The posed question arises for instance in regional gravity field modelling using weighted least-squares techniques if the gravity field functionals are synthesised from the spherical harmonic coefficients of a satellite-only global gravity model (GGM), and are used as one of the noisy datasets. The associated noise covariance matrix, appeared to be extremely ill-conditioned with a singular value spectrum that decayed gradually to zero without any noticeable gap. We analysed three methods to deal with the ill-conditioned noise covariance matrix: Tihonov regularisation of the noise covariance matrix in combination with the standard formula for the weighted least-squares estimator, a formula of the weighted least-squares estimator, which does not involve the inverse noise covariance matrix, and an estimator based on Rao’s unified theory of least-squares. Our analysis was based on a numerical experiment involving a set of height anomalies synthesised from the GGM GOCO05s, which is provided with a full noise covariance matrix. We showed that the three estimators perform similar, provided that the two regularisation parameters each method knows were chosen properly. As standard regularisation parameter choice rules do not apply here, we suggested a new parameter choice rule, and demonstrated its performance. Using this rule, we found that the differences between the three least-squares estimates were within noise. For the standard formulation of the weighted least-squares estimator with regularised noise covariance matrix, this required an exceptionally strong regularisation, much larger than one expected from the condition number of the noise covariance matrix. The preferred method is the inversion-free formulation of the weighted least-squares estimator, because of its simplicity with respect to the choice of the two regularisation parameters.

中文翻译：

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如何处理单卫星全球重力场模型合成的重力场泛函噪声协方差矩阵的高条件数？

例如，如果重力场函数是从仅卫星全球重力模型 (GGM) 的球谐系数合成的，并用作噪声数据集之一，则所提出的问题会出现在使用加权最小二乘法技术的区域重力场建模中. 相关的噪声协方差矩阵似乎非常病态，奇异值谱逐渐衰减到零，没有任何明显的差距。我们分析了三种处理病态噪声协方差矩阵的方法：噪声协方差矩阵的Tihonov正则化结合加权最小二乘估计量的标准公式，加权最小二乘估计量的一个公式，其中不涉及逆噪声协方差矩阵，以及基于 Rao 的最小二乘统一理论的估计器。我们的分析基于数值实验，该实验涉及一组从 GGM GOCO05s 合成的高度异常，并提供完整的噪声协方差矩阵。我们展示了三个估计器的性能相似，前提是每个方法知道的两个正则化参数都被正确选择。由于标准正则化参数选择规则在这里不适用，我们提出了一个新的参数选择规则，并展示了它的性能。使用此规则，我们发现三个最小二乘估计之间的差异在噪声范围内。对于具有正则化噪声协方差矩阵的加权最小二乘估计量的标准公式，这需要非常强的正则化，比噪声协方差矩阵的条件数所预期的要大得多。首选方法是加权最小二乘估计量的无反演公式，因为它在选择两个正则化参数方面很简单。