Page 182 - MATLAB Recipes for Earth Sciences

7.9 Geostatistics (by R. Gebbers) 177

in a one or multidimensional space. While usually we do not know the true
variogram of the spatial process we have to estimate it from observations.
This procedure is called variography. Variography starts with calculating
the experimental variogram from the raw data. In the next step, the experi-
mental variogram is summarized by the variogram estimator. Variography

fi nishes with fitting a variogram model to the variogram estimator. The ex-
perimental variogram is calculated as the difference between pairs of the
observed values depending on the separation vector h (Fig. 7.11). The clas-
sical experimental variogram is given by the semivariance,

where z is he observed value at location x and z is he observed value at
x x+h
another point within a distance h. The length of the separation vector h is
called lag distance or simply lag. The correct term for γ(h) is semivariogram
(or semivariance), where semi refers to the fact that it is half of the variance
of the difference between z and z . It is, nevertheless, the variance per
x x+h
point when points are considered as in pairs (Webster and Oliver, 2001).
Conventionally, γ(h) is termed variogram instead of semivariogram and so
we do at the end of this chapter. To calculate the experimental variogram we
first have to build pairs of observations. This is done by typing

[X1,X2] = meshgrid(x);
[Y1,Y2] = meshgrid(y);
[Z1,Z2] = meshgrid(z);
The matrix of separation distances D between the observation points is

D = sqrt((X1 - X2).^2 + (Y1 - Y2).^2);

x = x + h
j i

h

x
i
Fig. 7.11 Separation vector h between two points.