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The Cryosphere An interactive open-access journal of the European Geosciences Union
The Cryosphere, 10, 1477-1494, 2016
https://doi.org/10.5194/tc-10-1477-2016
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.
Research article
13 Jul 2016
Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model
Hongyu Zhu1, Noemi Petra2, Georg Stadler3, Tobin Isaac4, Thomas J. R. Hughes1,5, and Omar Ghattas1,6,7 1Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA
2Applied Mathematics, School of Natural Sciences, University of California, Merced, CA 95343, USA
3Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
4Computation Institute, University of Chicago, Chicago, IL 60637, USA
5Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, USA
6Jackson School of Geosciences, The University of Texas at Austin, Austin, TX 78712, USA
7Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA
Abstract. We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since 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; together they determine 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 partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov–Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We 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 – i.e., 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. This is 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.

Citation: Zhu, H., Petra, N., Stadler, G., Isaac, T., Hughes, T. J. R., and Ghattas, O.: Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model, The Cryosphere, 10, 1477-1494, https://doi.org/10.5194/tc-10-1477-2016, 2016.
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Short summary
We study how well the basal geothermal heat flux can be inferred from surface velocity observations using a thermomechanically coupled nonlinear Stokes ice sheet model. The prospects and limitations of this inversion is studied in two and three dimensional model problems. We also argue that a one-way coupled approach for the adjoint equations motivated by staggered solvers for forward multiphysics problems can lead to an incorrect gradient and premature termination of the optimization iteration.
We study how well the basal geothermal heat flux can be inferred from surface velocity...
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