the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Oblique geographic coordinates as covariates for digital soil mapping

### Anders Bjørn Møller

### Amélie Marie Beucher

### Nastaran Pouladi

### Mogens Humlekrog Greve

Decision tree algorithms, such as random forest, have
become a widely adapted method for mapping soil properties in geographic
space. However, implementing explicit spatial trends into these algorithms
has proven problematic. Using *x* and *y* coordinates as covariates gives
orthogonal artifacts in the maps, and alternative methods using distances as
covariates can be inflexible and difficult to interpret. We propose instead
the use of coordinates along several axes tilted at oblique angles to
provide an easily interpretable method for obtaining a realistic prediction
surface. We test the method on four spatial datasets and compare it to
similar methods. The results show that the method provides accuracies better
than or on par with the most reliable alternative methods, namely kriging
and distance-based covariates. Furthermore, the proposed method is highly
flexible, scalable and easily interpretable. This makes it a promising tool
for mapping soil properties with complex spatial variation.

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Machine learning has become a frequently applied means for mapping soil properties in geographic space. The most common approach is to train models from soil observations and covariates in the form of geographic data layers. The models can often provide reliable predictions of soil properties. Many researchers have used decision tree algorithms as they are computationally efficient, do not rely on assumptions about the distributions of the input variables, and can use both numeric and categorical data (Quinlan, 1996; Mitchell, 1997; Rokach and Maimon, 2005; Tan et al., 2014). Additionally, they effectively handle nonlinear relationships and complex interactions (Strobl et al., 2009).

However, a disadvantage of decision tree models is that they do not explicitly take into account spatial trends in the data. Unlike geostatistical methods, such as kriging, the predictions can therefore contain spatial bias.

A number of studies have applied regression kriging (RK) as a solution (Knotters et al., 1995; Odeh et al., 1995; Hengl et al., 2004). By kriging the residuals of the predictive model and adding the kriged residuals to the prediction surface, this approach can account for spatial trends and achieve higher accuracies. A disadvantage of RK is that the combination of two models hinders the combination of spatial trends with the other covariates. Spatial trends therefore remain disconnected from other statistical relationships in the analysis, leading to difficulties in interpreting the model and its associated uncertainties.

An obvious solution to this problem would be to use the *x* and *y* coordinates
of the soil observations as covariates. However, results have shown that
this approach can lead to unrealistic orthogonal artifacts in the output
maps when used in conjunction with decision tree algorithms (Behrens et
al., 2018; Hengl et al., 2018; Nussbaum et al., 2018). The cause of this
problem lies in the splitting procedure of decision tree algorithms, as they
use only one covariate for each split. Therefore, a dataset containing only
the *x* and *y* coordinates will force the algorithm to make orthogonal splits
in geographic space.

Several researchers have proposed solutions to this problem.
Behrens et al. (2018) proposed the use of Euclidean distance fields
(EDFs) in the form of distances to the corners and middle of the study area
and the *x* and *y* coordinates. Their results showed that this approach
efficiently integrated spatial trends and that accuracies were better than
or on par with other methods for integrating spatial context.

On the other hand, Hengl et al. (2018) suggested an approach referred to as spatial random forest (RFsp). This method consists of calculating data layers with buffer distances to each of the soil observations in the training dataset. It then trains a random forest model, using the buffer distances as covariates, either combined with auxiliary data or on their own. One of the main advantages of this approach is that it incorporates distances between observations in a similar manner to geostatistical models. The authors assessed the use of RFsp on a large number of spatial prediction problems and showed that it effectively eliminated spatial trends in the residuals.

Although these two methods are able to integrate spatial trends in machine-learning models, they can be difficult to interpret. The distances used in EDFs depend on the geometry of the study area, and for RFsp, they depend on the locations of the soil samples. The meaning and interpretation of the distances therefore varies depending on the study area and the soil observations.

EDFs and RFsp also have limited flexibility as both methods specify the number of geographic data layers a priori. For EDFs, the number of distance fields is seven, and for RFsp, the number of buffer distances is equal to the number of soil observations. This means that there is no straightforward way to increase the number of spatially explicit covariates if the number is insufficient to account for spatial trends. Conversely, there is no way to decrease the number of spatially explicit covariates, even if a smaller number would suffice. The latter is especially relevant for RFsp as the method is computationally unfeasible for datasets with a large number of observations (Hengl et al., 2018).

In this study, we propose an alternative method for including spatially
explicit covariates for mapping soil properties. With the method, we aim to
directly address the cause of the orthogonal artifacts produced with *x* and *y* coordinates as covariates in decision tree models. Furthermore, we aim to
improve upon the shortcomings of previous methods by developing a method
that is both flexible and easily interpretable.

We refer to the method as oblique geographic coordinates (OGCs). In short, it
works by calculating coordinates for the observations along a series of
axes tilted at several oblique angles relative to the *x* axis. By including
oblique coordinates as covariates, we enable the decision tree algorithm to
make oblique splits in geographic space. As this is not possible with only
*x* and *y* coordinates as covariates, this addition should allow the model to
produce a more realistic prediction surface. Furthermore, the number of
oblique angles is adjustable, and soil mappers can therefore choose a number
that suits their purpose. Some mapping tasks may require a higher number of
oblique angles than others, and soil mappers can therefore increase the
number as necessary. Alternatively, if a small number of oblique angles
suffices, soil mappers can reduce their number and thereby shorten
computation times.

