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S U M M A R Y Constraining laterally varying structures in planetary interiors is important for understanding both the composition and the internal dynamics of a planet. Recognizing that seismic imaging technique is currently only viable for studying the Earth’s interior structures, methods that can be supported by advanced space geodetic techniques may become alternatives to ‘image’ the interiors of other planets. The method of tidal tomography is one possibility, and it relies on high precision measurement of the response of a planet to its body tide. However, it is essential to develop an efficient analytical tool that computes the dependence of tidal response to 3-D interior structures. In this paper, we present a complete formulation of such an analytical tool, which calculates to high accuracy the tidal response of a terrestrial planet with lateral heterogeneities in its elastic and density structures. We treat the lateral heterogeneities as small perturbations and derive the governing equations based on the perturbation theory. In a spherical harmonic representation, equations at each order of perturbation are reduced into multiple matrix equations at harmonics that are allowed by mode couplings, and the total response equals the sum of all those single-harmonic responses, which can be solved semi-analytically. We test our perturbation method by applying it to the Moon with a harmonic degree-1 mantle structure for which the perturbation solutions of the tidal response are compared with those from a fully numerical method. The remarkable agreement between results from these two methods validates the perturbation method. As an example, we then use the perturbation method to evaluate the impact of lunar crustal thickness variations on tidal response of the Moon. We find that lunar crust produces much smaller degree-3 tidal responses than a relatively weak degree-1 structure in the deep lunar mantle. Our calculations show that degree-3 tidal response measurements may hold key constraints on possible degree-1 mantle structure of the Moon, as suggested from previous modelling results.

Elastic tidal response of a laterally heterogeneous planet: a complete perturbation formulation