The ill-posed models are widely encountered in various inversions of geodesy and remote sensing. The regularization approaches can significantly stabilize the solution to ill-posed models since the high-frequency noise is effectively suppressed. Although the famous Tikhonov regularization and truncated singular value decomposition (TSVD) regularization have been widely applied in various geodetic applications, there still remain theoretical drawbacks for either single regularization. For Tikhonov regularization, given a regularization parameter, the low-frequency terms are over-regularized, and high-frequency terms are under-regularized. For TSVD regularization, some medium-frequency terms will be mistaken for high-frequency terms to be truncated and the hidden signals will be lost. For this reason, we propose an adaptive regularized solution in spectral form, which adaptively divides the terms of different frequencies into three kinds: 1) the low-frequency terms are not regularized; 2) the medium-frequency terms are regularized by the Tikhonov method; and 3) the high-frequency terms are regularized by the TSVD method. The analytical conditions for determining the term sets are derived based on the criteria that the introduced biases should be smaller than the reduced errors; in other words, the mean square error (MSE) should be reduced. The two examples are presented to demonstrate the performance of our adaptive regularization. The first numerical example is solving the Fredholm integral equation of the first kind, which is widely encountered in remote sensing inversions. The simulations clearly demonstrate that the adaptive regularized solution can improve the MSE of ordinary Tikhonov and TSVD regularized functions by 25.00% and 9.09%, respectively. In the second example, we apply the new method to investigate the mass variation of the Yangtze River Basin based on the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field model. The Tongji-Gr...

中文翻译：

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逆病态模型的自适应正则化解


不适定模型在大地测量和遥感的各种反演中广泛遇到。由于高频噪声被有效抑制，正则化方法可以显着稳定不适定模型的解。尽管著名的吉洪诺夫正则化和截断奇异值分解（TSVD）正则化已广泛应用于各种大地测量应用中，但任一单一正则化仍然存在理论上的缺陷。对于吉洪诺夫正则化，给定正则化参数，低频项被过度正则化，高频项被欠正则化。对于 TSVD 正则化，一些中频项会被误认为是高频项而被截断，隐藏信号将会丢失。为此，我们提出了谱形式的自适应正则化解决方案，自适应地将不同频率的项分为三种：1）低频项不被正则化； 2）中频项采用Tikhonov方法进行正则化； 3)通过TSVD方法对高频项进行正则化。确定术语集的分析条件是根据引入的偏差应小于减少的误差的标准得出的；换句话说，均方误差（MSE）应该减少。这两个例子是为了展示我们的自适应正则化的性能。第一个数值例子是求解第一类Fredholm积分方程，该方程在遥感反演中广泛遇到。仿真结果清楚地表明，自适应正则化解可以将普通 Tikhonov 和 TSVD 正则化函数的 MSE 分别提高 25.00% 和 9.09%。 在第二个例子中，我们应用新方法基于重力恢复和气候实验（GRACE）时变重力场模型来研究长江流域的质量变化。同济大学...