The total radionuclide 40K amount was measured by the absorption energy (1.46 MeV). Thorium (232Th) and uranium (238U) were quantified by absorption energy (approximately 2.62 and 1.76 MeV, respectively). This quantification was indirectly performed through thallium (208Tl) and bismuth (214Bi), derived by radioactive decay, respectively, for 232Th and 238U, which are expressed as eTh and eU (equivalent thorium and uranium, respectively).

For soil gamma spectrometric characterization, we used the near-gamma-ray spectrometer (GM) model Radiation Solution RS 230 (Radiation Solution Inc., Ontario, Canada) (Fig. 1a). The sensor can quantify the eTh and eU concentrations in parts per million (ppm), whereas 40K is quantified in percentage due to its major content in the pedosphere. Conventionally, radionuclides are expressed in mg kg-1 for eU and eTh, whereas for 40K, percentage is used. The GM detects the gamma-ray radiation emission down to a depth of 30–60 cm, which varies mainly with soil bulk density and moisture content (Wilford et al., 1997; Taylor et al., 2002; Beamish, 2015).

First, the GM was automatically calibrated by switching on and leaving the sensor on the ground surface for 5 min until readings of eU, eTh, and 40K contents stabilized (Radiation Solutions, 2009). The measurements of radionuclides were taken in the “assay-mode” of the highest precision for quantification, in which the GM was kept at the soil surface for 2 min in each sampling point (79 total collection points) (Fig. 1). The geographic position was taken by a GPS coupled to the GM (GPS, Radiation Solution Inc., Ontario, Canada; precision of 1 m). The data collected from all points were concatenated with their respective information from the soil physicochemical analyses for later geoprocessing. The same methodology has been applied by Mello et al. (2021) for gamma-ray spectrometric data acquisition.

For soil magnetic susceptibility (κ) characterization, surface readings were recorded at all 79 points, using a geophysical susceptibility meter sensor (KT10, Terraplus) (Fig. 1b). This sensor can measure κ to a depth of 2 cm below the soil surface, with a precision of 10-6 in SI units, expressed in m3 kg-1. To perform the readings, the sensor was first calibrated by determining the frequency of the outdoor oscillator. Subsequently, we followed the sequence required to obtain the measurements performed in three steps: (1) determining the frequency and amplitude of the oscillator in free air; (2) measuring the frequency and amplitude of the oscillator with the coil placed directly on the soil surface (sample) outcrop; and (3) repeating step 1 and displaying the results. For more information about these procedures, see Sales, Support and Cusomisation (2021). We performed the readings in scanner mode, which uses the best geometric correlation to direct κ readings, providing fast and accurate quantification. We performed three readings in triangulation around each collection point and used the mean value of κ in all our analyses. This procedure was adopted to reduce noise. The same methodology for κ readings has been performed by Mello et al. (2020).

The ECa measurements were performed using the conductivity meter Geonics EM38 (Geonics Ltd., Mississauga, Ontario, Canada) (McNeill, 1986) (Fig. 1c). The EM38 provides measurements of the quad-phase (conductivity) without any requirement for soil-to-instrument contact (Geonics, 2002); the unit is m mS-1.

First, the EM38 was calibrated following the instructions of Heil and Schmidhalter (2019), Sect. 3.1.1. The values of ECa are a function of calibration, coil orientation, and coil separation (Heil and Schmidhalter, 2019). More details about the EM38 operation are provided in Hendrickx and Kachanoski (2002). After calibration, the ECa readings were performed at all 75 collection points (Fig. 1), using the EM38 at vertical dipole orientation, which provided data from an effective soil depth at 1.5 m. Data were collected in the field during the dry season, on bare soil, and at the same intervals to reduce the impacts of environmental variables. Also, all metal objects were kept away from the EM 38 to avoid reading interferences.

We developed our research and analysis by using three geophysical sensors (near-gamma-ray spectrometer RS 230, near-magnetic susceptibility sensor KT10, and conductivimeter Geonics EM38) due to the following reasons: these sensors are available in our institution and for our research partners, they are easy to operate, and the obtained data are highly accurate. In addition, the EM38 (conductivimeter) and RS 230 (gamma-ray spectrometer) provide information for the depth at which most of the pedogenetic processes occur. In addition, information obtained with EM38 and RS 230 can be associated with KT10 (susceptibilimeter) on the soil surface to provide additional information about some soil attributes related to soil subsurface horizons, which is also related to the other geophysical variables used (gamma-ray and apparent electrical conductivity).

