which terrain attributes were derived. Areas exhibiting evidence of erosion caused by overland water flow sufficient to justify the placement of grassed waterways were identified. The terrain attributes were used as predictor variables and models were fit using the field assessments of soil erosion. Leave-one-field-out validation analysis was conducted to assess the quality of predictions maps. For the models created with logistic regression, an average of 14% of the 4 by 4-m grid cells in noneroded areas were incorrectly classified as being eroded and 16% of cells in eroded areas were incorrectly classified as noneroded. For neural network analysis, these error rates were 15 and 19%, respectively. Most of these errors occurred because the analyses did not exactly define the shapes of the eroded features; however, both logistic regression and neural networks identified most waterway features in all fields. The proposed three-variable logistic regression model for erosion prediction should only be tested with datasets constructed using identical procedures for DEM creation and terrain analysis. This approach could improve the efficiency and accuracy of field site assessments for conservation planning.

Abbreviations: CREAMS, chemicals, runoff and erosion from agricultural management systems • CRP, conservation reserve program • DEM, digital elevation model • EGEM, ephemeral gully erosion model • GeoWEPP, Geo-spatial interface for the Water Erosion Prediction Project • GIS, Geographic Information Systems • GPS, global positioning system • HEL, highly erodible land • LIDAR, light detecting and ranging • NRC, National Research Council • NRCS, Natural Resource Conservation Service • RTK, real-time kinematic • RUSLE, revised universal soil loss equation • USGS, United States Geological Survey • VRT, variable rate technologies

Received for publication November 24, 2008.
   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
EPHEMERAL GULLIES are erosion features that result from concentrated water flow and tend to occur in the same locations year after year, even if they are temporarily repaired with tillage (Foster, 1986). This type of erosion is a serious problem in the United States and throughout the world. Bennett et al. (2000) examined erosion rates for 19 U.S. states published by the NRCS (1997). On average, ephemeral gully erosion accounted for approximately 38% of total erosion. In a Belgian study, ephemeral gully erosion accounted for 41% of total soil loss (Vandaele and Poesen, 1995).

Grassed waterways are effective conservation tools for reducing ephemeral gully erosion (NRC, 1986) because they trap sediment (Hayes et al., 1984) and reduce runoff (Briggs et al., 1999; Fiener and Auerswald, 2003). Chow et al. (1999) found that grassed waterways combined with terraces reduced erosion by 95%. In a German study, grassed waterways reduced erosion by 97% when properly designed and installed (Fiener and Auerswald, 2003). If eligibility requirements are met, the USDA provides payments for installing grassed waterways and maintaining them for 10 to 15 yr under the continuous sign-up provision of the conservation reserve program (CRP). Specifically, landowners receive annual rental payments, up to 50% cost share, and an additional incentive payment for installing waterways (NRCS, 2008a). To determine whether CRP-eligible grassed waterways can be established in agricultural fields, an NRCS conservationist must first make an on-farm site assessment. This involves walking across fields to find areas with evidence of erosion resulting from concentrated water flow. For large farms or regions, this can be a slow and time-consuming process and eroded channels scattered across large fields can easily be missed.

Potentially, precision conservation techniques could be used to improve the efficiency of identifying waterways. Precision conservation involves using geospatial technologies to ensure that soil and water resources will be sustained for future generations (Delgado and Berry, 2008). These technologies include GPS, geographic information systems (GIS), remote sensing, variable rate technologies (VRT), soil and crop sensors, and terrain attributes. Terrain attributes describe the shape of the landscape and are calculated from DEMs which can be downloaded from the USGS website. Better quality DEMs can be developed with elevation measurements obtained using survey-grade RTK GPS and light detecting and ranging (LIDAR).

Several studies have demonstrated that terrain attributes can be used to identify eroded channels. For example, Thorne et al. (1986) found that ephemeral gullies occurred in Mississippi fields when the product of upslope drainage area, slope, and plan curvature exceeded soil-specific threshold values. Two other studies found that ephemeral gullies were observed when threshold values of the topographic wetness index and the product of the specific catchment area and slope were exceeded (Moore et al., 1988; Srivastava and Moore, 1989). The thresholds for the topographic wetness index values varied between these two studies likely because of differences between soils (Srivastava and Moore, 1989).

