A Mini‐Batch Stochastic Optimization‐Based Adaptive Localization Scheme and Its Implementation in NLS‐i4DVar

Abstract

Abstract This paper proposes a mini‐batch stochastic optimization‐based adaptive localization scheme for computing the “optimal” localization radius in data assimilation (DA) applications. After constructing a cost function of the localization radius by estimating forecast and observation error statistics, a mini‐batch stochastic gradient descent method with a novel sampling strategy is proposed to minimize the cost function. The proposed stochastic optimization algorithm is further incorporated into the DA method NLS‐i4DVar (the nonlinear least squares integral correcting four‐dimensional variational DA method), which was developed by the authors in Tian et al. (2021, https://doi.org/10.1029/2021EA001767). It is utilized to compute the “optimal” covariance localization radii adaptively and flow‐dependently inside NLS‐i4DVar. The computational cost of NLS‐i4DVar with the proposed adaptive localization scheme only increases slightly due to the use of the mini‐batch stochastic optimization algorithm. Numerical experimental results using the shallow‐water equations demonstrate that NLS‐i4DVar with the proposed adaptive localization scheme shows substantial performance improvement over the standard NLS‐i4DVar method.