Interpolation: Kriging, Cokriging, Factorial Kriging, and Splines
In the previous two chapters, we discussed the meaning of the geostatistical model and of its parameters. We will now discuss how this model can be applied. We will start with deterministic techniques, known under the generic name of kriging. Here, “deterministic” should be understood in the sense of “providing only one solution.” We will see that, although the model is probabilistic, kriging produces only one solution. Kriging covers a wide range of applications. The first one consists of interpolating one single variable in one, two, or three dimensions, and the second one consists of interpolating one variable but using the extra information provided by another variable that is related, of course, to the first one.
Recall the probabilistic model we defined in the previous chapter. The variable Z(x) is interpreted as the sum of a polynomial trend, m(x), plus a residual, R(x), of mean zero. Under this model, universal kriging (Matheron, 1970) addresses the problem of interpolating a variable on the basis of a number of scattered data. This can be the interpolation of layer-averaged porosity from well data or the interpolation of seismic times from a 2D seismic campaign.
To understand kriging, let us consider the variogram from another perspective (Fig. 3-1). Suppose that layer-averaged porosity has been calculated at a well and that we want to estimate porosity 1 km away from that well by using the value at the well. Obviously, we will make an error, which can be directly read from
Figures & Tables
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.