The geochemistry of river sediments is routinely used to obtain information about geologic and environmental processes occurring upstream. For example, downstream samples are used to constrain chemical weathering and physical erosion rates, as well as the locations of mineral deposits or contaminant sources. Previous work has shown that, by assuming conservative mixing, the geochemistry of downstream samples can be reliably predicted given a known source region geochemistry. In this study we tackle the inverse problem and 'unmix' the composition of downstream river sediments to produce geochemical maps of drainage basins (i.e., source regions). The scheme is tested in a case study of rivers draining the Cairngorms, UK. The elemental geochemistry of the < 150 µm fraction of 67 samples gathered from the beds of channels in this region is used to invert for concentrations of major and trace elements upstream. We solve this inverse problem using the Nelder-Mead optimisation algorithm and by seeking only spatially smooth maps. The best-fitting source region geochemistry for 20 elements of different affinities (e.g., Be, Li, Mg, Ca, Rb, U, V) is assessed using independent geochemical survey data. The inverse approach makes reliable predictions of the major and trace element concentration in first order river sediments. We suggest this scheme could be a novel means to generate geochemical baselines across drainage basins and within river channels.