We test the method on four spatial datasets. Firstly, we test it for
predicting soil organic matter contents in a densely sampled agricultural
field in Denmark, located in northern Europe. Secondly, we test it on three
publicly available spatial datasets (*meuse*, *eberg* and Swiss rainfall). We hypothesize
that OGCs can provide accuracies on par with previous methods for including
explicitly spatial covariates. We also hypothesize that it is possible to
adjust the number of oblique angles in order to optimize accuracy and that
the results allow for meaningful interpretations.

## 2.1 Study areas

We test OGCs and compare them to other methods based on four spatial datasets. Firstly, we test them for a predicting soil organic matter (SOM) for an agricultural field in Denmark (Vindum). Secondly, we test them on three publicly available datasets. For Vindum, we will present methods and results in detail. For the other three datasets, we will present methods and results in brief, while Appendix A contains a detailed presentation of the methods and results for these datasets.

### 2.1.1 Vindum

This study area is a 12 ha agricultural field located in Denmark in northern
Europe (9.568^{∘} E, 56.375^{∘} N; European Terrestrial Reference System (ETRS89), 1989; Fig. 1). It lies in a kettled moraine landscape
55–66 m above sea level (a.s.l.). The parent materials in the field include clay
till, glaciofluvial sand and peat. The climate is temperate coastal, with
mean monthly temperatures ranging from 1 ^{∘}C in January to
17 ^{∘}C in July and a mean annual precipitation of 850 mm
(Wang, 2013). The field contains 285 measurements of SOM from the depth interval 0–25 cm located in a 20 m grid.

The SOM contents of the topsoil in the field range from 1.3 % to 38.8 %, with a mean value of 3.5 % and a median of 2.2 %. The values have a strong positive skew of 4.7 and are leptokurtic with a kurtosis of 26.9. Logarithmic transformation reduces skewness (2.9) and kurtosis (11.1). Pouladi et al. (2019) described the spatial structure of the data, with a stable variogram with 139 m range, nugget of 0 and sill of 23.8.

### 2.1.2 Additional datasets

For additional analyses, we included the *meuse* dataset, the *eberg* dataset and the
Swiss rainfall dataset. The *meuse* dataset, available through the R package *sp*
(Pebesma et al., 2020), contains 155 measurements of soil heavy-metal concentrations from a 5 km^{2} flood plain of the Meuse River near
the village of Stein in the Netherlands. For this dataset, we mapped zinc
concentrations. The *eberg* dataset, available through the R package *plotKML* (Hengl
et al., 2020), contains 3670 soil observations from a 100 km^{2} area in
Ebergötzen near the city of Göttingen in Germany. For this dataset, we
mapped soil types. Lastly, the *Swiss rainfall* dataset contains 476 rainfall measurements
from 8 May 1986 in Switzerland (Dubois et al., 2003). Although this is
not a soil dataset, we included it because of the high anisotropy of the
data, which makes it useful for comparing methods on their ability to
account for anisotropic spatial problems. We describe these three datasets
in more detail in Appendix A.

## 2.2 Oblique geographic coordinates

The method that we propose consists of calculating coordinates along a
number of axes titled at various oblique angles, relative to the *x* axis. In
the following, we show that it is possible to calculate the coordinate of a
point (*b*_{1}, *a*_{1}) along an axis tilted at an angle *θ* relative to
the *x* axis, based on *θ* and the *x* and *y* coordinates of the point (*b*_{1}, *a*_{1}).
Equation (1) shows the calculation of the oblique geographic coordinate, using
Fig. 2 for illustration.

where *θ* is the angle of the titled axis relative to the
*x* axis; *a*_{1} is the *y* coordinate of (*b*_{1}, *a*_{1}); *b*_{1} is the
*x* coordinate of (*b*_{1}, *a*_{1}); and *b*_{2} (or “OGCs”) is the coordinate of
(*b*_{1}, *a*_{1}) along an axis tilted with the angle *θ* relative to the
*x* axis.

As the *x* and *y* coordinates of soil observations are known, and *θ* is
given, it is possible to calculate coordinates at oblique angles for all
soil observations in a dataset. Likewise, as the *x* and *y* coordinates of the
cells in a geographic raster layer are known, it is possible to calculate
oblique coordinates for the cells. Our approach relies on calculating
coordinates along *n* axes tilted at angles ranging from zero to $\mathit{\pi}\left(\right(n-\mathrm{1})/n$) with
increments of *π*∕*n* between the angles. *θ* should not be *π* or
greater, as coordinates along axes tilted at these angles will correlate
with coordinates along axes tilted at angles of zero to $\mathit{\pi}\left(\right(n-\mathrm{1})/n$). For example,
coordinates along an axis with *θ*= 0.25*π* (northeast) perfectly
correlate with coordinates along an axis with *θ*=1.25*π*
(southwest). Figure 3 shows coordinates along axes
tilted at six different angles relative to the *x* axis for the Vindum study
area. The coordinate rasters (a) and (d) are equivalent to the *x* and *y* coordinates, respectively, while the coordinate rasters (b), (c), (e) and (f) show
coordinates at oblique angles.

