FWI: addressing the nonlinearity once and for all
Traveltime inversion focusses on the geometrical features of the wavefrom (traveltimes), which is generally smooth, and thus, tends to provide averaged (smoothed) information of the model. On other hand, general waveform inversion uses additional elements of the wavefield including amplitudes to extract higher resolution information, but this feature comes at the cost of introducing complex nonlinearity to the inversion operator complicating the convergence process. We use unwrapped-phase-based objective functions in waveform inversion as a link between the two general types of inversions in a domain in which such contributions to the inversion process can be easily identified and controlled. The instantaneous traveltime is a measure of the average traveltime of energy in a trace as a function of frequency. It unwraps the phase of wavefields yielding far less nonlinearity in the objective function than those experienced with the conventional wavefield, and yet it still holds most of the critical wavefield information in its frequency dependency. Such information is packaged in a frequency dependent traveltime that can be easily manipulated at different frequencies (including the infinite frequency traveltime). However, it suffers from nonlinearity introduced by the model (or reflectivity), as events interact with each other. This is caused by the sinusoidal nature of the band-limited reflectivity of the Earth model. Unwrapping the phase for such a model can mitigate this nonlinearity as well. Specifically, a simple modification to the inverted domain (or model), can reduce the effect of the model-induced nonlinearity and, thus, make the inversion more convergent. Unlike, wavefields in the frequency domain, phase functions can be smoothed in a way that reduces the effect of the nonlinearity in the data.
The plot below shows the objective function using the conventional misfit (a), using the unwrapped phase data (b), using the unwrapped phase data and model (c) for a simple 1-D problem that contains two events.