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

## 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 fluxmore »

- Authors:

- Univ. of Texas, Austin, TX (United States). Inst. for Computational Engineering and Sciences
- Univ. of California, Merced, CA (United States). Applied Mathematics, School of Natural Sciences
- New York Univ. (NYU), NY (United States). Courant Inst. of Mathematical Sciences
- Univ. of Chicago, IL (United States). Computation Inst.
- Univ. of Texas, Austin, TX (United States). Inst. for Computational Engineering and Sciences; Univ. of Texas, Austin, TX (United States). Dept. of Aerospace Engineering and Engineering Mechanics
- Univ. of Texas, Austin, TX (United States). Inst. for Computational Engineering and Sciences; Univ. of Texas, Austin, TX (United States). Jackson School of Geosciences; Univ. of Texas, Austin, TX (United States). Dept. of Mechanical Engineering

- Publication Date:

- Research Org.:
- Univ. of Texas, Austin, TX (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1435724

- Grant/Contract Number:
- SC0002710; SC0009286

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- The Cryosphere (Online)

- Additional Journal Information:
- Journal Name: The Cryosphere (Online); Journal Volume: 10; Journal Issue: 4; Journal ID: ISSN 1994-0424

- Publisher:
- European Geosciences Union

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 15 GEOTHERMAL ENERGY

### Citation Formats

```
Zhu, Hongyu, Petra, Noemi, Stadler, Georg, Isaac, Tobin, Hughes, Thomas J. R., and Ghattas, Omar.
```*Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model*. United States: N. p., 2016.
Web. doi:10.5194/tc-10-1477-2016.

```
Zhu, Hongyu, Petra, Noemi, Stadler, Georg, Isaac, Tobin, Hughes, Thomas J. R., & Ghattas, Omar.
```*Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model*. United States. doi:10.5194/tc-10-1477-2016.

```
Zhu, Hongyu, Petra, Noemi, Stadler, Georg, Isaac, Tobin, Hughes, Thomas J. R., and Ghattas, Omar. Wed .
"Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model". United States. doi:10.5194/tc-10-1477-2016. https://www.osti.gov/servlets/purl/1435724.
```

```
@article{osti_1435724,
```

title = {Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model},

author = {Zhu, Hongyu and Petra, Noemi and Stadler, Georg and Isaac, Tobin and Hughes, Thomas J. R. and Ghattas, Omar},

abstractNote = {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. Here, 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.},

doi = {10.5194/tc-10-1477-2016},

journal = {The Cryosphere (Online)},

number = 4,

volume = 10,

place = {United States},

year = {Wed Jul 13 00:00:00 EDT 2016},

month = {Wed Jul 13 00:00:00 EDT 2016}

}

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