We perform joint nonlinear inversions of glacial isostatic adjustment (GIA) data, including the following: postglacial decay times in Canada and Scandinavia, the Fennoscandian relaxation spectrum (FRS), late‐Holocene differential sea level (DSL) highstands (based on recent compilations of Australian sea level histories), and the rate of change of the degree 2 zonal harmonic of the geopotential, J2. Resolving power analyses demonstrate the following: (1) the FRS constrains mean upper mantle viscosity to be ∼3 × 1020 Pa s, (2) postglacial decay time data require the average viscosity in the top ∼1500 km of the mantle to be 1021 Pa s, and (3) the J2 datum constrains mean lower mantle viscosity to be ∼5 × 1021 Pa s. To reconcile (2) and (3), viscosity must increase to 1022–1023 Pa s in the deep mantle. Our analysis highlights the importance of accurately correcting the J2 observation for modern glacier melting in order to robustly infer deep mantle viscosity. We also perform a large series of forward calculations to investigate the compatibility of the GIA data sets with a viscosity jump within the lower mantle, as suggested by geodynamic and seismic studies, and conclude that the GIA data may accommodate a sharp jump of 1–2 orders of magnitude in viscosity across a boundary placed in a depth range of 1000–1700 km but does not require such a feature. Finally, we find that no 1‐D viscosity profile appears capable of simultaneously reconciling the DSL highstand data and suggest that this discord is likely due to laterally heterogeneous mantle viscosity, an issue we explore in a companion study.

We present a method for calculating the derivatives of measurements of glacial isostatic adjustment (GIA) with respect to the viscosity structure of the Earth and the ice sheet history. These derivatives, or kernels, quantify the linearised sensitivity of measurements to the underlying model parameters. The adjoint method is used to enable efficient calculation of theoretically exact sensitivity kernels within laterally heterogeneous earth models that can have a range of linear or non-linear viscoelastic rheologies. We first present a new approach to calculate GIA in the time domain, which, in contrast to the more usual formulation in the Laplace domain, is well suited to continuously varying earth models and to the use of the adjoint method. Benchmarking results show excellent agreement between our formulation and previous methods. We illustrate the potential applications of the kernels calculated in this way through a range of numerical calculations relative to a spherically symmetric background model. The complex spatial patterns of the sensitivities are not intuitive, and this is the first time that such effects are quantified in an efficient and accurate manner.

• We present a method for determining the planetary tidal response using laboratorybased viscoelastic models and apply it to Mars. • Maxwellian rheology results in considerably biased (low) viscosities and should be used with caution when studying tidal dissipation. • Mars' rheology and interior structure will be further constrained from InSight measurements of tidal phase lags at distinct periods.

We consider a new approach to both the forward and inverse problems in post-seismic deformation. We present a method for forward modelling post-seismic deformation in a self-gravitating, heterogeneous and compressible earth with a variety of linear and nonlinear rheologies. We further demonstrate how the adjoint method can be applied to the inverse problem both to invert for rheological structure and to calculate the sensitivity of a given surface measurement to changes in rheology or time-dependence of the source. Both the forward and inverse aspects are illustrated with several numerical examples implemented in a spherically symmetric earth model.

S U M M A R YThe motion of a self-gravitating hyperelastic body is described through a time-dependent mapping from a reference body into physical space, and its material properties are determined by a referential density and strain-energy function defined relative to the reference body. Points within the reference body do not have a direct physical meaning, but instead act as particle labels that could be assigned in different ways. We use Hamilton's principle to determine how the referential density and strain-energy functions transform when the particle labels are changed, and describe an associated 'particle relabelling symmetry'. We apply these results to linearized elastic wave propagation and discuss their implications for seismological inverse problems. In particular, we show that the effects of boundary topography on elastic wave propagation can be mapped exactly into volumetric heterogeneity while preserving the form of the equations of motion. Several numerical calculations are presented to illustrate our results.