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About this book

Engineers and earth scientists are increasingly interested in quantitative methods for the analysis, interpretation, and modeling of data that imperfectly describe natural processes or attributes measured at geographical locations. Inference from imperfect knowledge is the realm of classical statistics. In the case of many natural phenomena, auto- and cross- correlation preclude the use of classical statistics. The appropriate choice in such circumstances is geostatistics, a collection of numerical techniques for the characterization of spatial attributes similar to the treatment in time series analysis of auto-correlated temporal data. As in time series analysis, most geostatistical techniques employ random variables to model the uncertainty that goes with the assessments. The applicability of the methods is not limited by the physical nature of the attributes.
Geostatistics for Engineers and Earth Scientists presents a concise introduction to geostatistics with an emphasis on detailed explanations of methods that are parsimonious, nonredundant, and through the test of time have proved to work satisfactorily for a variety of attributes and sampling schemes. Most of these methods are various forms of kriging and stochastic simulation. The presentation follows a modular approach making each chapter as self-contained as possible, thereby allowing for reading of individual chapters, reducing excessive cross-referencing to previous results and offering possibilities for reviewing similar derivations under slightly different circumstances. Guidelines and rules are offered wherever possible to help choose from among alternative methods and to select parameters, thus relieving the user from making subjective calls based on an experience that has yet to be acquired.
Geostatistics for Engineers and Earth Scientists is intended to assist in the formal teaching of geostatistics or as a self tutorial for anybody who is motivated to employ geostatistics for sampling design, data analysis, or natural resource characterization. Real data sets are used to illustrate the application of the methodology.

Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract
Geostatistics can be regarded as a collection of numerical techniques that deal with the characterization of spatial attributes, employing primarily random models in a manner similar to the way in which time series analysis characterizes temporal data. The French engineer Georges Matheron—at the time with the Bureau de Recherches Géologiques et Minières—coined the word géostatistique, inspired by the clear meaning and success of the older terms geochemistry and geophysics in which the prefix geo- was added to the name of some classical body of knowledge to denote an application of such knowledge to the modeling and understanding of processes of interest in earth sciences and technology (Matheron, 1962, p. 22; Journel and Huijbregts, 1978, p. vi). Both geostatistics and time series analysis are conceptually and historically related and primarily address the situation in which inferences must be drawn from autocorrelated data that are insufficient for obtaining precise results. Unlike time series analysis, however, geostatistics was not originally developed by mainstream statisticians, which accounts for its terminology and a notation that at first sight may be unfamiliar to scientists with training in classical statistics (Christensen, 1991, p. 262).
Ricardo A. Olea

Chapter 2. Simple Kriging

Abstract
There are several forms of kriging, all of them initially formulated for the estimation of a continuous, spatial attribute at an unsampled site, preferably inside the convex hull defined by the location of the data. Figure 2.1 illustrates the case for a two-dimensional point sampling. Although extrapolations outside the convex hull are possible, they are unreliable. This points out a significant difference between kriging and time series analysis wherein the interest lies in predictions beyond the time span containing the data.
Ricardo A. Olea

Chapter 3. Normalization

Abstract
We have seen in the previous chapter that simple kriging is the best linear unbiased estimator in a minimum mean square error sense. No distributional assumptions were necessary to reach that level of optimality, but if one is willing to assume multinormality, the method can be taken to new heights.
Ricardo A. Olea

Chapter 4. Ordinary Kriging

Abstract
Historically, ordinary kriging was the first improvement of simple kriging, whose roots predate geostatistics (Goldberg, 1962; Cressie, 1990). We have seen that simple kriging requires knowledge of the mean to solve the problem of finding weights that minimize the variance of the estimation error. Ordinary kriging elegantly discards the requirement by filtering out the mean, taking advantage of Corollary 2.11. And by removing the mean from the estimator, the entire formulation becomes independent from the mean, as both the weights and the estimation variance in the simple kriging Algorithm 2.1 were already independent from the mean.
Ricardo A. Olea

Chapter 5. The Semivariogram

Abstract
Three good reasons may be cited to explain why the semivariogram is important in geostatistics:
1.
The semivariogram is a statistic that assesses the average decrease in similarity between two random variables as the distance between the variables increases, leading to some applications in exploratory data analysis.
 
