Multiscale seismic tomography

Seismic traveltime tomography is commonly discretized by a truncated expansion of the pursued model in terms of chosen basis functions. Whether parametrization affects the actual resolving power of a given data set as well as the robustness of the resulting earth model has long been seriously debated. From the perspective of the model resolution, however, there is one important aspect of the parametrization issue of seismic tomography that has yet to be systematically explored, that is, the space–frequency localization of a chosen parametrization. In fact, the two most common parametrizations tend to enforce resolution in each of their own particular domains. Namely, parametrization in terms of spherical harmonics with global support tends to emphasize spectral resolution while sacrificing the spatial resolution, whereas the compactly supported pixels tend to behave in the opposite manner. Some of the significant discrepancies among tomographic models are very likely to be manifestations of this effect, when dealing with data sets with non-uniform sampling. With an example of the tomographic inversion for the lateral shear wave heterogeneity of the D″ layer using S–SKS traveltimes, we demonstrate an alternative parametrization in terms of the multiresolution representation of the pursued model function. Unlike previous attempts of multiscale inversion that invoke pixels with variable sizes, or overlay several layers of tessellation with different grid intervals, our formulation invokes biorthogonal generalized Harr wavelets on a sphere. We show that multiresolution representation can be constructed very easily from an existing block-based discretization. A natural scale hierarchy of the pursued model structure constrained by the resolving power of the given sampling is embedded within the solution obtained. It provides a natural regularization scheme based on the actual ray-path sampling and is thus free from a priori prejudices intrinsic to most regularization schemes. Unlike solutions obtained through spherical harmonics or spherical blocks that tend to collapse structures onto ray paths, our parametrization imposes regionally varying Nyquist limits, that is, robustly resolvable local wavelength bands within the obtained solution.