Evaluating landscape complexity and the contribution of non‐locality to geomorphometry

A long-standing question in geomorphology concerns the extent that statistical models of terrain elevations have adequate characteristics with respect to the known scaling properties of landscapes. In previous work, it has been challenging to ascribe statistical significance to metrics adopted to measure landscape properties. In this paper, we use a recently developed surrogate data algorithm to generate synthetic surfaces with identical elevation values to the source dataset, while also preserving the value of the H¨older exponent at any point (the underpinning characteristic of a multifractal surface). Our primary source data are from a laboratory experiment on landscape evolution. This allows us to examine how the statistical properties of the surfaces evolve through time and the extent to which they depart from the simple (multi)fractal formalisms. We show that there is a strong departure that is driven by the diffusive processes in operation. The number of sub-basins of a given channel order (for orders sufficiently small relative to the basin order) exhibit a clear increase in complexity after a steady-state for sediment flux is established. We also study elevation data from Florida and Washington State where the relative departure from simple multifractality is even more strongly expressed but is similar for two very different locations. Our results show that at the very least, the minimum complexity for a stochastic model for terrain statistics with appropriate geomorphic scalings needs to incorporate a conditioning between the pointwise H¨older exponents and elevation.