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A Recursive Regularized Solution to Geophysical Linear Ill-Posed Inverse Problems

Ji, Kunpu, Shen, Yunzhong, Sneeuw, Nico, Zhang, Lin, and Chen, Qiujie, 2025. A Recursive Regularized Solution to Geophysical Linear Ill-Posed Inverse Problems. IEEE Transactions on Geoscience and Remote Sensing, 63:TGRS.2025, doi:10.1109/TGRS.2025.3528367.

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@ARTICLE{2025ITGRS..6328367J,
       author = {{Ji}, Kunpu and {Shen}, Yunzhong and {Sneeuw}, Nico and {Zhang}, Lin and {Chen}, Qiujie},
        title = "{A Recursive Regularized Solution to Geophysical Linear Ill-Posed Inverse Problems}",
      journal = {IEEE Transactions on Geoscience and Remote Sensing},
     keywords = {Gravity Recovery and Climate Experiment (GRACE), inversion of geophysical and remote sensing, linear ill-posed models, regularization method},
         year = 2025,
        month = jan,
       volume = {63},
          eid = {TGRS.2025},
        pages = {TGRS.2025},
     abstract = "{Linear ill-posed models are widely encountered in various problems in
        geophysics and remote sensing. The regularization technique can
        significantly improve the accuracy of the estimates since the
        biases introduced by the regularization are much smaller than
        the errors reduced by regularization. However, from the spectral
        point of view, certain low-frequency terms with large singular
        values might become over-regularized, whereas other high-
        frequency terms with small singular values might be
        insufficiently regularized for a given regularization parameter.
        For this reason, we developed a recursive regularization
        approach to further improve the regularized solution via
        additional regularization of some high-frequency terms and
        restricted regularization of some low-frequency terms. The
        analytical conditions to determine the terms to be further
        regularized are derived based on the criterion that the
        introduced biases should be smaller than the reduced errors; in
        other words, the mean squared error (mse) should be reduced.
        Furthermore, the universal form of the recursive regularized
        solution is derived. Two examples from remote sensing are
        designed to demonstrate the performance of the developed
        approach. The first example involves solving the Fredholm
        integral equation of the first kind, a fundamental mathematical
        model used in many inverse problems in remote sensing. The
        results indicate that the proposed method outperforms the
        ordinary Tikhonov regularization, partial regularization, and
        adaptive regularization, with roots of mse reduced by 25.8\%,
        14.5\%, and 8.1\%, respectively. Subsequently, we apply the
        proposed method to estimate regional mass anomalies based on the
        mascon modeling using the Gravity Recovery and Climate
        Experiment (GRACE) time-variable gravity field models. The
        results demonstrate that the proposed method preserves more
        signal than conventional regularization methods.}",
          doi = {10.1109/TGRS.2025.3528367},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2025ITGRS..6328367J},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

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