In geodesy, Tikhonov regularization and truncated singular value decomposition (TSVD) are commonly used to derive a well-defined solution for ill-conditioned observation equations. However, as single-parameter regularization methods, they may face some limitations in application due to their lack of flexibility. In this contribution, a kind of multiparameter regularization method is considered, called generalized ridge regression (GRR). Generally, GRR projects observations into several orthogonal spectral domains and then uses different regularization parameters to minimize the mean squared error of the estimated parameters in corresponding spectral domains. To find suitable regularization parameters for GRR, an iterative and shrinking generalized ridge regression (IS-GRR) is proposed. The IS-GRR procedure starts by introducing a predetermined approximation of unknown parameters. Subsequently, in each spectral domain, the signal and noise of the observations are estimated in an iterative and shrinking manner, and the regularization parameters are updated according to the estimated signal-to-noise ratio. Compared to conventional regularization schemes, IS-GRR has the following advantages: Tikhonov regularization usually oversmooths signals in the low-spectral domains and undersuppresses noise in the high-spectral domains, whereas TSVD usually undersuppresses noise in the low-spectral domains and oversmooths signals in the high-spectral domains. However, IS-GRR strikes a balance between retaining signals and suppressing noise in different spectral domains, thereby exhibiting better performance. Two experiments (simulation and mascon modelling examples) verify the effectiveness of IS-GRR for solving ill-conditioned equations in geodesy.

在大地测量学中，吉洪诺夫正则化和截断奇异值分解 (TSVD) 通常用于为病态观测方程导出明确的解。然而，作为单参数正则化方法，由于缺乏灵活性，它们在应用中可能面临一些限制。在这篇文章中，考虑了一种多参数正则化方法，称为广义岭回归（GRR）。一般来说，GRR将观测值投影到几个正交的谱域中，然后使用不同的正则化参数来最小化相应谱域中估计参数的均方误差。为了找到适合 GRR 的正则化参数，提出了迭代收缩广义岭回归（IS-GRR）。 IS-GRR 过程首先引入未知参数的预定近似值。随后，在每个谱域中，以迭代和收缩的方式估计观测值的信号和噪声，并根据估计的信噪比更新正则化参数。与传统的正则化方案相比，IS-GRR具有以下优点：Tikhonov正则化通常对低谱域中的信号过度平滑，对高谱域中的噪声抑制不足，而TSVD通常对低谱域中的噪声抑制不足，对高谱域中的信号过度平滑。高光谱域。然而，IS-GRR在不同谱域的保留信号和抑制噪声之间取得了平衡，从而表现出更好的性能。两个实验（模拟和 mascon 建模示例）验证了 IS-GRR 在求解大地测量学中病态方程方面的有效性。