Moore and Nieber (1989) suggested that terrain attributes could be used to identify highly erodible land (HEL) to identify areas that could be enrolled in government conservation programs. In the United States, eligibility for general (not continuous) CRP payments is based in part on the amount of HEL. Our interest was in identifying eroded land that could be eligible for continuous CRP payments if approved grassed waterways were installed. Berry et al. (2005) developed an innovative approach for calculating erosion index maps from terrain attribute data. They classified slope and flow into classes and assigned integer values to each class. Then they multiplied these class layers together and developed a scheme to interpret this layer into erosion potential maps. While this approach was effective, considerable trial and error is required to determine the most appropriate thresholds. Erosion modeling software can also be useful for conservation planning but most models do not predict the occurrence and location of ephemeral gully erosion. For example, the GeoWEPP program accounts for ephemeral gully erosion but only if the boundaries of the gullies are known before prediction (Renschler and Flanagan, 2008). Other erosion modeling software (e.g., CREAMS, EGEM) can predict ephemeral gully erosion rates but not the locations (Nachtergaele et al., 2002). Mueller et al. (2005) used logistic regression to combine terrain attribute, remote sensing, and soil electrical conductivity data into probability maps indicating the severity of erosion. To fit the models, they used soil survey erosion phase information describing the level of sheet and rill erosion (i.e., slight, moderate, and severe erosion phases).

In this study, we have used an approach similar to the one proposed by Mueller et al. (2005). Our objective was to determine how neural networks and logistic regression could be employed to develop models that predict the probability of erosion resulting from concentrated water flow. Rather than using soil survey erosion phase information to fit the models, we relied on field observations of erosion occurring in concentrated flow channels.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Site Description
This study was conducted in five fields on Worth and Dee Ellis Farms in Shelby County located in the Outer Bluegrass physiographic region of Kentucky. Field A (38°17' N, 85°9' W), B (38°18' N, 85°11' W), C (38°20' N, 85°12' W), D (38°20' N, 85°11' W), and Field E (38°20' N, 85°14' W), were 23, 36, 11, 57, and 33 ha in size, respectively. These fields had been in a no-till, corn (Zea mays L.), wheat (Triticum aestivum L.), and double-crop soybean [Glycine max (L.) Merr.] or corn-wheat rotation for more than 20 yr. Soils in this region developed primarily from limestone residuum overlain with pedisediment from limestone weathered materials and loess (Soil Conservation Service, 1980).

Delineation of Erosion Features
One of the co-owners of the Worth and Dee Ellis Farms was informally trained by a previous NRCS conservationist in Shelby Co. to identify eroded zones in some of the fields on the farm that would qualify for continuous CRP payments for the installation of grassed waterways. The farmer relied on this training to identify eroded channels in many of the other fields on the farm including Fields A through E. The farmer delineated and mapped the boundaries of these eroded channels with GPS and subsequently installed waterways in these zones. The boundaries of the waterways mainly included areas with evidence of past erosion resulting from concentrated flow. This is an important clarification because waterways created by the NRCS would have also included grassed buffers that flank one or both sides of eroded channels (NRCS, 2008b). The waterways in these fields were also not reshaped or graded as would have been required by the NRCS. Reshaping of the waterways would likely have changed the terrain attribute values in the eroded channels and confounded the results of this study.

One important question was whether the waterways the farmer established would have been eligible for enrollment in the continuous CRP program. Therefore, the NRCS district conservationist assessed the boundaries of the waterways in Fields A through E in February of 2008. The conservationist found that each of the grassed waterways examined (42% of all waterways) would have been eligible to be enrolled in the continuous CRP program. However, the waterways would have required widening to include buffers and reshaping in many cases to be eligible for conservation payments. If precision spraying and planting equipment had not been used, the district conservationist would have attempted to dissuade the farmer from enrolling the smaller waterways (e.g., those <0.1 ha) because these areas would have been logistically too difficult to manage using conventional equipment.

