Reconstructing landscapes: an adjoint model of the stream power and diffusion equation.

Reconstructing the evolution of a landscape provides insights into its geological and/or climatic history and into processes shaping the earth's surface: what was the configuration of the drainage network before a specific geological or climatic event, what are the areas that are currently most sensitive to fluvial incision or hillslope processes, or to which extent lithological contrasts influenced landscape evolution are frequent questions. Most of the time, these questions are addressed with forward models in which only a small part of the parameter space is explored.

Landscape evolution can be simulated using a diffusion-advection equation where the diffusive term represents hillslope erosion-deposition processes and advection simulates river incision. In this case, the advection velocity can be calculated from drainage area and erodibility parameters of the Stream Power Law.  The model can also include a source term, which simulates tectonic uplift. This approach permits to solve a PDE and formulate an adjoint model that can be used for parameter inversion and sensitivity analysis. In this study, we use the Firedrake package which includes automatic differentiation procedures for building the adjoint model. Our results illustrate different model sensitivities to diffusion and erodibility coefficients, and show that it is able to reconstruct spatial variations of these coefficients.

We then apply the adjoint model to sensitivity analysis and to parameter inversion in real-world cases. The first one is the southeastern border of the French Massif Central, for which we seek at understanding what was the topography prior to a major incision by tributaries of the Rhone River. The second case is the footwall topography along a segment of the Wasatch normal fault, USA, for which we aim at quantifying temporal uplift rate variations.

In the French Massif Central, inversion of the initial conditions reveals that the pre-incision topography consists of a relatively smooth and flat footwall delimited by a well-defined and linear fault escarpment that corresponds to a Mesozoic normal fault system. In the Wasatch range, the model indicates a significant increase in the uplift rate of the Wasatch Range, from 0.2 to 1 mm.yr^-1, since approximately 2 Ma, aligning well with recently published estimates.