The forward sensitivity and adjoint-state methods of glacial isostatic adjustment

Abstract

S U M M A R YIn this study, a new method for computing the sensitivity of the glacial isostatic adjustment (GIA) forward solution with respect to the Earth's mantle viscosity, the so-called the forward sensitivity method (FSM), and a method for computing the gradient of data misfit with respect to viscosity parameters, the so-called adjoint-state method (ASM), are presented. These advanced formal methods complement each other in the inverse modelling of GIArelated observations. When solving this inverse problem, the first step is to calculate the forward sensitivities by the FSM and use them to fix the model parameters that do not affect the forward model solution, as well as identifying and removing redundant parts of the inferred viscosity structure. Once the viscosity model is optimized in view of the forward sensitivities, the minimization of the data misfit with respect to the viscosity parameters can be carried out by a gradient technique which makes use of the ASM. The aim is this paper is to derive the FSM and ASM in the forms that are closely associated with the forward solver of GIA developed by Martinec. Since this method is based on a continuous form of the forward model equations, which are then discretized by spectral and finite elements, we first derive the continuous forms of the FSM and ASM and then discretize them by the spectral and finite elements used in the discretization of the forward model equations. The advantage of this approach is that all three methods (forward, FSM and ASM) have the same matrix of equations and use the same methodology for the implementation of the time evolution of stresses. The only difference between the forward method and the FSM and ASM is that the different numerical differencing schemes for the time evolution of the Maxwell and generalized Maxwell viscous stresses are applied in the respective methods. However, it requires only a little extra computational time for carrying out the FSM and ASM numerically. An straightforward approach to compute the gradient of the data misfit is the brute-force method, whereby the partial derivatives of the misfit with respect to model parameters are approximated by the centred difference of two forward model runs. Although the brute-force method is useful for computing the gradient of the data misfit with respect to a small number of model parameters, it becomes expensive for a viscosity model with a large number of parameters. The ASM offers an efficient alternative for computing the gradient of the misfit since the computational time of the ASM is independent of the number of viscosity parameters. The ASM is thus highly efficient for calculating the gradient of the misfit for models with large numbers of parameters. However, the forward-model solution for each time step must be stored, hence the memory demands scale linearly with the number of time steps. This is the main drawback of the ASM.Key words: Numerical solutions; Inverse theory; Sea level change; Rheology: mantle. I N T RO D U C T I O NStudies on glacial isostati...

Quantifying the sensitivity of post-glacial sea level change to laterally varying viscosity

“…We note that this equation is somewhat similar to equation (21) in Martinec et al (2015), but their result is incorrect.…”

“…The first application of adjoint methods to GIA was Al-Attar & Tromp (2014), who considered the sensitivity of viscoelastic deformation to surface loads and the viscosity structure. Since then, Martinec et al (2015) applied the method in order to calculate the sensitivity of measurements to viscosity for GIA including sea level. Furthermore, Larour et al (2017) have recently calculated the sensitivity of specific locations to present-day mass loss, but address only elastic, and not viscoelastic, deformation.…”

“…In order to do this, we build on the work of Al-Attar & Tromp (2014) by including gravitationally self-consistent sea level change. Our formulation is entirely independent of that of Martinec et al (2015), and is more general and and convenient for practical GIA applications. In particular, we present kernels for three-dimensional viscosity perturbations and with respect to ice sheet history.…”

GIA Model Statistics for GRACE Hydrology, Cryosphere, and Ocean Science

“…Combined Global Positioning System (GPS) and space gravimetry now impact current model constraint paradigms (Bevis et al, ; Blewitt et al, ; Ivins et al, ; Khan et al, ; Tamisiea et al, ). The advancements have led modelers to consider ever more sophisticated approaches to assimilating data (e.g., Caron et al, ; Martinec et al, ). We propose such an advancement here using Bayesian framework with a wealth of new GPS time series from 459 sites and 11,451 RSL data.…”

Short Comment in response to Reviewer’s comments

“…Martinec, 2000;Tanaka et al, 2011), and finite volume (e.g. Latychev et al, 2005b) approaches have also been used, while approaches that use the adjoint method are under development (Al-Attar and Tromp, 2014;Martinec et al, 2015). The equations used to represent solid Earth deformation may differ between these approaches, and in particular the finite element approach was originally developed to permit consideration of power-law rheology (Wu, 1992).…”