Distinguishing spatially correlated random variation in soil from a ‘pure nugget’ process

Abstract

In most spatial analysis of soil variation it is assumed that the random variation not captured by fixed effects (class means or continuous covariates) is spatially dependent. It is proposed that this should be tested formally, both to justify the kriging component in subsequent spatial prediction and as evidence of the extent to which the included fixed effects have succeeded in accounting for soil variation that is spatially dependent at the scales resolved by the soil sampling. A formal test is possible by computing the log ratio of the likelihoods for a full model with spatially dependent random effects and a null model which is pure nugget. It is shown that the sampling distribution of the log likelihood-ratio under the null model is not χ 2 (p) where p is the number of additional random effects parameters in the model with spatial dependence. This is because, while the null model is nested in the full model, parameters of the full model take bounding values in the null case. The sampling distribution may be computed by Monte Carlo simulations. It is shown that the power to reject the null model by the log likelihood-ratio test depends on the importance of the nugget effect in the underlying model, and on the sampling scheme. In many circumstances it may be hard to demonstrate spatial dependence. The recommended procedure was applied to some data on the organic carbon content of the topsoil and subsoil of a field inEngland. This was modelled either with the overall mean the only fixed effects, or with separate means for different soil map units as fixed effects. There was significant evidence for spatial dependence in the random effects at both depths when the overall mean was the only fixed effect. When map unit means were used as fixed effects there was significant, though weaker, spatial dependence in the topsoil, but the null model could not be rejected for the subsoil. This has implications for any further sampling to map organic carbon in the subsoil.

Ideas and perspectives: Tracing terrestrial ecosystem water fluxes using hydrogen and oxygen stable isotopes – challenges and opportunities from an interdisciplinary perspective

“…If soil-water isotopic composition cannot explain the observed isotopic signature in plant (xylem) water, then other factors (e.g. plant physiological processes such as fractionation at the soil-root interface, uptake of dew, mist and/or fog via leaves and bark, evaporation through the bark, or mixing of xylem and phloem water) may explain these patterns (Eller et al, 2013;Berry et al, 2014;Martín-Gómez et al, 2016;Sprenger et al, 2017Sprenger et al, , 2018bLehmann et al, 2017;Dawson and Goldsmith, 2018).…”

A statistical assessment of the uncertainty in a 3-D geological framework model

“…These conditions hold for all the comparisons that we make in this study except for comparisons between a model which has the spatially autocorrelated random variable η and a null model with only an i.i.d. residual (Lark, 2012). In these cases the distribution of L under the null model was found by Monte Carlo simulation, as described by Lark (2012).…”

“…residual (Lark, 2012). In these cases the distribution of L under the null model was found by Monte Carlo simulation, as described by Lark (2012).…”

“…Likelihood ratio tests were conducted to decide whether the modeller effects (α i ) and spatially-dependent random effects (η) were required, and so to select an appropriate random effects statistical model for the framework model errors. Inferences about η were based on sample distributions for L under the null model obtained by simulation, as described by Lark (2012). In all cases two models were fitted.…”

“…has a modeller effect, but no spatially correlated random effect. These models can be compared by the log-likelihood ratio, L, which is compared to thresholds which correspond to a P -values of 0.05, 0.01 and 0.001 obtained by simulation using the method of Lark (2012). A P -value is the probability of observing, through random variation, a value of L as large or larger than the one actually observed if the true model for the data were the second model (A2 etc.)…”