A total of 5 farm fields located in the southeastern of the Buenos Aires province were selected to explore the relationships among effective soil depth and those predictor variables related to factors of soil formation (Figure 1, Table 1). The area studied was 280 hectares. The soils are classified as Typic Argiudoll and Petrocalcic Argiudoll (USDA Taxonomy) ().

Figure 1. Location Source: (c)OpenStreetMap

The ESD up to the tosca layer was the variable to predict. To validating the model efficiency, the ESD was measured in each field. A total of 3035 soil depth measurements were carried out between 2010 and 2012. Each measurement was carried out using a soil hydraulic sampler (Giddings Machine Co., Fort Collins, CO) on a 30-m grid resolution. Table 1, shows the description of site variables.

The numerical calculation of predictor variables was carried out using different geomatic tools. Elevation was measured using a DGPS (Trimble R3 Navigation Limited, USA). A digital elevation model (DEM) was obtained from elevation data. From each DEM, geomorphometric indices were calculated using SAGA-GIS v2.0(tm) Terrain Analysis tool (). The geomorphometric indices were selected according to their capability for representing relief geometry, hydrological features and spatial topographical context in each field. On the one hand, slope and aspect are the primary geomorphometric indices related to basic relief parameters. On the other hand, catchment slope, drainage areas, topographic wetness index (TWI), SAGA wetness index (SWI), stream power index (SPI) and loss soil factor (LS_Factor), are secondary geomorphometric indices related to surface hydrology and spatial topographical context. A detailed description of these algorithms can be found in Hengl & Reuter (2008).

ECa data were collected using Veris 3100(r) (Veris Technologies, Salina, KS, USA). This measurement was carried out in a series of parallel transects spaced 20 m apart (Corwin & Lesch, 2003). The six electrodes configuration is referred to as Wenner array, which is a commonly used on a measurement of soil electrical resistivity (). Veris 3100, have allowed georeference measurements at 0-30 cm (ECa_30) and 0-90 cm (ECa_90), simultaneously.

Table 1. Description

ESD and predictors were computed with ordinary kriging method using ArcGIS v10. Based on this interpolation, a map was carried out at a spatial resolution of 10 m.

A dataset was ensemble using all predictors. To facility the comparison among fields, elevation parameter was transformed. The transformation consisted of subtracting at all elevation values, the lowest one for each field. The dataset was prepared as follows: (i) design and confection of a 20 m regular grid for each field, using spatial analyst extension in ArcGis 10.0(tm), (ii) values extraction of variable predictor at each grid point, using the function -extract multi values to point- spatial analyst extension in ArcGIS 10.0.and (iii) unify all points forming the dataset.

The dataset was ensemble using 6145 training dataset, obtained from predictors for all fields. Then, the training dataset was randomly divided into two subsets: a calibration dataset (CD, n=4300) and a validation dataset (VD, n=1845). The CD was used to model the relationship among ESD and predictors, while the VD was used to test the accuracy of the models prediction.

A variable selection procedure and the model simplification involved: (i) calibration of "Random Forest" (RF) algorithm, (ii) quantification of the predictors importance, (iii) variables selection and (iv) simplification of predictive model. This procedure is usually used for variable selection in high dimensional issues (Genuer et al., 2010).

The aim of RF calibration is to adjust the parameters of the algorithm in order to develop an optimal predictive model (Guyon & Elisseeff, 2003). This calibration was carried out in two steps. Firstly, correlation among predictors was calculated. Secondly, the parameters of RF were estimated: number of trees in the forest (ntree) and number of predictors evaluated at each node (mtry). The RF calibration procedure was carried out using "RandomForest" library of R v3.1.2. (tm). RF calibrated ntree and mtry parameters using "bootstrap" as method for error estimation (Breiman, 2001). This consists of using several random samples (bootstrap samples) from the CD. Then, RF builds a tree for each bootstrap sample. At each bootstrap sample Xi, one-third of data are left out of it and not used in the construction of the k-th tress, so-called out of bag (out-of-bag (OOB)). OOB estimate of error rate (OOB_EER ) is estimated from aggregating trees in the forest. This OOB_EER is calculated following equation 1.

Where, y_i is the initial ensemble error classification on OOB. As a bootstrap sample is added, an OOB error (y_OOB (X_i )) is estimated and repeating k-times. Those mtry and ntree which have the lowest OOB_EER should be selected for the model (Breiman, 2001).

The importance classification for predictors was carried out using the procedure known as "permutation accuracy importance". Details are described in Strobl et al. (2009). "permutation accuracy importance" is the most adequate procedure for classifying importance variables when RF is used (Breiman, 2001). Results were graphically represented using "party" R package v3.1.2.

