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Random Field Models in Earth Sciences

Random Field Models in Earth Sciences

by George Christakos

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This graduate-level text surveys the problems of earth and environmental sciences by means of theoretical models that have as an essential basis a purely random element. In addition to introducing students to spatiotemporal modeling as a fundamental methodology in the earth sciences, this volume illustrates the role of spatiotemporal modeling in the general


This graduate-level text surveys the problems of earth and environmental sciences by means of theoretical models that have as an essential basis a purely random element. In addition to introducing students to spatiotemporal modeling as a fundamental methodology in the earth sciences, this volume illustrates the role of spatiotemporal modeling in the general framework of the scientific method.
Starting with discussions of the science of the probable, the various theories of probability, and the physical significance of the random field model, the text explores a variety of problems in earth sciences in which the random field model constitutes an effective approach. A critical and concise summary of the fundamental concepts and results of the general random field theory is followed by considerations of the intrinsic spatial random field model, the factorable random field model, the spatiotemporal random field model, and space transformations of random fields.
Additional topics include random field modeling of natural processes, the simulation of natural processes, estimation in space and time, and sampling design.

Editorial Reviews

Explores the application of random field models and stochastic data processing to problems in hydrogeology, geostatistics, climate modeling, and oil reservoir engineering, among others. Researchers in the geosciences who work with models of natural processes will find discussion of: spatiotemporal random fields, space transformation, multidimensional estimation, simulation, sampling design, and stochastic partial differential equations. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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Dover Publications
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Random field models in Earth Sciences


Dover Publications, Inc.

Copyright © 1992 George Christakos
All rights reserved.
ISBN: 978-0-486-16091-7



"Common sense is the layer of prejudice laid down in the mind prior to the age of eighteen."

A. Einstein

1. The Science of the Probable and the Random Field Model

There are numerous phenomena in the physical world a direct (deterministic) study of which is not possible. In fact, physics, geology, meteorology, hydrology, and environmental engineering have introduced us to a realm of phenomena that cannot give rise to certainty in our knowledge. However, a scientific knowledge of these phenomena is possible by replacing the study of individual natural processes by the study of statistical aggregates to which these processes may give rise. A statistical aggregate is a configuration of possibilities relative to a certain natural process. The properties of such aggregates are expressed in terms of the concept of probability, more specifically, under the form of a probability law. It is important to recognize that the probability law is a perfectly determined concept. The difference between a probabilistic and a deterministic law is that, while in the deterministic law the states of the system under consideration directly characterize an individual natural process, in a probabilistic law these states characterize a set of possibilities regarding the process.

The application of the mathematical theory of probability to the study of real phenomena is made through statistical concepts. Therefore, it is essentially in the form of statistical knowledge that a science of the probable is constituted. This implies that the science of the probable replaces a direct study of natural processes by the study of the set of possibilities to which these processes may give rise.

Since knowledge regarding these processes is achieved only indirectly by way of statistical concepts, it will be characterized as probable or stochastic knowledge. Modern science provides convincing evidence that such probable knowledge is no less exact than certain knowledge. Naturally, the constitution of a science of the probable raises two fundamental philosophical problems:

(a) The elucidation of the content of the concept of probability.

(b) The foundation of probable or stochastic knowledge as a concept directly related to the application of the concept of probability to the study of real phenomena.

Clearly, these two problems are closely related to each other. For example, the concept of probability must be such that it can be used in the empirical world. Each one of these problems, however, possesses certain distinct aspects that deserve to be studied separately.

