CNNs are based on the concept of a layer of convolving windows which move along a data array in order to detect features (e.g. edges) of the data by using different filters (Fig. ). When stacked together, convolutional layers are capable of extracting features of increasing complexity and abstraction . Since CNNs have the capacity to leverage the spatial structure of the data, they have been widely and effectively used in computer vision for image recognition or extraction .

A CNN has a number of three-dimensional hidden layers, with each layer learning to detect different features of the input images . In our case, each of the layers can perform one of the two types of operations: convolution or pooling. Convolution takes the input images through a set of convolutional filters (e.g. a 3×3 size filter), each of which detects and enhances certain features from the images. Units in a convolutional layer are organised in feature maps (here we used 3×3). Each unit of the feature map is connected to local patches in the feature maps of the previous layer through a set of weights. This local weighted sum is then passed through a non-linear transfer function.

A pooling operation merges similar features by performing non-linear down-sampling. Here we used max-pooling layers which combine inputs from a small 2×2 window. Pooling also makes the features robust against noise. All the convolutional and pooling layers are finally “flattened” to the fully connected layer. In effect, the fully connected layer is a weighted sum of the previous layers.

To obtain optimal weights for the network, we train the network using a training dataset. Weights were adjusted based on a gradient-based algorithm to minimise the error using an Adam optimiser . We refer to a review by on the details of CNN.

CNNs have the capacity to predict multiple properties simultaneously. By doing so, a multi-task CNN is capable of sharing learned representations between different targets and also using the other targets as “clues” during the prediction process. In consequence, the error of the simultaneous prediction is generally lower compared with a single prediction for each target . An additional advantage of using a multi-task CNN is the reported reduction on the risk of overfitting .

In DSM, where the combination of large extents, high resolution, and bootstrap routines leads to running multiple model realisations on billions of pixels, combined with the fact that CNNs use a group of pixels around the soil observation instead of a single pixel, the time and computational resources required for training and inference are an important factor. Due to the simultaneous training and inference of multiple targets, a multi-task CNN presents the advantage of reducing both training and inference time compared with a single-task model.

The data used in this work correspond to Chilean soil information. Since most observations are distributed on agricultural lands, we complemented that information with a second small data collection compiled from the literature and collaborators. We selected soil organic carbon (SOC) content (%) at depths 0–5, 5–15, 15–30, 30–60, and 60–100 cm as our target attribute. In total, 485 soil profiles were used after excluding soil profiles with total depth lower than 100 cm (in order to assure that all the profiles have observations at all depth intervals). For more details about the data and depth standardisation we refer the reader to .

As covariates, we used (a) a digital elevation model (HydroSHEDS, ), which is provided at 3 arcsec resolution, in addition to its derived slope and topographic wetness index, calculated using SAGA ; and (b) long-term mean annual temperature and total annual rainfall derived from information provided by WorldClim at 30 arcsec resolution. All data layers were standardised to a 100 m grid size.

Deep learning techniques are described as “data-hungry” since they usually work better with large volumes of data. The direct effect of data augmentation is to generate new samples by modifying the original data without changing its meaning . To achieve this, we rotated the 3-D array shown in Fig.  by 90, 180, and 270∘, hence quadruplicating the number of observations. It is important to note that the central pixel preserves its initial position.

A secondary effect of data augmentation is regularisation, reducing the variance of the model and overfitting . Data augmentation also induces rotation invariance by generating alternative situations (rotated data) where the model response should be similar to the original data (e.g. a soil profile next to a gully is expected to be similar to a profile next to the opposite side of the gully, ceteris paribus).

The multi-task CNN used in this study (Fig. ; Table ) consists of an input layer passed through a series of convolutional and pooling layers with a ReLU (rectified linear unit) activation function, which adds non-linearity by passing the learned weights through the function f(x)=max⁡(0,x). The initial common/shared network has a function of extracting features shared between the five target depth ranges. Next, the common features are propagated through independent branches, one per depth range, of three fully connected layers.

The multiple connection between the layers generates a high number of parameters. In order to reduce the risk of overfitting, we introduce a dropout rate. In between the layers, 30 % of the connections were randomly disconnected . We added this dropout operation in the shared layer and another dropout before the output.

As explained in Sect. , our method uses a window around a soil observation which encloses a group of pixels instead of the single pixel that coincides with the observation. Most likely, the extent or size of that window will affect the model performance. To assess this effect, we compared the results of different models trained with a window size of 3, 5, 7, 9, 15, 21, and 29 pixels.

As the vicinity size increases, so does the number of parameters of the network (considering a fixed network architecture) and the risk of overfitting. To minimise overfitting, we modified the architecture of the network depending on the vicinity size (Table ).