To reduce the number of predictors without losing predictive accuracy, the model is simplified (Xiong et al., 2012). This procedure is running in two steps. First, a prediction model was computed considering all predictors. The purpose was to determine the effect on predictive RF model performance by reducing the number of variables. Second, it was evaluated the RF model simplified with CD and VD.

We used iterative elimination to describe the predictive accuracy of RF model (Xiong et al., 2012). To do this, we used "caret" R package v3.1.2. Coefficient of determination (R²⁾ and Root mean square error (RMSE) for each model were calculated to compare the overall accuracy of RF models while applying iterative elimination. The RF simplified model was applied to estimate the effective soil depth for each field. Then, a regression analysis was carried out among effective soil depth observed and predictive values. This regression was applied to CD and VD. Finally, the prediction performance of RF simplified was compared with a visual analysis of effective soil depth maps generated for each field.

Figure 2, summarizes the simultaneous evaluation OOB_EER from mtry and ntree tested. It was evident the effect of mtry increment on OOB_EER . When mtry increased, OOB_EER decreased from 50% (mtry=1) to 32% (mtry=9). Changes in OOB_EER indicate the importance of additional variables. RM model with mtry higher than 9 incremented OOB_EER and can generate an over-fit model (Breiman, 2001). The effect of ntree was lower than mtry. RF models with ntree higher than 500 showed more stability.

Figure 2. Relationship

In general, our results indicated that mtry equal to 9 and ntree equal to 500 are those parameters of RF model which fit best, in order to achieve an improvement in overall accuracy of ESD.

Figure 3, shows the results of "permutation accuracy importance" use.

Figure 3. Predictors

"Permutation accuracy importance" classifies the variables importance according to difference in prediction before and after a variable was permuted to all ntree. The average of this difference in all ntree, determine the absolute value of predictor importance (Strobl et al., 2009).

Our results suggested that the limit absolute value between relevant and irrelevant variables is 45 (dashed line - Figure 3). According to that, those most important predictors of effective depth were ECa_90, catchment slope, elevation and ECa_30.

Our results showed that ECa could have better predictive accuracy than geomorphometric indices to model ESD. The complexity of spatial pattern of predictors may be related to abrupt changes at short distances for depth, strengthen and thickness in petrocalcic and argillic horizons. According to Dietrich et al. (2014), it is possible that ECa would have affected by: (i) high spatial variability of ESD and (ii) changes in CaCO₃ density in the pretrocalcic horizon. Generally, depth tosca present lower CaCO₃ density (Pazos & Mestelan, 2002). These changes affect soil hydrological dynamic and thus soil electrical conductance.

Argillic horizon may affect ECa values in mainly two ways. The first, due to the percentage relation between argillic horizon and those underlying and overlying; the second would be attributed to the simultaneous presence of petrocalcic and argillic horizons. This was assessed by Pazos & Mestelan (2002), who reported that the presence of tosca affects either depth and thickness of argillic horizons.

In that context, ECa may possibly be affected by the dynamic of spatial patterns of petrocalcic and argillic horizons. Therefore, ECa can be a potential predictor for determining spatial patterns of ESD.

The importance of catchment slope in the ESD prediction was expectable. This is related to water holding capacity, runoff rate and water drainage (Wilson & Gallant, 2000). It is widely known that these hydrological characteristics are directly related to ESD. Despite that, none evidence was found which had mentioned catchment slope as predictor of ESD. Therefore, this can be a novel result.

Figure 4, shows the changes on prediction performance of RF model whereas predictors were iteratively eliminated. That RF model which only uses ECa_90, catchment slope, elevation and ECa_30 had a similar performance as one with all predictors. However, this is not indicating that the rest of predictor lack of prediction accuracy. It would be probably that prediction accuracy could decrease since ECa_90, catchment slope, elevation and ECa_30 had a better prediction performance in the model (Xiong et al., 2012).

Figure 4. Coefficient

Figure 5, shows the prediction performance of simplified RF model applied to CD and VD. Our results suggested that simplified RF model demonstrated a potential application to ESD prediction. However, its use in others fields is limited for different reasons: (i) a high variability in depth, thickness and CaCO₃ density of tosca at short distance, (ii) the relationship between tosca and argillic horizon and (iii) the anthropic effect. It is widely known that the southeast of Buenos Aires province is an outstanding zone due which income depends on agricultural production. Due to that, common suitability land processes such as terraces, crop management zones and drainages had been valuable. Generally, these could affect the spatial pattern of ECa as well as geomorphometric indices and therefore, increase error of simplified RF model developed.

Figure 5. Random

Figure 6. Visual

Figure 6, shows maps of effective depth made from simplified RF model development and ESD interpolation. Simplified RF model determined similar spatial patterns of ESD as interpolation technique did. Based on our result, we could validate that RF may be an effective and practical tool to select and predict ESD. However, although this RF model would not have the same performance for all fields, further studies will improve that.