The modeling process is demonstrated in the flowchart (Fig. 3) and can be divided into two parts: the selection of covariates and the training/testing of the data. In the selection phase, the algorithm tries to produce the ideal set of covariates, following the principle of parsimony. This is performed by removing highly correlated variables, evaluating the importance of covariables, and removing variables that have a minor importance in training the model in the prediction process of each algorithm. Darst et al. (2018) considered the joint application of the methods for the selection of covariates by correlation and importance (RFE) since the use of RFE only reduces the effect of highly correlated covariates but does not eliminate it.

The correlation selection process was used to calculate the correlation of the set of covariates and covariables, which were evaluated with a correlation greater than the limit (Pearson test >95%). The pairs that showed higher values were evaluated due to their correlation with the complete set of covariates, eliminating that with the highest value of the sum of the absolute correlation with the other covariables that started in this process. For this phase, we applied the “cor” and “find correlation” functions of the “stats” (Hothorn, 2021) and “caret” (Kuhn et al., 2020) packages, in the R software, respectively (Kuhn and Johnson, 2013). In this phase, the covariables curv_cross_secational and curv_longitudinal were eliminated for all tested sensor sets. The set of covariables that passed this phase joined the samples followed by the separation of samples from training and testing.

The separation of training and testing was performed using the “nested” leave-one-out (nested-LOOCV) method (Clevers et al., 2007; Honeyborne et al., 2016; Rytky et al., 2020). It is important to highlight that the number of soil samples and readings with geophysical sensors was small (75) due to several difficulties encountered in the field during data collection (high sugar cane size, sloping terrain, dense forest, etc.). In this sense, the nested LOOCV method is indicated for small sample sets (values near 100 samples) to which other validation/testing methods (such as holdout validation) would not be viable due to the small sample set in the testing and/or training group (Ferreira et al., 2021). This is one of the main innovations of this research.

The nested LOOCV method is a double-loop process. In the first loop, the model is trained with a data set of size n-1, and the test is done in the second loop with the missing sample to validate the training performance (Jung et al., 2020; Neogi and Dauwels, 2022). The final results of the performance of the machine learning algorithm will be the mean performance indicators for all points (training/testing). This is a robust method to evaluate the performance of the algorithm and to detect possible samples with problems in the collections or outliers. The training set generated in each loop went through the process of selecting covariates for importance and subsequent training.

The selection of covariates by importance is performed using the back forward method, applying the recursive feature elimination (RFE) function contained in the caret package (Kuhn and Johnson, 2013). The RFE is unique for each algorithm, with the result being the set of selected covariates used in the prediction of the final model in the same algorithm. The RFE is a selection method that eliminates the variables that least contribute to the model, based on a measure of importance for each algorithm (Kuhn and Johnson, 2013). The algorithm will be applied to complete sets of data (variable by the set of tested sensors) and 18 more subsets with 5, 6, 7, … 19, 20, and 30 covariables. Reaching a set of fewer variables (more parsimonious) results in a better prediction performance. The optimization of the ideal covariate subset was based on LOOCV, a repetition, and four values of each of the internal hype parameters of each tested algorithm (“tune length”). The hyperparameters of each algorithm are described in the caret package manual in chapter 6, “Models described”, available at https://topepo.github.io/caret/train-models-by-tag.html (last access: 1 February 2022). The metric for choosing the best subset for each model was R2. For this work, five algorithms were tested: random forest (RF), Cubist (C), support vector machines (SVMs), and generalized linear models (LMs). The choice was made with the use of families of different algorithms in mind, using linear and non-linear algorithms. The algorithms used are commonly applied in soil attribute mapping studies. At the end of the selection phase by importance, the most optimized set of covariates for training was generated for each algorithm.

