What Spherically Symmetric Viscosity Structure Produces the same PGR as a Realistic 3D Earth?

Observations of isostatic adjustment of the earth's surface due to transient loading provide important constraints on the mantle viscosity structure. However, most studies of this response have assumed a spherically symmetric (1D) earth. This study is motivated by the following question: when a one-dimesional viscosity model is derived from post-glacial rebound (PGR) observations, how does this 1D structure correspond to the three-dimensional structure of the earth? Using the 3D spherical finite element software CitcomSVE [Zhong et al., 2002], we are able to compute the earth's response to realistic glacial loading when the earth has a truly 3D viscosity structure. We generate a realistic viscosity structure by converting seismic shear wave tomography models to a 3D temperature field, and then to viscosity. The isostatic response of this earth is computed when subjected to the glacial history Ice-3G (Tushingham & Peltier, 1991). We then measure typical PGR observables: relative sea level change and ˙ {J_2}. These measurements are treated as synthetic data, and we search for 1D (radially stratified) viscosity models, forced with the same glaciation history, that best fit these synthetic PGR observations. We find that the viscosity structure beneath the ice load may serve as a reasonable 1D proxy, even when observing sea levels away from the loaded region. We also perform a Monte Carlo type inversion for a 1D viscosity by running thousands of forward models and minimizing the misfit to observations from the 3D earth. The 1D structure found to best fit the PGR observations does not, in general, look very much like the true viscosity. This may be due to uniquely three-dimensional effects. We also find that attempting to invert for too many parameters (for example, viscosity in more than four layers) may yield a worse recovery of the true viscosity structure than an inversion for fewer parameters.