## 2.3 Method comparison

### 2.3.1 Vindum

We use the 285 SOM observations from the Vindum study area in order to test the accuracy of predictions made by random forest models using OGCs as covariates. In addition to OGCs, we also employed 19 data layers with auxiliary data, which Pouladi et al. (2019) derived from a 1.6 m digital elevation model (DEM), satellite imagery and electromagnetic induction. Topographic variables included the sine and cosine of the aspect, depth of sinks, plan and profile curvature, elevation, flow accumulation, valley bottom flatness, midslope position, standard and modified topographic wetness index, slope gradient, slope length, and valley depth. Satellite imagery included normalized difference, absolute difference, ratio and soil-adjusted vegetation indices. Lastly, we used the apparent electrical conductivity from a DUALEM-1 sensor in perpendicular mode.

In order to optimize the number of raster layers for OGCs, we generated
datasets with 2–100 coordinate rasters. We then trained random forest
models from each dataset, both with and without auxiliary data. In order to
assess predictive accuracy, we used 100 repeated splits on the SOM
observations, each using 75 % of the observations for model training and a
25 % holdout dataset for accuracy assessment. We trained models using the
R package *ranger* (Wright and Ziegler, 2015) and parameterized the models
using the R package *caret* (Kuhn, 2008). For each split, we tested five
different values for *mtry*, with minimum node sizes of 1, 2, 4 and 8 and two
different splitting rules, namely *variance* and *extratrees*. We mainly adjusted *mtry* and the minimum node
size in order to avoid overfitting. We tested *mtry* values at even intervals
between 2 and the total number of covariates, including both auxiliary data
and spatially explicit covariates. The tested *mtry* values therefore varied,
depending on the number of covariates. The *extratrees* splitting rule generates random
splits as opposed to the *variance* splitting rule, which chooses optimal splits. By
default, *extratrees* generates one random split for each covariate and then chooses the
random split that gives the largest variance reduction (Geurts et
al., 2006). It therefore leads to a greater degree of randomization. We
selected the setup that provided the lowest root mean square error (RMSE) for the out-of-bag
predictions on the training data and used this setup for predictions on the
25 % holdout dataset.

We used the same 100 repeated splits for each number of coordinate rasters,
with and without auxiliary data. We calculated accuracy based on Pearson's
*R*^{2}, RMSE and Lin's concordance criterion (ccc) and subsequently used
the number of coordinate rasters that yielded the lowest RMSE. We selected a
different number of coordinate rasters with and without auxiliary data.

We then compared the accuracies obtained with the optimal numbers of
coordinate rasters, with and without auxiliary data, to the accuracies
obtained with other methods. We tested kriging, random forest models trained
only on the auxiliary data and random forest models trained using EDFs and
RFsp, with and without auxiliary data. We trained the random forest models
using the same procedure outlined above. For kriging, we used variograms
automatically fitted on logarithmic-transformed SOM observations using the
*autofitVariogram* function of the R package *automap* (Hiemstra, 2013). A previous study using
the same dataset showed that kriging predicted SOM more accurately than
regression kriging (Pouladi et al., 2019). We therefore omitted
regression kriging from the analysis although, without this previous
finding, it would have been relevant to include it.

We used the same 100 repeated splits for assessing the accuracies of all
methods. This allowed us to carry out pairwise *t* tests between the
accuracies of the methods. We used the results of the pairwise *t* tests to
rank the methods according to their accuracies according to each of the
metrics. If there was no statistical difference (*p*>0.05)
between the accuracies of two or more methods, these methods received the
same rank. We calculated separate ranks for the methods for each accuracy
metric, resulting in three different sets of ranks. In order to illustrate
the results, we produced maps of SOM with each method, using models trained
from all the data.

We also investigated the covariate importance of models trained with OGCs and
tested all methods for spatially autocorrelated residuals using experimental
variograms. To produce sample variograms of the residuals, we produced maps
with each method using all observations. We converted both observations and
predictions to a natural logarithmic scale. We then subtracted the predictions
from the observations and calculated variograms for these residuals. For
this purpose, we used the function *variogram* from the R package *gstat* (Pebesma and
Graeler, 2020) with its default parameters.

### 2.3.2 Additional datasets

We also compared OGCs to other methods based on the three additional datasets
*meuse*, *eberg* and Swiss rainfall. The methods in the comparison depended on the
dataset. For the *meuse* dataset, we tested all the methods tested on the Vindum
dataset, with the addition of RK using random forest models for regression.
For the *eberg* dataset, we tested random forest models based on auxiliary data
(AUX), EDFs and OGCs, and the combined methods (EDFs + AUX and OGCs + AUX). For the Swiss rainfall dataset, we tested only purely spatial methods,
including ordinary kriging (OK), EDFs, RFsp and OGCs. As for the Vindum
dataset, we tested each method based on 100 splits into training and test
data and carried out pairwise *t* tests on the resulting accuracies. Appendix
A gives additional details on the methods for each dataset. For the three
additional datasets, we focused on the accuracies and maps produced with
each method. We therefore omitted analyses of the residuals and covariate
importance of these datasets.

## 3.1 Optimal number of coordinate rasters

### 3.1.1 Vindum

For the Vindum dataset, accuracies of predictions obtained with OGCs, without
auxiliary data, increased with the number of coordinate rasters up to an
optimum at seven coordinate rasters (Fig. 4).
However, with more than seven coordinate rasters, accuracies deteriorated
slightly with the number of coordinate rasters. This pattern was the same
for all three metrics. On the other hand, with OGCs in combination with
auxiliary data, accuracies generally increased with the number of coordinate
rasters. The increase was greatest when the number of coordinate rasters was
small, while the effect of more coordinate rasters decreased for larger
numbers of coordinate rasters. With auxiliary data, the optimal number of
coordinate rasters was 94 for Pearson's *R*^{2}, 80 for RMSE and 89 for ccc.
Accuracies with auxiliary data were almost invariably higher than accuracies
achieved without auxiliary data.

