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

Abstract

S U M M A R YIn the conventional frequency-domain waveform inversion, either multifrequency simultaneous inversion or sequential single-frequency inversion has been implemented. However, most conventional frequency-domain waveform inversion methods fail to recover background velocity when low-frequency information is missing. Recently, new waveform inversion techniques in the Laplace and Laplace-Fourier domain have been proposed to recover background velocity structure from data with insufficient low-frequency information. In such techniques, however, all frequencies are inverted simultaneously, and this requires large computational resources and long computation times.In this paper, we propose a sequentially ordered single-frequency 2-D acoustic waveform inversion using the logarithmic objective function in the Laplace-Fourier domain. Our algorithm sequentially inverts single-frequency data in the Laplace-Fourier domain, thus reducing computational resources. Unlike most conventional waveform inversion methods requiring an initial velocity model close to the true model, we propose a one-step waveform inversion method in seeking to find a final velocity structure from the simple initial model through a hybrid combination of the Laplace domain inversion and the Fourier domain inversion.We adopt and evaluate the multiloop algorithm by modifying the double-loop algorithm commonly used in the conventional frequency-domain waveform inversion. Using the multiloop algorithm repeating loop over frequencies, the quality of the inversion results can be improved and the decision problem of the number of iterations for each frequency can be overcome effectively. Because the sequential order of the Laplace-Fourier frequencies in a 2-D plane should be assigned for inverting Laplace-Fourier frequency data consecutively, we present three different sequential orders of Laplace-Fourier frequencies while considering the multiscale and layer-stripping approach, and we compare the inversion results from the numerical experiments.We applied the sequentially ordered single-frequency 2-D acoustic waveform inversion in the full Laplace-Fourier domain to the synthetic seismic data produced from complex structure model and field data. A realistic model could be recovered in an efficient and robust manner, even using the two-layer homogeneous velocity model as an initial model. The inverted velocity model from the field data was validated by examining the migrated image from the pre-stack depth migration and the flattening of the common-image gathers or by comparing the synthetic shot gather with the real shot gather. The proposed one-step waveform inversion algorithm can be easily extended to the sequential inversion of 3-D acoustic or elastic data in the full Laplace-Fourier domain.

Laplace-domain full-waveform inversion of seismic data lacking low-frequency information

“…Because the data were acquired on 3D elastic media, the elastic waves and the 3D geometric spreading of the amplitude can serve as the noise. However, it has been shown that the Laplace-domain inversion can yield an accurate large-scale velocity model from field data sets (Shin and Cha, 2008;Shin et al, 2010) and synthetic marine data with elastic waves (Ha et al, 2010b).…”

“…Those inversion results can be used as an initial model for conventional full-waveform inversions (Shin and Ha, 2008). It can produce high-resolution velocity models when combined with the frequency-domain inversion method (Shin and Cha, 2009;Shin et al, 2010). The method is very sensitive to the noise before the first arrival due to the damping function in the Laplace transformation (Shin and Cha, 2008).…”

Source-independent elastic waveform inversion using a logarithmic wavefield

“…Numerous studies have been devoted to develop a robust waveform inversion algorithm. Among them, Shin and Min (2006) proposed the logarithmic waveform inversion in the frequency domain, and it has been widely applied in both the frequency and Laplace domains Cha, 2008, 2009;Shin et al, 2007;Shin et al, 2010). Taking the logarithm of a wavefield separates the amplitude (real part) and phase (imaginary part) of the Fourier transformed wavefield, where the amplitude and phase are related to the energy and kinematic properties of the wavefield, respectively.…”

Dispersion analysis of an average-derivative optimal scheme for Laplace-domain scalar wave equation

“…The success of Laplace-domain FWI lies in that (1) the objective function in Laplace domain is much smoother than its frequency-domain counterpart, thus considerably mitigating the problem of local minima (Shin and Ha, 2008); (2) Lapace-domain FWI is less sensitive to the lack of low-frequency information in real seismic data in comparison with Fourier-domain FWI . Combined with conventional FWI, Lapace-domain FWI has made important contributions to the successful application of FWI to real data (Shin and Cha, 2009;Shin et al, 2010;.…”