Elevation Data
The RTK GPS data was collected in 2000 for Field D (Mueller et al., 2003), 2004 for Fields E and B (Pike et. al., 2006) and in 2007 for Fields C and A. To create surveys for Fields A, B, C, and E, a Trimble (Trimble, Ltd., Sunnyvale, CA) duel-frequency AgGPS 214 receiver was used as a base station, and a dual duel-frequency Trimble 5800 receiver was used as a rover. Two single-frequency Trimble 4600 receivers (one base and one as a rover) were used to create the survey for Field D and postprocessing of the data was required. Relative elevation measurements were logged each second along parallel passes with approximately 3 (Field D) and 4 m (Fields A, B, C, and E) between consecutive measurements and 7.5 (Field D) and 12 m (Fields A, B, C, and E) between passes. The vertical errors for both Trimble 4800 and 5800 receivers were expected to be <2.2 cm because the baselines were <1200m (Trimble, 1997, 2008).

USGS 9.1-m elevation data was obtained from the Kentucky Division of Geographic Information for the area outside the field. The relative RTK elevation measurements were corrected and then merged with the USGS data. The original purpose for combining these datasets was to account for upslope area outside of the field and to allow us to create 3-D maps of the field and surrounding area. However, this step turned out to be unnecessary because at nearly all field boundaries, water flows off rather than onto the fields.

Digital Elevation Model Creation
The DEMs were created using the TOPOTORASTER ArcGIS 3D analyst command (ESRI, Redlands, CA). Hutchinson and Gallant (2000) described the proprietary splining technique used by the software and recommended that the drainage enforcement option be used to remove sinks that are rare in nature and often occur because of errors in elevation data. This option was not used in this investigation because sinkholes often occur due to the karst geology of this area. However, there were no depressions in the DEMs as determined with the ESRI ArcINFO Sink command presumably because sinkholes were avoided by the rover operator when creating the RTK survey.

The terrain surfaces had substantial noise. This was very apparent in the derived terrain attributes such as plan and profile curvature where there were streaks that occurred in the ATV's direction of motion when the RTK GPS survey was created (terrain attributes will be described in detail in the next subsection). Much of this noise was removed by first creating 1-m contour maps from the DEMs as described by Sears et al. (2005) and Pike et al. (2006); then the TOPOTORASTER command was used to create new 4-m DEMs from the contour map. This processing of the DEMs substantially reduced the magnitude and variability of the terrain attributes. Therefore, the models that will be presented in this study cannot be applied to fields where different terrain modeling procedures were used to create the DEMs.

Terrain Modeling
The TAPESG for Windows 7.1 software (University of Southern California, Los Angeles, CA) was used to calculate the various terrain attributes used in this study. Primary terrain attributes included slope (β), aspect, plan and profile curvature, upslope contributing area, and specific catchment area. Flow direction, calculated by the FD8 algorithm, was used to estimate upslope contributing area for each grid in the DEM. Flow was distributed in multiple directions until the upslope contributing area exceeded the maximum cross-grading area (Gallant and Wilson, 2000), which was hardcoded to a value of 50,000 m2 by the authors of the TAPESG for Windows software. For areas with upslope contributing areas exceeding 50,000 m2, the algorithm switched to single direction flow (D8). The finite difference algorithm was used to calculate slope. The specific catchment area is an adjustment of the upslope contributing area to account for flow width and was calculated as follows:

Formula 1[1]
For D8 flow routing, flow width (also referred to as contour width) was set to a value of 1 grid cell width (i.e., 4.0 m) for the four cardinal directions and to the length of the grid cell diagonal (i.e., 5.7 m) for the other four ordinal (intercardinal) directions. The calculation of flow width for the FD8 algorithm is more complicated and described in detail by Gallant and Wilson (2000). Secondary terrain attributes (i.e., those computed from two or more primary attributes) were also determined with TAPESG. This included the topographic wetness index calculated as

Formula 2[2]
the stream power index estimated as

Formula 2
and the length-slope factor calculated as

Formula 3[3]
This terrain attribute (Eq. [3]) is an estimate of the length-slope factor from the Revised Universal Soil Loss Equation (RUSLE) model even though it has some limitations particularly for longer and steeper slopes (Wilson and Gallant, 2000). Upslope contributing area and specific catchment area are predictors of runoff volume, whereas length-slope factor, stream power index and channel initiation threshold are indices of erosion (Wilson and Gallant, 2000).