Problem (a) can be considered under the light of either a subjectivist explanation of the notion of probability, taken in itself, or an objectivist explanation. More specifically, according to the subjectivist explanation, probability is a measure attached to the particular state of knowledge of the subject. It may correspond to a degree of certitude or to a degree of belief (say, about where the actual state of nature lies); or to the attitude with which a rational person will approach a given situation that is open to chance (say, the attitude with which one is willing to place a bet on an event whose outcome is not definitely known in advance). According to explanations of the objectivist type, probability is a measure attached to certain objective aspects of reality. Such a measure may be regarded as the ratio of the number of particular outcomes, in a specific type of experiment, over the total number of possible outcomes; or as the limit of the relative frequency of a certain event in an infinite sequence of repeated trials; or as a characteristic property (particularly, the propensity) of a certain experimental arrangement. The main point of the last view is that it takes as fundamental the probability of the outcome of a single experiment with respect to its conditions, rather than the frequency of outcomes in a sequence of experiments. (See, e.g., Keynes, 1921; von Mises, 1928; Popper, 1934, 1972; Jeffreys, 1939; Savage, 1954; Byrne, 1968.)

In conclusion, there exist more than one meaning of probability. Figure 1.1 merely represents a skeleton outline of various complex and diversified analyses of the notion of probability considered over the years by several eminent mathematicians, scientists, and philosophers (see Poincaré, 1912, 1929; Borel, 1925, 1950; Kolmogorov, 1933; Reichenbach, 1935; Nagel, 1939; Boll, 1941; Gendre, 1947; Servien, 1949; Carnap, 1950; Polya, 1954; Polanyi, 1958; Fisher, 1959; Russel, 1962; Jaynes, 1968; de Finetti, 1974).

A subjective (sociological, Psc, or psychological, Py) concept of probability is of importance in social and psychological sciences, but it cannot serve as a basis for inductive logic or a calculus of probability applicable as a general tool of science. An objective (logical relationship, Pr, or propensity Pp) notion of probability is based on the assumption that objectivity is equivalent to formalization. It is, however, open to question whether formal logic can achieve the goals of the Pr concept, as has been demonstrated by Graig's theorem regarding empirical logic and by Godel's theorem on the limitation of formalization. Moreover, modern approaches to logic argue that the world obeys a nonhuman kind of reasoning and, hence, to cope with natural processes we must scrap our mode of human reasoning in favor of a new so-called quantum logic. The Pp concept, on the other hand, has been seriously criticized on the basis of the argument that what people understand by probability is broader than mathematical formalization. Last, to restrict probability to a mathematical meaning (Pm) is for many philosophers an ineffective approach, because the notion of probability transcends the bounds of mathematics (e.g., Byrne, 1968; Benenson, 1984).

Naturally, this variability of theoretical viewpoints reflects to an analogous variability in the practical implementation of the theory of probability. It seems that in the various fields of science and engineering, people do not stick to a unique meaning of probability. Occasionally, they prefer to choose what they consider to be the most appropriate meaning for the specific problem at hand.

The problem of probable knowledge [problem (b) above] is closely related to important modern scientific areas such as, for example, artificial intelligence and expert systems. With regard to this problem, two types of answers have been given: One is related to a subjectivist interpretation of probable knowledge, and the other is related to an objectivist interpretation. According to the former, the phenomena we are studying with the aid of probability theory are in themselves entirely determined and, therefore, they could be, ideally, the object of certain knowledge. And, if we are obliged to restrict ourselves to a probable knowledge of these phenomena, it is merely because we have at our disposal only incomplete information. The limitation of our information can be conceived either as purely contingent (due to insufficient sources of knowledge, inadequate measuring instruments and computers, etc.), or as a limitation in principle (because our capacities are inherently limited). The former point of view is used by classical statistical mechanics, while the latter is used in the context of the so-called orthodox theory of quantum mechanics. On the other hand, the objective interpretation of probable knowledge assumes that the incompleteness of our information is due to the object itself. Probable knowledge is then the expression of an objective contingency of the real phenomena. This contingency reflects either a principle of chance that exists in the very elementary components of the physical phenomena, or the lack of access to the various processes that determine these phenomena.

Evidently, there is a mutual relationship between the two aforementioned sets of problems: the subjective (objective) explanation of probability is well suited to the subjective (objective) explanation of the foundation of probable knowledge. But this is not always the case. For example, a subjective explanation of probability can well be used in the context of an objective interpretation of probable knowledge. Therefore, it is necessary that these two sets of problems be distinguished one from the other.