Training was performed with the variables selected in the previous step and each tested algorithm by using LOOCV and 10 repetitions. Four values of each of the internal hype parameters of each tested algorithm were also tested (tune length). At the end of the training phase, a sample prediction was made that was not used in the training, and the result was saved for the performance study. The performance of the prediction of the algorithms and the set of sensors was determined with a set of samples from the outer loop of the nested-LOOCV method. Three evaluation parameters were used: R squared, R2 (Eq. 1); root mean squared error, RMSE (Eq. 2); mean absolute error, MAE (Eq. 3). 1R2=∑Qpred-Qpred‾×Qobs-Qobs‾2∑Qpred-Qpred‾2×∑Qobs-Qobs‾2, 2RMSE=1n×∑Qobs-Qpred2,3MAE=1n×∑Qpred-Qobs, where Qpred denotes predicted samples, Qobs denotes observed samples, and n denotes the number of samples.

For comparison purposes, null model values (NULL_RMSE and NULL_MAE) were also calculated. The null model considers using the average value quantified by the collected samples (Eqs. 4 and 5). The null model (NULL_RMSE and NULL_MAE) emulates other model-building functions but returns the simplest model possible given a training set: a single mean for numeric outcomes. The percentage of the training set samples with the most prevalent class is returned when class probabilities are requested. The null model can be considered the simplest model that can be adjusted and that serves as a reference. Models that present similar or worse performances compared to the null model should be discarded. The best models had lower RMSE and MAE results than those found for NULL_MAE and NULL_RMSE. This shows that the final model is better than using the mean values, which also demonstrates a better quality in creating the models.

Given the above, the null model considers using the mean value quantified by the collected samples (Eqs. 4 and 5). This methodology is widely used, as well as spatialization processes in kriging when the variable in which spatialization is desired has spatial dependence (pure nugget effect). The equations are as follows: 4NULL_RMSE=1N∑i=1NQtrain‾i-Qobsi212,5NULL_MAE=1n×∑Qtrain‾i-Qobsi, where Qtrain denotes the mean of the training samples, Qobsi denotes the validation sample, N denotes the number of samples (loop).

Here NULL_RMSE and NULL_MAE values lower than those observed in the prediction of the algorithm in the validation phase show that the use of means of the samples of the desired propriety agrees with the model created by the algorithms of the machine learning. The NULL_RMSE and NULL_MAE were calculated using the “null mode” function of the caret package (Kuhn et al., 2020).

The final result of the performance of the algorithms of each attribute was obtained using the 75 loops, with the training results being the average of the performance and the results of the test samples calculated from the 75 external loops results using Eqs. (1)–(3). The importance of the algorithms was calculated by the caret package (Kuhn and Johnson, 2013); each model presents its creation methodology. The final importance for each algorithm and attribute was determined from the importance created in the loop, being the average of the importance of the 75 repetitions.

In general, for all geophysical sensor combinations, the majority of terrain attributes used did significantly influence sand and clay content prediction (Figs. 4–6 and 8). However, in most cases, parent material and magnetic susceptibility strongly influenced clay content prediction, except for G+C (Fig. 7). Ließ et al. (2012) found that the best performance was obtained using the RF model, with elevation and overland flow distance strongly affecting the model performance. According to Bauer (2010), the greater sand/clay ratio upslope is explained by the selective transport of fine material downslope, whereas in the present study, the clay content increased because of the influence of parent material (diabase), as also demonstrated by Mello et al. (2020).

Magnetic susceptibility (κ), followed by DEM and parent material, were the key variables that contributed to sand and clay content prediction by RF and SVM, respectively, for G+S (Fig. 8). Siqueira et al. (2010) and Mello et al. (2020) found a positive correlation between soil magnetic susceptibility and clay content and a negative correlation between magnetic susceptibility and sand content. In fact, the mineralogical composition of the parent material strongly affects soil magnetic susceptibility (Ayoubi et al., 2018), mainly in tropical soils under the top of basalt spills (Da Costa et al., 1999), where our study was undertaken.

