Time domain elastic-wave full waveform inversion based on first-order approximate instantaneous frequency

The weak signals of artificial seismic records contain the subsurface medium information that is required in the inversion. But in the full waveform inversion (FWI), the weak signals contribute less to the objective functions. Therefore, how to improve the contribution of the weak signals in the objective functions of FWI is the problem that needs to be solved urgently. The research shows (Ren, D. 1980. Preliminary research on seismic record and instantaneous frequency. Oil Geophysical Prospecting 15 no. 1: 7–21) that instantaneous frequency attributes, which are very sensitive to the changes in subsurface velocity, have the potential to extract the weak signals from the seismic records. However, this frequency can only be estimated from the complex seismic signals. Empirical mode decomposition (EMD) method has been widely used in signal analysis so as to estimate the instantaneous frequency, but it is difficult to be applied in FWI due to the huge computation. In order to solve this problem, the instantaneous frequency is replaced with the first-order approximation of the exponential frequency in FWI. In this paper, the objective functions of the first-order approximate exponential frequency FWI (FRE-EFWI) in elastic waves and the source terms of its back propagation formula were derived. Besides, the FRE-EFWI method was proved to improve the contribution of the weak signals in the objective functions of FWI. In addition, the correctness and effectiveness of the method were demonstrated by the examples of FWI.