The terrain attributes were converted from a raster to vector format (i.e., to point files). These point data were then combined into a single database file for each field. Each 4-m grid point was assigned a value indicating whether or not the point was located within or outside the boundary of an eroded zone. Specifically, a value of 0 was assigned to each point located outside of the eroded zones and a value of 1 to each point that was in an eroded zone.

Data Preparation
The complete dataset included each 4 by 4 m-grid cell within the experimental areas of each of the five fields. Some areas within the boundary files were excluded from the analysis that contained field buffers, ponds, or sink holes. The numbers of observations from noneroded and eroded areas were as follows: 13,967 and 501 for Field A, 21,038 and 1450 for Field B, 6500 and 145 for Field C, 31,447 and 4148 for Field D, and 18,709 and 1800 for Field E.

Six separate datasets were created to be used for the five leave-one-field-out validation analyses and the one all-five-fields analysis (to be described in later subsections). An equal number of 0's and 1's were selected from each field to prevent oversampling of noneroded features. Our preliminary analyses revealed that this step was critical and allowed us to identify eroded features that would have otherwise been difficult to identify. The datasets included an equal number of observations from each field. For each dataset, 70% of the observations were randomly assigned to be used for training and 30% for validation.

Neural Network and Logistic Regression Modeling
The analyses were conducted with the "neural network" and "regression" nodes in SAS Enterprise Miner 4.3. The neural networks were trained as feedforward networks using the Dual Quasi-Newton method, which is the default in SAS Enterprise Miner for networks with 100 to 500 weights. The number of weights in the networks for our analyses was 121. For larger networks, the Quasi-Newton is impractical because the intensive memory requirements (Priddy and Keller, 2005) and other methods (e.g., Levenberg-Marquardt) are more appropriate for smaller networks (Matignon, 2005).

The training data set was used to estimate model coefficients for each iteration. The neural network coefficients selected were those that resulted in the lowest misclassification errors determined for the validation datset. The coefficients for the final models were used to estimate the probability of an eroded feature at a given location in fields that were not used to generate models (see the Leave-One-Field-Out Validation subsection)(Bishop, 2002; Priddy and Keller, 2005).

The logistic regression procedure models the logit of the probability (p) of an eroded feature as a linear function of the predictor variables (x1,..., xk) as described here:

Formula 3
where β 0, ..., β k represent fitted model parameters. The estimated probability of an eroded feature for given values of the independent variables can be then be derived as follows:

Formula 3

Leave-One-Field-Out Validation
Our interest was to perform model validation using data that was not used to generate the model. This is often achieved by randomly setting aside a portion of the complete data to be used for testing. However, this approach would not have allowed the models to be evaluated outside the boundaries of the fields on which the models were based. For this study, a leave-one-field-out validation was conducted instead. The models were developed for data from four of the five fields, and then used to generate predictions for the fifth field. Then, this process was repeated five times so each field was left out once.

For each leave-one-field-out validation analysis, there were 809 training and 351 validation observations randomly selected when either Field A, B, D, or E were left out of the analyses and used as a test dataset. There were 2802 and 1206 observations selected from Fields A, B, D, and E when Field C was used as a test dataset. This difference occurred because Field C was the smallest of the five fields and the smallest field used to develop the model limited the size of the training and validation datasets since the same number of observations were used from each field.

Misclassification Errors
The calculation of misclassification errors involved creating a new variable and assigning it a value of 0 for predicted probabilities ≤0.5 and a value of 1 for all probabilities >0.5. Therefore, if an observation located in an actual eroded zone had a predicted probability ≤0.5, it would be considered incorrectly classified, but correctly classified if values were >0.5.

The average misclassification errors for the validation datasets were used by SAS as the criteria for model selection. The predicted probabilities were also used to determine how well the general locations of the eroded areas could be identified (e.g., presence or absence of a waterway) and how well their shapes (e.g., widths) could be defined. Ideally, both should be predicted well. However, a model that only predicts the general location of eroded areas would still have tremendous utility for planners. Therefore, our model interpretations and evaluations include visual map analyses as well.