Figure 5 shows SOM contents mapped for Vindum with
increasing numbers of coordinate rasters, without auxiliary data. The
predictions with only two coordinate rasters showed a pattern very typical
of predictions with *x* and *y* coordinates with very visible orthogonal
artifacts. As the number of coordinate rasters increased, the patterns of
the artifacts changed. With coordinate rasters at three different angles,
the artifacts had a hexagonal pattern, and with coordinate rasters at four
different angles, the artifacts gained an octagonal pattern. Furthermore, as
the number of coordinate rasters increased, the artifacts became less
pronounced. Although some artifacts were visible with coordinate rasters at
seven different angles, they were much less visible than the artifacts in
the map produced with only two coordinate rasters.

With auxiliary data, the effect of increasing the number of coordinate rasters was less clearly visible for the Vindum dataset (Fig. 6). Even with only two coordinate rasters, the predictions had no orthogonal artifacts. However, they contained noisy patterns and sharp boundaries in some areas. This is most likely an artifact from the auxiliary data. For example, using a high-resolution DEM may have created noise in the predictions. However, with coordinate rasters at 80 different angles, the spatial pattern of the predicted SOM contents became substantially smoother, with a reduction in both noise and sharp boundaries. Furthermore, some areas with moderately high SOM contents became more clearly visible and coherent, for example, in the area approximately one-third of the way from the western to the northern corner of study area. The predicted patterns with a higher number of coordinate rasters were therefore not only more accurate but also more realistic.

### 3.1.2 Additional datasets

For the three additional datasets, the effect of increasing the number of
coordinate rasters without auxiliary data was generally the same as for the
Vindum dataset. In all three cases, there was relatively little, if any,
increase in accuracy after an initially very steep increase. For the *meuse*
dataset, the optimal number of coordinate rasters was six or eight,
depending on the accuracy metric (Fig. A1 in the Appendix). For
the *eberg* dataset, the optimal number was 91, but there was only limited
improvement in accuracy with more than five coordinate rasters
(Fig. A3). For the Swiss rainfall dataset, the
optimal number of coordinate rasters was 33 or 50, depending on the accuracy
metric (Fig. A5).

As for the Vindum dataset, the optimal number of coordinate rasters was
generally larger in combination with auxiliary data than without auxiliary
data. For the *meuse* dataset, the optimal number of coordinate rasters in
combination with auxiliary data was 11 or 13, depending on the accuracy
metric. For the *eberg* dataset, the optimal number of coordinate rasters in
combination with auxiliary data was 22. However, unlike the results for the
Vindum dataset, accuracies for these two datasets gradually decreased when
the number of coordinate rasters was larger than the optimal value.

In summary, the combination of OGCs with auxiliary data generally increased
the optimal number of coordinate rasters. Furthermore, in some cases,
accuracy deteriorates when the number of coordinate rasters surpasses an
optimal value, while in other cases it reaches a plateau. The decrease in
accuracy past the optimum may be due to the correlation between the coordinate
rasters. Coordinates *x* and *y* are perfectly uncorrelated, but the coordinate
rasters become increasingly correlated as their number increases. The
optimal value may therefore be a trade-off between the increased ability of
the model to account for spatial trends and the adverse effect of
increasingly correlated covariates. It is therefore likely that it depends
on the complexity of the spatial distribution of the target variable and the number of observations.

With OGCs in combination with auxiliary data, the process-based covariates in
the auxiliary data most likely help to reduce the effect of correlation
between the coordinate rasters. Furthermore, in this case, the number of
coordinate rasters also affects the relative weighting between the auxiliary
data and the coordinate rasters. When *mtry* is smaller than the total number of
covariates, a higher number of coordinate rasters increases the chance that
a coordinate raster will be available for a split. The optimal number of
coordinate rasters may therefore depend on the optimal weighting between
process-based and explicitly spatial covariates. This optimal weighting may
depend on the number of covariates in the auxiliary data and the
strength of the relationship between the target variable and the auxiliary
data.

At present, several factors could therefore explain the optimal number of coordinate rasters for each dataset with and without auxiliary data. The exact interplay between these factors is unclear, and the best option may therefore be to experiment with different numbers of coordinate rasters.

## 3.2 Method comparison

### 3.2.1 Predictive accuracy

For all four datasets, there were large overlaps in the accuracies of the
methods, as accuracies varied across the 100 repeated splits
(Figs. 7, A2, A4 and A6).
However, an analysis on the Vindum dataset revealed that the accuracies
generally correlated between the methods across the repeated splits. The
mean correlation coefficient (Pearson's R) was 0.52 (0.19–0.88) for
*R*^{2}, 0.71 (0.65–0.71) for RMSE and 0.65 (0.41–0.89) for ccc. This
shows that some holdout datasets yielded consistently high accuracies, while
others yielded consistently low accuracies. Furthermore, especially for
*R*^{2} and ccc, a few holdout datasets yielded much lower accuracies than
the other holdout datasets, leading to long negative tails
(Figs. 7, A2 and A6).

