AlAttar, David and Tromp, Jeroen (2014) Sensitivity kernels for viscoelastic loading based on adjoint methods. Geophysical Journal International, 196 (1). pp. 3477. ISSN ISSN: 0956540X, ESSN: 1365246X DOI 10.1093/gji/ggt395

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Abstract
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 nonideal, solution is to use gradientbased 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 selfconsistent ocean loading is neglected. The earth model considered is nonrotating, selfgravitating, compressible, hydrostatically prestressed, 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 firstorder perturbation can be written δJ=∫MSKηδlnηdV+∫t1t0∫∂MKσ˙δσ˙dSdt, where δ ln η = δη/η denotes relative viscosity variations in solid regions MS, dV is the volume element, δσ˙ is the perturbation to the time derivative of the surface load which is defined on the earth model's surface ∂M and for times [t0, t1] and dS is the surface element on ∂M. The ‘viscosity kernel’ Kη determines the linearized sensitivity of J to viscosity perturbations defined with respect to a laterally heterogeneous reference earth model, while the ‘rateofloading kernel’ Kσ˙ determines the sensitivity to variations in the time derivative of the surface load. By restricting attention to spherically symmetric viscosity perturbations, we also obtain a ‘radial viscosity kernel’ K¯¯¯η such that the associated contribution to δJ can be written ∫ISK¯¯¯ηδlnηdr, where IS denotes the subset of radii lying in solid regions. In order to illustrate this theory, we describe its numerical implementation in the case of a spherically symmetric earth model using a 1D spectral element method, and calculate sensitivity kernels for a range of realistic observables.
Item Type:  Article 

Additional Information:  This article has been accepted for publication in Geophysical Journal International © The Authors 2013. Published by Oxford University Press on behalf of The Royal Astronomical Society. All rights reserved. 
Uncontrolled Keywords:  2013AREP; IA66; OAGE 
Subjects:  02  Geodynamics, Geophysics and Tectonics 
Divisions:  02  Geodynamics, Geophysics and Tectonics 08  Green Open Access 
Journal or Publication Title:  Geophysical Journal International 
Volume:  196 
Page Range:  pp. 3477 
Identification Number:  10.1093/gji/ggt395 
Depositing User:  Sarah Humbert 
Date Deposited:  14 Feb 2014 18:33 
Last Modified:  10 Oct 2014 18:34 
URI:  http://eprints.esc.cam.ac.uk/id/eprint/2946 
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