Estimating variograms of soil properties by the method-of-moments and maximum likelihood

Variograms of soil properties are usually obtained by estimating the variogram for distinct lag classes by the method‐of‐moments and fitting an appropriate model to the estimates. An alternative is to fit a model by maximum likelihood to data on the assumption that they are a realization of a multivariate Gaussian process. This paper compares the two using both simulation and real data. The method‐of‐moments and maximum likelihood were used to estimate the variograms of data simulated from stationary Gaussian processes. In one example, where the simulated field was sampled at different intensities, maximum likelihood estimation was consistently more efficient than the method‐of‐moments, but this result was not general and the relative performance of the methods depends on the form of the variogram. Where the nugget variance was relatively small and the correlation range of the data was large the method‐of‐moments was at an advantage and likewise in the presence of data from a contaminating distribution. When fields were simulated with positive skew this affected the results of both the method‐of‐moments and maximum likelihood. The two methods were used to estimate variograms from actual metal concentrations in topsoil in the Swiss Jura, and the variograms were used for kriging. Both estimators were susceptible to sampling problems which resulted in over‐ or underestimation of the variance of three of the metals by kriging. For four other metals the results for kriging using the variogram obtained by maximum likelihood were consistently closer to the theoretical expectation than the results for kriging with the variogram obtained by the method‐of‐moments, although the differences between the results using the two approaches were not significantly different from each other or from expectation. Soil scientists should use both procedures in their analysis and compare the results.     

The toxicity of Castanospermum australe seeds for cattle

Two calves given a mean of 16.1 g and 16.4 g ripe Castanospermum australe seeds/kg body weight daily for 13 and 16 days respectively developed haemorrhagic gastroenteritis. The first calf died. The second calf had mild myocardial degeneration and necrosis and mild nephrosis at necropsy. Two calves given a mean of 16.8 g unripe C. australe seeds/kg body weight daily for 18 days remained clinically normal and had mild gastritis at necropsy. The activity of alpha-glucosidase was reduced in the mononuclear cells of peripheral blood and in skeletal muscle. This was attributed to the presence of the indolizidine alkaloid, castanospermine, in the seeds. The toxin causing the gastroenteritis and other lesions is unknown.     

Estimation of Linear Models of Coregionalization by Residual Maximum Likelihood

Observations of ancillary soil properties spatially correlated to a soil property of interest may be used to increase the precision and reduce the sampling costs of a geostatistical survey. The relationship between such coregionalized properties must be expressed as a linear model of coregionalization but the conventional estimator of the linear model of coregionalization is biased unless the mean value of each property is constant across the study region. However, the mean value of a soil property may vary according to a spatial trend or a deterministic relationship with other factors which vary within the study region. We therefore propose that a linear mixed model should be fitted to coregionalized soil properties by residual maximum likelihood. This approach simultaneously fits spatial trends or deterministic relationships and random effects to the observations with minimum bias. We implement a residual maximum likelihood estimator for coregionalized properties and suggest a criterion to decide what order of spatial trend and which deterministic relationships should be included in the model. The effectiveness of the estimator is proved upon simulated data and upon observations of zinc and cadmium concentrations from the Swiss Jura.     

The Matérn variogram model: Implications for uncertainty propagation and sampling in geostatistical surveys

The Matérn variogram model has been advocated because it is flexible and can represent varied behaviour at small lags. We show how the constraints on the spherical and exponential variogram at short lags ignore a possible source of uncertainty in the variogram and so in kriging surveys, that the Matérn model can describe. Matérn, spherical and exponential variogram models were fitted by maximum likelihood to a set of log₁₀(K) observations made on a regular grid at Broom's Barn Farm, Suffolk, England. The likelihood profiles of the Matérn parameter estimates were asymmetric. Thus the uncertainty of these estimates could only be adequately assessed by a Bayesian approach. The uncertainty of estimated parameters of the Matérn variogram was larger than for the exponential variogram. This is an indication that the assumption of an exponential model limits the behaviour that may be described by the variogram. Thus uncertainty analyses where an exponential variogram is assumed may underestimate the uncertainty of kriged estimates. Bayesian analysis of the kriged estimates of log₁₀(K) at Broom's Barn Farm using the Matérn variogram revealed an observable component of uncertainty due to variogram uncertainty. When an exponential variogram model was used, the estimate of this component of uncertainty was negligible. The Matérn variogram should therefore be used rather than the exponential model when assessing the adequacy of a variogram estimate. A method of designing sample schemes which is suitable for both estimating a Matérn variogram and interpolation is suggested.     

