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Mapping of environmental variables often relies on map accuracy assessment through cross-validation with the data used for calibrating the underlying mapping model. When the data points are spatially clustered, conventional cross-validation leads to optimistically biased estimates of map accuracy. Several papers have promoted spatial cross-validation as a means to tackle this over-optimism. Many of these papers blame spatial autocorrelation as the cause of the bias and propagate the widespread misconception that spatial proximity of calibration points to validation points invalidates classical statistical validation of maps. We present and evaluate alternative cross-validation approaches for assessing map accuracy from clustered sample data. The first method uses inverse sampling-intensity weighting to correct for selection bias. Sampling-intensity is estimated by a two-dimensional kernel approach. The two other approaches are model-based methods rooted in geostatistics, where the first assumes homogeneity of residual variance over the study area whilst the second accounts for heteroscedasticity as a function of the sampling intensity. The methods were tested and compared against conventional k-fold cross-validation and blocked spatial cross-validation to estimate map accuracy metrics of above-ground biomass and soil organic carbon stock maps covering western Europe. Results acquired over 100 realizations of five sampling designs ranging from non-clustered to strongly clustered confirmed that inverse sampling-intensity weighting and the heteroscedastic model-based method had smaller bias than conventional and spatial cross-validation for all but the most strongly clustered design. For the strongly clustered design where large portions of the maps were predicted by extrapolation, blocked spatial cross-validation was closest to the reference map accuracy metrics, but still biased. For such cases, extrapolation is best avoided by additional sampling or limitation of th

Spatial soil applications frequently involve binomial variables. If relevant environmental covariates are available, using a Bayesian generalized linear model (BGLM) might be a solution for mapping such discrete soil properties. The geostatistical extension, a Bayesian generalized linear geostatistical model (BGLGM), adds spatial dependence and is thus potentially better equipped. The objective of this work was to evaluate whether it pays off to extend from a BGLM to a BGLGM for mapping binary soil properties, evaluated in terms of prediction accuracy and modelling complexity. As motivating example, we mapped the presence/absence of the Pleistocene sand layer within 120 cm from the land surface in the Dutch province of Flevoland, using the BGLGM implementation in the R-package geoRglm. We found that BGLGM yields considerably better statistical validation metrics compared to a BGLM, especially with – as in our case – a large (n = 1,000) observation sample and few relevant covariates available. Also, the inferred posterior BGLGM parameters enable the quantification of spatial relationships. However, calibrating and applying a BGLGM is quite demanding with respect to the minimal required sample size, tuning the algorithm, and computational costs. We replaced manual tuning by an automated tuning algorithm (which eases implementing applications) and found a sample composition that delivers meaningful results within 50 h calculation time. With the gained insights and shared scripts spatial soil practitioners and researchers can – for their specific cases – evaluate if using BGLGM is feasible and if the extra gain is worth the extra effort.

If a map is constructed through prediction with a statistical or non-statistical model, the sampling design used for selecting the sample on which the model is fitted plays a key role in the final map accuracy. Several sampling designs are available for selecting these calibration samples. Commonly, sampling designs for mapping are compared in real-world case studies by selecting just one sample for each of the sampling designs under study. In this study, we show that sampling designs for mapping are better compared on the basis of the distribution of the map quality indices over repeated selection of the calibration sample. In practice this is only feasible by subsampling a large dataset representing the population of interest, or by selecting calibration samples from a map depicting the study variable. This is illustrated with two real-world case studies. In the first case study a quantitative variable, soil organic carbon, is mapped by kriging with an external drift in France, whereas in the second case a categorical variable, land cover, is mapped by random forest in a region in France. The performance of two sampling designs for mapping are compared: simple random sampling and conditioned Latin hypercube sampling, at various sample sizes. We show that in both case studies the sampling distributions of map quality indices obtained with the two sampling design types, for a given sample size, show large variation and largely overlap. This shows that when comparing sampling designs for mapping on the basis of a single sample selected per design, there is a serious risk of an incidental result. Highlights: We provide a method to compare sampling designs for mapping. Random designs for selecting calibration samples should be compared on the basis of the sampling distribution of the map quality indices.

It is commonly accepted that an estimated soil variogram can be transferred to another similar area for deriving the tolerable spacing of a sampling grid or, more generally, the sample size, given a requirement on the quality of the soil property map of the recipient area. The quality of the derived tolerable grid spacing depends on how similar the population variograms of the donor area and recipient area are. In practice we are uncertain about the variograms of both areas due to sampling errors. Ideally, the uncertainty about the variogram of the donor area is accounted for in deriving the tolerable grid spacing. To assess the transferability, we should also account for uncertainty in the estimated variogram of the recipient area. In this study the transferability of variograms of soil pH, P, Mg and K is analysed for three grassland fields in Ireland, which are similar in soil-forming factors. One field served as donor area, the other two as recipient area. For all three fields and for each soil property, 500 variograms were sampled from the posterior distribution of the variogram parameters. Results showed that the estimated variogram parameters of the recipient fields differed largely from those of the transferred variograms. The ranges of estimated mean kriging variance values for the various grid spacings, as obtained with the two sets of variograms (one set of the donor field, one set of the recipient field), did not overlap. Even after scaling the transferred variogram with an estimate of the variance of the recipient field, the transferred variogram was of no use for determining the tolerable grid spacing. The difference in the variograms can possibly be explained by the difference in historical land use. Highlights: Transferability of variograms to derive tolerable grid spacing for mapping grassland fields is assessed Transferability should be based on the uncertainty distributions of the tolerable grid spacings Due to difference in historical land use, lo

Several misconceptions about the design-based approach for sampling and statistical inference, based on classical sampling theory, seem to be quite persistent. These misconceptions are the result of confusion about basic statistical concepts such as independence, expectation, and bias and variance of estimators or predictors. These concepts have a different meaning in the design-based and model-based approach, because they consider different sources of randomness. Also, a population mean is still often confused with a model mean, and a population variance with a model-variance, leading to invalid formulas for the variance of an estimator of the population mean. In this paper the fundamental differences between these two approaches are illustrated with simulations, so that hopefully more pedometricians get a better understanding of this subject. An overview is presented of how in the design-based approach we can make use of knowledge of the spatial structure of the study variable. In the second part, new developments in both the design-based and model-based approach are described that try to combine the strengths of the two approaches.

Area-to-point kriging (ATPK) is a geostatistical method for creating high-resolution raster maps using data of the variable of interest with a much lower resolution. The data set of areal means is often considerably smaller (<50 observations) than data sets conventionally dealt with in geostatistical analyses. In contemporary ATPK methods, uncertainty in the variogram parameters is not accounted for in the prediction; this issue can be overcome by applying ATPK in a Bayesian framework. Commonly in Bayesian statistics, posterior distributions of model parameters and posterior predictive distributions are approximated by Markov chain Monte Carlo sampling from the posterior, which can be computationally expensive. Therefore, a partly analytical solution is implemented in this paper, in order to (i) explore the impact of the prior distribution on predictions and prediction variances, (ii) investigate whether certain aspects of uncertainty can be disregarded, simplifying the necessary computations, and (iii) test the impact of various model misspecifications. Several approaches using simulated data, aggregated real-world point data, and a case study on aggregated crop yields in Burkina Faso are compared. The prior distribution is found to have minimal impact on the disaggregated predictions. In most cases with known short-range behaviour, an approach that disregards uncertainty in the variogram distance parameter gives a reasonable assessment of prediction uncertainty. However, some severe effects of model misspecification in terms of overly conservative or optimistic prediction uncertainties are found, highlighting the importance of model choice or integration into ATPK.