Two-dimensional asymptotic iterative elastic inversion

Abstract

S U M M A R Y An asymptotic linearized iterative elastic inversion method is proposed to invert 2-D Earth parameters from multicomponent data and is tested numerically. The forward problem is solved by a combination of the Born approximation and ray theoretical methods. We express the perturbed seismogram in terms of perturbations of Pand S-wave impedances and density. The inversion method is based on generalized least squares. We introduce a special form of the l2 norm with a weighting function that corrects for geometrical spreading and obliquity of the reflectors. The Hessian for this norm could be estimated in a closed form that is asymptotically valid at high frequencies. We propose a quasi-Newtonian iterative method for the solution of the inverse problem. The first iteration of this inversion method resembles the operator proposed by Beylkin (1985) and Beylkin & Burridge (1990) for the asymptotic inversion of seismic data. O u r method is more general than theirs because it can handle arbitrary discrete distributions of sources and receivers. Elastic inversion is generally ill-posed because the problem is overdetermined but undersampled. We study the resolution of the asymptotic inversion method for general sets of sources and receivers. We show that simultaneous inversion for both Pand S-wave impedance is generally ill-conditioned if data for a single scattering mode are available. In particular, it seems that only one parameter can be reliably resolved from marine data. Simultaneous inversion for a finite set of parameters can be resolved only for multicomponent elastic data containing both P-wave and S-wave information. Inversion tests using synthetic data calculated by finite-differences demonstrates that it is possible to invert simultaneously for P and S impedances.

Full-waveform inversion in acoustic orthorhombic media and application to a North Sea data set

“…The effect of the background medium on the perturbation is expected to be small (He and Plessix, 2017). The resolution of the scattering point is governed by the diffraction tomography principles formula, which relates the model wavenumber k m to the frequency ω and the scattering angle θ (Devaney, 1984;Miller et al, 1987;Jin et al, 1992), and is given by…”

A recipe for practical full-waveform inversion in orthorhombic anisotropy

“…The azimuth is measured from the x 1 (or x) axis of the orthorhombic anisotropy. The resolvability of the desired scattering object is furnished by di↵raction tomography principles in which the model wavenumber vector magnitude is proportional to the frequency, !, but also proportional to the cosine of half of the scattering angle (Miller et al, 1987;Jin et al, 1992;Thierry et al, 1999), given: |k m| = 2 ! As mentioned, the scattering potential of the central velocity parameter is isotropic over scattering angles and azimuths (Figure 1d).…”

Full-model wavenumber inversion: An emphasis on the appropriate wavenumber continuation

“…Based on the adjoint Born scattering approximation, which involves the interaction of the two locally planer wavefields using a crosscorrelation process (often used to obtain a velocity gradient), the resulting wavenumber vector is given by (Miller et al, 1987;Jin et al, 1992;Thierry et al, 1999) …”