Explicit models were obtained for each of the component variables needed to calculate SOCstock indirectly and for predicting SOCstock directly. Models for predicting component variables used a higher quantity of predictors for each of the respective models than the direct modelling approach (Table 3). With the exception of SOC%, the models for component variables included a combination of DTA and spectral variables. The SOC% models relied solely on DTA predictors for both stocks, but with additional spatial partitioning by geologic map units for the topsoil model. The models for directly predicting the SOCstock used only three DTA predictors for the topsoil and only four Landsat predictors for the subsoil.

Fitting performances for the component variable models were better than the fitting performances for the direct modelling of SOCstock (Table 4). For the component variables, R2 values of subsoil models were only slightly less than the topsoil models. SOC% was the exception by having the lowest fitting performance for the subsoil stock (R2=0.55), while the model for the SOC% topsoil was able to fit observations with an R2 of 0.86. However, it was the aim of this research to examine whether the performance of the models was maintained through the calculation of SOCstock.

Comparison of the SOCstock predictions by the indirect approach to observed values showed better performance for the topsoil stock (R2=0.73) than for the subsoil stock (R2=0.34). Fitting performance for directly modelling SOCstock showed the same pattern, but was lower than the indirect approach for both stocks. Analysis of the direct approach's ability to fit observed values yielded an R2 of 0.58 for the topsoil and 0.19 for the subsoil.

In general, calculated model efficiencies (ME) showed that the respective models reduced the mean absolute error (MAE) to about half the MAE that would result from simply using the mean of all points as the prediction. The SOC% model for the topsoil improved upon the mean model more than the other MLR models with a ME of 0.34. However, an intriguing result is the lack of model efficiency for the indirect modelling of the subsoil's SOCstock. Despite the component models all having MEs well below 1, the indirect approach did not improve upon the mean model for predicting the subsoil SOCstock. Although the ME of the direct model for subsoil SOCstock was also not as good as the other models, it was still an improvement over the mean model.

It is common for digital soil mapping models to be evaluated by cross-validation procedures. However, in the context of this study, the meaning of such an analysis has less utility. Higher sample density increases the robustness of the model (Minasny et al., 2013); thus the popularity of cross-validation procedures over independent validation procedures in order to maintain more points in the calibration set. However, the model generated for each cross-validation run is different because of differences in calibration sets. The performance of each run is dependent on the randomly selected calibration points' ability to represent the variation in the remaining validation points. For a simple data trend, a single outlier would have minimal effect because only the runs in which it is included in the validation set – and not used in calibrating the model – would have lower performance values. However, in a complex landscape where similar soil properties can result from different combinations of factors, the concept of an outlier has many more dimensions (Johnson et al., 1990; Phillips, 1998). A point with a similar value can be an outlier by being a product of a different set of factors. In other words, the problem of induction continues to apply in predictive soil mapping. Further, in the context of error propagation, the error estimation from the actual model used seems more appropriate than the mean of error estimations from a series of less robust models.

Nonetheless, the models in this study were cross-validated using the k-fold method with 10 iterations. The R2 was naturally reduced in the cross-validation analysis, but the MAE was not as severely affected (Table 5). The R2 values for the respective models all decreased greatly in the cross-validation, except for the topsoil SOC% and the subsoil SOCstock models. The subsoil SOCstock model already had a low R2 value for the internal fit. In contrast, the MAEs for the cross-validation of the models were not increased enough to present a practical problem. The relative stability of the MAEs also suggests that the estimated uncertainties are also robust. For example, the MAE for both stocks of BD only increased 0.03 g cm-3. Also, the MAE for SOC% only increased 0.13 and 0.03 % for the topsoil and subsoil, respectively. Similarly, the MAE for the direct SOCstock model increased 0.67 and 0.05 kg m-2 for the topsoil and subsoil, respectively. The MAE for the models of stock H and SK did increase more in cross-validation. However, they had a minor impact on the indirect modelling of SOCstock. The increase of 5.9 cm for the topsoil H MAE was only a shift of the depth estimated by topsoil or subsoil models. The larger MAE for SK was more of an issue for the subsoil. However, the majority of the samples had SK below 5 %, leaving most of the error due to the difficulty in predicting the limited areas of high SK. While it was possible that a different sampling design could have improved the R2 values for cross-validation, they are not always practical for landscape-scale mapping.