For the Vindum dataset, kriging achieved the highest rank for *R*^{2}
(Table 2). For RMSE, kriging shared the highest rank
with EDFs, RFsp and OGCs in combination with auxiliary data. Lastly, OGCs and
RFsp in combination with auxiliary data shared the highest rank for ccc. In
short, kriging, RFsp with auxiliary data and OGCs with auxiliary data all had
the highest rank for two accuracy metrics out of three. We therefore regard
these three methods as the most accurate methods for the Vindum dataset.
Furthermore, we regard these three methods as equally accurate for this
dataset, as none of them was universally more accurate than the other two
methods.

Auxiliary data used on their own, and RFsp without auxiliary data, had the lowest rank for all three accuracy metrics on the Vindum dataset. Furthermore, OGCs without auxiliary data had the same rank as EDFs without auxiliary data for all three accuracy metrics.

Pouladi et al. (2019) tested several methods for predicting SOM on the Vindum dataset, including kriging and the machine-learning algorithms cubist and random forest, with and without kriged residuals. The authors found that kriging provided the most accurate predictions of SOM. The results for Vindum affirm the high accuracy of kriging predictions, but they also show that random forest models combining auxiliary data with spatial trends can achieve similar accuracies.

For the *meuse* dataset, OGCs in combination with auxiliary data achieved the
highest rank for *R*^{2} and RMSE (Table 3). For
ccc, OGCs in combination with auxiliary data shared the highest rank with EDFs
in combination with auxiliary data. Without auxiliary data, OGCs received the
third rank for RMSE and the fourth rank with *R*^{2} and ccc. OGCs without
auxiliary data were generally on par with models based only on auxiliary data
and with EDFs. They were less accurate than combined methods and OK (*R*^{2} and
ccc). RFsp without auxiliary data was the least accurate method.

For the *eberg* dataset, OGCs in combination with auxiliary data were the most
accurate method (Table 4). Without auxiliary data,
OGCs had the third rank. They were less accurate than EDFs combined with
auxiliary data but more accurate than EDFs without auxiliary data and models
based only on auxiliary data. Models based only on auxiliary data yielded
the lowest accuracies.

For the Swiss rainfall dataset, OGCs were the most accurate method for all three metrics (Table 5). RFsp was the second most accurate method, followed by EDFs. OK was the least accurate method.

In summary, for Vindum, *meuse* and *eberg*, OGCs combined with auxiliary data were either
the most accurate method or one of the most accurate methods. Without
auxiliary data, OGCs were not one of the most accurate methods for these
datasets. However, for the Swiss rainfall dataset, OGCs were the most accurate
method, even though we used no auxiliary data.

It is important to consider that in most cases all methods yielded acceptable accuracies. Although the differences between the accuracies of the methods were in many cases statistically significant, they were generally small. However, the results show that OGCs compare well with other methods for integrating spatial trends in machine-learning models.

### 3.2.2 Maps

For the Vindum dataset, kriging produced a smooth prediction surface, which
is very common for this method (Fig. 8a). The
prediction surface with EDFs was mostly smooth, but it also contained a
distinct “rings in the water” artifact caused by the raster with the
distance to the middle of the study area (Fig. 8b). The prediction surface with RFsp was smoother than the prediction
surface produced by kriging (Fig. 8c). The
predictions with only auxiliary data were very similar to the predictions
made with *x* and *y* coordinates in combination with auxiliary data (compare
Figs. 8c and 6a). In
combination with auxiliary data, both EDFs and RFsp produced smoothing
effects similar to the effect seen with OGCs in combination with auxiliary
data (compare Figs. 8e–f to 6b).
However, for EDFs the smoothing was less visible than with OGCs and for RFsp
it was more visible than with OGCs.

For the *meuse* dataset, OK, EDFs and RFsp produced smooth prediction surfaces
(Fig. 9). However, OGCs without auxiliary data
produced a prediction surface with several abrupt, angular artifacts. The
accuracy of OGCs without auxiliary data was on par with some of the other
methods, but the maps revealed that the predictions were not realistic.
Predictions with the combined methods (RK, EDFs + AUX, RFsp + AUX and OGCs + AUX) were mostly similar to predictions with only auxiliary data.
However, in some places these methods smoothed out the spatial patterns
produced with only auxiliary data (for example, in the northern part of the
study area), and in other places they made them more distinct (for example,
southwest of the middle of the study area). In this regard, the results are
similar to the results from Vindum.

For the *eberg* dataset, predictions based only on auxiliary data showed a very
noisy spatial pattern with many soil types occupying small, incoherent areas
(Fig. 10c). The spatial patterns produced with OGCs
and especially EDFs were much smoother and contained several large, rounded
areas with little internal variation in soil types
(Fig. 10a and b).
The predictions obtained with the combined methods were similar to the
spatial pattern obtained with only auxiliary data. However, they were much
smoother as the soil types occupied mostly coherent areas. The effect for
predictions of soil types therefore appears similar to the effect for
numeric variables seen for Vindum and *meuse*.

For the Swiss rainfall dataset, OK produced a smooth, highly anisotropic prediction surface (Fig. 11a). The prediction surfaces of EDFs, RFsp and OGCs also showed anisotropy, but they were generally smoother and more rounded. For example, with OK, some individual observations showed an effect on the prediction surface as elongated spots in the direction of the anisotropy. With the other three methods, a few individual observations showed an effect in the prediction surface, but their effects are more rounded and less distinct. The predictions with EDFs, RFsp and OGCs therefore appear more general than the OK predictions. Moreover, the prediction surfaces of EDFs, RFsp and OGCs appear very similar.