First, 10 % (n=49) of the total dataset was randomly selected and used as a test set. The remaining 90 % of the samples (n=436) were augmented (see Sect. ), obtaining a total of 1744 samples. Following the data augmentation, we performed a bootstrapping routine with 100 repetitions, where the training set is obtained by sampling with replacement, generating a set of size 1744. The samples which were not selected, about one-third, correspond to the out-of-bag validation set.

As a control, we compared our results with a previous study by where we used a Cubist regression tree model to predict soil organic carbon at a national extent. The Cubist model has been used in many other DSM studies due to its interpretability and robustness. In that study, we used the same set of soil observations and covariates described in Sect. .

In this work (and in ), the uncertainty is represented as the 90 % prediction interval derived from the 100 bootstrap iterations. To estimate the upper and lower prediction interval limits, we used the following formula: PIL=x‾±1.645σ2+MSE, where x‾ and σ2 are the mean and variance of the 100 iterations, and MSE is the mean square error of the 100 fitted models.

The CNN was implemented in Python (v3.6.2; ) using Keras (v2.1.2; ) and Tensorflow (v1.4.1; ) back end. Computing was done using the University of Sydney's Artemis high-performance computing facility.

To generalise and improve the CNN model, we created new data using only information from the training data by rotating the original image input. Data augmentation was effective at reducing model error and variability (Fig. ). It was possible to observe a decrease in the error, by 10.56 %, 10.56 %, 11.25 %, 14.51 %, and 24.77 % for the 0–5, 5–15, 15–30, 30–60, and 60–100 cm depth ranges, respectively. The results are in accordance with image classification studies which generally showed that data augmentation increased the accuracy of classification tasks . It is hypothesised that, by increasing the amount of training data, we can reduce overfitting of CNN models.

In terms of the data spatial autocorrelation, we need to consider that after augmenting the data we have four samples in the same locations with exactly the same SOC content, therefore assuming that there is no variance when distance is equal to 0. That is theoretically true if we consider that the distance is exactly equal to 0. In practice, when calculating the semivariogram, the semivariance value of the first bin will be lower, but that does not significantly affect the final model.

To incorporate contextual information for DSM prediction, we represent the input as an image. The image is represented as observation in the centre, with surrounding pixels in a square format. The size of the neighbourhood window (vicinity) has a significant effect on the prediction error (Fig. ). There is no significant difference when using a vicinity size of 3, 5, 7, or 9 pixels, but sizes above 9 pixels showed an increase in the error. It is possible to observe a lower error in the test dataset, compared with the training and validation, due to the slight differences in the dataset distributions (Fig. ). Since the SOC distribution is right-skewed, the random sampling used to generate the training dataset does not completely recreate the original distribution, excluding samples with very high SOC values. This should not significantly affect the conclusions given that the error for the samples with hight SOC values is accounted for during the bootstrapping routine and reflected in the training and validation curves of Fig. . In this example, for a country-scale mapping of SOC at 100 m grid size, information from a 150 to 450 m radius is useful. A similar influence distance was obtained by and , who reported a medium-scale spatial correlation range for SOC in China of around 300 and 550 m, respectively; by , who reported a range of around 190 m for a coastal forest in Tanzania; and by , who reported a range of around 200 m in two grassland sites in Germany. A similar spatial correlation range was reported for croplands in an review by , where, based on 41 variograms, the authors estimated an average spatial correlation range of around 400 m.

As described in Sect. , we slightly modified the architecture of the network as input window size increased, in order to minimise the risk of overfitting and isolate the effect of the vicinity size. As we increase the vicinity size, we give the model a broader spatial context. Our results show that just a small amount of extra context provides enough information to improve the model predictions, and a larger amount of neighbouring information acts as noise, impairing the generalisation of the model. Since we used the relatively large resolution of 100 m, it is hard to tell specifically what the minimum amount of context needed to improve SOC predictions is. We believe that using higher resolutions (<10 m) could produce more insights about this matter.

Soil-forming factors interact in complex ways and affect soil properties with different strength. At the local scale, a broader context (i.e. larger vicinity size) does not necessarily provide extra information to the model, for instance when one of the factors is relatively homogeneous. The extra information could be even detrimental if the vicinity size is well beyond the area of influence of a factor, which is what probably happened when we increased the vicinity size above 9 pixels (radius ≈450 m). Representing this complexity in numerical terms would imply varying the size of the input array, such as a different vicinity size for each forming factor, most likely also varying depending on the spatial location of the soil observation (e.g. smaller vicinity for homogeneous areas and larger for heterogeneous areas). This is technically possible but considerably increases the complexity of the modelling.

