Discretized Adjoint State Time and Frequency Domain Full Waveform Inversion: A Comparative Study

This study derives the discretized adjoint states full waveform inversion (FWI) in both time and frequency domains based on the Lagrange multiplier method. To achieve this, we applied adjoint state inversion on the discretized wave equation in both time domain and frequency domain. Besides, in this article, we introduce reliability tests to show that the inversion is performing as it should be expected. Reliability tests comprise of objective function descent test and Jacobian test. The influence of data imperfections is also being studied. We define data imperfection as any factor that causes deterioration in FWI results. Some of these factors are coherent and incoherent noises in data, source wavelet inaccuracy in phase and amplitude, and the existence of gaps in the seismic survey. We compare time and frequency domain inversion methods sensitivity to data imperfection. In all cases, we found that time domain full waveform inversion is more sensitive to imperfections in the data. In general, we find that time domain FWI result shows more deterioration than frequency domain FWI. All tests have been done using 2D full waveform inversion codes. We employ the multi-scale inversion and finite difference scheme (FDM) for discretization, and the misfit function is minimized via limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method.