### 3.2.3 Residuals

For the Vindum dataset, the residuals of the SOM predictions had some degree of spatial dependence for all methods except kriging (Fig. 12). This finding contrasts with Hengl et al. (2018), who found that there was no spatial trend in the residuals of predictions with RFsp. EDFs, RFsp and OGCs used without auxiliary data had the most spatially dependent residuals. However, the residuals of the combined methods (EDFs + AUX, RFsp + AUX and OGCs + AUX) had less spatial dependence than the residuals of models based only on auxiliary data. OGCs + AUX was the machine-learning method with the least spatially dependent residuals, although the residuals still had more spatial dependence than kriging residuals.

## 3.3 Covariate importance

For the Vindum dataset, the most important covariate from the auxiliary data was the depth of sinks (Table 6). The most likely reason for its high importance is the presence of a large sink with very high SOM contents northwest of the middle of this study area (Fig. 1). As sinks trap surface runoff, they often have wet conditions, which give rise to peat accumulation.

When used in combination with the auxiliary data, the importance of the
individual coordinate rasters varied from 0.6 % to 3.1 % of the
importance of the depth of sinks, with mean value of 1.7 %. The most
important coordinate raster had *θ*=0.48*π* (close to a
north–south axis) and was the 12th most important covariate. The sum of
the importance of the coordinate rasters was equal to 134.3 % of the
importance of the depth of sinks (Table 6).
Therefore, with coordinate rasters at 80 different angles, the effect of the
individual rasters on the predictions was subtle, but their combined effect
was strong.

Figure 13 shows the importance of the coordinate
rasters relative to *θ* in a way that is similar to a wind rose. The plots
repeat the bars for *θ*≥*π*, as the importance of a given
angle is directionless. For example, the importance of *θ*=0
(east) is equal to the importance of *θ*=*π* (west).

Without auxiliary data, the most important coordinate rasters had a general
northwestern to southeastern angle (Fig. 13). On the
other hand, the coordinate rasters with angles between a north–south and a
northeast–southwest axis had low importance. The most likely reason for this
pattern is the location of the sink with very high SOM contents to the
northwest of the middle of this study area. This creates a large difference
in the SOM contents of the northwestern and southeastern parts of the study
area, giving large importance to covariates that can explain this
difference. Additionally, the northwestern side of the sink has a very steep
slope, creating a steep gradient in SOM contents in this direction. A stable
variogram showed anisotropy along a north–northeast to south–southwest
axis (*θ*=0.34*π*), with a major range of 136 m and a minor
range of 118 m. The direction of the anisotropy therefore coincided with the
direction of the least important coordinate rasters.

On the other hand, with OGCs in combination with auxiliary data, the most
important coordinate rasters had tilt angles close to a north–south axis
(*θ*=0.5*π*). At the same time, the least important coordinate
rasters had tilt angles close to a northeast–southwest axis (*θ*=0.25*π*). The residuals from the predictions with only auxiliary data
also displayed a degree of anisotropy. A stable variogram showed anisotropy
along a northeast to southwest axis (*θ*=0.21*π*), with a major
range of 52 m and a minor range of 38 m. Again, the angle of the anisotropy
coincided with the angle of the least important coordinate rasters. The
spatial pattern of the residuals therefore differed from the spatial pattern
of the SOM contents in the Vindum study area. Apparently, there are
unaccounted for processes decreasing the spatial variation along a
northeast–southwest axis relative to other angles.

A possible cause of the anisotropy in the residuals may be the plowing
direction. The main plowing direction in the Vindum study area is along an
east–northeast to west–southwest axis (*θ*=0.18*π*). This
angle is nearly parallel to the angle of the least important coordinate
rasters (Fig. 14). The plowing direction,
combined with the topography, has a large impact on soil movement, as
plowing displaces soil both along and across its direction (Lindstrom
et al., 1990; De Alba, 2003; Heckrath et al., 2006). Most of the study area
has the same plowing direction, irrespective of the topography, resulting
in up-, down- and cross-slope plowing in various parts of the field. This
creates in a complex pattern of soil redistribution, which likely affects
the SOM contents of the topsoil. As downslope soil movement is strongest in
the plowing direction, variation in soil properties parallel to this
direction is likely to be smaller than the variation perpendicular to the
plowing direction. This corresponds to the low importance of coordinate
rasters with angles close to the plowing direction. However, none of the
auxiliary data accounted for the plowing direction. This indicates that
OGCs can add information on the most likely processes affecting soil
properties in an area.

## 3.4 Choice of method

At Vindum, the three most accurate methods were kriging, RFsp with auxiliary
data and OGCs with auxiliary data. For *meuse*, OGCs and EDFs combined with auxiliary
data were most accurate and for *eberg,* OGCs combined with auxiliary data were most
accurate. For the Swiss rainfall dataset, OGCs were the most accurate method.

Although kriging was in most cases less accurate than other methods, some soil mappers would probably still choose it for mapping soil properties due to its computational efficiency and conceptual simplicity. However, aside from accuracy, an advantage of methods based on machine learning lies in the fact that they provide larger amounts of information than geostatistical models. Kriging in itself does not provide information on the processes that control spatial variation in soil properties, but machine-learning models can include covariates related to soil processes, providing information on the processes that are most likely to affect the spatial distribution of a soil property.

With spatial approaches such as EDFs, RFsp and OGCs, researchers can
incorporate feature space and geographic space in a machine-learning model.
Of the previously used approaches, OGCs are most similar to EDFs, as they used
the *x* and *y* coordinates, and the distances to the corners of the study area
resemble the coordinates. On the other hand, RFsp is more similar to
geostatistical models, as it relies on distances between observations.
However, this similarity comes at the cost of calculating a large number of
distance rasters.

