Abstract: In a recent paper, Kent and Wood (1997) investigated some new increment-based estimators of the fractal dimension of a stationary Gaussian process. In the present paper, we extend this work by constructing increment-based estimators based on two-dimensional sampling of surface data (as opposed to the one dimensional, or line transect, sampling previously considered). Much of our attention is focussed on two new estimators based on the ``square increment''. The practical performance of these estimators is examined in the study of several real datasets and via simulation. We also provide a detailed theoretical study of their properties in both Gaussian and non-Gaussian settings. Perhaps surprisingly, it turns out that there are differences in the limit theory in the Gaussian and non-Gaussian cases.
Key words and phrases: Gaussian random field, generalised least squares, self-similarity, stationary.