A long-standing question in geomorphology concerns the applicability of statistical models for elevation data based on fractal or multifractal representations of terrain. One difficulty with addressing this question has been the challenge of ascribing 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 as the source dataset, while also preserving the value of the Hölder exponent at any point (the underpinning characteristic of a multifractal surface). Our primary data are from an experimental study of 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 also study elevation data from Florida and Washington State. We are able to show that the properties of the experimental and actual terrains depart from the simple statistical models. Of particular note is that 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 flux steady-state is established in the experimental study. The actual number of basins is much lower than occur in the surrogates. The imprint of diffusive processes on elevation statistics means that, at the very least, a stochastic model for terrain based on a local formalism needs to consider the joint behavior of the elevations and their scaling (as measured by the pointwise Hölder exponents).

Understanding the complex interplay between erosional and depositional processes, and their relative roles in shaping landscape morphology is a question at the heart of geomorphology. A unified framework for examining this question can be developed by simultaneously considering terrain elevation statistics over multiple scales. We show how a long-standing tool for landscape analysis, the elevation-area or hypsometry, can be complemented by an analysis of the elevation scalings to produce a more sensitive tool for studying the interplay between processes, and their impact on morphology. We then use this method, as well as well-known geomorphic techniques (slope-area scaling relations, the number of basins and basin size as a function of channel order) to demonstrate how the complexity of an experimental landscape evolves through time. Our primary result is that the complexity increases once a flux equilibrium is established as a consequence of the role of diffusive processes acting at intermediate elevations. We gauge landscape complexity by comparing results between the experimental landscape surfaces and those produced from a new algorithm that fixes in place the elevation scaling statistics, but randomizes the elevations with respect to these scalings. We constrain the degree of randomization systematically and use the amount of constraint as a measure of complexity. The starting point for the method is illustrated in the figure, which shows the original landscape (top-left) and three synthetic variants generated with no constraints to the randomization. The value quoted in these panels is the root-mean-squared difference in the elevation values for the synthetic cases relative to the original terrain. This value is greatest where the original ridge becomes a valley. All these landscapes contain the same elevation values (i.e. the same probability distribution functions), and the same elevation scalings at a point. The differences emerge because the elevations themselves are distributed randomly across the surface.