The Forward Sensitivity and Adjoint-State Methods of Glacial Isostatic Adjustment

martinec, zdenˇek ; sasgen, ingo ; vel´ımsk´ y, jakub;

In 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 solve rofGIA 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 straight forward 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 aviscosity 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.