Glacial-Isostatic Adjustment-II. The Inverse Problem

Abstract

The principle of correspondence is invoked to determine Laplace transform domain solutions to the surface mass loading problem for radially stratified visco-elastic (Maxwell) spheroids. These Laplace transform domain solutions are expressed in terms of visco-elastic analogues of the conventional surface load Love numbers of elasticity. These visco-elastic Love numbers may be approximately transformed to the time domain using an extremal technique. Application of this technique shows that the Love number time histories may be well approximated by the linear superposition of a discrete set of purely exponential relaxations plus a constant term. Alternatively the discrete spectrum of relaxation times involved in the synthesis of each Love number time history may be found exactly by solving the associated homogeneous problem. Such solutions determine the set of eigen-decay times associated with the normal modes of viscous gravitational relaxation of the visco-elastic planetary model. The solution of the inhomogeneous problem may be expressed in the form of a normal mode expansion. This normal mode expansion is employed as the basis for the construction of a rigorous first-order perturbation theory for the inference of the viscosity of the deep interior of the planet. A variational principle is derived which determines to first order that shift in position of a free decay pole in the relaxation spectrum which is forced by the addition of a radially-distributed perturbation of viscosity to the starting model. This determines the differential kernels required for the solution of the inverse problem. The uniqueness of the state of isostatic equilibrium for the viscously incompressible Maxwell models employed in this analysis is demonstrated and this uniqueness allows us to calculate the shift in the residue which is associated with the shift in position of a free decay pole for the inhomogeneous problem. The inhomogeneous problem is thus closed to first order. A formula is derived for the differential kernels appropriate to the inhomogeneous problem. The time domain form of these kernels may be calculated analytically. The structure of the full inverse theory is sufficiently simple that it may be employed to rigorously test the compatibility of the simple Maxwell model with the observed relaxation of the Earth's shape which accompanied deglaciation at the end of the last ice age.