We compared our approach with the Cubist model used in our previous study , where we did not use any contextual information. We observed a significant decrease in the error (Fig. ) by 23.0 %, 23.8 %, 26.9 %, 35.8 %, and 39.8 % for the 0–5, 5–15, 15–30, 30–60, and 60–100 cm depth ranges, respectively. Most current DSM studies rely on punctual observations without contextual information, and, given the improvements shown by our approach, we believe there is a big potential for CNNs to be used in operational DSM.

To compare our results with a method that uses contextual information, we ran a test using wavelet decomposition as per . In addition to the five covariates, we used their approximation coefficients from the first, second, and third levels of a Haar decomposition . The results including wavelet-decomposed variables were similar to ones obtained with the Cubist model. The CNN approach reduced the error by 24.8, 24.7, 28.5, 28.6, and 23.5 for the 0–5, 5–15, 15–30, 30–60, and 60–100 cm depth ranges, respectively. reported an average improvement of 1 % for the prediction of clay content. In our case the wavelet decomposition reduced the error of the SOC content by 5.1 %, on average, compared with the Cubist model, but the reduction was only observed in depth, where SOC content is low (2.4 %, 1.2 %, 2.3 %, -10.1 %, and -21.4 % error change for the 0–5, 5–15, 15–30, 30–60, and 60–100 cm depth ranges, respectively), hence reducing the effect in applications such as carbon accounting. Our approach showed greater error reductions and through the whole profile.

Our approach uses a multi-task CNN to predict multiple depths simultaneously in order to produce a synergistic effect. Compared with predicting each depth range in isolation by training a network with the same structure (Sect. ) but with only one output, our approach reduced the error by 1.5 %, 6.7 %, 6.6 %, 8.9 %, and 13.0 % for the 0–5, 5–15, 15–30, 30–60, and 60–100 cm depth ranges, respectively. In this case, the reduction was modest and we believe the effect can be greater when more soil observations are available.

In DSM, there are two main approaches to deal with the vertical variation of a soil attribute: 2.5-D and 3-D modelling. In the first one, an independent model is fitted for each depth range. The latter explicitly incorporates depth in order to obtain a single model for the whole profile. Interestingly, both approaches show a decrease in the variance explained by the model as the prediction depth increases. In a 3-D mapping of SOC for a 125 km2 region in the Netherlands, presented R2 values of 0.75, 0.23, and 0.09 for the 0–30, 30–60, and 60–90 cm depth ranges, respectively. In our previous study , the 2.5-D mapping showed R2 values of 0.39, 0.39, 0.27, 0.19, and 0.17 for the 0–5, 5–15, 15–30, 30–60, and 60–100 cm depth ranges, respectively. Similar studies show the same trend , independent of the models used or the soil attribute predicted. This is expected as the information used as covariates usually represents surface conditions. Our multi-task network presented the opposite trend (Fig. ), showing an increase in the explained variance as the prediction depth increases.

The prediction of the adjacent layers served as guidance, producing a synergistic effect. A soil attribute through a profile usually has a predictable behaviour (unless there are lithological discontinuities), which has been described by many authors in the form of depth functions . A CNN is capable of generating an internal representation of the vertical distribution of the target attribute, which resembles the observed pattern (Fig. ).

Visually, the maps generated with the Cubist tree model and our multi-task CNN showed differences (Fig. ). In an example for an area in southern Chile (around 72.57∘ S), the map generated with the Cubist model (Fig. a) shows more details related with the topography, but also presents some artefacts due to the sharp limits generated by the tree rules. On the other hand, the map generated with the multi-task CNN using a 7×7 window (Fig. b) shows a smoothing effect, which is an expected behaviour consequence of using neighbour pixels.

A recommended DSM practice is to present a map of a predicted attribute along its associated uncertainty . Our multi-task CNN significantly reduced the prediction interval width (PIW, Table ) compared with the Cubist model. On average, we observed a reduction of 13.8 % and 13.1 % for the CNN model generated with and without data augmentation pretreatment, respectively, for the first three depth intervals. Our multi-task CNN model showed a slightly lower prediction interval coverage, but all were wider than the proposed 90 % coverage.

In terms of the spatial patterns of the uncertainty (Fig. ), the greater reductions of the PIW were observed in elevated areas of the Andes, followed by the central valleys. A slight increase, on the order of 6 %–8 %, was observed in the western coastal ranges. The reduction of the PIW in the Andes is most likely due to a more reserved extrapolation by the CNN models compared with the Cubist model. It is worth noting that the central valleys are where most of the agricultural lands are located and the uncertainty reduction observed in these areas could have important implications.