Several geophysical flows, such as ice flows or lava flows, are described by a gravity-driven low Reynolds number movement of a free surface viscoplastic fluid over a bedrock. Their modeling involves constitutive laws, typically describing their rheological behavior or interactions with their bedrock, that lean on empirical parameterizations. Otherwise, the thorough observation of this type of flows is rarely possible; data associated to the observation of these flows, mainly remote-sensed surface data, can be sparse, missing or uncertain. They are also generally indirect : unknown parameters such as the basal slipperiness or the rheology are difficult to measure on the field. This PhD work focuses on the direct and inverse modeling of these geophysical flows described by the power-law Stokes model, specifically dedicated to ice flows, using variational methods. The solution of the direct problem (Stokes non-linear) is based on the principle of minimal dissipation that leads to a variational four-field saddle-point problem for which we ensure the existence of a solution. In this context, the incompressibility condition and the constitutive rheological law represent constraints associated to the minimization problem. The critical points of the corresponding Lagrangian are determined using an augmented Lagrangian type algorithm discretized using three- field finite elements. This algorithm provides an important time and memory saving compared to classical algorithms. We then focus on the inverse numerical modeling of these fluids using the adjoint model through two main associated tools : sensitivity analysis and data assimilation. We first study the rheological modeling through the two principal input parameters (fluid consistency and rheological exponent). Sensitivity analyses with respect to these locally defined parameters allow to quantify their relative weights within the flow model, in terms of surface velocities. Identification of these parameters is also performed. The results are synthetized as a methodology towards "virtual rheometry" that could help and support rheological measurements. The basal slipperiness, major parameter in ice dynamics, is investigated using the same approach. Sensitivity analyses demonstrate an ability to see beyond the "filtered" and non-local transmission of the basal variability to the surface. Consequently these sensitivities can be used to help defining areas of interest for observation and measurement. This basal slipperiness, empirical modeling of a multiscale complex process, is then used to carry on a comparison with a so called "self-adjoint" method, common in glaciology (neglecting the dependency of the viscosity on the velocity, i.e. the non-linearity). The adjoint model, obtained by automatic differentiation and evaluated by reverse accumulation, leads to define this approximation as a limit case of the complete inverse method. This formalism allows to generalize the process of the numerical evaluation of the adjoint state into an incomplete adjoint method, adjustable in time and accuracy depending on the quality of the data and the level of detail required in the identification. All this work is associated to the development of DassFlow-Ice software dedicated to the direct and inverse numerical simulation of free-surface viscoplastic fluids. This bidimensional prospective software, distributed within the glaciological com- munity, serves as a model for the current development of the tridimensional version.
