Sensitivity kernels for viscoelastic loading based on adjoint methods
Observations of glacial isostatic adjustment (GIA) allow for inferences to be made about mantle viscosity, ice sheet history and other related parameters. Typically, this inverse problem can be formulated as minimizing the misfit between the given observations and a corresponding set of synthetic da...
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fthighwire:oai:open-archive.highwire.org:gji:196/1/34 2023-05-15T16:41:26+02:00 Sensitivity kernels for viscoelastic loading based on adjoint methods Al-Attar, David Tromp, Jeroen 2014-01-01 00:00:00.0 text/html http://gji.oxfordjournals.org/cgi/content/short/196/1/34 https://doi.org/10.1093/gji/ggt395 en eng Oxford University Press http://gji.oxfordjournals.org/cgi/content/short/196/1/34 http://dx.doi.org/10.1093/gji/ggt395 Copyright (C) 2014, Oxford University Press Geodynamics and tectonics TEXT 2014 fthighwire https://doi.org/10.1093/gji/ggt395 2018-04-07T06:24:45Z Observations of glacial isostatic adjustment (GIA) allow for inferences to be made about mantle viscosity, ice sheet history and other related parameters. Typically, this inverse problem can be formulated as minimizing the misfit between the given observations and a corresponding set of synthetic data. When the number of parameters is large, solution of such optimization problems can be computationally challenging. A practical, albeit non-ideal, solution is to use gradient-based optimization. Although the gradient of the misfit required in such methods could be calculated approximately using finite differences, the necessary computation time grows linearly with the number of model parameters, and so this is often infeasible. A far better approach is to apply the ‘adjoint method’, which allows the exact gradient to be calculated from a single solution of the forward problem, along with one solution of the associated adjoint problem. As a first step towards applying the adjoint method to the GIA inverse problem, we consider its application to a simpler viscoelastic loading problem in which gravitationally self-consistent ocean loading is neglected. The earth model considered is non-rotating, self-gravitating, compressible, hydrostatically pre-stressed, laterally heterogeneous and possesses a Maxwell solid rheology. We determine adjoint equations and Fréchet kernels for this problem based on a Lagrange multiplier method. Given an objective functional J defined in terms of the surface deformation fields, we show that its first-order perturbation can be written <f>$\delta J = \int _{M_{\mathrm{S}}}K_{\eta }\delta \ln \eta \,\mathrm{d}V +\int _{t_{0}}^{t_{1}}\int _{\partial M}K_{\dot{\sigma }} \delta \dot{\sigma } \,\mathrm{d}S \,\mathrm{d}t$</f>, where δ ln η = δη/η denotes relative viscosity variations in solid regions M S , d V is the volume element, <f>$\delta \dot{\sigma }$</f> is the perturbation to the time derivative of the surface load which is defined on the earth model's surface ∂ M ... Text Ice Sheet HighWire Press (Stanford University) Eta ENVELOPE(-62.917,-62.917,-64.300,-64.300) Lagrange ENVELOPE(-62.597,-62.597,-64.529,-64.529) Geophysical Journal International 196 1 34 77 |
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Open Polar |
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HighWire Press (Stanford University) |
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fthighwire |
language |
English |
topic |
Geodynamics and tectonics |
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Geodynamics and tectonics Al-Attar, David Tromp, Jeroen Sensitivity kernels for viscoelastic loading based on adjoint methods |
topic_facet |
Geodynamics and tectonics |
description |
Observations of glacial isostatic adjustment (GIA) allow for inferences to be made about mantle viscosity, ice sheet history and other related parameters. Typically, this inverse problem can be formulated as minimizing the misfit between the given observations and a corresponding set of synthetic data. When the number of parameters is large, solution of such optimization problems can be computationally challenging. A practical, albeit non-ideal, solution is to use gradient-based optimization. Although the gradient of the misfit required in such methods could be calculated approximately using finite differences, the necessary computation time grows linearly with the number of model parameters, and so this is often infeasible. A far better approach is to apply the ‘adjoint method’, which allows the exact gradient to be calculated from a single solution of the forward problem, along with one solution of the associated adjoint problem. As a first step towards applying the adjoint method to the GIA inverse problem, we consider its application to a simpler viscoelastic loading problem in which gravitationally self-consistent ocean loading is neglected. The earth model considered is non-rotating, self-gravitating, compressible, hydrostatically pre-stressed, laterally heterogeneous and possesses a Maxwell solid rheology. We determine adjoint equations and Fréchet kernels for this problem based on a Lagrange multiplier method. Given an objective functional J defined in terms of the surface deformation fields, we show that its first-order perturbation can be written <f>$\delta J = \int _{M_{\mathrm{S}}}K_{\eta }\delta \ln \eta \,\mathrm{d}V +\int _{t_{0}}^{t_{1}}\int _{\partial M}K_{\dot{\sigma }} \delta \dot{\sigma } \,\mathrm{d}S \,\mathrm{d}t$</f>, where δ ln η = δη/η denotes relative viscosity variations in solid regions M S , d V is the volume element, <f>$\delta \dot{\sigma }$</f> is the perturbation to the time derivative of the surface load which is defined on the earth model's surface ∂ M ... |
format |
Text |
author |
Al-Attar, David Tromp, Jeroen |
author_facet |
Al-Attar, David Tromp, Jeroen |
author_sort |
Al-Attar, David |
title |
Sensitivity kernels for viscoelastic loading based on adjoint methods |
title_short |
Sensitivity kernels for viscoelastic loading based on adjoint methods |
title_full |
Sensitivity kernels for viscoelastic loading based on adjoint methods |
title_fullStr |
Sensitivity kernels for viscoelastic loading based on adjoint methods |
title_full_unstemmed |
Sensitivity kernels for viscoelastic loading based on adjoint methods |
title_sort |
sensitivity kernels for viscoelastic loading based on adjoint methods |
publisher |
Oxford University Press |
publishDate |
2014 |
url |
http://gji.oxfordjournals.org/cgi/content/short/196/1/34 https://doi.org/10.1093/gji/ggt395 |
long_lat |
ENVELOPE(-62.917,-62.917,-64.300,-64.300) ENVELOPE(-62.597,-62.597,-64.529,-64.529) |
geographic |
Eta Lagrange |
geographic_facet |
Eta Lagrange |
genre |
Ice Sheet |
genre_facet |
Ice Sheet |
op_relation |
http://gji.oxfordjournals.org/cgi/content/short/196/1/34 http://dx.doi.org/10.1093/gji/ggt395 |
op_rights |
Copyright (C) 2014, Oxford University Press |
op_doi |
https://doi.org/10.1093/gji/ggt395 |
container_title |
Geophysical Journal International |
container_volume |
196 |
container_issue |
1 |
container_start_page |
34 |
op_container_end_page |
77 |
_version_ |
1766031867599912960 |