Extended Data Fig. 7: A synthetic dataset to illustrate our method of reconstructing erosion rates from pairwise bank displacement observations.

a, Continuous discharge time series for the Koyukuk River from the USGS station at Hughes (1961–1982). b, Average annual discharge cycle based on the data in a. c, A simple synthetic time series for erosion rate based on equation (6) (the entrainment-limited endmember). The ‘instantaneous’ erosion rate gives the total annual erosion that would occur if that erosion rate were sustained for a 365-day period. The grey lines depict the times for which we have PlanetScope images (see Supplementary Table 1). d, The cumulative erosion from the synthetic curve in d. e, A pairwise displacement matrix computed from the synthetic cumulative erosion curve in d. f, An example real (noisy) displacement record. g, Stacking and bracketing of the displacement matrix leads to less noisy cumulative displacement records. Stacking refers to averaging the differential displacement time series along each column of the matrix in e. Bracketing refers to computing the cumulative displacement from every second column, every third column, every fourth column and so on. Stacking (averaging over the rows) makes the cumulative displacement estimates less sensitive to errors in the co-registration of the template image (rows of E(x, y)), whereas bracketing (skipping columns) makes the cumulative displacement estimates less sensitive to errors in the co-registration of the search image (columns of E(x, y)). h,i, Remaining noise in the stacked and bracketed cumulative erosion record (g) is reduced by imposing the constraint that the cumulative displacement time series should be a monotonic function of time; in most cases, an eroding riverbank should not switch from eroding to accreting over the course of our approximately 6-year analysis. Thus, temporary back-stepping of the bank position (h) is probably an error. i, We use MCMC to construct the most probable monotonic path through the cumulative displacement time series. j, Differentiating the record in i with respect to time yields an estimate for the instantaneous erosion rate. The green curve shows the synthetic curve used to generate the displacement matrix (e) and the grey curve gives the reconstructed erosion rate (shown as a stair-step plot rather than a continuous curve because our temporal observations are limited to the roughly ten cloud-free PlanetScope mosaics each year (Supplementary Table 1).