In a degrading landscape, does the episodic nature of erosion affect observed erosion rates in a systematic way, and can one account for the effects? We present a null hypothesis for surface change rate variations based on minimal assumptions about the processes of topographic evolution. Variance in erosion along with censorship of topographic change distributions combine to act as a first-order control on time-scale dependence of erosion rates. In general, censorship is ubiquitous, and as a result, time-scale dependence of rates is likely to apply to almost every system. In detail, this occurs because at short time scales, surface changes can be censored from field measurements that become inherently incorporated by natural surface evolution at longer time scales. Additionally, the granularity of systems implies minimum time scales below which specific rate measurements have no physical interpretation. Finally, we show the existence of a crossover time scale at which the short-term rate dependence of a process gives way to the long-term rate that is no longer subject to censoring.
We demonstrate these points by applying the proposed framework to a growing seepage channel network in Florida, United States. Using a dendrogeomorphic approach in concert with cosmogenic radionuclide methods, we estimate denudation rates averaged over annual to multimillennial time scales. Exposure of hundreds of tree roots combined with tree ages supplies surface change rates along approximately one third of the valley bottom within the studied area. Erosion rates are distributed approximately between 1 mm/yr and 10 mm/yr. This erosion exhibits variance that depends linearly on time scale and is interpreted to represent a process well modeled as normal diffusion. As a result, the erosion rates show an inverse-square dependence on time scale. We used the 10Be contents from nine quartz sand samples to estimate long-term erosion rates and then coupled them with existing estimates of long-term erosion. These estimates of erosion are of the order of 0.01 mm/yr to 0.1 mm/yr and correspond to time scales of roughly 105 yr. Based on theory derived from censored measurements of rate distributions, the short-term rates can be used to infer expected long-term rates. In this case, distributions of measured and expected long-term rates do not overlap, and an explicit test of a null hypothesis is not necessary. We interpret the large-magnitude, short-time-scale rates as a product of recent influences on the ravines, and we discuss how this framework could be applied in other settings.