# Inversion of geothermal flux in a thermo-mechanically coupled nonlinear Stokes ice sheet model

We propose a formal optimization method for inferring the geothermal flux based on a thermo-mechanical ice sheet model and observations of surface velocity. The method is demonstrated on an idealized problem using synthetic observations (surface velocity) and geothermal flux fields.

Via its impact on the ice sheet thermal state, and in turn on ice rheology, the poorly observed geothermal flux field is a source of uncertainty in ice sheet modeling. Methods for better constraining it can also be expected to add confidence to model-based projections of ice sheet mass loss and sea-level rise.

We address the inverse problem of inferring the basal geothermal flux from surface velocity observations using an instantaneous, thermo-mechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem because the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection-diffusion equation, which couples to the nonlinear Stokes ice flow equations through the temperature dependent, flow-law rate factor, which impacts the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to render the problem well-posed. We derive adjoint-based gradient and Hessian expressions for the resulting PDE-constrained optimization problem and propose an inexact Newton method for its solution. Using two- and three-dimensional model problems, we study the prospects for (and limitations of) the inference of the geothermal flux field from surface velocity observations. As expected, the results show that the reconstruction improves as the noise level in the observations decreases, and that small wavelength variations in the geothermal heat flux are difficult to recover. We also analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split, or “staggered” solvers for forward multiphysics problems – those that drop two-way coupling terms to yield a one-way coupled forward Jacobian – we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. We show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.