Wavelet whitened patch-based diffusion prior for velocity models

Many physical properties, such as subsurface velocity models, exhibit fractal-like 1/ f^α power spectra with α > 1, implying a power-law decay of spectral energy with frequency, where low-frequency components dominate and high-frequency con- tributions decay polynomially. This induces long-range spatial correlations and complicates the estimation of statistical quan- tities, most notably yielding ill-conditioned score functions central to score-based generative models. Recent patch-based training strategies have been introduced to mitigate the chal- lenges of high-dimensional learning; however, for data with strong long-range correlations, patching further exacerbates ill-posedness in score estimation. To address this challenge, we propose a scale- and orientation-normalized invertible wavelet transform tailored for patch-based training of score-based gener- ative models. Additionally, we introduce a sampling correction that compensates for numerical inconsistencies induced by the multiscale transform during inference. Empirical results show that, relative to baseline patch-based models, the proposed ap- proach achieves an average improvement of approximately 20% in power spectral fidelity, while reducing training and inference time by 20% and 50%, respectively.