Comparison between time domain and frequency domain least-squares reverse time migration

Lei Yang, Daniel O. Trad

In this report, we compare the algorithms of least-squares reverse time migration (LSRTM) in both time and frequency domain and propose a full waveform inversion (FWI) based LSRTM method in the frequency domain. First we show the mathematical equivalence between the gradient of the FWI objective function and the reverse time migration (RTM) imaging condition. Then, we use the FWI formulation with the truncated Newton’s method, to solve the linear equation which relates Hessian, model perturbation and the gradient by linear conjugate gradient method. We use simple layer models to compare the two formulations, LSRTM in time and frequency domain. Because of convergence problems that we have not solved yet, we get lower resolution images with the frequency domain FWILSRTM method. On the other hand, when the model is inaccurate, the reflector depth seems less affected in the frequency domain. The FWI-based LSRTM method seems to be more robust to velocity errors even if we don’t correct the background model as usually done in FWI. Low frequencies seem to be less affected by the inaccurate velocities, and by model smoothness than the high frequencies, suggesting using methods from low frequencies to constraint the high frequencies can help to develop a more robust LSRTM.