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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.

A B S T R A C TLogistic regression, neural networks and support vector machines are tested for their effectiveness in isolating surface waves in seismic shot records. To distinguish surface waves from other arrivals, we train the algorithms on three distinguishing features of surface-wave dispersion curves in the k − ω domain: spectrum coherency of the trace's magnitude spectrum, local dip and the frequency range for a fixed wavenumber k in the spectrum. Numerical tests on synthetic data show that the kernel-based support vector machines algorithm gives the highest accuracy in predicting the surface-wave window in the k − ω domain compared to neural networks and logistic regression. This window is also used to automatically pick the fundamental dispersion curve. The other two methods correctly pick the low-frequency part of the dispersion curve but fail at higher frequencies where there is interference with higher-order modes.

We present a dispersion‐inversion method which inverts for the P‐velocity model from guided waves propagating in wave guides of arbitrary geometry. Its misfit function is the squared summation of differences between the predicted and observed dispersion curves of guided P waves, and the inverted result is a high‐resolution estimate of the near‐surface P‐velocity model. We denote this procedure as wave‐equation dispersion inversion of guided P waves (WDG), which is valid for near‐surface waveguides with irregular layers and does not require a high‐frequency approximation. It is more robust than full waveform inversion and can sometimes provide velocity models with higher resolution than wave‐equation traveltime tomography. Both the synthetic‐data and field data results demonstrate that WDG for guided P waves can accurately invert for complex P‐velocity models at the near surface of the Earth.

We present a skeletonized inversion method that inverts surface-wave data for the Q s quality factor. Similar to the inversion of dispersion curves for the S-wave velocity model, the complicated surface-wave arrivals are skeletonized as simpler data, namely the amplitude spectra of the windowed Rayleigh-wave arrivals. The optimal Q s model is the one that minimizes the difference in the peak frequencies of the predicted and observed Rayleigh wave arrivals using a gradient-based wave-equation optimization method. Solutions to the viscoelastic waveequation are used to compute the predicted Rayleigh-wave arrivals and the misfit gradient at every iteration. This procedure, denoted as wave-equation Q s inversion (WQ s), does not require the assumption of a layered model and tends to have fast and robust convergence compared to full waveform inversion (FWI). Numerical examples with synthetic and field data demonstrate that the WQ s method can accurately invert for a smoothed approximation to the subsurface Q s distribution as long as the V s model is known with sufficient accuracy.

We have developed a skeletonized inversion method that inverts the S-wave velocity distribution from surface-wave dispersion curves. Instead of attempting to fit every wiggle in the surface waves with predicted data, it only inverts the picked dispersion curve, thereby mitigating the problem of getting stuck in a local minimum. We have applied this method to a synthetic model and seismic field data from Qademah fault, located at the western side of Saudi Arabia. For comparison, we have performed dispersion analysis for an active and controlled noise source seismic data that had some receivers in common with the passive array. The active and passive data show good agreement in the dispersive characteristics. Our results demonstrated that skeletonized inversion can obtain reliable 1D and 2D S-wave velocity models for our geologic setting. A limitation is that we need to build layered initial model to calculate the Jacobian matrix, which is time consuming.

Full-waveform inversion of seismic data is often plagued by cycle-skipping problems such that an iterative optimization method often gets stuck in a local minimum. To avoid this problem, we simplify the objective function so that the iterative solution can quickly converge to a solution in the vicinity of the global minimum. The objective function is simplified by only using parsimonious and important portions of the data, which are defined as skeletonized data. We have developed a mostly nonmathematical tutorial that explains the theory of wave-equation inversion of skeletonized data. We also demonstrate its effectiveness with examples.

Two robust imaging technologies are reviewed that provide subsurface geologic information in challenging environments. The first one is wave-equation dispersion (WD) inversion of surface waves and guided waves (GW) for the shear-velocity (S-wave) and compressional-velocity (P-wave) models, respectively. The other method is traveltime inversion for the velocity model, in which supervirtual refraction interferometry (SVI) is used to enhance the signal-to-noise ratio of far-offset refractions. We have determined the benefits and liabilities of both methods with synthetic seismograms and field data. The benefits of WD are that (1) there is no layered-medium assumption, as there is in conventional inversion of dispersion curves. This means that 2D or 3D velocity models can be accurately estimated from data recorded by seismic surveys over rugged topography, and (2) WD mostly avoids getting stuck in local minima. The liability is that WD for surface waves is almost as expensive as full-waveform inversion (FWI) and, for Rayleigh waves, only recovers the S-velocity distribution to a depth no deeper than approximately 1/2 to 1/3 wavelength of the lowest-frequency surface wave. The limitation for GW is that, for now, it can estimate the P-velocity model by inverting the dispersion curves from GW propagating in near-surface low-velocity zones. Also, WD often requires user intervention to pick reliable dispersion curves. For SVI, the offset of usable refractions can be more than doubled, so that traveltime tomography can be used to estimate a much deeper model of the P-velocity distribution. This can provide a more effective starting velocity model for FWI. The liability is that SVI assumes head-wave first arrivals, not those from strong diving waves.