https://www.clw.csiro.au/aclep/documents/McBratney%20et%20al.%202003.pdf

On digital soil mapping

An overview of model-based geostatistics.- Gaussian models for geostatistical data.- Generalized linear models for geostatistical data.- Classical parameter estimation.- Spatial prediction.- Bayesian inference.- Geostatistical design.

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Model-based Geostatistics

Geographical information systems could be improved by adding procedures for geostatistical spatial analysis to existing facilities Most traditional methods of interpolation are based on mathematical as distinct from stochastic models of spatial variation Spatially distributed data behave more like random variables, however, and regionalized variable theory provides a set of stochastic methods for analysing them Kriging is the method of interpolation deriving from regionalized variable theory It depends on expressing spatial variation of the property in terms of the variogram, and it minimizes the prediction errors which are themselves estimated We describe the procedures and the way we link them using standard operating systems We illustrate them using examples from case studies, one involving the mapping and control of soil salinity in the Jordan Valley of Israel, the other in semi-arid Botswana where the herbaceous cover was estimated and mapped from aerial photographic survey

Kriging: a method of interpolation for geographical information systems

A framework is provided for scaling and scale issues in hydrology. The first section gives some basic definitions. This is important as researchers do not seem to have agreed on the meaning of concepts such as scale or upscaling. ‘Process scale’, ‘observation scale’ and ‘modelling (working) scale’ require different definitions. The second section discusses heterogeneity and variability in catchments and touches on the implications of randomness and organization for scaling. The third section addresses the linkages across scales from a modelling point of view. It is argued that upscaling typically consists of two steps: distributing and aggregating. Conversely, downscaling involves disaggregation and singling out. Different approaches are discussed for linking state variables, parameters, inputs and conceptualizations across scales. This section also deals with distributed parameter models, which are one way of linking conceptualizations across scales. The fourth section addresses the linkages across scales from a more holistic perspective dealing with dimensional analysis and similarity concepts. The main difference to the modelling point of view is that dimensional analysis and similarity concepts deal with complex processes in a much simpler fashion. Examples of dimensional analysis, similarity analysis and functional normalization in catchment hydrology are given. This section also briefly discusses fractals, which are a popular tool for quantifying variability across scales. The fifth section focuses on one particular aspect of this holistic view, discussing stream network analysis. The paper concludes with identifying key issues and gives some directions for future research.

/pdf/scale-issues-in-hydrological-modelling-a-review-1x61lpwqjr.pdf

Scale issues in hydrological modelling: A review

Geostatistics in soil science: state-of-the-art and perspectives

SUMMARY
The variogram is central in the spatial analysis of soil, yet it is often estimated from few data, and its precision is unknown because confidence limits cannot be determined analytically from a single set of data. Approximate confidence intervals for the variogram of a soil property can be found numerically by simulating a large field of values using a plausible model and then taking many samples from it and computing the observed variogram of each sample. A sampling distribution of the variogram and its percentiles can then be obtained. When this is done for situations typical in soil and environmental surveys it seems that variograms computed on fewer than 50 data are of little value and that at least 100 data are needed. Our experiments suggest that for a normally distributed isotropic variable a variogram computed from a sample of 150 data might often be satisfactory, while one derived from 225 data will usually be reliable.

Sample adequately to estimate variograms of soil properties

Earth scientists and land managers often wish to group sampling sites that are both similar with respect to their properties and near to one another on the ground. This paper outlines the geostatistical rationale for such spatial grouping and describes a multivariate procedure to implement it. Sample variograms are calculated from the original data or their leading principal components and then the parameters of the underlying functions are estimated. A dissimilarity matrix is computed for all sampling sites, preferably using Gower's general similarity coefficient. Dissimilarities are then modified using the variogram to incorporate the form and extent of spatial variation. A nonhierarchical classification of sampling sites is performed on the leading latent vectors of the modified dissimilarity matrix by dynamic clustering to an optimum. The technique is illustrated with results of its application to soil survey data from two small areas in Britain and from a transect. In the case of the latter results of spatially weighted classifications are compared with those of strict segmentation. An appendix lists a Genstat program for a spatially constrained classification using a spherical variogram as an example.

A geostatistical basis for spatial weighting in multivariate classification

Many geomorphic properties can be treated as spatially-dependent random variables. Some are second-order stationary, others appear to vary without bound. In these circumstances their variation is best described by the semi-variogram. In most instances the semi-variogram can be modelled by a simple mathematical function, which itself is bounded for a stationary variable and unbounded otherwise. The function must be conditional negative semi-definite to be permissible. More complex variation can be represented by combining two or more permissible models.

Sample semi-variograms of several landform and soil properties illustrate the common types of semi-variogram. Their form and parameters are interpreted in physical terms.

Semi‐variograms for modelling the spatial pattern of landform and soil properties

. Geostatistics is principally the application of regionalized variable theory. The methods it embodies are applicable throughout the earth sciences for investigating the spatial variation of, and for estimating continuous random variables. The semi-variogram is the central tool of geostatistics. It can quantify the scale and intensity of spatial variation and it provides the essential spatial information for local estimation by kriging and for optimizing sample intensity. It can also be used in an exploratory manner to try to discover underlying causes of the variation. Geostatistical methods have been widely applied in the mining industry and there are many examples of their application in soil science. Their use is illustrated by a case study of soil spatial variation in the Wyre Forest of England.

M. A. Oliver

Papers

Kriging: a method of interpolation for geographical information systems

Sample adequately to estimate variograms of soil properties

A geostatistical basis for spatial weighting in multivariate classification

Semi‐variograms for modelling the spatial pattern of landform and soil properties

Geostatistics and its application to soil science