A Complete Formalism For the Inversion of Post-Glacial Rebound Data: Resolving Power Analysis

Abstract

SUMMARY A formal inverse theory for mantle viscosity based upon the data of glacial isostatic adjustment is here formulated and applied to synthetic data. In this theory full account has been taken of the normal mode nature of the forward problem for realistic viscoelastic (Maxwell) models of the planet. Since it is impossible to accurately infer the excitation and decay constants of the individual normal modes from the observations, the formalism is cast in terms of the observed gross Earth data in the time domain. In this analysis expressions are required for the first-order perturbations in both the modal amplitudes and relaxation times that are induced by an arbitrary radial perturbation to the starting viscosity profile. A numerical technique is developed which enables us to accurately determine differential kernels for the modal amplitudes. the analogous kernels for the modal decay times are derived analytically (Peltier 1976), and the complete set of kernels is shown to satisfy the physical constraint imposed by the uniqueness of the state of isostatic equilibrium for the viscously incompressible Maxwell models that we employ. When the problem is parametrized in terms of the logarithm of viscosity, the kernels are capable of accurately predicting shifts in the normal mode characteristics for at least an order of magnitude variation in mantle viscosity. Using Bayesian statistics a formal inversion is applied to a set of synthetically generated data. These data, chosen to reproduce the space-time coverage of the actual observables, include a subset related to the global gravity field and a large sequence of idealized relative sea level (RSL) curves. It is found that even very weak a priori constraints can provide a stable and accurate inversion. A resolving power analysis indicates a spatial resolution of approximately 1200km near the core-mantle boundary (CMB) with a gradual improvement to better than 350km in the middle of the upper mantle. Subsets of the synthetic data are inverted in order to examine conditions on stability and accuracy, and to determine their relative contributions to the spatial resolution. Data from progressively older beaches are shown to contribute most to the spatial resolution at all depths, though the improvement in lower mantle resolution converges for data obtained from beaches formed within the last 5000 yr. Furthermore, the RSL curves in the vicinity of the peripheral bulge of the ancient Laurentide ice sheet are significant contributors to lower mantle resolution (as demonstrated in previous analyses of the forward problem). the inversion of this subset of the data also appears to be encouragingly stable.