Multiscale Terrain and Topographic Modelling with the Implicit TIN

It is often assumed that real land surfaces demonstrate the statistically self-affine scaling behaviour of fractional Brownian surfaces. Tests of this assumption against empirical data, however, show many deviations. Estimates of fractal properties vary between methods and over different scale ranges. So far, this empirical evidence has come from the analysis of variograms for DEMs representing areas up to tens of kilometres in diameter. Here we report results obtained by using variograms to analyse land surface DEMs at the continental scale, with a grid resolution of 30 arc seconds. Results reveal variogram curvature and breaks of slope, but also linear sections over distance lags of hundreds of kilometres. The estimated mean fractal dimension calculated from these sections is 2.66, substantially higher for all continents at these broad scales (around 200 km) than values calculated at the erosional landscape scale (around 200 m). Thus the land surface is not self-affine, and it is not clear that it follows any simple multifractal model. At the longest wavelengths, patterns found in the variograms appear to be related to broad tectonic features of the Earth's surface. For the reader to assess their quality and generality, estimates of fractal dimension should always be accompanied by statements of the scale range covered and the goodness of fit to a log-linear relationship.