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

Abstract

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.

Variational formulation of marine ice-sheet and subglacial-lake grounding-line dynamics

“…2016; Zhu et al. 2016; Helanow & Ahlkrona 2018). In contrast to existing approaches to full-Stokes grounding-line dynamics, we resolve the contact conditions directly during solution of the variational problem (Durand et al.…”

“…While approximations to the Stokes equations have dominated ice-sheet modelling for decades, full-Stokes ice-sheet models have become more common in prognostic and diagnostic studies (Durand et al 2009b;Zhang et al 2011;Petra et al 2012;Seddik et al 2012;Seroussi et al 2012;Isaac, Stadler & Ghattas 2015;Gagliardini et al 2016;Zhu et al 2016;Helanow & Ahlkrona 2018). In contrast to existing approaches to full-Stokes grounding-line dynamics, we resolve the contact conditions directly during solution of the variational problem (Durand et al 2009a;Favier et al 2012;Cheng et al 2020).…”

Learning physics-based models from data: perspectives from inverse problems and model reduction

“…Rapid spectral decay has been demonstrated, not just for the model problems of this section, but for a broad set of inverse problems arising in science and engineering, either explicitly through low-rank approximation of the Hessian of the data misfit (Section 4) or implicitly through rapid convergence of conjugate gradients for the Hessian system (Section 3). These include ice sheet dynamics (Petra et al 2012, Petra, Martin, Stadler and Ghattas 2014, Isaac, Petra, Stadler and Ghattas 2015, Zhu et al 2016b, Babaniyi, Nicholson, Villa and Petra 2021, shape and medium acoustic and electromagnetic scattering (Akçelik, Biros and Ghattas 2002, Chaillat and Biros 2012, Ambartsumyan et al 2020, O'Leary-Roseberry, Villa, Chen and Ghattas 2020, Chen, Haberman and Ghattas 2021, seismic wave propagation (Akçelik et al 2003a, Epanomeritakis, Akçelik, Ghattas and Bielak 2008, Martin, Wilcox, Burstedde and Ghattas 2012, Bui-Thanh, Ghattas, Martin and Stadler 2013, Zhu et al 2016a, mantle convection (Worthen et al 2014), viscous incompressible flow (Biros and Ghattas 1999, 2005a,b, Yang, Stadler, Moser and Ghattas 2011, atmospheric transport (Akçelik et al 2003b, Bashir et al 2008, Flath et al 2011, Alexanderian, Petra, Stadler and Ghattas 2014, Wu, Chen and Ghattas 2020, Villa, Petra and Ghattas 2021, ocean dynamics (Kalmikov and Heimbach 2014), turbulent combustion (Chen, Villa and Ghattas 2019a), poroelasticity (Hesse and Stadler 2014, Alghamdi, Hesse, Chen and Ghattas 2020, Alghamdi et al 2021, infectious disease spread …”

Constraining the geothermal heat flux in Greenland at regions of radar-detected basal water

“…Studies have attempted to formulate and solve the inverse problem for GHF (e.g. Zhu and others, 2016). However, their analysis is limited to a cold ice sheet with a no-slip basal condition.…”