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Low-rank regularization is an effective technique for addressing ill-posed inverse problems when the unknown variable exhibits low-rank characteristics. However, global low-rank assumptions do not always hold for seismic wavefields; in many practical situations, local low-rank features are instead more commonly observed. To leverage this insight, we propose partitioning the unknown variable into tiles, each represented via low-rank factorization. We apply this framework to regularize multidimensional deconvolution in the frequency domain, considering two key factors. First, the unknown variable, referred to as the Green's function, must maintain symmetry according to the reciprocity principle of wave propagation. To ensure symmetry within the tile-based low-rank framework, diagonal tiles are formulated as the product of a low-rank factor and its transpose if numerically rank-deficient. Otherwise, they are represented by preconditioned dense forms. Symmetry in off-diagonal elements is achieved by parameterizing sub-diagonal tiles as the product of two distinct low-rank factors, with the corresponding super-diagonal tiles set as their transposes. Second, the rank of the Green's function varies with frequency; in other words, the Green's function has different ranks at different frequencies. To determine the numerical rank and optimal tile size for each frequency, we first solve the multidimensional deconvolution problem using a benchmark solver. Based on these results, we estimate the optimal tile size and numerical rank for our proposed solver. |