It is difficult to compare results between SOC mapping studies due to differences in study areas and strategies for defining SOCstock (i.e. map extent and resolution, sampling density, and consideration of depth). Further, the differences between and variability within methods for estimating component variables for calculating SOCstock can have a large impact on results, especially bulk density (Liebens and VanMolle, 2003; Schrumpf et al., 2011) and SOC% (Lowther et al., 1990; Soon and Abboud, 1991; Sutherland, 1998; Bowman et al., 2002). Also, because model performance is dependent upon the provided predictors, results of different studies can vary based on the predictors available to and derived by the modeller (Miller et al., 2015). However, because the area in this study has been used for several previous studies, some comparisons between methods can be made.

Kühn et al. (2009) examined many of the same samples used in this study and found a coefficient of determination between soil electrical conductivity and soil organic matter to a 1 m depth (kg m-2) of R2 = 0.59. Although a slightly different calculation, that coefficient of determination is similar to this study's direct model of topsoil SOCstock (R2=0.58), which used three DTA predictors. However, for the topsoil, the indirect approach in this study produced a SOCstock model with less estimated error and an R2 of 0.73. The Kühn et al. (2009) study usually included depths that this study defined as subsoil, where the models in this study did not perform as well (direct R2=0.19, indirect R2=0.34).

For the same area as this study, Selige et al. (2006) compared MLR and partial least-squares regression for predicting SOC% from hyperspectral data with a 6 m spatial resolution. Although the study by Selige et al. (2006) utilized a higher spectral resolution, the MLR models produced by both that study and the present study had an R2 of 0.86 for the topsoil SOC%. In the present study, Cubist was able to compensate for the limited spectral information by utilizing several DTA predictors that were available at a high spatial resolution.

Spectral predictors from satellites such as Ikonos and Landsat have most commonly been used to detect characteristics of land use, vegetation, and soil water content (Bannari et al., 1995; Xie et al., 2008). However, they have also been used to detect mineralogy on sparsely vegetated areas (Mulder et al., 2011). Although Ikonos has a finer spatial resolution, it is limited to three bands (band 1: blue; band 2: green; and band 3: red) in the visible spectrum, plus a near-infrared band (band 4: NIR). Landsat provides additional bands in the shortwave infrared (band 5: SWIR-1; band 7: SWIR-2) and thermal infrared (band 6: TIR). The relative reflectance of a single band can be used to distinguish landscape conditions. For example, the green band can be used to distinguish different vegetation from bare soil. However, combinations of bands – particularly including the red and NIR bands – have been even more useful for distinguishing the spectral signature of different land uses (Richards, 2006) and the condition of the vegetation (Ashley and Rea, 1975; Myneni et al., 1995; Rasmussen, 1998; Daughtry, 2001; Hatfield et al., 2008). Additional use of TIR emission would resemble methods such as the surface temperature/vegetation index for estimating soil moisture (Bartholic et al., 1972; Heilman et al., 1976; Carlson et al., 1994; Li et al., 2009; Petropoulus et al., 2009). Similarly, use of SWIR wavelengths in concert with red and infrared bands would be a way of compensating for the changing effect of soil reflection in dry to wet conditions (Huete, 1988; Lobell and Asner, 2002). Relationships between bands in the visible to SWIR range have also been used to predict SOC% and its biochemical composition (Bartholomeus et al., 2008; Gomez et al., 2008; Stevens et al., 2010).

Spectral predictors have been used for both classification of discrete phenomenon and quantification of continuous phenomenon on the landscape. Because of the rule-based MLR structure of the Cubist models, spectral predictors used for conditional rules were more likely to be distinguishing discrete features (e.g. vegetation/land use type) than when used within an MLR equation. Continuous features (e.g. vegetation health) were more likely to be represented in MLR equations.

