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

Abstract

A model of the earth can be described using a Fourier basis represented by its wavenumber content. In full-waveform inversion (FWI), the wavenumber description of the model is natural because our Born-approximation-based velocity updates are made up of wavefields. Our objective in FWI is to access all the model wavenumbers available in our limited aperture and bandwidth recorded data that are not yet accurately present in the initial velocity model. To invert for those model wavenumbers, we need to locate their imprint in the data. Thus, I review the relation between the model wavenumber buildup and the inversion process. Specifically, I emphasize a focus on the model wavenumber components and identified their individual influence on the data.Missing the energy for a single vertical low-model wavenumber from the residual between the true Marmousi model and some initial linearly increasing velocity model produced a worse least-squares fit to the data than the initial model itself, in which all the residual model wavenumbers were missing. This stern realization validated the importance of wavenumber continuation, specifically starting from the low-model wavenumbers, to higher (resolution) wavenumbers, especially those attained in an order dictated by the scattering angle filter. A numerical Marmousi example determined the important role that the scattering angle filter played in managing the wavenumber continuation from low to high. An application on the SEG2014 blind test data set with frequencies lower than 7 Hz muted out further validated the versatility of the scattering angle filtering.

Full waveform inversion using oriented time‐domain imaging method for vertical transverse isotropic media

“…). Since far‐offset data only recover the low‐wavenumber part of the model, the inverted η has lower resolution (Alkhalifah ). Another difficulty is the variable influence of different parameters on the waveform.…”

Target‐level waveform inversion: a prospective application of the convolution‐type representation for the acoustic wavefield

“…It is well known that waveform inversion is a non‐linear inverse problem (Virieux and Operto ) and that its degree of non‐linearity varies with different components of the model and data set considered (Sirgue ; Alkhalifah ). In order to have success in the inversion process, this non‐linearity should be mitigated when possible.…”

Centered Differential Waveform Inversion with Minimum Support Regularization

“…The W 1 1.2 norm is used as the stabilizing functional when inverting for the average model and baseline models. This naturally blends features of Tikhonov and TV regularizations to allow for smooth updates prior to the sharp ones, and thus, establish better spatial model wavenumber continuity (Alkhalifah, 2016;Kazei et al, 2016). Figure 3 shows the baseline model estimate after multiscale inversion with 20 iterations per frequency and offset at 3Hz and 20 iterations of L-BFGS per frequency, otherwise.…”