In general, for Fe2O3 and TiO2, the most important variables were parent material, magnetic susceptibility, and DEM, which, in most cases, contributed 100 % (Figs. 4–8). In fact, the mineralogical composition of the parent material and the pedoenvironmental conditions strongly influence the amount of Fe / Ti oxides in soils (Schwertmann and Taylor, 1989; Kämpf and Curi, 2000; Bigham et al., 2002) and accelerate redistribution by downslope erosion (Mello et al., 2020). Also, the mineralogical composition of the parent material (Mullins, 1977; Ayoubi et al., 2018) and the landform evolution (Blundell et al., 2009; Sarmast et al., 2017) control the magnetic susceptibility of soil. Since the sensors used record the surface response and topography effect, it is expected that the most important variables indicated by the models would be related to surface processes. For the best combination of sensors (G+S), magnetic susceptibility and standardized height were more important variables in the prediction of Fe2O3 (100 %) and TiO2 (55 %) contents (Fig. 8), corroborating the expected surface processes and materials in the magnetic susceptibility of the soil (Shenggao, 2000; Damaceno et al., 2017) and the relief in the distribution of these materials (De Jong et al., 2000).

For SiO2, the most important variable was DEM, which, in most cases, contributed 100 % (Figs. 4–7). The level of SiO2 in soil is directly related to the nature of the parent material and the erosion processes at different topographic positions (Bockheim et al., 2014; Breemen and Buurman, 2003). This can explain the greater contribution of the DEM in the prediction models. For the best sensor combination (G+S), the variable that most contributed was mid-slope position, which also is related to topographic features.

For CEC, the variables DEM and magnetic susceptibility were the most important ones, contributing 100 % in most of the cases (Figs. 4–8). This can be explained by the high correlation between magnetic susceptibility, clay content, and CEC (Siqueira et al., 2010; de Souza Bahia et al., 2017; Mello et al., 2020). These variables vary with parent material and surface geomorphic processes, concentrating ferrimagnetic minerals (Frihy et al., 1995; Mello et al., 2020).

Considering that the gamma-ray spectrometer sensor is composed of three channels (eU, eTh, and 40K), it can be called “three sensors”. Thus, considering the combination of sensors used, it is possible to create a modeling performance graph using the number of sensors used through learning curves (Fig. 9). Such a learning curve shows a measure of the predictive performance of a given domain as a function of some measurements of varying amounts of learning effort (Perlich, 2010). In our case, the varying amounts were the number of sensors: non-use of geophysical sensors (zero sensors), S+C (two sensors), G+S (four sensors), and S+G+C (five sensors). In this analysis, the combination of G+C sensors will not be used because they present the same number of G+S sensors (four sensors). However, the combination G+C presented lower results than G+S.

For five soil properties (clay, sand, CEC, Fe2O3, and SiO3), the best results did not occur with a greater number of sensors, showing that increasing the number of covariables can lead to a lower performance (Fig. 9). This fact is associated with the addition of a new sensor as a covariate, which may provide conflicting information for the set of the other sensors found, where the ECa may have presented conflicting values with the sensors generated by the gamma-ray spectrometry channels, which generates a loss of performance when sensors are combined. The application of the RFE importance selection method was able to amortize this, making it a reliable method to reduce this effect.

0-NU denotes non-use of geophysical sensors; 2-S+C denotes two channels corresponding to susceptibilimeter + conductivimeter; 3-G denotes three channels corresponding to eU, eTh and 40K from gamma-ray spectrometer; 4-G+S denotes four channels corresponding to eU, eTh and 40K from gamma-ray spectrometer + susceptibilimeter.

For this study, the independent RMQS data set was not large enough (75 sites). Therefore, validation using 74 sites provided erratic and inconsistent results, mainly when comparing different pedoenvironmental indicators, even considering that this data set, in theory, provides “unbiased” estimates of forecast performance (Loiseau et al., 2020). Similarly, Lagacherie et al. (2019) showed that the location and number of samples used for independent assessment can significantly impact the values of these indicators. This indicates that the greatest variations were observed for evaluation sets with less than 100 samples.

Modeling soil attributes using relief and geophysical data presented promising results for geoscience studies and soil scientists. The use of several algorithms from different “families”, as well as the training and validation method, also made the study more robust and more reliable. In addition, machine learning models allow the importance of covariates to be defined, which is, sometimes, not possible when using ordinary spatialization methods, such as kriging and the inverse square of distance.

The “nested leave-one-out validation” method was useful with small sample sizes, being a potential tool to be used in geoscience studies. However, the academic community still knows little about the potential applicability of machine learning techniques.