DTA predictors in this study were all derived from the lidar data for elevation. The land-surface derivatives (e.g. slope gradient, relative elevation) described the surface geometry with which the climate interacts. For example, aspect has been shown to influence the amount of solar insolation a hillslope receives (Hunckler and Schaetzl, 1997; Beaudette and O'Geen, 2009). The surface geometry is also known to direct water flow, which affects erosion processes and groundwater recharge (Huggett, 1975; Zevenbergen and Thorne, 1987). Hydrologic predictors (e.g. flow accumulation, catchment slope) provided additional information about the relative volume and energy that the water flow may have (Moore et al., 1991; Wilson and Gallant, 2000).

All of the topsoil models generated by Cubist relied on DTA predictors the most. Of those predictors, different a-scales of relative elevation, topographic position index (TPI), and aspect were the most commonly used. With the exception of the direct SOCstock model, every topsoil model also included one or two predictors indicative of flow accumulation (i.e. flow path length, SAGA wetness index, or modified catchment area).

Aspect at different a-scales influenced predictions for three of the indirect topsoil models. The Cubist-generated model identified decreasing topsoil SOC% on more north facing slopes (155 m a-scale), which corresponds to a potential decrease in plant productivity due to less solar insolation. Aspect (215 m a-scale) was also used to predict higher topsoil BD on south- to west-facing slopes, especially on topographic (2000 m a-scale) and microtopographic (20 m a-scale) highs. Additionally, aspect at a variety of a-scales was used to predict decreasing topsoil SK for low-TPI areas facing southeast to southwest. Together, these models suggested a pattern of increased erosion and deposition along the southern sides of hillslopes. This type of pattern has been observed before in other landscapes and has been attributed to topo-climatic differences such as exposure to storms, differences in temperature regime, rainfall effectiveness, or vegetation density (Kennedy, 1976; Churchill, 1981; Cuff, 1985; van Breda Weaver, 1991).

Although DTA parameters dominated the topsoil models, their predictions were often modified by spectral variables. For example, the primary distinction for predicting topsoil H was between low and high relative elevations. Low relative elevations had a mean topsoil H that was about 20 cm thicker than high relative elevations (1100 m a-scale). Within most MLR equations, however, predictions were increased by less blue and more green reflectance in early July. This combined use of blue and green bands indicated increasing topsoil H with more productive vegetation on wetter soils. In summary, the dominant pattern identified by the model matched the pattern of high and low ground (Bushnell, 1943; Sommer et al., 2008), but the degree of topsoil thinning or thickening was predicted by the vegetation's response to soil conditions.

Cubist selected a much simpler combination of only DTA predictors to directly model the topsoil SOCstock. In general, the model predicted increasing SOCstock with decreasing vertical distance to channel. Areas low in relative elevation (1100 m a-scale) and not far above the channel network were predicted to have the largest SOCstock. However, for areas low in relative elevation, but sufficiently above the DEM based channel network, the model predicted the opposite trend of the SOCstock decreasing with decreasing vertical distance to channel. This pattern identified by the model may be explained by a corresponding pattern observed in the model for the topsoil H. In that model, areas low in relative elevation (1100 m a-scale) were predicted to have some of the thickest topsoil stocks. However, within a few of those zones the modelled topsoil H decreased with decreasing relative elevation and TPI. This trend in the observed data, as detected by Cubist, was potentially caused by an eroding-out of topsoil sediments closer to the centre of drainageways, in which case, the vertical distance to channel – used in the topsoil SOCstock model – may have been more an indicator of proximity to the channel than wetness; the threshold was only 0.5 m above the channel modelled from the DEM. Predictors related to surface flow energy would have been expected to be better predictors of this kind of process. However, the upslope drainage network for much of the map area extended beyond the boundaries of the available data. Thus the use of local elevation data may have been a better proxy in this case compared to the predictors calculated from truncated watersheds.

With the exception of SOC%, the subsoil models all used several predictors from Landsat. Selection of Landsat predictors for subsoil models suggested that vegetation characteristics or surface soil moisture at different times of the year indicated subsoil conditions. In contrast, the subsoil SOC% model's complete dependence on DTA predictors suggested that soil property was mostly related to hydrology and that vegetation had little response to or effect on the SOC content in the subsoil.

