Limitations on the inversion for mantle viscosity from postglacial rebound

Observations of postglacial rebound (PGR) can provide important constraints on mantle viscosity structure. In this study, we investigate how well PGR observations are able to constrain the spherically symmetric (1-D) viscosity structure of the Earth. We generate synthetic PGR data by calculating the response of an earth model with realistic 3-D viscosity. The viscosity model is constructed starting from seismic tomography models. We generate synthetic PGR data from this model including relative sea levels, exponential relaxation times,. J(2), polar wander, and GRACE time-variable gravity measurements, where most of the data is concentrated in the Laurentide region. We then attempt an inversion for a 1-D (spherically symmetric) viscosity structure based on minimizing the misfit to these PGR data. Using a Monte Carlo algorithm to invert for two layers of viscosity [upper and lower mantle (UM and LM)], we obtain well-constrained values which correspond to the two-layer average of the logarithm of the 3-D viscosity structure in the vicinity of Laurentia. We then attempt to invert for four layers of viscosity. In this case we find a 'trade-off effect' in which neighbouring layers may have highly variable viscosities while maintaining a constant average between them. Since the PGR data are insensitive to this trade-off in neighbouring layers, the viscosity of any one of the four layers cannot be well constrained. By repeating the inversion with synthetic data derived from an Earth model which itself is 1-D with four viscosity layers, we demonstrate that it is the insensitivity of the PGR data, not complications due to 3-D structure, that allow the trade-off effect and the resulting failure of the inversion. We also perform a resolution test to study the extent of the insensitivity of PGR to these viscosity trade-offs, finding that the limit of resolution in the UM is about the size of the entire UM, and similarly for the LM. This reinforces our findings that only two layers are obtainable in an inversion for viscosity from PGR.