Spectral Sensitivity Analysis of FWI in a Constant-gradient Background Velocity Model

Abstract

SUMMARYFull waveform inversion suffers from local minima, due to a lack of low frequencies in the data. A reflector below the zone of interest may help in recovering the long-wavelength components of a velocity perturbation, as demonstrated in a paper by Mora. Because smooth models are more popular as initial guesses for FWI, we consider the Born approximation for a perturbation in a reference model with a constant velocity gradient. Analytic expressions are found that describe the spatial wavenumber spectrum of the recorded seismic signal as a function of the spatial spectrum of the inhomogeneity. We study this spectrum in more detail in terms of sensitivities. Since the velocity model is inhomogeneous near the perturbation, we need to specify its depth. We compare these sensitivities and find that low frequencies are extremely useful for the first stages of inversion -a well-known fact. However, also the high-frequency data contain some information about the low spatial wavenumbers in the perturbation, which offers opportunities for inversion in the absence of low frequencies in the data. We observe that the longer wavenumbers are better resolved in the deeper parts of the model if large enough offsets are available.

Scattering Radiation Pattern Atlas: What Anisotropic Elastic Properties Can Body Waves Resolve?

“…Scattering functions for surface waves are sometimes called interaction matrices and can be found in Snieder (), and an analysis of their numerical kernels can be found in Sieminski et al (). If they occur in the background medium, multiples can enhance the illumination of certain wavenumbers in the spectrum of the model perturbation, leading to sinusoidal artifacts in FWI (Kazei, Kashtan, et al, ; Kazei et al, ). If the multiples are of a certain order and separated from other waves, then they can actually improve the resolution of the migration (Schuster & Huang, ).…”

“…This technique was later applied to acoustic FWI scenarios for surface seismic acquisition (Mora, ) and vertical seismic profiling (Wu & Toksöz, ). Kazei, Troyan, et al () introduced the amplitude of the spectral sensitivity of the data to the perturbations of various background media and applied it to models with diving waves and multiples (Kazei et al, ; Kazei, Kashtan, et al, ). An elastic isotropic‐background case with VTI perturbations was considered by Podgornova et al ().…”

Waveform inversion for orthorhombic anisotropy with P waves: feasibility and resolution

“…In monoparameter inversion, the lack of low frequencies can to some extent be compensated by long offsets (Sirgue & Pratt 2004;Kazei et al 2013b), and vise versa. In multiparameter inversion, the increase of the offset will lead to the blending of parameters.…”

“…The technique was later applied in acoustic FWI scenarios for surface seismic acquisition (Mora 1989) and vertical seismic profiling (Wu & Toksöz 1987). Kazei et al (2013a) introduced amplitude of the Resolution of P −wave orthorhombic inversion 9 spectral sensitivity of data to the perturbations of various background media, and applied it to models with diving waves and multiples (Kazei et al 2015(Kazei et al , 2013b. The elastic isotropic background case with VTI perturbations was considered by Podgornova et al (2015).…”

“…The offset is assumed to be effectively infinite in this case, e.g. diving waves illuminate the whole region of interest (Kazei et al 2013b). Diving waves, propagating horizontally at the depth of perturbation, activate the "transmission" part of the radiation patterns with wide opening angles Alkhalifah & Plessix (2014), equivalent to low vertical wavenumbers in spectral sensitivities.…”