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首页> 美国卫生研究院文献>Springer Open Choice >How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model?
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How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model?

机译：如何处理仅基于卫星的全球重力场模型合成的重力场函数的噪声协方差矩阵的高条件数？

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摘要

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.
机译：如果重力场功能是由仅卫星全球重力模型（GGM）的球谐系数合成的，并且被用作嘈杂的数据集之一，那么在使用加权最小二乘技术进行区域重力场建模时就会出现这个问题。 。相关的噪声协方差矩阵似乎处于极端病态，其奇异值频谱逐渐衰减到零而没有任何明显的间隙。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正则化与加权最小二乘估计器的标准公式，加权最小二乘估计器的公式（不涉及）逆噪声协方差矩阵，以及基于Rao最小二乘统一理论的估计器。我们的分析基于一项数值实验，涉及一组由GGM GOCO05合成的高度异常，并提供了完整的噪声协方差矩阵。我们证明了，只要正确选择了每种方法知道的两个正则化参数，这三个估计量的性能相似。由于标准正则化参数选择规则不适用于此处，因此我们建议了一种新的参数选择规则，并演示了其性能。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之内。对于具有正则化噪声协方差矩阵的加权最小二乘估计器的标准公式，这需要异常强的正则化，远大于从噪声协方差矩阵的条件数预期的正则化。优选的方法是加权最小二乘估计量的无逆公式，因为它相对于两个正则化参数的选择简单。

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