Comparison of scaling methods for waveform inversion

Abstract

A B S T R A C TWaveform inversion can lead to faint images for later times due to geometrical spreading. The proper scaling of the steepest-descent direction can enhance faint images in waveform inversion results. We compare the effects of different scaling techniques in waveform inversion algorithms using the steepest-descent method. For the scaling method we use the diagonal of the pseudo-Hessian matrix, which can be applied in two different ways. One is to scale the steepest-descent direction at each frequency independently. The other is to scale the steepest-descent direction summed over the entire frequency band. The first method equalizes the steepest-descent directions at different frequencies and minimizes the effects of the band-limited source spectrum in waveform inversion. In the second method, since the steepest-descent direction summed over the entire frequency band is divided by the diagonal of the pseudoHessian matrix summed over the entire frequency band, the band-limited property of the source wavelet spectrum still remains in the scaled steepest-descent directions. The two scaling methods were applied to both standard and logarithmic waveform inversion. For standard waveform inversion, the method that scales the steepest-descent direction at every frequency step gives better results than the second method. On the other hand, logarithmic waveform inversion is not sensitive to the scaling method, because taking the logarithm of wavefields automatically means that results for the steepest-descent direction at each frequency are commensurate with each other. If once the steepest-descent directions are equalized by taking the logarithm of wavefields in logarithmic waveform inversion, the additional equalizing effects by the scaling method are not as great as in conventional waveform inversion.

Sparse-promoting full-waveform inversion based on online orthonormal dictionary learning

“…Unfortunately, due to the large number of N ω , N s , N r and N z , N x , it is prohibitive to compute H −1 k directly with all data in industrialscale FWI problems. In order to reduce the computational complexity of FWI, it has been widely reported that for the cases with large acquisition aperture and wide frequency bandwidth, H k is almost diagonally dominant and H −1 k can be approximated with a diagonal matrix [Beylkin, 1985, Shin et al, 2001, Plessix and Mulder, 2004, Jang et al, 2009, Ren et al, 2013, Pan et al, 2015. The computational complexity can also be reduced by approximating H k with quasi-Newton methods such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno (l-BFGS) algorithm [Nocedal, 1980, Nocedal andWright, 2006].…”

Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators

“…Note that data-adaptive scaling methods such as those cited in the previous paragraph differ in an essential way from scaling by an approximate diagonal of the Hessian [20,21]. As noted by Rickett [14], a dip-independent scaling cannot well approximate an inverse Hessian in general, as the appropriate amplitude correction (from migrated image to inversion) depends on dip.…”

Sequentially ordered single-frequency 2-D acoustic waveform inversion in the Laplace-Fourier domain

“…However, the simultaneous inversion process requires large computational resources, such as a high‐performance parallel or a massively parallel system, to handle the large number of shots or frequencies effectively. In addition, to balance the contributions from the different frequency components, proper scaling of the steepest‐descent direction or data‐weighting scheme should be applied (Hu et al 2007; Jang et al 2009). Based on the facts that (1) a finite region in the wavenumber domain can be obtained from a single‐frequency component, depending on the acquisition geometry and (2) the inherent nonlinearity of the waveform inversion problem can be mitigated when the low‐frequency data are inverted initially because the low‐frequency data are less nonlinear with the model than high‐frequency data, a sequential single‐frequency inversion has been attempted that successively inverts from low‐ to high‐frequency data (Wu & Toksöz 1987; Yokota & Matsushima 2004; Sirgue & Pratt 2004; Operto et al 2006; Ben‐Hadj‐Ali et al 2008).…”