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Towards adjoint-based inversion for rheological parameters in nonlinear viscous mantle flow

Worthen, Jennifer and Stadler, Georg and Petra, Noemi and Gurnis, Michael and Ghattas, Omar (2014) Towards adjoint-based inversion for rheological parameters in nonlinear viscous mantle flow. Physics of the Earth and Planetary Interiors, 234 . pp. 23-34. ISSN 0031-9201. doi:10.1016/j.pepi.2014.06.006.

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We address the problem of inferring mantle rheological parameter fields from surface velocity observations and instantaneous nonlinear mantle flow models. We formulate this inverse problem as an infinite-dimensional nonlinear least squares optimization problem governed by nonlinear Stokes equations. We provide expressions for the gradient of the cost functional of this optimization problem with respect to two spatially-varying rheological parameter fields: the viscosity prefactor and the exponent of the second invariant of the strain rate tensor. Adjoint (linearized) Stokes equations, which are characterized by a 4th order anisotropic viscosity tensor, facilitates efficient computation of the gradient. A quasi-Newton method for the solution of this optimization problem is presented, which requires the repeated solution of both nonlinear forward Stokes and linearized adjoint Stokes equations. For the solution of the nonlinear Stokes equations, we find that Newton’s method is significantly more efficient than a Picard fixed point method. Spectral analysis of the inverse operator given by the Hessian of the optimization problem reveals that the numerical eigenvalues collapse rapidly to zero, suggesting a high degree of ill-posedness of the inverse problem. To overcome this ill-posedness, we employ Tikhonov regularization (favoring smooth parameter fields) or total variation (TV) regularization (favoring piecewise-smooth parameter fields). Solution of two- and three-dimensional finite element-based model inverse problems show that a constant parameter in the constitutive law can be recovered well from surface velocity observations. Inverting for a spatially-varying parameter field leads to its reasonable recovery, in particular close to the surface. When inferring two spatially varying parameter fields, only an effective viscosity field and the total viscous dissipation are recoverable. Finally, a model of a subducting plate shows that a localized weak zone at the plate boundary can be partially recovered, especially with TV regularization.

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Gurnis, Michael0000-0003-1704-597X
Additional Information:© 2014 Elsevier B.V. Received 11 October 2013. Received in revised form 17 March 2014. Accepted 3 June 2014. Available online 27 June 2014. This work has been supported by NSF grants, CMMI-1028978, EAR-1247022, and ARC-0941678.
Group:Seismological Laboratory
Funding AgencyGrant Number
Subject Keywords:Mantle flow; Inverse problem; Adjoint method; Mantle viscosity; Nonlinear Stokes equations; Tikhonov and total variation regularization; Newton’s method
Record Number:CaltechAUTHORS:20140918-141101245
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Official Citation:Jennifer Worthen, Georg Stadler, Noemi Petra, Michael Gurnis, Omar Ghattas, Towards adjoint-based inversion for rheological parameters in nonlinear viscous mantle flow, Physics of the Earth and Planetary Interiors, Volume 234, September 2014, Pages 23-34, ISSN 0031-9201, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:49831
Deposited By: Ruth Sustaita
Deposited On:18 Sep 2014 21:25
Last Modified:10 Nov 2021 18:48

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