Impact of noise on landscapes and metrics generated with stream power models

Abstract. The Stream Power Model (SPM) has become a cornerstone of quantitative geomorphology, widely used to predict landscape evolution including the generation, moderation, and lowering of Earth's topography, sedimentary flux and biogeochemical processes. It is well known that landscape geometries predicted by the SPM can be strongly influenced by noise. However, its impact on the uncertainties or probabilities of, for instance, drainage planform geometries and widely used metrics is poorly understood. Noise can be incorporated into SPM simulations in a variety of ways. For instance, random, low amplitude, topographic anomalies are often inserted into starting conditions to enhance the realism of calculated drainage networks. Spatio-temporal or quenched (frozen) noise also influence the trajectories of evolving landscapes. Our goal with this paper is to establish how noise impacts the probabilities of landscape geometries and the reliability of tectonic and erosional information recovered from them. A series of landscape evolution models are run in which different arrangements, distributions, and implementations of noise are added to models evolving under the same tectonic and erosional forcings to an equilibrium state. We quantify uncertainties that arise from incorporating different arrangements of typical (uniform; white) and naturalistic initial, quenched and spatio-temporal noise. We focus on three conclusions. First, tectonic rates and values of erosional-geometric parameters (e.g., concavity and steepness indices, Hack exponents) recovered via metrics-based approaches (e.g., slope-area, χ, length-area) are uncertain in the presence of noisy initial conditions. Recovered values from individual landscapes generated with the same distribution but different specific arrangement of noise are at least as uncertain as ranges attributed to, for instance, changes in aridity. In fact, even noise with amplitudes that are <1 % of cumulative uplift can cause tectonic rates to no longer be recoverable to within a factor of two of true values. These results emphasise the sensitivity of metrics that rely on calculating derivatives (e.g., slope-area, χ) to noise. Secondly, whilst noise can make landscape geometries highly uncertain (different in different simulations), the distributions of their geomorphic properties (e.g., hypsometries, channel length-area relationships, Hack exponents) appear to have well defined statistical properties (e.g., expected values and variance). Finally, we suggest that a useful way to assess the impact of noise on SPM predictions is to generate ensembles of hundreds to thousands of models in which different arrangements of the chosen distribution of noise are inserted. Doing so can provide means to quantify uncertainty in predicted geometries and derived metrics, which can be substantial.