Incremental dimensionality reduction for efficiently solving Bayesian inverse problems

The inversion of large sparse matrices poses a major challenge in geophysics, particularly in Bayesian seismic inversion, significantly limiting computational efficiency and practical applicability to large-scale datasets. Existing dimensionality reduction methods have achieved partial success in addressing this issue. However, they remain limited in terms of the achievable degree of dimensionality reduction. An incremental deep dimensionality reduction approach is proposed herein to significantly reduce matrix size and is applied to Bayesian linearized inversion (BLI), a stochastic seismic inversion approach that heavily depends on large sparse matrices inversion. The proposed method first employs a linear transformation based on the discrete cosine transform (DCT) to extract the matrix's essential information and eliminate redundant components, forming the foundation of the dimensionality reduction framework. Subsequently, an innovative iterative DCT-based dimensionality reduction process is applied, where the reduction magnitude is carefully calibrated at each iteration to incrementally reduce dimensionality, thereby effectively eliminating matrix redundancy in depth. This process is referred to as the incremental discrete cosine transform (IDCT). Ultimately, a linear IDCT-based reduction operator is constructed and applied to the kernel matrix inversion in BLI, resulting in a more efficient BLI framework. The proposed method was evaluated through synthetic and field data tests and compared with conventional dimensionality reduction methods. The IDCT approach significantly improves the dimensionality reduction efficiency of the core inversion matrix while preserving inversion accuracy, demonstrating prominent advantages in solving Bayesian inverse problems more efficiently.