Uncertainty in quantitative analyses of topographic change: Error propagation and the role of thresholding

Abstract

Topographic surveys inevitably contain error, introducing uncertainty into estimates of volumetric or mean change based on the differencing of repeated surveys. In the geomorphic community, uncertainty has often been framed as a problem of separating out real change from apparent change due purely to error, and addressed by removing measured change considered indistinguishable from random noise from analyses (thresholding). Thresholding is important when quantifying gross changes (i.e. total erosion or total deposition), which are systematically biased by random errors in stable parts of a landscape. However, net change estimates are not substantially influenced by those same random errors, and the use of thresholds results in inherently biased, and potentially misleading, estimates of net change and uncertainty. More generally, thresholding is unrelated to the important process of propagating uncertainty in order to place uncertainty bounds around final estimates. Error propagation methods for uncorrelated, correlated, and systematic errors are presented. Those equations demonstrate that uncertainties in modern net change analyses, as well as in gross change analyses using reasonable thresholds, are likely to be dominated by low‐magnitude but highly correlated or systematic errors, even after careful attempts to reduce those errors. In contrast, random errors with little to no correlation largely cancel to negligible levels when averaged or summed. Propagated uncertainty is then typically insensitive to the precision of individual measurements, and is instead defined by the relative mean error (accuracy) over the area of interest. Given that real‐world mean elevation changes in many landscape settings are often similar in magnitude to potential mean errors in repeat topographic analyses, reducing highly correlated or systematic errors will be central to obtaining accurate change estimates, while placing uncertainty bounds around those results provides essential context for their interpretation.