Comparison between RTM gradient and PSPI gradient in the process of FWI

Sergio J. Romahn, Marcelo Guarido, Kristopher A. Innanen

Full waveform inversion (FWI) can be described as an iterative cycle of four steps. Firstly, we generate synthetic seismic data (modelled shots) from a smoothed initial model and obtain the difference among observed and modelled shots (data residuals). Secondly, we migrate the data residual (using the current velocity model) and stack. This step produces the gradient. Thirdly, we scale the gradient in order to create a velocity update. And finally, we obtain a new velocity model by adding the velocity update to the current velocity model. We start another cycle by using the new velocity model. This report is focused in the second step of the cycle. Standard FWI uses reverse time migration (RTM) to obtain the gradient. On the other hand, iterative modelling, migration and inversion (IMMI) opens the door to use any type of migration method to produce the gradient. In this report, we compare the performance of the phase shift plus interpolation (PSPI) migration and RTM to obtain the gradient. We start pointing out the fundamental difference between these two methods: the fact that the first one is a one-way and the second one is a two-way wave operator. Then, we analyze the migration response and highlight the consequences of the previous point. Finally, we compare the inversion result by applying both methods. The PSPI and RTM gradients were scaled by applying the well calibration technique. We used synthetic data in an acoustic frame in this experiment. We found that both methods are suitable for producing the FWI gradient. However, the PSPI gradient is more sensitive to the initial velocity model than RTM, because its one-way wave operator does not recover long wavelengths as RTM does. This characteristic allows RTM producing a better inversion. PSPI would be a good option providing that the initial velocity model incorporates enough low frequency information. PSPI also showed a great sensitivity to the well interval coverage that is used to calibrate the gradient, while the RTM gradient is quite stable. We found that a hybrid inversion by using both methods is feasible and saves computational time.