Vertical profile of effective turbidity reconstructed from broadening of incoherent body-wave pulses—I. General approach and the inversion procedure

A method is developed for the reconstruction of a non-uniform distribution of scattering properties in the upper layers of the Earth using data on broadening of an incoherent body-wave group or pulse along a number of rays. The theoretical basis for this reconstruction is a linear integral formula after Bocharov (1985, 1988), which is employed to design a linear inversion procedure. The inversion is performed in terms of a single scalar parameter of effective turbidity. This parameter presents an adequate generalization of the common turbidity parameter used in the isotropic scattering case; it describes, simultaneously, scattering attenuation, pulse broadening and backscattering or coda formation. As a preliminary step, necessary conditions of applicability of the transport equation approach for the analysis of regional high-frequency seismic waves are verified. A new compact derivation of Bocharov's formula is then presented. A linear least-squares inversion procedure for recovering a layered turbidity structure is proposed and tested on synthetic data of onset-to-peak delays of incoherent body-wave pulses. A few practical aspects of the application of the general approach to seismological data are analysed, including the correctness of the low-angle approximation, the use of peak delay observations instead of pulse centroid, the effects of a realistic spatial spectrum of inhomogeneity field, the potential bias produced by intrinsic loss, and the distortions produced by a non-spherical (double dipole) source radiation pattern. The latter point is considered as critically important, as one can expect significant data contamination by nodal arrivals. An efficient robust estimation procedure is designed and tested that is capable of suppressing distortions from nodal and near-nodal data.