Heterogeneity spectrum inversion from a stochastic analysis of global surface-wave delay-time data

The wavelength of mantle heterogeneity reflects the nature of the planet's dynamics, and constraining it on the basis of seismic data helps us to evaluate the likelihood of different proposed models of mantle convection. We neglect the geographic distribution of mantle heterogeneity, inverting global delay-time data to determine directly the heterogeneity spectrum of the Earth. Inverting for the spectrum is in principle (fewer unknowns) much cheaper and robust than inverting for the three-dimensional (3D) structure of a planet: as a result, this approach should ultimately help us to constrain the properties of planetary structure at wavelengths shorter than those of current 3D models. The linearized algorithm that we employ is based on the work of Gudmundsson and co-workers in the early 1990s: seismic rays starting at close sources and arriving at close receivers are collected, and the variance of the associated delay times is calculated; this exercise is repeated for a range of values of maximum distance between close sources and between close receivers. The dependence of calculated variance on the inter-source and inter-receiver distance can then be linked to the heterogeneity spectrum of the planet via a linearized least-squares inversion. For the time being, we limit ourselves to a two-dimensional problem, analyzing surface-wave dispersion in the membrane-wave approximation. Besides inverting global seismic data, we have conducted a number of synthetic tests to evaluate the resolution power of the method and its robustness, and the dependence of inversion results on the amount of inverted data and on the level of complexity that we allow for. Synthetic data were generated in different theoretical frameworks (numerical membrane waves, ray theory) and inverted with different algorithms (Cholesky Factorization, Non-Negative Least Squares). The resolution obtained at this point is generally limited in comparison with "classical" global tomography, suggesting that the simplifications required to linearize the problem compromise the method's accuracy. We infer that fully nonlinear inversion might be necessary to directly constrain the Earth's spectrum up to relatively high harmonic degrees.