We couple deep generative priors with observed shot gathers to perform posterior sampling that honors both the wave physics and geological plausibility. Concretely, a diffusion-model prior—trained unconditionally on velocity patches—anchors a sampling-based FWI update, yielding velocity models that are data-consistent and structurally credible. On several demanding 2D benchmarks, a towed-streamer field line, and a 3D upscaling test, we observe improved data fit, better velocity quality along stratigraphic and salt boundaries, and uncertainty that concentrates where illumination is poor. Computationally, encoded-shot batching reduces wave-equation cost and parallelizes cleanly over samples and source sets, making larger 3D runs tractable on current clusters. This opens the possibility of fully utilizing the observed shot gathers to do inference at significantly reduced cost.