The Inference of Mantle Viscosity From an Inversion of the Fennoscandian Relaxation Spectrum

Summary A formal inverse theory for mantle viscosity is here applied to a relaxation spectrum derived from the post‐glacial uplift of Fennoscandia. the spectrum represents the set of eigenfrequencies (or inverse decay times) for the fundamental mode of viscous gravitational relaxation between the spherical harmonic degrees 14 to 45 and 65 to 80. Theoretical predictions of the eigenfrequencies are based upon the determination of the zeroes of the secular determinant function derived for a spherically symmetric, self‐gravitating, visco‐elastic planet. Differential kernels relating shifts in the eigenfrequencies to arbitrary perturbations in the radial viscosity profile (i.e. Fréchet kernels) are computed using the variational principle derived by Peltier (1976). the inversions are performed within the framework of non‐linear Bayesian inference, and the problem has been parameterized in terms of the logarithm of viscosity. The inversions have yielded a set of robust constraints which all models for the radial viscosity profile below Fennoscandia must satisfy. the a posteriori estimates and variance reduction are found to be insensitive to the a priori variance ascribed to the model layers. the constraints have, furthermore, been summarized into a set of a posteriori estimates of the average model viscosity value in radial regions consistent with the resolving power of the data (which decreases from a radial length scale of approximately 120km at the base of the lithosphere to 1200km at 1000km depth; the data provide essentially no information regarding the mantle rheology below 1200km depth). For example, for Earth models with a lithospheric thickness (LT) of 100 km, the volumetric average logarithm of viscosity in regions in the depth ranges 1040‐400 km, 670‐210 km and 235‐100 km is constrained to be, respectively, 21.03±0.09, 20.70±0.08 and 20.37±0.19. We have repeated the inversions for a number of assumed lithospheric thicknesses and have found that a relatively low‐viscosity layer in the sublithospheric region (with respect to the underlying upper mantle) is required for LT ≤ 120km. In this respect we have quantified the previously described trade‐off between a decrease in the viscosity of this region and a decrease in LT (Cathles 1975). In forward analyses of the glacial isostatic adjustment data set it is common to use Earth models with isoviscous upper and lower mantle regions. to investigate this ‘two‐layer’ case we have also performed inversions which assume perfect correlation amongst the model layers in the upper and, separately, the lower mantle. Under this strict model space limitation, the inversions yield models with upper and lower mantle viscosities in the range 3.7 × 10 20 ‐4.5 × 10 20 Pa s and 2.2 × 10 21 ‐1.9 × 10 21 Pa s, respectively. (The ranges are obtained from a suite of inversions using lithospheric thickness from 70 km to 145 km.) The a posteriori constraints generated from the Bayesian inversions are used together with a statistic based on the computed misfit to the Fennoscandian relaxation spectrum, to rule out a number of previously published viscosity models.