Nonperturbational surface-wave inversion: A Dix-type relation for surface waves

Abstract

We extend the approach underlying the well-known Dix equation in reflection seismology to surface waves. Within the context of surface wave inversion, the Dix-type relation we derive for surface waves allows accurate depth profiles of shear-wave velocity to be constructed directly from phase velocity data, in contrast to perturbational methods. The depth profiles can subsequently be used as an initial model for nonlinear inversion. We provide examples of the Dix-type relation for under-parameterized and over-parameterized cases. In the under-parameterized case, we use the theory to estimate crustal thickness, crustal shear-wave velocity, and mantle shear-wave velocity across the Western U.S. from phase velocity maps measured at 8-, 20-, and 40-s periods. By adopting a thin-layer formalism and an over-parameterized model, we show how a regularized inversion based on the Dix-type relation yields smooth depth profiles of shearwave velocity. In the process, we quantitatively demonstrate the depth sensitivity of surface-wave phase velocity as a function of frequency and the accuracy of the Dix-type relation. We apply the over-parameterized approach to a nearsurface data set within the frequency band from 5 to 40 Hz and find overall agreement between the inverted model and the result of full nonlinear inversion.

Perturbational and nonperturbational inversion of Rayleigh-wave velocities

“…Xia et al (1999) conduct numerical tests and find the Rayleigh-wave maximum sensitivity depth to be well-described by 0.63l, where l is the wavelength. Haney and Tsai (2015) further show that, although sensitivity is spread out among all depths, a good rule of thumb in vertically inhomogeneous velocity profiles is that the fundamental mode Rayleigh wave is sensitive to a depth of approximately 0.5l. This concept can be generalized to higher modes by taking the deepest sensitivity as 0.5ml, where m is the mode number and m ¼ 1 is the fundamental mode, m ¼ 2 is the first overtone, and so on.…”

“…Recently, Haney and Tsai (2015) show that, to a good approximation, a linear matrix-vector relationship exists between the squared phase velocities and the squared shear velocities for a set of layers:…”

“…In the simplest case, the kernel is formulated based on the assumption that, at each frequency, a Rayleigh wave is propagating in a different homogeneous medium. A more accurate formulation, discussed in Haney and Tsai (2015), approximates the Rayleigh-wave eigenfunctions over a wide range of shear velocity profiles described by power laws. As pointed out in Haney and Tsai (2015), the relation in equation 27 is analogous to the Dix relation between squared layer velocities and squared stacking velocities in reflection seismology (Dix, 1955).…”

“…We use a nonperturbational type of Rayleigh-wave inversion recently introduced by Haney and Tsai (2015) that bears similarities to the Dix equation in reflection seismology. In one of the software examples included with this paper, we show how to use the Dix-type relation from Haney and Tsai (2015) to construct an initial model for noisy synthetic data from the near-surface MODX model (Xia et al, 1999). The initial model is then further refined using iterative perturbational inversion until a final model is found that fits the synthetic data to within the noise level.…”

Time-average velocity estimation through surface-wave analysis: Part 1 — S-wave velocity

“…Weighting factors are determined for each propagating wavelength as a function of layer thicknesses. Haney and Tsai (2015) proposed a Dix-type relationship to obtain a depth profile directly from the DC. Their approach is based on the simplified assumption that each frequency component propagates in a homogeneous half-space and phase velocities are computed using a first-order approximation eigenfunction in the limit of a weakly heterogeneous velocity profile.…”