Abstract High dimensional error covariance matrices and their inverses are used to weight the contribution of observation and background information in data assimilation procedures. As observation error covariance matrices are often obtained by sampling methods, estimates are often degenerate or ill-conditioned, making it impossible to invert an observation error covariance matrix without the use of techniques to reduce its condition number. In this paper, we present new theory for two existing methods that can be used to ‘recondition’ any covariance matrix: ridge regression and the minimum eigenvalue method. We compare these methods with multiplicative variance inflation, which cannot alter the condition number of a matrix, but is often used to account for neglected correlation information. We investigate the impact of reconditioning on variances and correlations of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users regarding method selection, and choice of target condition number. The new theory shows that, for the same target condition number, both methods increase variances compared to the original matrix, with larger increases for ridge regression than the minimum eigenvalue method. We prove that the ridge regression method strictly decreases the absolute value of off-diagonal correlations. Theoretical comparison of the impact of reconditioning and multiplicative variance inflation on the data assimilation objective function shows that variance inflation alters information across all scales uniformly, whereas reconditioning has a larger effect on scales corresponding to smaller eigenvalues. We then consider two examples: a general correlation function, and an observation error covariance matrix arising from interchannel correlations. The minimum eigenvalue method results in smaller overall changes to the correlation matrix than ridge regression but can increase off-diagonal correlations. Data assimilation experiments reveal that reconditioning corrects spurious noise in the analysis but underestimates the true signal compared to multiplicative variance inflation.

中文翻译：

---

改进估计协方差矩阵的条件数

摘要 高维误差协方差矩阵及其逆矩阵用于对观测资料和背景信息在资料同化过程中的贡献进行加权。由于观测误差协方差矩阵通常是通过抽样方法获得的，因此估计通常是退化或病态的，因此如果不使用减少条件数的技术就不可能对观测误差协方差矩阵求逆。在本文中，我们提出了两种现有方法的新理论，这些方法可用于“重新调整”任何协方差矩阵：岭回归和最小特征值方法。我们将这些方法与乘法方差膨胀进行比较，乘法方差膨胀不能改变矩阵的条件数，但通常用于解释被忽略的相关信息。我们在理论和实践环境中研究了重新调节对一般协方差矩阵的方差和相关性的影响。改进的理论理解为用户提供了有关方法选择和目标条件数选择的指导。新理论表明，对于相同的目标条件数，与原始矩阵相比，两种方法都增加了方差，岭回归的增加幅度大于最小特征值方法。我们证明岭回归方法严格降低了非对角相关的绝对值。再调节和乘法方差膨胀对数据同化目标函数影响的理论比较表明，方差膨胀在所有尺度上统一改变信息，而修复对对应于较小特征值的尺度有更大的影响。然后我们考虑两个例子：一个通用的相关函数和一个由通道间相关引起的观测误差协方差矩阵。与岭回归相比，最小特征值方法导致相关矩阵的整体变化更小，但可以增加非对角线相关性。数据同化实验表明，与乘法方差膨胀相比，重新调节校正了分析中的虚假噪声，但低估了真实信号。与岭回归相比，最小特征值方法导致相关矩阵的整体变化更小，但可以增加非对角线相关性。数据同化实验表明，与乘法方差膨胀相比，重新调节校正了分析中的虚假噪声，但低估了真实信号。与岭回归相比，最小特征值方法导致相关矩阵的整体变化更小，但可以增加非对角线相关性。数据同化实验表明，与乘法方差膨胀相比，重新调节校正了分析中的虚假噪声，但低估了真实信号。