Spatial Correlation and the Variogram

Nearby Is Similar

Porosity at one well tells you something about porosity a short distance away, but almost nothing about a well on the far side of the field. This is spatial correlation, and it is the reason we can map a property from scattered samples at all. Geostatistics makes that intuition quantitative with the variogram, the single most important tool in this chapter.

The Variogram

The variogram $\gamma(h)$ is half the average squared difference between every pair of samples a distance $h$ apart: $\gamma(h) = \tfrac{1}{2N(h)} \sum [z(x) - z(x+h)]^2$. At short lags the paired values are similar and $\gamma$ is small; as the lag grows the pairs become less alike and $\gamma$ rises, until beyond a certain distance the pairs are unrelated and it levels off. In the widget we generate a field with a known structure, drop samples on it, and watch the experimental variogram climb to meet the model that produced it.

Sill and Range

Two features carry the meaning. The sill is the plateau the variogram reaches, equal to the overall variance of the property. The range is the lag at which it gets there: pairs closer than the range are correlated, pairs farther apart are effectively independent. Read together, the sill says how variable the property is and the range says how far that variability stays organized, and those two numbers drive every interpolation and simulation that follows.