One advantage of using spatially explicit covariates (EDFs, RFsp or OGCs) is that researchers can interpret local and spatial effects at once. In this regard, OGCs have an advantage over EDFs and RFsp, as it is clear what the coordinate rasters represent. It is less clear how researchers should interpret distances to the corners of the study area or the distance to a specific observation. We have also shown that it is straightforward to illustrate covariate importance of OGCs.

Furthermore, an advantage of OGCs relative to RFsp is that OGCs required fewer
covariates to achieve the same accuracy. In fact, without auxiliary data,
OGCs achieved a higher accuracy with a smaller number of covariates for the
datasets of Vindum, *meuse* and Swiss rainfall. This demonstrates a clear advantage
of OGCs, as it is possible to adjust the number of coordinate rasters. EDFs
and RFsp do not presently have similar options.

We will stress that, as a rule, soil mappers should not use machine-learning
models relying only on spatial trends, as EDFs, RFsp and OGCs all yielded
lower accuracies without auxiliary data for the soil datasets (Vindum,
*meuse* and *eberg*). Moreover, surprisingly, these methods had the most spatially
autocorrelated residuals for the Vindum dataset, although they relied
exclusively on spatial trends. The results therefore suggest that soil
mappers should primarily use these methods in combination with auxiliary
data and not on their own. If no auxiliary data are available, kriging is
likely to be a better option. However, results from the Swiss rainfall
dataset show that, for other spatial problems, auxiliary data may be
unnecessary.

We have shown in this study that the use of oblique geographic coordinates
(OGCs) is a reliable method for integrating auxiliary data with spatial
trends for modeling and mapping soil properties. In most cases, the method
eliminated the orthogonal artifacts that arise from the use of *x* and *y* coordinates and achieved higher accuracies than maps created with only two
coordinate rasters. However, for *meuse*, without auxiliary data, OGCs still
produced abrupt angular artifacts in the final map. Soil mappers should
therefore combine OGCs with auxiliary data, as this gives higher accuracies
and spatial patterns with a higher degree of realism.

OGCs are more interpretable than previous similar approaches, and more flexible, as it is possible to adjust the number of coordinate rasters. This should allow soil mappers to find a good compromise between accuracy and computational efficiency for mapping soil properties, as the optimal number of coordinate rasters may vary depending on the study area and the soil property in question.

At this point, we have only tested the method for three soil datasets and one meteorological dataset. It will therefore be highly relevant to test the method for other soil properties and areas. It will especially be relevant to test the method in larger, less densely sampled areas. Previous studies have shown that machine learning is likely to provide higher accuracies in such areas (Zhang et al., 2008; Greve et al., 2010; Keskin et al., 2019), and it will be relevant to test if this is also the case for oblique geographic coordinates. Results from the Vindum and the Swiss rainfall datasets also suggest that the method can be useful for mapping variables with anisotropic spatial distributions, and it will therefore be relevant to test it on datasets with a high degree of anisotropy. Lastly, one should note that we carried out this study for relatively small areas using “flat” coordinate systems. Using OGCs for larger areas and other coordinate systems may require alterations to the method.

We call upon researchers within digital soil mapping to aid us in testing oblique geographic coordinates as covariates for additional datasets, and we have therefore made the function for generating oblique geographic coordinates available as an R package. Moreover, to allow other researchers to test methods on the Vindum dataset, we have made it available and part of the same package.

## A1 Methods

### A1.1 *meuse*

We mapped zinc contents for the *meuse* dataset (155 points). The *meuse* dataset contains
covariates including the flooding frequency and the distance to the river.
We added two covariates in the form of a digital elevation model (DEM,
https://www.ahn.nl/, last access: 9 July 2020) and surface water occurrence
(Pekel et al., 2016). We converted the categorical
raster of flooding frequency to indicator variables and transformed all the
covariates to principal components. This resulted in six principal
components.

We tested all the methods applied to the Vindum dataset, with the addition
of regression kriging (RK). We used random forest models trained on the
auxiliary data for regression and then kriged the residuals using the
function *krige.conv* from the R package *geoR* (Ribeiro et al., 2020). As for the
Vindum dataset, we tested each method with 100 repeated splits into training
(75 %) and test (25 %) data. For each split, we calculated Pearson's
*R*^{2}, RMSE and ccc. We carried out pairwise *t* test on the accuracies
obtained with each method in order to assess if the differences between their
accuracies were statistically significant. We also produced maps with each
of the nine methods in order to compare results.

### A1.2 *eberg*

We mapped soil types for the *eberg* dataset. The *eberg* dataset contains 3670 soil
observations. We removed points outside the coverage of the covariates and
points without a soil type classification. Furthermore, we removed the soil
types “Moor” and “HMoor”, as only one observation was available for each
soil type. This reduced the dataset to 2552 observations.

The *eberg* dataset contains covariates including the parent material, a DEM, the
SAGA GIS topographic wetness index and the thermal infrared reflectance from
satellite imagery. We converted the parent material classes to indicators
and converted all covariates to principal components. This resulted in 11
principal components.

The dataset is highly clustered, which is likely to affect accuracy
assessments, as some areas have much higher point densities than others. To
counter this effect, we organized the data in 100 groups using *k* means
clustering on their coordinates. We then produced 100 splits into training
and test data based on these groups. In each split, the training data
contained observations from 75 groups, and the test data contained
observations from the remaining 25 groups.