An example of spectral predictors detecting vegetation characteristics that likely reflected subsoil conditions was the subsoil SK model. All of the MLR equations were strongly influenced by the predictors of stream power, catchment slope, or SAGA wetness index. However, the SK predictions were modified by green reflectance in June and additional Landsat predictors collected at different times of the year that related to the vigour of the vegetation. The weaker or drier the vegetation appeared, the higher the prediction of SK content in the subsoil. Assuming soil moisture conditions did not reach detrimental levels that year, these patterns fit known relationships between particle size, soil drainage, and timing to crop maturity (Day and Intalap, 1970; Rawls et al., 1982).

The generated model for subsoil BD most likely utilized a relationship with soil moisture as detected by spectral predictors. In all areas, the MLR equations decreased predictions of subsoil BD with increasing reflectance in the blue and SWIR-1 bands along with increasing emission in the TIR band. Increases in the normalized difference vegetation index (NDVI) were used to slightly increase predictions of subsoil BD. The use of the NDVI to offset the decreasing BD predicted by the other Landsat predictors suggested those variables were indicating soil moisture conditions. Locations that are wetter due to surface runoff would have a greater potential for organic material to be translocated deeper in the soil profile (Schaetzl, 1986, 1990). Also, the association of wetter environments with cooler temperatures and anaerobic conditions would also inhibit decomposition (Gates, 1942; Krause et al., 1959; Frazier and Lee, 1971).

The subsoil SOC% model was different than the other subsoil models generated. Instead of selecting spectral predictors, the subsoil SOC% model relied solely on DTA predictors. The model predicted the highest subsoil SOC% on steeper mid-slopes. The pattern of increasing subsoil SOC% from the upper to middle slope fit the landscape translocation model proposed by Sommer et al. (2000). In that study, the SOC% in the Bh horizon increased from the upper slope to the midslope due to lateral translocation. Different than the pattern identified in the present study, the data in Sommer et al. (2000) showed a continued increase in the SOC% of Bh horizons in the downslope position. However, this contradiction may be partially explained by aggradation where the slope gradient declines and the topsoil stock has been overthickened by developmental upbuilding (McDonald and Busacca, 1988; Almond and Tonkin, 1999). Also, lateral flow would be expected to return closer to the surface at downslope positions. In Sommer et al. (2000), while the upslope and midslope profiles had E horizons separating the Bh from A horizons, the downslope Bh horizons were exceptionally thick, with little to no division between them and the A horizon. In that situation, the definition of topsoil used in the present study would have grouped the downslope Bh horizons into the topsoil stock. Therefore, the Cubist-generated model may have been a simplification of the complex interaction between topography and lateral flow depth and direction.

The rule groups for subsoil SOC% also differentiated for the plan curvature where the slope gradient was not too high and the stream power index (SPI) was not too low. Concave plan curvatures (138 m a-scale) were predicted to have increasingly higher and convex plan curvatures were predicted to have increasingly lower subsoil SOC%. This relationship with plan curvature matches patterns of water movement identified to be important to soil formation by Huggett (1975), where convergent footslopes have the highest deposition rates (Pennock and De Jong, 1987). Assuming the absence of any restrictive layer below, areas with the highest sediment deposition rates would be expected to also have the highest volume of water infiltration.

The Cubist-generated model for predicting the subsoil SOCstock was simpler than any of the indirect component models. It used only one MLR equation to relate red and infrared predictors to subsoil SOCstock. This model predicted more SOCstock storage with increasing reflectance in the red and SWIR-2 bands along with increasing emission in the TIR band – primarily captured on 6 July. Of these variables, model predictions were dominated by increasing reflectance in the red band increasing the estimated subsoil SOCstock. This suggested less productive vegetation corresponding with larger subsoil SOCstock. This trend was counter to the patterns observed in the topsoil models, but was sensible in the context of how the subsoil stock was defined for this study. Although the total SOCstock was less in areas with lower plant productivity, the subsoil SOCstock was larger relative to other subsoil areas due to the inverse relationship between topsoil and subsoil H used in this study. A thicker topsoil stock would mean a thinner subsoil stock – and vice versa – due to the 2 m depth limit. Regarding the other predictors in this model, increases in SWIR-2 reflectance could have indicated more plant productivity. However, its use with the TIR band suggested that together they were indicators of wetter soil conditions.