Variogram Did You Mean Variogram?

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Three functions are used in geostatistics for describing the spatial or the temporal correlation of observations: these are the correlogram, the covariance and the semivariogram. The last is also more simply called variogram.

The variogram is the key function in geostatistics as it will be used to fit a model of the spatial/temporal correlation of the observed phenomenon. One is thus making a distinction between the experimental variogram that is a visualisation 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 like the linear, the spherical, the gaussian and the exponential models, to name only those that are the most frequently used.

When the variogram is used to describe the correlation of different variables and is speaking about cross-variograms. Cross-variograms are used in co-kriging. Should the variable be binary or represent classes of values, one is then talking about indicator variograms. Indicator variogram is used in indicator kriging.

The experimental variogram is computed by measuring the mean-squared difference of a value of interest z evaluated at two points x and x+h. This mean squared difference is the semivariance and is assigned to the value h, which is known as the lag. A plot of the semivariance versus h is the variogram.

Reference

Principles of Geographical Information Systems by Peter A. Burrough & Rachael A. McDonnell

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