As we aimed to predict a categorical variable, we did not use kriging. Furthermore, due to the large size of the dataset, we did not use RFsp, as this would require us to produce more than 2000 raster layers with buffer distances. Hengl et al. (2018) avoided this by calculating only buffer distances to each soil type. However, we did not choose this solution as it would create problems for accuracy assessment. If a raster layer contains distances to test observations and training observations, the result would be circular logic, invalidating the accuracy assessment. Buffer distances based only on the training data would be less problematic. However, as we used 100 repeated splits, this was not an option.

We therefore tested only five methods for the *eberg* dataset, namely models based on
auxiliary data (AUX), Euclidean distance fields (EDFs), OGCs, and EDFs
and OGCs combined with auxiliary data.

Due to the large size of the dataset, model training was slower than for the
other datasets. We therefore tuned a random forest model only once for each
method and used the resulting parameterization for all 100 data splits. For
each split, we calculated the accuracy on the test data as the fraction of
observations correctly predicted. We carried out pairwise *t* tests on the
accuracies obtained with each method in order to assess if the differences
between their accuracies were statistically significant.

We produced maps of soil types with each of the five methods in order to compare results.

### A1.3 Swiss rainfall

The Swiss rainfall dataset contains 467 rainfall observations from
Switzerland from 8 May 1986. We did not use any covariates for this
dataset, and we therefore tested only purely spatial methods. We tested
ordinary kriging with correction for anisotropy, EDFs, RFsp and OGCs. As for
the Vindum dataset, we tested each method with 100 repeated splits into
training data (75 %) and test data (25 %). For each split, we calculated
Pearson's *R*^{2}, RMSE and ccc. We carried out pairwise *t* tests on the
accuracies obtained with each method in order to assess if the differences
between their accuracies were statistically significant. Lastly, we produced
maps of rainfall with each of the four methods in order to compare results.

## A2 Results

### A2.1 *meuse*

For the *meuse* dataset, the accuracy of OGCs combined with auxiliary data was
consistently higher than the accuracy of OGCs without auxiliary data,
irrespective of the accuracy metric and the number of coordinate rasters
(Fig. A1). The accuracy of OGCs initially increased
quickly with the number of coordinate rasters up to an optimum, after which
there was no further improvement. For OGCs + AUX, the increase in accuracy
was more gradual, up to an optimum, after which the accuracy deteriorated
slightly. The optimal number of coordinate rasters without auxiliary data
was six for RMSE and ccc and eight for *R*^{2}. With auxiliary data, the
optimal number of coordinate rasters was 11 for RMSE and 13 for *R*^{2} and
ccc. In the subsequent analysis, we used six coordinate rasters for OGCs
without auxiliary data and 11 coordinate rasters for OGCs with auxiliary
data.

For the *meuse* dataset, as for the Vindum dataset, the differences between the
accuracies of the methods were generally small relative to the variation in
accuracy between the test splits (Fig. A2).
Furthermore, most methods had long tails with lower accuracies.

### A2.2 *eberg*

For the *eberg* dataset, accuracy for OGCs without auxiliary data first increased
sharply up to five coordinate rasters. Past this point, there was little
improvement in accuracy and some numbers of coordinate rasters produced
sharp, irregular drops in accuracy (Fig. A3).
Combined with auxiliary data, the accuracy of OGCs increased up to 22
coordinate rasters, after which they gradually declined. Without auxiliary
data, the optimal number of coordinate rasters was 91. However, the highly
irregular pattern of the accuracies did not justify any number past the
initial increase, and we therefore used only five coordinate rasters. For
OGCs combined with auxiliary data, we used 22 coordinate rasters.

For the *eberg* dataset, as for the Vindum dataset, variation in accuracy between
the splits into training and test data was in most cases greater than
variation between the methods (Fig. A4). However,
unlike the other datasets, the distributions of the accuracies were mostly
symmetric.

### A2.3 Swiss rainfall

For the Swiss rainfall dataset, the accuracy of OGCs generally increased with
the number of coordinate rasters (Fig. A5). The
increase in accuracy was steep at first, then gradual. For Pearson's
*R*^{2}, the optimal number of coordinate rasters was 33 and for RMSE and
ccc it was 50. There was little change in accuracy past the optimal number
of coordinate rasters.

As for the other datasets, variation in accuracies on the Swiss rainfall
dataset was greater between the splits into training and test data than
between the methods (Fig. A6). The distributions
of RMSE were mostly symmetric, but the distributions of *R*^{2} and ccc had
long negative tails, as some splits yielded much lower accuracies than other
splits.

The function for generating oblique geographic coordinates is available as an R package at https://bitbucket.org/abmoeller/ogc/src/master/rPackage/OGC/ (Møller, 2019). The package also contains the SOM observations and auxiliary data from the Vindum dataset. Furthermore, we have made the R code used in this study available in a public repository at http://dx.doi.org/10.5281/zenodo.3820068 (Møller et al., 2020).

ABM and NP prepared the data. ABM carried out the analyses and prepared the paper with inputs from all coauthors.

The authors declare that they have no conflict of interest.

We are obliged to the two anonymous referees and to Alexandre Wadoux, who provided vital feedback on the paper. Their comments and advice have greatly improved the paper, and we give them our thanks.

This paper was edited by Kristof Van Oost and reviewed by two anonymous referees.

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