Application of first- and second-order adjoint methods to glacial isostatic adjustment incorporating rotational feedbacks

TL;DR

This paper extends the adjoint theory for glacial isostatic adjustment by incorporating rotational feedbacks and introducing second-order adjoint methods, simplifying implementation and enabling advanced sensitivity and uncertainty analyses.

Contribution

It introduces the first application of second-order adjoint methods to GIA and reformulates the theory for easier implementation in heterogeneous Earth models.

Findings

Incorporation of rotational feedbacks into GIA adjoint models.

First and second-order adjoint methods enable sensitivity and Hessian kernel computations.

Simplified reformulation facilitates implementation in existing GIA codes.

Abstract

This paper revisits and extends the adjoint theory for glacial isostatic adjustment (GIA) of Crawford et al. (2018). Rotational feedbacks are now incorporated, and the application of the second-order adjoint method is described for the first time. The first-order adjoint method provides an efficient means for computing sensitivity kernels for a chosen objective functional, while the second-order adjoint method provides second-derivative information in the form of Hessian kernels. These latter kernels are required by efficient Newton-type optimisation schemes and within methods for quantifying uncertainty for non-linear inverse problems. Most importantly, the entire theory has been reformulated so as to simplify its implementation by others within the GIA community. In particular, the rate-formulation for the GIA forward problem introduced by Crawford et al. (2018) has been replaced with the conventional equations for modelling GIA in laterally heterogeneous earth models. The implementation of the first- and second-order adjoint problems should be relatively easy within both existing and new GIA codes, with only the inclusions of more general force terms being required.