Trends and Stationarity

The Stationarity Assumption

The variogram assumes the property is stationary: its statistics do not change systematically across the field. Often they do. Porosity may fall steadily with depth, or a property may ramp across a structure. That systematic drift is a trend, and it breaks the assumption the variogram rests on.

How a Trend Fools the Variogram

With a trend present, two far-apart samples differ mostly because of the trend, not because of random geological variability. So the raw variogram keeps climbing with lag and never reaches a sill; its apparent range and sill are artifacts of the trend rather than measurements of spatial structure. Modeling that variogram would be modeling the trend, and badly.

Fit, Remove, Model the Residual

The fix is to fit the trend as a smooth function of position, subtract it to leave a residual, and compute the variogram of the residual. The residual is approximately stationary, so its variogram does reach a clean sill, and that is the one to model; the trend is added back at the end. Watching the raw variogram diverge while the residual one stays flat is the clearest sign a trend is present.