A background velocity model containing the correct low-wavenumber information is desired for both the quality of the migration image and the success of waveform inversion. To achieve this goal, the velocity is updated along the reflection wavepaths, rather than along both the reflection ellipses and transmission wavepaths as in conventional FWI. This method allows for reconstructing the low-wavenumber part of the background velocity model, even in the absence of long offsets and low-frequency component of the data. Moreover, in gradient-based iterative updates, instead of forming the data error conventionally, we propose to exploit the phase mismatch between the observed and the calculated data. The phase mismatch emphasizes a kinematic error and varies quasi-linearly with respect to the velocity error. The phase mismatch is computed (1) in the frequency-wavenumber (f-k) domain replacing the magnitudes of the calculated common shot gather by those of the observed one, and (2) in the temporal-spatial domain to form the difference between the transformed calculated common-shot gather and the observed one. The background velocity model inverted according to the proposed methods can serve as an improved initial velocity model for conventional waveform inversion. Tests with synthetic and field data show both the benefits and limitations of this method.
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