# The Covariance and the Variogram

## Abstract

Nature often behaves in a very complicated way, and geology is no exception. In petro-leum applications, where we are dealing with reservoirs at depths of several kilometers that are recognized by only a few wells and some seismic, we need to simplify the description of these reservoirs by means of models. A model is a simplification of nature and should never be identified with the natural phenomenon it seeks to describe. However, a model has the advantage of reducing our understanding of the reservoir to the estimation of a few parameters. The best approach is to explain the concepts using a 1D example.

Fig. 2-1 shows, on the left, a variable that varies around a constant mean. At any location, the behavior of the variable, although complicated, can be qualified as “homo-geneously heterogeneous.” On the average, it behaves the same everywhere, in the sense that we would make the same kind of error at any location if we were to predict the value of the variable from the value of the horizontal line. This will be discussed later as the stationarity hypothesis.

Fig. 2-1 shows, on the left, a variable that varies around a constant mean. At any location, the behavior of the variable, although complicated, can be qualified as “homo-geneously heterogeneous.” On the average, it behaves the same everywhere, in the sense that we would make the same kind of error at any location if we were to predict the value of the variable from

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# Geostatistics for Seismic Data Integration in Earth Models

In this introduction, we would like to highlight what appear to be the important landmarks in the history of geostatistical applications in the petroleum industry. What do we mean by "geostatistics?" In this course, this term will cover the petroleum applications resulting from the pioneering work of Prof. Georges Matheron and his Research Group at the Centre de Géostatistique de l'Ecole des Mines de Paris. As far as this course is concerned, the main pillars of this work are the developments of variogrambased modeling applications.

Variogram-based modeling applications can be classified in two broad categories, the first of which can be called deterministic geostatistics and is essentially all the development around kriging. We will see later that this covers a very wide number of techniques, including external drift kriging, error cokriging, factorial kriging, and collocated cokriging. Although kriging is a technique based on a stochastic model, it generates one single model as a result, and it is deterministic in that sense.

The second category can be called stochastic geostatistics, and it covers the numerous techniques developed around the conditional simulation concept. Conditional simulation is stochastic in the sense that, as with the Monte-Carlo simulation, it generates a family of "realizations" of 1D, 2D, or 3D models, all compatible with the a priori model and the existing data. With regard to kriging, conditional simulation includes several techniques, such as indicator simulation, collocated cosimulation, or geostatistical inversion. This explains why this one-day course is subdivided in two half-days, the first half-day presenting the basic concepts and the deterministic family of applications, the second half-day covering the stochastic applications (Fig. 1-1). The most complete synthesis of Matheron's work can be found in Chilès and Delfiner (1999). Isaaks and Srivastava (1989), Hohn (1988), and Deutsch (2002) are also other excellent presentations of geostatistics.

Following the work of Matheron, petroleum applications went through different episodes (Fig. 1-2). The first one could be qualified as deterministic mapping. This was the first development of kriging for mapping applications; see, for instance, the papers of Haas and Viallix (1974) or Haas and Jousselin (1976). This period saw the development of commercial mapping applications, such as Bluepack (Renard, 1990). Another important step in the development of 2D mapping applications was Doyen's (1988) paper showing the potential of cokriging for mapping porosity using seismic-derived information and well data.

The mid-1980s to mid-1990s saw the explosion of 3D stochastic (simulationbased) reservoir modeling.