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

P>In 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.