Instantaneous phase inversion based on an unwrapping algorithm

The full-waveform inversion method is a high-precision inversion method based on the minimization of the misfit between the synthetic seismograms and the observed data. However, this method suffers from cycle skipping in the time domain or phase wrapping in the frequency because of the inaccurate initial velocity or the lack of low-frequency information. furthermore, the object scale of inversion is affected by the observation system and wavelet bandwidth, the inversion for large-scale structures is a strongly nonlinear problem that is considerably difficult to solve. In this study, we modify the unwrapping algorithm to obtain accurate unwrapped instantaneous phase, then using this phase conducts the inversion for reducing the strong nonlinearity. The normal instantaneous phases are measured as modulo 2 pi, leading the loss of true phase information. The path integral algorithm can be used to unwrap the instantaneous phase of the seismograms having time series and one-dimensional (1D) signal characteristics. However, the unwrapped phase is easily affected by the numerical simulation and phase calculations, resulting in the low resolution of inversion parameters. To increase the noise resistance and ensure the inversion accuracy, we present an improved unwrapping method by adding an envelope into the path integral unwrapping algorithm for restricting the phase mutation points, getting accurate instantaneous phase. The objective function constructed by unwrapping instantaneous phase is less affected by the local minimum, thereby making it suitable for full-waveform inversion. Further, the corresponding instantaneous phase inversion formulas are provided. Using the improved algorithm, we can invert the low-wavenumber components of the underneath structure and ensure the accuracy of the inverted velocity. Finally, the numerical tests of the 2D Marmousi model and 3D SEG/EAGE salt model prove the accuracy of the proposed algorithm and the ability to restore large-scale low-wavenumber structures, respectively.