A spectral formalism for computing three‐dimensional deformations due to surface loads: 1. Theory

We outline a complete spectral formalism for computing high spatial resolution three‐dimensional deformations arising from the surface mass loading of a spherically symmetric planet. The main advantages of the formalism are that all surface mass loads are always described using a consistent mathematical representation (expansions using spherical harmonic basis functions) and that calculations of deformation fields for various spatial resolutions can be performed by simply altering the spherical harmonic degree truncation level of the procedure. The latter may be important when incorporating improved observational constraints on a particular surface mass load, when considering potential errors in the computed field associated with mass loading having a spatial scale unresolved by the observational constraints, or when treating a number of global surface mass loads constrained with different spatial resolutions. These advantages do not extend to traditional “Green's function” approaches which involve surface element discretizations of the global mass loads. In the practical application of the Green's function approach the discretization is subjective and dependent on the particular surface mass load being considered. Furthermore, treatment of mass loads with higher spatial resolutions can require tedious rediscretization of the surface elements. Another advantage of the spectral formalism, over the Green's function approach, is that a posteriori analyses of the computed deformation fields, such as degree correlations, power spectra, and filtering analyses, are easily performed. In developing the spectral formalism, we consider specific cases where the Earth's mantle is assumed to respond as an elastic, slightly anelastic, or linear viscoelastic medium. In the case of an elastic or slightly anelastic mantle rheology the spectral response equations incorporate frequency dependent Love numbers. The formalism can therefore be used, for example, to compute the potentially resonant deformational response associated with the free core nutation and Chandler wobble eigenfunctions. For completeness, the spectral response equations include both body forces, as arise from the gravitational attraction of the Sun and the Moon, and surface mass loads. In either case, and for both elastic and anelastic mantle rheologies, we outline a pseudo‐spectral technique for computing the ocean adjustment associated with the total gravitational perturbation induced by the external forcing. Three‐dimensional deformations computed using the usual Love number approach are generally referenced to an origin located at the center of mass of the undeformed planet. We derive a spectral technique for transforming the results to an origin located at the center of mass of the deformed planet.