Resolvability of the 3D density structure of the Earth's mantle using normal mode theory

Abstract

We performed an investigation of the large scale seismic wave speeds and density structure of the Earth’s mantle using free oscillations. Seismic free oscillations, or normal modes, are convenient for analysing low-frequency seismograms in a hetero- geneous Earth. To use these, we must address how to calculate exact seismograms using normal modes, and how to formulate the inverse problem to infer Earth’s 3D structure. The most important findings of this research are: • In order for seismograms to be theoretically exact, full mode coupling calcula- tions must involve an infinite set of modes. In practice, only a finite subset of modes can be used, introducing an error into the seismograms. We found that coupling modes 1-2 mHz above the highest frequency of interest is essential for having sufficiently accurate signals to infer density. • Observations of free oscillations provide important constraints on the heteroge- neous structure of the Earth. This inference problem has usually been addressed by the measurement and interpretation of splitting functions. These can be seen as secondary data extracted from low frequency seismograms. The measurement step necessitates the calculation of synthetic seismograms, but current imple- mentations rely on approximations referred to as self- or group-coupling and do not use fully accurate seismograms. We therefore investigated whether a systematic error might be present in currently published splitting functions. As is well known, the density signal is weak in low-frequency seismograms. Our results suggest this signal is of similar magnitude to the realistic uncertainties associated with currently published splitting functions. Thus, great care must be taken in any attempt to robustly infer details of Earth’s density structure using current splitting functions. • We investigated the problem of inferring density using currently published split- ting functions with properly calibrated uncertainties together with a novel prob- abilistic inversion technique, Hamiltonian Monte Carlo. Models are strongly dependent on damping. We found that shear wave speed models are statisti- cally significant in terms of misfit change, while density and compressional wave speeds are not. Therefore any interpretation of Earth’s mantle density based on splitting functions might be inaccurate. • A promising approach is the direct spectral inversion, which uses spectra di- rectly without the need of splitting functions. We found that misfit changes corresponding to the inferred models are statistically significant even for den- sity and compressional wave speed, but depend on a good starting model. We only used group coupling and relatively low frequency spectra for computa- tional reasons. Full coupling together with high frequencies might solve this long-lasting problem to infer density contrasts in the Earth’s mantle.