Large-scale Versus Small-scale Variation Decomposition, Followed by Kriging Based on a Relative Variogram, in Presence of a Non-stationary Residual Variance
Laurent Raty , Marius Gilbert
Journal of Geographic Information and Decision Analysis, Vol. 2., No. 2, pp. 91-115, 1998.
We propose here an interpolation method based on a decomposition of the data in large- and small-scale variation. This decomposition was performed using a two-way directional decomposition, similar to the decomposition used by Cressie in his median-polish kriging (1993), though we applied a decomposition by means instead of medians. We considered the effects isolated by the decomposition as associated to intervals along the two directions, and not to a row or column co-ordinate. We replaced the "plating" used by Cressie to interpolate large-scale variation between grid rows and columns by the fit of a more continuous surface, with surface mean over each interval accounting for its associated mean effect. Residuals were predicted by kriging. However, as residual variance appeared to be strongly linked to large-scale-variation structure, we did not assume variance stationarity. Rather, we built a variance predictor based on this structure and modelled the data continuity through a relative variogram, relative to the sum of predicted variances at the tail and head of the lag vector. In the kriging equations, the variogram value used to characterize the variation between two locations was the relative variogram corresponding to the lag separating these two locations, rescaled by the sum of variance predictors at these two locations.
KEYWORDS: trend, two-way decomposition, kriging, non-stationary variance, relative variogram.