Different mean squared errors and log-likelihood derived statistics were calculated but not reported, because they have limited interpretive value since the residuals were spatially auto-correlated. Instead, misclassification statistics were reported for the fields that were left out of the leave-one-field-out validation analysis.

Preliminary Analyses
For the neural network procedure, preliminary sensitivity analyses (Pike et al., 2008) were conducted to evaluate the effects of different numbers of neurons (4, 8, 12, 16, 20, and 24), activation functions (hyperbolic, logistic, Gaussian, and Elliot), standardization procedures (none, mid-range, range, and standard deviation), numbers of preliminary runs (0, 1, 3, 5, 10, and 40), and the number of hidden layers (1, 2, and 3). The type of activation function, type of standardization procedure, the number of neurons, number of preliminary runs, and number of hidden layers had little impact on the results of the neural network analysis. The prediction maps were very similar no matter which techniques were used. Specifically, the same eroded features were defined regardless of which settings were used, even though the shapes of features changed to some extent.

Our approach in selecting parameter settings to use for subsequent analyses was to choose those that produced the lowest misclassification rates unless the analyses were too computationally resource intensive to be practical. Accordingly, all of the neural network analyses described from this point forward were performed with 20 neurons, the hyperbolic activation function, normalization by the standard deviation, five preliminary runs, and one hidden layer.

Next, variable selection was considered. All of the terrain attributes described in the Terrain Modeling subsection of this paper were tested. However, there were only small differences between misclassification rates (i.e., differences were <2.7% points) and prediction maps for models created using all terrain attributes and for those that were generated with only three or four attributes. The concern with using all of the terrain attributes was that multi-collinearity would be introduced and the logistic regression model parameters would not be biophysically meaningful. For example, the erosion indices (i.e., specific catchment area, stream power index, channel initiation threshold, and the length-slope estimate) would often have negative parameter estimates when seven variables were included in the models for predicting eroded grassed waterways. The predictive power of such a model (i.e., one with biophysically meaningless parameters) was in some cases excellent for the fields for which the model was developed. However, if another field on a nearby farm with similar soils occupied a part of the inference space that was not sampled when the original model was fit, the use of an over-fitted model with biophysically meaningless model parameters could lead to erroneous predictions. While it could be justifiably argued that predictions should never be made outside of the inference space, as a practical matter, in multi-dimensional inference space boundaries are particularly difficult to determine.

Only four variables were selected to be used for analysis: plan curvature, specific catchment area, the length-slope factor, and the topographic wetness index. These parameters were selected because they were not highly correlated among each other and consistently resulted in model parameter estimates that were biophysically meaningful (i.e., the coefficients for plan curvature were always negative and the values for the others were always positive). In addition, they all had t scores > 2 for at least some of the leave-one-field-out validation analyses. Finally, they are commonly used as tools to identify soil erosion. For example, Thorne et al. (1986) used plan curvature, Moore et al. (1988) and Srivastava and Moore (1989) used the topographic wetness index and the specific catchment area to identify ephemeral gully erosion. The length-slope factor estimates the contribution of topography to rill and inter-rill erosion (Moore and Wilson, 1992). It was used as a predictor variable because preliminary map analyses indicated that waterways tended to occur in areas where length-slope factor values were large.

All-Five-Fields Analysis
Next, two models were developed with the data from all five fields using logistic regression. Initially, plan curvature, specific catchment area, the length-slope factor, and the topographic wetness index were used to develop the first model. Then the variable with the smallest t score was removed and the three remaining factors were used to develop the second model. No more models were developed because all t scores were >2.

For the training and validation datasets, 1011 and 439 observations were used, respectively. The other settings used for the leave-one-field-out validation analysis were also used here. Model parameter estimates and t statistics were reported and interpreted.


