Average‐derivative optimized 21‐point and improved 25‐point forward modelling and full waveform inversion in frequency domain

Seismic wave forward modelling is a crucial method for studying the propagation characteristics of seismic waves in subsurface media and is a key component of full waveform inversion. Compared to time‐domain forward modelling, frequency‐domain forward modelling offers advantages such as not being constrained by stability limits and reducing the dimension of the solution space. However, forward algorithms based on the rotation coordinate system in the frequency domain cannot adapt to situations with unequal spatial sampling intervals. To enhance the adaptability of the forward modelling algorithm in the frequency domain, we derived a 21‐point finite‐difference scheme based on the average derivative method and calculated the difference coefficients and dispersion conditions. Additionally, to address the significant computational cost in frequency domain forward modelling, we developed an improved 25‐point finite‐difference scheme. The improved 25‐point format is more accurate than the conventional 25‐point format. Building on this foundation, we applied the two derived differential schemes to full waveform inversion to synthesize the shot records of the inversion data. Additionally, we introduced a frequency compensation factor into the gradient processing, which effectively compensates for the deep layer while suppressing noise in the shallow gradient field. Finally, we demonstrated the effectiveness of our approach through a full waveform inversion application on the Marmousi model showcasing its capability in invertig fine subsurface structures.