Unsupervised learning of background error covariance matrix for variational data assimilation

 The background error covariance matrix (B) plays a central role in variational data assimilation by controlling how observational and background information are combined and is therefore essential for producing an accurate analysis. However, explicitly constructing B and computing its inverse are severely constrained in practice due to the extremely high dimensionality and associated computational cost.

 To mitigate this limitation, this study proposes an unsupervised learning approach that directly estimates the inverse square root of the background error covariance (B⁻¹ᐟ²) from the forecast error patterns. A feedforward neural network that learns a linear matrix corresponding to B⁻¹ᐟ² is trained under whitening constraints, which are intrinsic properties of B⁻¹ᐟ². The learned operator satisfies symmetry and positive definiteness, and the inverse background error covariance (B⁻¹) is obtained in a numerically stable manner by squaring the learned B⁻¹ᐟ².

 The performance of the learned B⁻¹ᐟ² is evaluated through verification of its whitening properties and comparison with a reference B⁻¹ᐟ² constructed by the pseudo inversion of B using singular value decomposition, demonstrating that it reproduces the dominant structural characteristics and leading modes of the reference. The learned B⁻¹ is further implemented within a three-dimensional variational data assimilation (3D-Var) framework, where it stably controls the spatial structure of analysis increments without numerical instability. These results indicate that the proposed unsupervised approach provides a practical and effective alternative for estimating and applying the B⁻¹ in variational data assimilation.