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78 CH 5 MODELING SPATIAL CONTINUITY
The non-randomness of Earth Sciences phenomena entails that values measured close
to each other are more “alike” than values measured further apart, in other words a spa-
tial relationship exists between such values. The term “spatial relationship” incorporates
all sorts of relationships, such as relationships among the available spatial data or be-
tween the unknown values and the measured data. The data may be of any type, pos-
sibly different from that of the variable or property being modeled. Therefore, in order
to quantify uncertainty about an unsampled value, it is important to first and foremost
quantify that spatial relationship, that is, quantify through a mathematical model the un-
derlying spatial continuity. In this book such a model is termed a “spatial continuity
model”. The simplest possible quantification consists of evaluating the correlation coef-
ficient between any datum value measured at locations u = (x,y,z) and any other mea-
sured a distance h away. Providing this correlation for various distances h will lead to
the definition of a correlogram or variogram, which is one particular spatial continuity
model discussed.
In the particular case of modeling the subsurface, spatial continuity is often determined
by two major components: one is the structural features such as fault and a horizon sur-
faces; the second is the continuity of the properties being studied within these structures.
Since these two components present themselves so differently, the modeling of spatial
continuity for each is approached differently, although it should be understood that there
may be techniques that apply to both structure and properties. Modeling of structures is
treated in Chapter 8.
In terms of properties, this chapter presents various alternative spatial continuity
models for quantifying the continuity of properties whether discrete or continuous and
whether they vary in space and or time. This book deals mostly with spatial processes
though, understanding that many of the principles for modeling spatial processes apply
also to time processes or spatio-temporal processes (both space and time). The models
developed here are stochastic models (as opposed to physical models) and are used to
model so-called “static” properties such as a rock property or a soil type. They are rarely
used to model dynamic properties (those that follow physical laws) such as pressure or
temperature, unless it is required to interpolate pressure and temperature on a grid or do a
simple filtering operation or statistical manipulation. Dynamic properties follow physical
laws and their role in modeling uncertainty is the topic of Chapter 10.
Three spatial models are presented in this chapter: (i) the correlogram/variogram
model, (ii) the object (Boolean) model, and (iii) the 3D training image models. As
with any mathematical model, “parameters” are required for such models to be com-
pletely specified. The variogram is a model that is built based on mathematical consid-
erations rather than physical ones. While the variogram may be the simplest model of
the set, requiring only a few parameters, it may not be easy to interpret from data when
there are few, nor can it deliver the complexity of real spatially varying phenomena. Both
the object-based and the training image-based models attempt to provide models from
a more realistic perspective, but call for a prior thorough understanding and interpreta-
tion of the spatial phenomenon and require many more parameters. Such interpretation is
subject to a great deal of uncertainty (if not one of the most important one).