Variogram

A variogram is the graphical representation of the spatial dependence between pairs of data points, commonly used in geostatistics and spatial statistics. The term is sometimes used synonymously with semivariogram, but the latter is also used by some authors to refer to half of a variogram, and should therefore be avoided. Likewise, the term semivariance can be misleading, since the values shown in a variogram are entire variances of observations at a given spatial separation (lag). The variogram is the key function in geostatistics as it will be used to fit a model of the temporal/spatial correlation of the observed phenomenon. One is thus making a distinction between the experimental variogram that is a visualization of a possible spatial/temporal correlation and the variogram model that is further used to define the weights of the kriging function. Note that the experimental variogram is an empirical estimate of the covariance of a Gaussian process. As such, it may not be positive definite and hence not directly usable in kriging, without constraints or further processing. This explains why only a limited number of variogram models are used: most commonly, the linear, the spherical, the Gaussian, and the exponential models. For example, in gold mining, a variogram will give a measure of how much two samples taken from the mining area will vary in gold percentage depending on the distance between those samples. Samples taken far apart will vary more than samples taken close to each other.

Definition

The semivariogram γ ( h ) {\displaystyle \gamma (h)} was first defined by Matheron (1963) as half the average squared difference between a function and a translated copy of the function separated at distance h {\displaystyle h} . Formally

γ ( h ) = 1 2 ∭ V [ f ( M + h ) − f ( M ) ] 2 d M , {\displaystyle \gamma (h)={\frac {1}{2}}\iiint _{V}\left[f(M+h)-f(M)\right]^{2}dM,}

where M {\displaystyle M} is a point in the geometric field V {\displaystyle V} , and f ( M ) {\displaystyle f(M)} is the value at that point. The triple integral is over 3 dimensions. h {\displaystyle h} is the separation distance (e.g., in meters or km) of interest. For example, the value f ( M ) {\displaystyle f(M)} could represent the iron content in soil, at some location M {\displaystyle M} (with geographic coordinates of latitude, longitude, and elevation) over some region V {\displaystyle V} with element of volume d V {\displaystyle dV} . To obtain the semivariogram for a given γ ( h ) {\displaystyle \gamma (h)} , all pairs of points at that exact distance would be sampled. In practice it is impossible to sample everywhere, so the empirical variogram is used instead. The variogram is twice the semivariogram and can be defined, differently, as the variance of the difference between field values at two locations ( s 1 {\displaystyle \mathbf {s} _{1}} and s 2 {\displaystyle \mathbf {s} _{2}} , note change of notation from M {\displaystyle M} to s {\displaystyle \mathbf {s} } and f {\displaystyle f} to Z {\displaystyle Z} ) across realizations of the field (Cressie 1993):

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