Lakes on fractal surfaces: A null hypothesis for lake-rich landscapes

A class of stochastic processes known as fractional Brownian motion (fBm) provides strikingly realistic simulations of certain types of terrain, particularly those which appear to be unmodified by geomorphological and geological processes. In addition to their less serious applications in video games and science fiction movies, fractal terrain simulations have proven useful in a number of areas of spatial analysis. For example, they can provide sample data sets for testing the efficiency of data structures and algorithms designed for topographic applications. Previous work has shown that stream networks simulated on fBm surfaces show the same deviations from accepted theories of channel network topology as do real stream networks, implying that such deviations originate in the geometrical constraints of packing channels onto surfaces, rather than from geological or other environmental controls. In effect, this work demonstrates the usefulness of fBm as a null hypothesis for terrain. One difficulty, however, stems from the abundant pits which occur in the simulations, because peaks and pits are equally likely. Flooding of pits on fBm surfaces was simulated to obtain lakes. Lake-rich stream networks were extracted and represented with a suitable integer code. The relative frequencies of various network topologies and groups of topologies were compared to known characteristics of channel networks on real lake-rich landscapes. 'Lake-string' topologies are significantly less abundant than in glaciated landscapes. Lake areas show good fits to hyperbolic distributions, but lake in-degrees do not fit the negative binomial model. fBm surfaces are appropriate null hypotheses of scale-free, lake-rich landscapes.