We describe and apply a linear inverse model which calculates spatial and temporal patterns of uplift rate by minimizing the misfit between inventories of observed and predicted longitudinal river profiles. Our approach builds upon a more general, nonlinear, optimization model, which suggests that shapes of river profiles are dominantly controlled by upstream advection of kinematic waves of incision produced by spatial and temporal changes in regional uplift rate. Here we use the method of characteristics to solve a version of this problem. A damped, nonnegative, least squares approach is developed that permits river profiles to be inverted as a function of uplift rate. An important benefit of a linearized treatment is low computational cost. We have tested our algorithm by inverting 957 river profiles from both Africa and Australia. For each continent, the drainage network was constructed from a digital elevation model. The fidelity of river profiles extracted from this network was carefully checked using satellite imagery. River profiles were inverted many times to systematically investigate the trade‐off between model misfit and smoothness. Spatial and temporal patterns of both uplift rate and cumulative uplift were calibrated using independent geologic and geophysical observations. Uplift patterns suggest that the topography of Africa and Australia grew in Cenozoic times. Inverse modeling of large inventories of river profiles demonstrates that drainage networks contain coherent signals that record the regional growth of elevation.