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## Hierarchical decomposition of variance with applications in environmental mapping based on satellite images

### Mathematical Geology (1996-05-01) 28: 385-405 , May 01, 1996

*A quadtree-based image segmentation procedure (HQ) is presented to map complex environmental conditions. It applies a hierarchical nested analysis of variance within the framework of multiresolution wavelet approximation. The procedure leads to an optimal solution for determining mapping units based on spatial variability with constraints on the arrangement and shape of the units. Linkages to geostatisiics are pointed out, but the HQ decomposition algorithm does not require any homogeneity criteria. The computer implementation can be parameterized by either the number of required mapping units or the maximum within-unit variance, or it can provide a “spectrum” of significances of nested ANOVA. The detailed mathematical background and methodology is illustrated by a salt-affected grassland mapping study (Hortobágy, Hungary), where heterogeneous environmental characteristics have been sampled and predicted based on remotely sensed images using these principles*.

## A case study of model selection and parameter inference by maximum likelihood with application to uncertainty analysis

### Nonrenewable Resources (1998-03-01) 7: 63-73 , March 01, 1998

One of the uses of geostatistical conditional simulation is as a tool in assessing the spatial uncertainty of inputs to the Monte Carlo method of system uncertainty analysis. Because the number of experimental data in practical applications is limited, the geostatistical parameters used in the simulation are themselves uncertain. The inference of these parameters by maximum likelihood allows for an easy assessment of this estimation uncertainty which, in turn, may be included in the conditional simulation procedure. A case study based on transmissivity data is presented to show the methodology whereby both model selection and parameter inference are solved by maximum likelihood.

## Efficient updating of kriging estimates and variances

### Mathematical Geology (1992-01-01) 24: 129-133 , January 01, 1992

This short note presents a method for efficiently updating ordinary kriging estimates and variances when one or more additional samples are incorporated into the kriging system. First, the foundation linear algebra result is presented. Then the update equations are derived. Finally, an illustrative application of updating is briefly discussed.

## The fractal character of photos of slabbed cores

### Mathematical Geology (1992-01-01) 24: 73-97 , January 01, 1992

Core photos have been analyzed statistically and found to be fractional Gaussian noise (fGn) in both horizontal and vertical directions. This fractal character, fGn, is the same as that previously found to describe vertical and horizontal well logs. An equation for the two-dimensional spectral density of core photos is presented that generates computer photos very similar to “real” core photos.

## Application of geostatistical methods to arsenic data from soil samples of the Cova dos Mouros mine (Vila Verde-Portugal)

### Environmental Geochemistry and Health (2005-09-01) 27: 259-270 , September 01, 2005

A total of 286 soil samples were collected in the Cova dos Mouros area. All samples were dry sieved into the <200 mesh size fraction and analysed for Fe, Cu, Zn, Pb, Co, Ni, Bi and Mn by atomic absorption spectrometry (AAS) and for As, Se, Sb and Te by atomic absorption spectrometry-hydrid generation (AAS-HG). Only the results of arsenic are discussed in this paper although the survey was extended to all analysed chemical elements. The purpose of this study was to make a risk probability mapping for arsenic that would allow better knowledge about the vulnerability of the soil to arsenic contamination. To achieve this purpose, the initial variable was transformed into an indicator variable using as thresholds the risk-based standards (intervention values) for soils, as proposed by [Swartjes 1999. Risk based assessment of soil and groundwater quality in the Netherlands: Standards and remediation. *J. Geochem. Explor*.73 1–10]. To account for spatial structure, sample variograms were computed for the main directions of the sampling grid and a spherical model was fitted to each sample variogram (arsenic variable and indicator variables). The parameters of the spherical model fitted to the arsenic variable were used to predict arsenic concentrations at unsampled locations. A risk probability mapping was also done to assess the vulnerability of the soil towards the mining works. The parameters of the spherical model fitted to each indicator variable were used to estimate probabilities of exceeding the corresponding threshold. The use of indicator kriging as an alternative to ordinary kriging for the soil data of Cova dos Mouros produced unbiased probability maps that allowed assessment of the quality of the soil.

## Additive models in mining and exploration

### Nonrenewable Resources (1997-03-01) 6: 11-25 , March 01, 1997

In this paper we present the use of additive models (AMs) for geostatistical applications. AMs are generalizations of linear regression models which hold the central place in the toolbox of applied statisticians. Generally speaking, the linear relationship between response and predictors is replaced with a general functional form. Recently such models were introduced in geostatistics. Especially, we give an approach for binary data. In this case we get generalized additive models (GAMs). Logistic regression is quite popular in medical and biological research. Using logit links also in GAMs we get so called additive logistic models. An application for geostatistical data is introduced. In a second approach we use AMs for spatial prediction and surface modelling. In both cases an advantage of multivariate data can be taken. The proposed applications can be used in the development of exploration strategies, especially in the early stage of exploration

## On the importance of choosing a change of support model for global reserves estimation

### Mathematical Geology (1988-11-01) 20: 1001-1019 , November 01, 1988

The practical problem considered here is: how can block distribution in an orebody be forecast from sample data? The task is arduous because information yielded by samples is too often insufficient to allow an accurate evaluation of blocks. In practice, necessary additional information is obtained via a model. Choosing that model is crucial; the value of results reflects the model, i.e., its adequacy to represent reality. In this paper, the importance of choosing the change of support model is illustrated with simulations and practical examples (especially deposits with a skewed sample distribution and a large spike at the origin). An attempt to quantify this importance is made also.

## A parametric study of robustness of kriging variance as a function of range and relative nugget effect for a spherical semivariogram

### Mathematical Geology (1986-07-01) 18: 477-488 , July 01, 1986

In geostatistics, an estimation of blocks of a deposit is reported along with the variance of error made in their estimation. This calculation is based on the model chosen for the semivariogram of the deposit so that mistakes in its estimation can manifest themselves in the perception of accuracy with which blocks are known. Changes in kriging variance resulting from various amounts of error in modeling the relative nugget effect and range of the semivariogram are investigated for an extensive set of spherical semivariograms.

## Bayesian kriging—Merging observations and qualified guesses in kriging

### Mathematical Geology (1987-01-01) 19: 25-39 , January 01, 1987

Frequently a user wants to merge general knowledge of the regionalized variable under study with available observations. Introduction of fake observations is the usual way of doing this. Bayesian kriging allows the user to specify a qualified guess, associated with uncertainty, for the expected surface. The method will provide predictions which are based on both observations and this qualified guess.

## Disjunctive kriging: Comparison of theory with actual results

### Journal of the International Association for Mathematical Geology (1980-08-01) 12: 305-320 , August 01, 1980

The method of disjunctive kriging is applied to the evaluation of blocks of ore in a gold mine, and the estimated block values are compared with the true block values. First the theory of disjunctive kriging as applicable to mine valuation is presented from the user's viewpoint. The theory is then applied to process sample values originating from a large gold mine. Using the computer simulation of a number of sampling programs, estimated values and true values are compared. The disjunctive kriging estimates are also compared with estimates obtained by other methods of evaluation, including logarithmic kriging. It is shown that the results obtained, using real world data, are in complete agreement with the results expected according to the geostatistical theory.