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Terrain Attributes
The boundaries of erosion features were related to the terrain attributes used to develop erosion probability models: plan curvature (Fig. 1 ), specific catchment area (Fig. 2 ), the topographic wetness index (Fig. 3 ), and the length-slope factor (Fig. 4 ). Specifically, negative values of plan curvature and positive values of the other terrain attributes were generally associated with these eroded waterways. A negative association was expected for plan curvature (Fig. 1) because areas with negative values are concave in shape (in the direction perpendicular to water flow) and water tends to concentrate in these zones. In contrast, convex ridge tops had positive plan curvature values. Specific catchment area values were generally larger in erosion channels (Fig. 2) because waterways generally had greater upslope catchment areas draining into them during storms and greater water flow has more erosive power. Larger topographic wetness values were associated with erosion channels (Fig. 3), an observation consistent with research documenting that ephemeral gullies tend to occur in areas with larger wetness index values (Moore et al., 1988; Srivastava and Moore, 1989; Daba et al., 2003). Although the DEM-derived length-slope index does not usually provide an exact estimate of the RUSLE erosion index (Wilson and Gallant, 2000), it was associated with many of the erosion features in this study (Fig. 4).


Figure 1
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  		Fig. 1. Plan curvature (radians m–) overlain by the boundaries of erosion features.

 

Figure 2
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  		Fig. 2. Specific catchment area (m2) overlain by the boundaries of erosion features.

 

Figure 3
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  		Fig. 3. Topographic wetness index overlain by the boundaries of erosion features.

 

Figure 4
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  		Fig. 4. Length-slope overlain by the boundaries of erosion features.

 
Clearly, the terrain attribute data provided a wealth of information that would be useful for conservation planning. However, no single terrain attribute was adequate for identifying all of the eroded features. For example, some of the eroded features on the northwestern side of Field D did not have distinctly negative plan curvature values (Fig. 1). The wishbone shaped feature in southern half of Field B and several of the waterways along the southern half of the western side of Field E did not have large positive specific catchment area values (Fig. 2). The waterways along the western side of Field E also did not have large wetness values (Fig. 3). The length-slope factor values were large for most waterways, but poorly delineated some of the eroded features such as those in Field D and E (Fig. 4).

Ideally, planners should bring all of these maps to the field while making site assessments. However, it would be difficult to interpret multiple terrain attribute maps in the field. For this reason, we considered whether index maps integrating information from several terrain attributes could be created in a way that would be useful for conservation planning.

Leave-One-Field-Out Validation Analysis
The erosion probability maps created with logistic regression (Fig. 5 ) and neural networks (Fig. 6 ) in the leave-one-field-out analyses were similar. Most of the same eroded features were identified with both analytical procedures. In some cases more irregular and abrupt features or artifacts occurred in the maps created with neural networks. For example there was a V shape feature on the southern side of Field E in the neural network probability map (Fig. 6) that was not apparent in the logistic regression map (Fig. 5).


Figure 5
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  		Fig. 5. Erosion probability maps created with logistic regression overlain by the boundaries of erosion features.

 

Figure 6
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  		Fig. 6. Erosion probability maps created with neural network analysis overlain by the boundaries of erosion features.

 
Neural network models are much more complex than logistic regression models and include many more parameters. Models should be only as complex as necessary for a given application (Schabenberger and Pierce, 2002). Logistic regression was more than adequate for describing the variability of the erosion features for the fields studied on this farm. Given the similarity between the model predictions (Fig. 5 and 6), the data also suggested that the "rules" governing erosion in this field were generally linear. However, for future, more global models, relationships may be more complex and may require nonlinear models (e.g., neural networks) to adequately describe variability.

Most waterways were identified by these probability maps based on leave-one-field-out analyses, which demonstrates that these models have predictive capacity outside the boundaries of the fields for which the models were developed. The geographic region of this study, however, was small (i.e., limited to the confines of the Worth and Dee Ellis Farm). It will be important to determine whether the parameter estimates for these types of models can be applied over larger geographic regions. For example, would the model parameters from this study be valid throughout the Outer Bluegrass physiographic region or across other geographic regions that were also primarily derived from relatively thin layers of loess overlaying limestone residuum such as soils in the Inner Bluegrass and Pennyroyal?

