Implementation of a Modified Adaptive Covariance Inflation Scheme for the Big Data‐Driven NLS‐4DVar Algorithm

Abstract

The adaptive inflation scheme is critical for avoiding underestimation of ensemble-estimated error variances. In this study, we expanded a spatially and temporally varying adaptive inflation scheme proposed within the framework of the local ensemble transform Kalman filter to a global version to facilitate its application in four-dimensional ensemble-variational (4DEnVar) data assimilation methods. We adopted an efficient local correlation matrix decomposition approach to enhance its computation efficiency and then implemented the modified adaptive inflation scheme using the big data-driven nonlinear least squares 4-D variational (BD-NLS4DVar) data assimilation method to improve its robustness. The ensemble analysis scheme of the BD-NLS4DVar method with the modified adaptive inflation scheme was able to adjust the ensemble spreads and error covariances more accurately. Several groups of observing system simulation experiments based on 2-D shallow-water equations demonstrated that the BD-NLS4DVar method implemented with the modified adaptive inflation scheme provides some performance improvement over the standard BD-NLS4DVar method with no inflation.

i4DVar: An Integral Correcting Four‐Dimensional Variational Data Assimilation Method

“…Both the i4DVar and 4DVar methods perform considerably well in the framework of 2-D shallow-water equation model (Tian et al, 2020;Tian & Zhang, 2019a, 2019b under the perfect model scenario (and large initial errors), as they produce significantly smaller RMS errors than those from the control simulations (Figures 2a and 2b). Obviously, the system-related errors are only the initial errors, which can be largely corrected by adding the 4DVar correction E  x at the initial time point through the ensemble nonlinear least squares-based approach (i.e., the NLS-4DVar approach) using only 60 ensemble samples.…”

An Adjoint‐Free Alternating Direction Method for Four‐Dimensional Variational Data Assimilation With Multiple Parameter Tikhonov Regularization

“…According to the following approximation (Tian & Zhang, 2019b), Equation can be rewritten as …”

“…The total initial MPs of BD‐NLS4DVar was composed of two ensembles, a preprepared historical big‐data ensemble ( ) and a small online ensemble ( ), where N h + N o = N ( N h > N o ). See Tian and Zhang (2019a, 2019b) for more details about the preparation and update of the ensemble for the BD‐NLS4DVar. The default parameter values were N o = 10, N h = 100, α = 11, β = 2, l max = 1, and j max = 4, and the covariance localization Schür radius r x = r y = 15 × 300 km for the x and y coordinates, respectively (Gaspari & Cohn, 1999).…”

“…We first evaluated the performance of BD‐NLS4DVar with multiple‐ (NLS‐TK_m) and single‐parameter (NLS‐TK_s) Tikhonov regularization and without regularization (referred to as NLS). The spatially and temporally averaged RMSEs of height ( ) and wind ( ) (see Tian & Zhang, 2019b for definitions of and ) for NLS‐TK_m/s and NLS were compared for each assimilation window under the perfect‐model assumption ( h 0 = 250 m) for all true, forecast, and assimilation runs (Figures 1a and 1c). Figures 1a and 1c show that both NLS‐TK_m and NLS‐TK_s performed significantly better than NLS.…”

“…It therefore enhances assimilation accuracy while maintaining an overall structure that still resembles traditional 4DEnVar algorithms (i.e., with no dependence on the adjoint model) and provides a flow‐dependent B e . Very recently, NLS‐4DVar was further upgraded to a big‐data‐driven version (BD‐NLS4DVar; Tian & Zhang, 2019a, 2019b), which has better performance than the standard NLS‐4DVar without increased computational costs.…”