# Surface-wave inversion for a P-velocity profile with a constant depth gradient of the squared slowness

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## Abstract

Surface waves are often used to estimate a near-surface shear-velocity profile. The inverse problem is solved for the locally one-dimensional problem of a set of homogeneous horizontal elastic layers. The result is a set of shear velocities, one for each layer. To obtain a P-wave velocity profile, the P-guided waves should be included in the inversion scheme. As an alternative to a multi-layered model, we consider a simple smooth acoustic constant-density velocity model, which has a negative constant vertical depth gradient of the squared P-wave slowness and is bounded by a free surface at the top and a homogeneous half-space at the bottom. The exact solution involves Airy functions and provides an analytical expression for the dispersion equation. If the vs/vp ratio is sufficiently small, the dispersion curves can be picked from the seismic data and inverted for the continuous P-wave velocity profile. The potential advantages of our model are its low computational cost and the fact that the result can serve as a smooth starting model for full-waveform inversion. For the latter, a smooth initial model is often preferred over a rough one. We test the inversion approach on synthetic elastic data computed for a single-layer P-wave model and on field data, both with a small vs/vp ratio. We find that a single-layer model can recover either the shallow or deeper part of the profile but not both, when compared with the result of a multi-layer inversion that we use as a reference. An extension of our analytic model to two layers above a homogeneous half-space, each with a constant vertical gradient of the squared P-wave slowness and connected in a continuous manner, improves the fit of the picked dispersion curves. The resulting profile resembles a smooth approximation of the multi-layered one but contains, of course, less detail. As it turns out, our method does not degrade as gracefully as, for instance, diving-wave tomography, and we can only hope to fit a subset of the dispersion curves. Therefore, the applicability of the method is limited to cases where the vs/vp ratio is small and the profile is sufficiently simple. A further extension of the two-layer model to more layers, each with a constant depth gradient of the squared slowness, might improve the fit of the modal structure but at an increased cost.