In any case, satisfactory answers to problems (a) and (b) above clearly belong to the field of epistemology and, thus, they require access to multidimensional philosophical considerations. In particular, any argument concerning the objectiveness or the subjectiveness of the concept of probability demands a deeper understanding of human nature and knowledge. To adopt the subjectivist or the objectivist interpretation of probable knowledge is a decision closely related to understanding of the nature of the world. In this regard, all the aforementioned attempts to answer the fundamental philosophical problems (a) and (b) are without doubt inadequate. It is far from being evident that we have at our disposal today the philosophical tools necessary to obtain a true understanding of the concepts of probability and probable knowledge. There is, perhaps, in the probabilistic concept the emergence of a type of knowledge very different from that considered by traditional schools of philosophy. In fact, people begin to realize that full consciousness of what is involved in knowledge of this sort is going to oblige us to modify fundamental concepts such as truth, knowledge, and experience.

In view of the above considerations, in this book we will not define probability as a concept in itself and will not indulge the epistemological problematics of probable knowledge. Had we decided to do so we would then have had the extremely difficult task of providing sound justification for a number of issues: If we had adopted the subjectivist explanation, we should explain why and how we have the right to suppose that the natural phenomena are entirely determined in themselves, and also why and how our knowledge, supposedly inadequate (be it in principle or merely in fact), turns out nevertheless to be quite adequate at the level of the statistical aggregates. If we had chosen the objectivist interpretation, we should justify why and how contingency appears in the physical phenomena, and why and how phenomena supposedly undetermined in themselves can give rise to statistical aggregates that are, for their part, entirely determined.

This book will focus attention on the language of probability, which is not at all constituted from some given epistemology, but from certain concrete problems that the traditional methods were not able to solve. Our concern will be on issues of application of the science of the probable in the context of the so-called random field (RF ) model. In particular, the RF model will be considered a statistical aggregate about which we will make two a priori assumptions:

(i) Randomness is not a property of reality itself but, instead, a property of the RF model used to describe reality.

(ii) Probable knowledge cannot be considered as an image of reality. Through it we aim at reality and we learn something about it, but the relationship between our knowledge and its object becomes indirect and remote.

Under the light of assumptions (i) and (ii), the concept of probability in all its richness of content is of far greater importance for real world applications than the words and terms used to express it. By using probable knowledge the real is approached only through an abstract construction that involves the possible and is rather like a detecting device through which one grasps certain aspects of reality. Thanks to the detecting device, one can register certain reactions of reality and thus know it, not through an image of it, but through the answers it gives to one's questions.

The RF formalism does not restrict our concept of probable science to a physical theory of natural phenomena, governed by "randomness" or "chance"; or to a logical theory of plausible reasoning in the presence of incomplete knowledge. These theories, as well as several others, are viewed as potential modeling tools and detecting devices, rather than as unique realities. Within the RF context, we are looking at real problems that are in principle very subtle and complex. Therefore, like many other human enterprises, the practice of the science of the probable requires a constantly shifting balance between a variety of theories and methods, such as stochastic calculus, probability and statistics, logic and information theory.

As a matter of fact, RF methods have proven themselves very useful, although the notion of probability itself has not been philosophically well defined. This is true for almost all scientific theories. For example, despite the fact that terms such as mass, energy, and atom are philosophically undefined or ill-defined, theories based on these terms have led to extremely valuable applications in science and engineering.

In this book we study the use of RF models in the context of stochastic data analysis and processing. More precisely, the term stochastic data analysis and processing refers to the study of a natural process on the basis of a series of observations measured over a sample region of space (spatial series), over a sample period of time (time series), or over a spatial region for a sample time period (space-time series). In general, the aim of such a study is to evaluate and reconstruct the properties of the underlying unique physical process from fragmentary sampling data, by accommodating the probabilistic notion of RF. Hence, before proceeding with the description of the stochastic data analysis and processing research program, it is appropriate to discuss the physical content implicit in the RF representation of a natural process.


Excerpted from Random field models in Earth Sciences by GEORGE CHRISTAKOS. Copyright © 1992 George Christakos. Excerpted by permission of Dover Publications, Inc..
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