  Full waveform inversion (FWI) has the potential to provide high-resolution
subsurface model estimations. However, due to limitations in observation, e.g.,
regional noise, limited shots or receivers, and band-limited data, it is hard
to obtain the desired high-resolution model with FWI. To address this
challenge, we propose a new paradigm for FWI regularized by generative
diffusion models. Specifically, we pre-train a diffusion model in a fully
unsupervised manner on a prior velocity model distribution that represents our
expectations of the subsurface and then adapt it to the seismic observations by
incorporating the FWI into the sampling process of the generative diffusion
models. What makes diffusion models uniquely appropriate for such an
implementation is that the generative process retains the form and dimensions
of the velocity model. Numerical examples demonstrate that our method can
outperform the conventional FWI with only negligible additional computational
cost. Even in cases of very sparse observations or observations with strong
noise, the proposed method could still reconstruct a high-quality subsurface
model. Thus, we can incorporate our prior expectations of the solutions in an
efficient manner. We further test this approach on field data, which
demonstrates the effectiveness of the proposed method.
