ALBERT

Notes: 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 × 1020-4.5 × 1020 Pa s and 2.2 × 1021-1.9 × 1021 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.

Notes: The majority of investigations of the glacial isostatic adjustment problem have proceeded by invoking the correspondence principle and solving for the (Laplace) transformed impulse response of the viscoelastic Earth model that is represented in terms of Love number spectra (Peltier 1974). This formulation requires a final inversion of the solution into the time domain, and the present paper is concerned with a comparison and assessment of the three techniques (pure collocation, full normal mode analysis using residue theory, and a hybrid technique which we term mixed collocation) that have been developed to perform it. On the basis of the analysis presented here we conclude that both the full normal mode analysis and mixed collocation can generate accurate inversions of the Love number spectra. We also derive clear guidelines on the choice of collocation points that ensure that the same accuracy is achieved using pure collocation. As a final point we stress anew that, regardless of the technique employed, the accuracy of the inversions can and should be checked by comparing the predicted infinite time-scale response for a Heaviside loading history with an independent calculation of the response for an inviscid Earth with a lithosphere of appropriate thickness.

Notes: 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.

Notes: We outline two parametrizations for post-glacial relative sea-level (RSL) histories associated with previously glaciated regions. The first parametrization is based on a site-dependent normalization of the RSL history, while the second involves the estimate of a site-dependent (logarithm of the) inverse decay time for the exponential-like form which characterizes these histories. Both parametrizations are shown to yield data sets which are relatively insensitive to the details of the late Pleistocene surface load history, and therefore inferences of mantle viscosity based upon them will be particularly robust. We apply the two parametrizations to consider the RSL record at a number of sites across the Hudson Bay region. In this regard our inferences (which are derived from both forward and inverse calculations) are based upon the actual RSL age-height pairs obtained by survey, rather than the highly subjective set of RSL ‘trends’adopted in previous studies. One of the main goals of the analysis is to assess the validity of a set of previously published and highly contradictory inferences of the radial profile of mantle viscosity based on the Hudson Bay RSL record. Forward analyses using models with isoviscous upper and lower mantle regions (as adopted in the vast majority of previous analyses) indicate that the parametrized versions of the RSL record in Hudson Bay, excluding data from the Cape Henrietta Maria site, are best fitted by a lower mantle viscosity near 1021 Pa s. The same conclusion holds when data from only northern Hudson Bay are considered. The RSL record in southern Hudson Bay is not self-consistent (if the error bars adopted herein are reasonable); however, the parametrized versions of the RSL curves from each site in this region can be reconciled by a model with a lower mantle viscosity somewhere in the rather moderate range 0.5–3.0 × 1021 Pa s. The value of 3 × 1021 Pa s represents a lower bound on the lower mantle viscosity required to fit the RSL records at Cape Henrietta Maria; this record is characterized by a longer decay time than those associated with other sites in the data base. This lower bound is in contrast with previous suggestions that the RSL record at Cape Henrietta Maria requires a lower mantle viscosity of 30 × 1021 Pa s. Inverse analyses described herein indicate that the RSL record from the entire Hudson Bay constrains the average viscosity in the radial region extending from the lower reaches of the upper mantle to mid-lower mantle depths; the inferences listed above are therefore more properly ascribed to this region (rather than the entire lower mantle). We have found that the data are sensitive to moderately deeper variations in the radial viscosity profile as one considers sites situated further north in Hudson Bay. This provides one means for a spherically symmetric model to yield decay times which vary across the Hudson Bay; however, the data do not provide an unambiguous requirement for such a variation.