Intrinsic connection of traditional FWI and travel-time inversion

We examine the theory behind traditional Full Waveform Inversion (FWI) using L2 norm minimization, with a particular focus on the challenge of cycle skipping. Cycle skipping typically arises when the initial velocity model is significantly inaccurate and lacks extremely low-frequency components in the data. When the time difference between predicted and observed seismic data exceeds half a wavelength, the residual error increases, leading to incorrect gradient directions in the inversion process and hindering FWI convergence. In contrast, travel-time-based methods have demonstrated effectiveness in addressing cycle skipping by providing accurate gradient calculations for velocity updates. Through our investigation, we discovered that a slight modification to the traditional FWI formulation can yield an equivalent travel-time inversion framework. This approach effectively resolves the cycle skipping problem while retaining the theoretical elegance of conventional FWI. We present a detailed derivation of this modified framework and demonstrate its capability using synthetic examples. Our results reveal that, even in the absence of low-frequency components in the input data, the proposed method ensures accurate and stable convergence for FWI.