The length‐scaling properties of topography

The scaling properties of synthetic topographic surfaces and digital elevation models (DEMs) of topography are examined by analyzing their “structure functions,” i.e., the q th order powers of the absolute elevation differences: Δ h q ( l ) = E {| h ( x + l ) ‐ h ( x )| q }. We find that the relation Δ h 1 ( l ) ≈ cl H describes well the scaling behavior of natural topographic surfaces, as represented by DEMs gridded at 3 arc sec. Average values of the scaling exponent H between ∼0.5 and 0.7 characterize DEMs from Ethiopia, Saudi Arabia, and Somalia over 3 orders of magnitude range in length scale l (∼0.1–150 km). Differences in apparent topographic roughness among the three areas most likely reflect differences in the amplitude factor c. Separate determination of scaling properties in the x and y coordinate directions allows us to assess whether scaling exponents are azimuthally dependent (anisotropic) or whether they are isotropic while the surface itself is anisotropic over a restricted range of length scale. We explore ways to determine whether topographic surfaces are characterized by simple or multiscaling properties. The difference between scaling exponents of Δ h 1 ( l ) and for the DEMs is small, but positive, and such divergence in the scaling exponents of the structure functions is consistent with multiscaling behavior. Exceedance and perimeter sets of fractional Brownian surfaces fail to yield the trivial fractal dimensions expected of sets of known monofractals, suggesting a practical limitation arising from the use of finite resolution data sets in the analysis. By comparing the hypsometry of “real” topography (represented as DEMs) with that of fractional Brownian surfaces, we show that synthetic surfaces based on Gaussian statistics are limited as models for natural topography. Hypsometric curves, which probably reflect the relative importance of tectonic and erosional processes in shaping topography, clearly show that statistical moments higher than the second are important in describing topographic surfaces. Scaling analysis is a valuable tool for assessing the quality and accuracy of DEM representations of the Earth's topography.