Comparison of misfit functions for phase-only inversion in the frequency domain

Abstract:

 Full waveform inversion suffers from non-uniqueness and non-linearity problems. By using kinematic property of wavefield rather than dynamic property, we can mitigate such problems because the phase is linear and robust (Kamei et al. 2013). For the phase-only inversion, several misfit functions were suggested. Bednar et al. (2007) compared the logarithmic phase-only inversion proposed by Shin and Min (2006) with the conventional phase-only inversion. On the other hand, Kamei et al. (2014) introduced another method that uses the exponential of phase by normalizing the wavefield with respect to the amplitude.

 In this study, we compare the aforementioned three phase-only inversion methods in the frequency domain: i) the logarithmic phase-only inversion, ii) the conventional phase-only inversion I (briefly conventional I method) that normalizes wavefield with respect to the amplitude variation, and iii) the conventional phase-only inversion II (briefly conventional II method) that replaces the amplitude of the modeled data with that of field data. In the cases of the logarithmic and conventional I methods, if the modeled signal function is close to 0 or becomes large, the gradients of the misfit function diverge to infinity or converge to 0, respectively. In contrast, the conventional II method does not suffer from these problems. For fair comparison, we removed extremely small or large values with Gaussian filtering to avoid the instability problem in the logarithmic and conventional I methods. In addition, we assumed that the phase of the field data is unwrapped to the same degree as the phase of the modeled data in all the cases. On the other hand, the logarithmic and conventional II methods require the additional assumption that amplitudes of the field data are the same as those of the modeled data. However, the conventional I method does not require such an assumption. Our numerical examples show that the conventional I method yields more robust and accurate results than the logarithmic and conventional II methods.

 Acknowledgements

 This work was supported by the Human Resources Development program (No. 20134010200510) of the KETEP grant funded by the Korean government MOTIE and by the “Development of Technology for CO₂ Marine Geological Storage” grant funded by the MOF of Korea.