2.
It has been demonstrated by the foregoing algorithms and exercises that kriging is not possible without knowledge of the semivariogram or the covariance. In the formulation of our exercises, the covariance has been a known analytical expression—which, incidentally, is what the rigorous application of the algorithms demands. Yet, in practice, neither the covariance nor the semivariogram is known. The way in which geostatistics sidesteps this impasse is by use of an estimate of the semivariogram or the covariance instead of the true moments of the random function model, an approximation for which the derivation of the normal equations does not account.
 
3.
In the previous chapter we have also seen that the practice is to solve the kriging system of equations in terms of covariances. This is primarily for convenience in the handling of the square matrices, despite the slight loss in generality. Yet in terms of determining the spatial correlation, the practice continues to be to estimate the semivariogram and then, provided that the covariance exists, to use the following Corollary 5.2 for converting semivariograms into covariances.
 
Ricardo A. Olea

Chapter 6. Universal Kriging

Abstract
In Chapter 4 we saw a first attempt to generalize simple kriging by removing the requirement that the constant mean of the random function be known and an effort to lift the constant mean requirement by moving from a global constant mean to a locally constant mean. There are several instances of attributes— water depth near the shore, temperature in the upper part of the earth’s crust, water table elevation in the High Plains aquifer of Kansas—that have a clear systematic variation. Models that presume constancy of the mean are inadequate for the characterization of such attributes.
Ricardo A. Olea

Chapter 7. Crossvalidation

Abstract
Upon selecting any estimator and its parameters and producing estimations for a given spatial attribute, there is curiosity and the need to know more about the quality of the job done.
Ricardo A. Olea

Chapter 8. Drift and Residuals

Abstract
In Chapter 6, we have formulated universal kriging for the modeling of an attribute with a systematic trend. In some instances, however, the characteristic of interest is the trend itself rather than the attribute or, even more commonly, it is the residuals resulting from subtracting the drift from the kriging modeling of the attribute.
Ricardo A. Olea

Chapter 9. Stochastic Simulation

Abstract
If one analyzes a regular grid of kriging estimates for a spatial attribute such as that shown in Figure 9.1, one finds that there is an uneven smoothing in the grid of estimates—smoothing that is inversely proportional to the data density. Such distortion can be visualized in several ways:
(a)
The experimental semivariogram of the estimates is different from the sampling experimental semivariogram. As illustrated in Figure 9.2, the experimental semivariogram for the grid has a smaller sill and a larger range than the experimental semivariogram for the sampling, denoting an exaggerated continuity in the estimated values.
 
(b)
The histogram of the sampling is different than the histogram of the estimated values. Relative to the sample histogram, the histogram for the estimated values has fewer values in the tails and a larger proportion close to the mean. In Figure 9.3, the quartile deviation of the sampling is 36 ft, while that for the grid values is only 26.1 ft.
 
(c)
Crossvalidation of the sampling reveals that there is a tendency of kriging to underestimate values above the sample mean and to overestimate those below the mean, which results in a regression line such as that shown in Figure 9.4. Its slope is less steep than the ideal slope of 1.0 for the main diagonal—a distortion called conditional bias in the estimation.
 
Ricardo A. Olea

Chapter 10. Reliability

Abstract
Characteristic of geostatistics and other stochastic methods is the ability to assign confidence intervals to the estimates. The confidence intervals are derived from cumulative distributions of random functions.
Ricardo A. Olea

Chapter 11. Cumulative Distribution Estimators

Abstract
In the previous chapter we have seen different ways to generate cumulative distributions, including a method based on the kriging estimate, when the assumption of multinormality is accepted.
Ricardo A. Olea

Chapter 12. Block Kriging

Abstract
In the Introduction I mentioned that kriging was formulated from the outset to generalize linear regression, partly in order to consider different supports for the estimate and the sampling. In Chapter 5 we defined support as the shape, size, and orientation of the volume associated with any observation. So far we have not used that potential of kriging.
Ricardo A. Olea

Chapter 13. Ordinary Cokriging

Abstract
Simple, ordinary, and universal kriging are not multivariate models in the usual statistical sense of the term. Despite the fact that they employ a random function model comprising an infinite number of random variables, they are all used for the modeling of a single attribute.
Ricardo A. Olea

Chapter 14. Regionalized Classification

Abstract
An important task for earth scientists is to solve inverse problems by measuring regionalized attributes. The results can then be used to postulate natural processes that may have generated the phenomena we observe today. Because natural processes are fairly complex and samplings are rarely large enough, simplifying reality by the imposition of a manmade order commonly plays an important part in inverse modeling.
Ricardo A. Olea

Backmatter

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