Research Note: Full‐waveform inversion of the unwrapped phase of a model

Abstract

Reflections in seismic data induce serious non‐linearity in the objective function of full‐ waveform inversion. Thus, without a good initial velocity model that can produce reflections within a half cycle of the frequency used in the inversion, convergence to a solution becomes difficult. As a result, we tend to invert for refracted events and damp reflections in data. Reflection induced non‐linearity stems from cycle skipping between the imprint of the true model in observed data and the predicted model in synthesized data. Inverting for the phase of the model allows us to address this problem by avoiding the source of non‐linearity, the phase wrapping phenomena. Most of the information related to the location (or depths) of interfaces is embedded in the phase component of a model, mainly influenced by the background model, while the velocity‐contrast information (responsible for the reflection energy) is mainly embedded in the amplitude component. In combination with unwrapping the phase of data, which mitigates the non‐linearity introduced by the source function, I develop a framework to invert for the unwrapped phase of a model, represented by the instantaneous depth, using the unwrapped phase of the data. The resulting gradient function provides a mechanism to non‐linearly update the velocity model by applying mainly phase shifts to the model. In using the instantaneous depth as a model parameter, we keep track of the model properties unfazed by the wrapping phenomena.

Full-model wavenumber inversion: An emphasis on the appropriate wavenumber continuation

“…On the other hand, the influence of the high wavenumbers is localized in the data as scattering, yet FWI has a hard time inverting for the low wavenumbers because the update process (the linear form) is based on scattering theory. The unwrapped phase of the depth model wavenumber allows us to move the influence of low wavenumbers to the sensitivity range of FWI (Alkhalifah, 2014a). Often missing in such arguments is a reasonable account of the true impact of various model wavenumbers in the data, and conversely, the source of such model wavenumbers within the data considering the multitude of inversion techniques available, including those that use the image.…”