Two robust estimators of the cross‐variogram for multivariate geostatistical analysis of soil properties

If we wish to describe the coregionalization of two or more soil properties for estimation by cokriging then we must estimate and model their auto‐ and cross‐variogram(s). The conventional estimates of these variograms, obtained by the method‐of‐moments, are unduly affected by outlying data which inflate the variograms and so also the estimates of the error variance of cokriging predictions. Robust estimators are less affected. Robust estimators of the auto‐variogram and the pseudo cross‐variogram have previously been proposed and used successfully, but the multivariate problem of estimating the cross‐variogram robustly has not yet been tackled. Two robust estimators of the cross‐variogram are proposed. These use covariance estimators with good robustness properties. The robust estimators of the cross‐variogram proved more resistant to outliers than did the method‐of‐moments estimator when applied to simulated fields which were then contaminated. Organic carbon and water content of the soil was measured at 256 sites on a transect and the method‐of‐moments estimator, and the two robust estimators, were used to estimate the auto‐variograms and cross‐variogram from a prediction subset of 156 sites. The data on organic carbon included a few outliers. The method‐of‐moments estimator returned larger values of the auto‐ and cross‐variograms than did either robust estimator. The organic carbon content at the 100 validation sites on the transect was estimated by cokriging from the prediction data plus a set of variograms fitted to the method‐of‐moments estimates and two sets of variograms fitted to the robust estimates. The ratio of the actual squared prediction error to the cokriging estimate of the error variance was computed at each validation site. These results showed that cokriging using variograms obtained by the method‐of‐moments estimator overestimated the error variance of the predictions. By contrast, cokriging with the robustly estimated variograms gave reliable estimates of the error variance of the predictions.     

Baseline values and change in the soil, and implications for monitoring

The soil is subject to change, and this must be monitored and understood. There are various circumstances in which the change in a soil property will depend in part on the baseline value that was first measured. Such a relationship may allow us to estimate change with greater precision, but it will also reflect in part the statistical phenomenon of regression to the mean. In this paper, we describe two approaches to the analysis of how change in a variable depends on its baseline values. These methods allow for the effect of regression to the mean. We discuss their applicability to the problems of soil monitoring, and we illustrate one of them by means of a case study on change in the organic carbon content of the soils of England and Wales.     

Analysing spatially intermittent variation of nitrous oxide emissions from soil with wavelets and the implications for sampling

Emissions of gases from the soil are known to vary spatially in a complex way. In this paper we show how such data can be analysed with the wavelet transform. We analysed data on rates of N₂O emission from soil cores collected at 4‐m intervals on a 1024‐m transect across arable land at Silsoe in England. We used a thresholding procedure to represent intermittent variation in N₂O emission from the soil as a sparse wavelet process, i.e. one in which most of the wavelet coefficients are not significantly different from zero. This analysis made clear that the rate of N₂O emission varied more intermittently on this transect than did soil pH, for which many more of the wavelet coefficients had to be retained. This account of intermittent variation motivated us to consider a class of random functions, which we call wavelet random functions, for the simulation of spatially intermittent variation. A wavelet random function (WRF) is an inverse wavelet transform of a set of random wavelet coefficients with specified variance at each scale. We generated intermittent variation at a particular scale in the WRF by specifying a binormal process for the wavelet coefficients at this scale. We showed by simulation that adaptive sampling schemes are more efficient than ordinary stratified random sampling to estimate the mean of a spatial variable that is intermittent at a particular scale. This is because the sampling can be concentrated in the more variable regions. When we simulated values that emulate the intermittency of our data on N₂O we found that the gains in efficiency from simple adaptive sampling schemes were small. This was because the emission of N₂O is intermittent over several disparate scales. More sophisticated adaptive sampling is needed for these conditions, and it should embody knowledge of the relevant soil processes.