The erosion index maps created from these terrain attributes (Fig. 5 and 6) could likely be effective tools for conservation planning. However, the shades of gray could potentially distract planners during site assessments, making them less efficient in the field. Therefore, we considered the potential effectiveness of discretized maps created by converting continuous probability values into binary maps (Fig. 7 and 8 ). These maps delineated most of the waterways in the field, were less "busy" than the probability maps, and were therefore easier to read. However, the some planners may find that the shades of gray provide useful information about the uncertainty of estimates. To ensure acceptance, end users should evaluate the effectiveness of different mapping techniques.


Figure 7
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  		Fig. 7. Discretized erosion probability map created with logistic regression overlain by the boundaries of erosion features.

 

Figure 8
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  		Fig. 8. Discretized erosion probability map created with neural network analysis overlain by the boundaries of erosion features.

 
Table 1 presents the number of observations (i.e., 4 by 4-m grid cells) that were correctly and incorrectly classified by model predictions for the leave-one-field-out validation analyses. The row misclassification statistics provide a fairer metric of model performance than field average misclassification rates. The first statistic represents the percentage of false negatives and the second false positives.


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  		Table 1. Confusion tables and misclassification statistics for logistic regression and neural network analyses with tables indicating the number of 4- by 4-m grid points in each category.

 
The misclassification rates were substantial in some cases (e.g., those values > 25%) (Table 1) because the shapes of waterways were not always well defined; however, the modeling procedures could identify most waterways (Fig. 6 and 7). These observations suggest that the proposed modeling approach should not be used to locate and place grassed waterways without field confirmation by experienced planners. False positives (Table 1) would not be a great concern because they could be easily detected by an experienced planner during field visits. The use of the proposed maps by conservation planners would likely allow them to more efficiently and accurately identify grassed waterways.

Models for All-Five Fields
The coefficients and t statistics for the two models developed with data from all five fields are given in Table 2 . As anticipated, parameter estimates were negative for plan curvature and positive for the other terrain attributes. The size of the t statistic is proportional to the relative importance of each given factor in the models.


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  		Table 2. Model variable parameter estimates and t scores for the logistic regression analysis conducted for all five fields.

 
The specific catchment area was removed as a variable for Model 2 because the t score for this variable was 0.8 in Model 1 (Table 1). The t scores were >2 for each terrain attribute in Model 2, indicating statistically significant contributions of the variables to the model. We propose Model 2 be used to predict the occurrence of channel erosion in these fields where the

Formula 4[4]
with LS = the estimated length-slope factor, WET = the topographic wetness index, and PLAN = plan curvature.

This model (Eq. [4]) should only be tested if the same methods will be used to create the DEMs and calculate terrain attributes. This is crucial because the process used to smooth elevation measurements in this study had a substantial impact on the sizes of the various terrain attributes (see DEM Creation Subsection). Additional factors that might affect the magnitude of the terrain attributes are the flow algorithm (i.e., FD8 method), as well as the source of elevation data (e.g., USGS DEMs and LIDAR vs. RTK GPS data).


   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
The findings of this study suggest that logistic regression and neural networks can be used to create empirical erosion models with excellent predictive capacity using RTK GPS derived terrain attributes as input variables. Neural networks did not produce substantially better maps indicating that relationships were generally linear for the fields in this study. A logistic model was developed to predict the probability of erosion as a function of the length-slope factor, the topographic wetness index, and plan curvature. More work will be necessary to understand how this general modeling approach can be applied to larger geographic areas. There is also a need to determine whether the procedures proposed in this study would be effective if terrain attributes were derived from USGS 10-m DEMs and DEMs created with LIDAR data. Ultimately, the utility of these models for conservation planning should be carefully evaluated by potential end-users.


   ACKNOWLEDGMENTS
 
We appreciate Mike Ellis's generosity in providing us access to his grassed waterway datasets and Mike, Bob, and Jim Ellis for allowing us to conduct this research on their farm. We are also grateful for the assistance of Randall Rock, Jack Kuhn, and Danny Hughes from the NRCS. Lastly, we appreciate statistical guidance from Cid Srinivasan and Chandramouli Viswanathan regarding neural network analysis.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
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   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 





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