Skeletonized wave equation of surface wave dispersion inversion

Abstract

SUMMARYWe present the theory for wave equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. Similar to wave-equation traveltime inversion, the complicated surface-wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the (k x , ω) domain. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2D or 3D velocity models. This procedure, denoted as wave equation dispersion inversion (WD), does not require the assumption of a layered model and is less prone to the cycle skipping problems of full waveform inversion (FWI). The synthetic and field data examples demonstrate that WD can accurately reconstruct the S-wave velocity distribution in laterally heterogeneous media.

Tutorial for wave-equation inversion of skeletonized data

“…It is obvious that there is a low-velocity zone on the downthrown side of the fault. This is consistent with the P-wave velocity tomogram in Figure 7b and the COG profile in Figure 7a for 150 m< x < 300 m (Li and Schuster, 2016).…”

“…Setting ߢ̃= Δߢ , says that the wavenumber shift aligns the predicted and observed spectra with one another for a specified ω; thus, the cross-correlation achieves the maximum value and its wavenumber derivative becomes zero. This method is denoted as skeletonized waveequation dispersion inversion (WD) (Li and Schuster, 2016;Li et al, 2017a), and is less prone to the cycle skipping problems of full waveform inversion (FWI).…”

Wave-equation dispersion inversion of surface waves recorded on irregular topography

“…In practice, these spectra are computed by applying a linear Radon transform (LRT) to the common shot gather in the frequency domain (Li and Schuster, 2016). Any order of dispersion curve can, in principle, be picked and inverted, but for the examples in this paper we only use the fundamental mode of Rayleigh waves.…”

Wave‐Equation Dispersion Inversion of Guided P Waves in a Waveguide of Arbitrary Geometry

“…The observed data D ( g , ω ) obs of all sources are backprojected, where the forward‐propagated source has the weighted source spectrum W ( ω )Δ κ ( ω ). The gradients of all shot gathers are summed together to get the P ‐velocity model update (J. Li & Schuster, ). In practice, the velocity update at each iteration is computed by reverse time migration of the weighted traces in the space‐time domain.…”

“…We now present the WDG method which eliminates the high‐frequency approximation of ray‐based tomography, removes the layered medium approximation of standard inversion of dispersion curves (Boiero et al, ; Campbell et al, ), and mitigates FWI's tendency of getting stuck in a local minimum. WDG is similar to the surface wave WD method of J. Li and Schuster () and J. Li, Feng, and Schuster (), which is typically used to obtain the shear‐wave velocity associated with Rayleigh‐mode dispersion curves and a layered medium with irregular interfaces. Now we invert for the P ‐wave velocity model from guided P waves that propagate exclusively in the shallow layers.…”