Extended Data Fig. 6: Methodology for measuring sub-pixel erosion along riverbanks.

a, An illustration of the workflow for the sub-pixel detection of riverbank erosion. b,c, An example of the two Sentinel-2 images used to compute the migration of the Koyukuk River (2016–2022) in Extended Data Fig. 9. The crops in b and c show a region of the Koyukuk River near Huslia (65.6966° N, 156.3824° W). Note that the river stage and sediment load are higher on 30 August 2016 compared with 13 July 2022, causing the river colour (RGB values) and the position of the land–water boundary to be different in the two images. We want to make sure that our algorithm records the net migration of the river as a result of bank erosion, rather than the variable exposure of sand on the riverbanks resulting from rising and falling river stage. To do so, we transform the multispectral satellite image to the dimensionless NDVI band ratio (equation (18)). The NDVI accentuates the spectral difference between the river water and the vegetated floodplain while collapsing the spectral difference between unvegetated sand and river water. The result is that the NDVI image is relatively insensitive to changes in water level (which expose or submerge unvegetated bars). Next, we extract an n × n-pixel chip, centred at the bank edge for the location of interest, from the image acquired at time 1. We extract an n × n-pixel chip at the same location in the image acquired at time 2. We use Fourier methods to take the 2D cross-correlation of the two image chips. The 2D cross-correlation spectrum, which we upsample by a factor of 10, peaks at a (Δx, Δy) value that records the estimated riverbank displacement between time 1 and time 2. Note that, given the relatively linear bank geometry (at least on the scale of the n × n-pixel chips), the cross-correlation spectrum has a ridge-like geometry rather than a sharp peak. Thus, when searching for the maximum in the 2D cross-correlation spectrum, we search along a vector that is perpendicular to the orientation of the riverbank (and therefore perpendicular to the ridge in the cross-correlation spectrum). d,e, Illustration of how we perform the methodology described in a for every position along the 450-km reach of the Koyukuk River shown in Fig. 1b.