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We address the inverse problem of inferring the basal geothermal heat fluxfrom surface velocity observations using a steady-state thermomechanicallycoupled nonlinear Stokes ice flow model. This is a challenging inverseproblem since the map from basal heat flux to surface velocity observables isindirect: the heat flux is a boundary condition for the thermaladvection–diffusion equation, which couples to the nonlinear Stokes ice flowequations; together they determine the surface ice flow velocity. Thismultiphysics inverse problem is formulated as a nonlinear least-squaresoptimization problem with a cost functional that includes the data misfitbetween surface velocity observations and model predictions. A Tikhonovregularization term is added to render the problem well posed. We deriveadjoint-based gradient and Hessian expressions for the resultingpartial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method forits solution. As a consequence of the Petrov–Galerkin discretization of theenergy equation, we show that discretization and differentiation do notcommute; that is, the order in which we discretize the cost functional anddifferentiate it affects the correctness of the gradient. Using two- andthree-dimensional model problems, we study the prospects for and limitationsof the inference of the geothermal heat flux field from surface velocityobservations. The results show that the reconstruction improves as the noiselevel in the observations decreases and that short-wavelength variations inthe geothermal heat flux are difficult to recover. We analyze theill-posedness of the inverse problem as a function of the number ofobservations by examining the spectrum of the Hessian of the cost functional.Motivated by the popularity of operator-split or staggered solvers forforward multiphysics problems – i.e., those that drop two-way coupling termsto yield a one-way coupled forward Jacobian – we study the effect on theinversion 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 equationscan lead to an incorrect gradient and premature termination of optimizationiterations. This is due to loss of a descent direction stemming frominconsistency of the gradient with the contours of the cost functional.Nevertheless, one may still obtain a reasonable approximate inverse solutionparticularly if important features of the reconstructed solution emerge earlyin optimization iterations, before the premature termination

Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model