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

Abstract

 The strongly constrained 4DVar has a hidden mechanism that can correct the model error at the initial time. i4DVar is proposed by treating the initial and model errors together as a whole and by correcting them simultaneously and indiscriminately. An adjoint-free nonlinear least squares fast algorithm is developed to solve the i4DVar problem.

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

“…By substituting the above assumptions and minimizing the i4DVar cost function by the Gauss‐Newton iteration method (Dennis & Schnabel, 1996), Tian et al. (2021) obtained $$${\mathbf{\beta }}_{\rho }^{i}={\mathbf{\beta }}_{\rho }^{i-1}+\sum\limits _{k=0}^{S}{\left({\mathbf{\rho }}_{y,k}< e > {\boldsymbol{P}}_{y,k}^{\ast }\right)}^{\text{T}}{L}_{k}^{{\rhook}}\left({\boldsymbol{x}}^{\prime ,i-1}\right)+\sum\limits _{k=0}^{S}{\left({\mathbf{\rho }}_{y,k}< e > {\boldsymbol{P}}_{y,k}^{\#}\right)}^{\text{T}}{R}_{k}^{-1}\left[{y}_{\text{obs},k}^{{\rhook}}-{L}_{k}^{{\rhook}}\left({\boldsymbol{x}}^{\prime ,i-1}\right)\right]$$$ $$${x}^{\prime ,i}=\left({\rho }_{x}< e > {P}_{x}\right){\mathbf{\beta }}_{\rho }^{i}$$$ for i = 1,⋯, i max , where i max is the maximum iteration number, $${\mathbf{P}}_{y,k}=\left({\mathbf{y}}_{1}^{\prime },{{\cdots}},{\mathbf{y}}_{N}^{\prime }\right)\in {\Re }^{{n}_{y,k}\times N}$$…”

“…Therefore, the observation operator H k is simply a bilinear interpolation operator. The ensemble number N is 70 and the default (optimal through sensitivity experiments) covariance localization radius is r 0 = 6 × 300 km (Tian et al., 2021) and the 4D moving strategy (Tian & Feng, 2015) is adopted to produce the initial MPs for NLS‐i4DVar. Finally, the following modified square root analysis scheme $$${\mathbf{P}}_{x}={\mathbf{P}}_{x}{\mathbf{V}}_{2}{{\Phi}}^{\mathrm{T}}$$$ is utilized for the online ensemble update in the assimilation cycles (see Tian et al., 2020 for the definitions of V 2 and Φ).…”

“…the assimilation window) by proposing the following i4DVar cost functional (for more details, seeTian et al, 2021): …”

Integral correction of initial and model errors in system of multigrid NLS‐4DVar data assimilation for numerical weather prediction (SNAP)

“…As traditional 4DVar‐based operational systems have successfully been run in several major NWP centers worldwide for decades despite existing model error (Rabier et al ., 2000; Bouttier and Kelly, 2001; Clayton et al ., 2013; Kazumori, 2014), Tian et al . (2021) demonstrated that the strong‐constraint 4DVar contains a hidden mechanism that corrects model error at the initial time. Inspired by this idea, an integral correction 4DVar (i4DVar) approach was developed by treating initial and model errors together to correct them simultaneously and indiscriminately; the potential merits of the i4DVar method were demonstrated theoretically, and the approach was evaluated using a shallow‐water equation model (Tian et al ., 2021).…”

“…(2021) demonstrated that the strong‐constraint 4DVar contains a hidden mechanism that corrects model error at the initial time. Inspired by this idea, an integral correction 4DVar (i4DVar) approach was developed by treating initial and model errors together to correct them simultaneously and indiscriminately; the potential merits of the i4DVar method were demonstrated theoretically, and the approach was evaluated using a shallow‐water equation model (Tian et al ., 2021).…”