A Big Data‐Driven Nonlinear Least Squares Four‐Dimensional Variational Data Assimilation Method: Theoretical Formulation and Conceptual Evaluation

Tian, Xiangjun; Zhang, Hongqin

doi:10.1029/2019ea000735

Cited by 13 publications

(25 citation statements)

References 31 publications

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“…The cost function (equations –) in terms of the new control β can be further transformed into the following format:

J ((), β) = \frac{1}{2} Q {(boldβ)}^{T} Q ((), β)

based on the approximation of

L_{k}^{'} ((), {boldx}^{'}) \approx H_{k}^{'} M_{k}^{'} ((), {boldx}^{'})

, where

H_{k}^{'}

and

M_{k}^{'}

are the tangent linear models of H k and

M_{t_{0} ⟶ t_{k}}

, respectively, after further calculations (see Tian & Zhang, , for details), where

Q ((), β) = ((), \begin{array}{c} \sqrt{N - 1} boldβ \\ R_{+, 0}^{- 1 / 2} ([], P_{y}^{0} boldβ - y_{obs, 0}^{'}) \\ ⋮ \\ R_{+, S}^{- 1 / 2} ([], P_{y}^{S} boldβ - y_{obs, S}^{'}) \end{array}),

…”

Section: Methodsmentioning

confidence: 99%

“…Thus, the Gauss‐Newton iteration scheme for the nonlinear least squares problem (equations and ) is defined by (Dennis & Schnabel, )

\begin{matrix} lefttrue \\ {boldβ}^{l} = {boldβ}^{l - 1} - {([], ((), N - 1) I_{N \times N} + \sum_{k = 0}^{S} {(P_{y}^{k})}^{T} R_{k}^{- 1} ((), {boldP}_{y}^{k}))}^{- 1} \\ \times \{(N - 1) {boldβ}^{l - 1} + {false\sum}_{k = 0}^{S} {((), {boldP}_{y}^{k})}^{normalT} {boldR}_{k}^{- 1} [L_{k}^{'} ({boldP}_{x} {boldβ}^{l - 1}) - {boldy}_{obs, k}^{'}]\} \end{matrix}

for l = 1,⋯, l max , where l max is the maximum iteration number and I N × N denotes the N × N identity matrix (see Tian et al, , for further details). Tian and Zhang () localize equation as follows:

\begin{matrix} lefttrue \\ {boldβ}_{ρ}^{l} = {boldβ}_{ρ}^{l - 1} + {false\sum}_{k = 0}^{S} {((), ρ_{y} < e > P_{y,}^{k *})}^{normalT} L_{k}^{'} (x^{', l - 1}) \\ + {false\sum}_{k = 0}^{S} {((), ρ_{y} < e > P_{y}^{k #})}^{normalT} {boldR}_{k}^{- 1} [{boldy}_{obs, k}^{'} ...] \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

“…Similar to Tian and Zhang (), we utilized the following 2‐D shallow‐water equations, with the f‐ plane formulation, as the forecast model for the OSSEs:

\frac{\partial u}{\partial t} = - u \frac{\partial u}{\partial x} - v \frac{\partial u}{\partial y} + italicfv - g \frac{\partial h}{\partial x}

\frac{\partial v}{\partial t} = - u \frac{\partial u}{\partial x} - v \frac{\partial u}{\partial y} - italicfu - g \frac{\partial h}{\partial y}

\frac{\partial h}{\partial t} = - u \frac{\partial ((), h - h_{s})}{\partial x} - v \frac{\partial ((), h - h_{s})}{\partial y} - ((), H + h - h_{s}) ((), \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y})

where f = 7.272 × 10 −5 s −1 is the Coriolis parameter, H = 3,000 m is the basic state depth, h s = h 0 sin(4 πx / L x )[sin(4 πy / L y )] 2 is the terrain height, h 0 = 250 m, and the lengths of the two sides of the computational domain are D = L x = L y = 44 d m, respectively (where d = ∆ x = ∆ y = 300 km is the uniform grid size). The domain was partitioned into a square mesh with 45 grid points in each coordinate direction, and periodic boundary conditions were imposed at x = (0, L x ) and y = (0, L y ).…”

Section: Osses Using the Shallow‐water Equation Modelmentioning

confidence: 99%

“…As in Tian and Zhang (), we first obtained the true initial fields for all OSSEs by integrating the perfect (i.e., h 0 = 250 m) shallow‐water equation model from the following initial conditions:

h = 360 {[\sin (\frac{πy}{D})]}^{2} + 120 \sin ((), \frac{2 πx}{D}) \sin ((), \frac{2 πy}{D})

u = - f^{- 1} g \frac{\partial h}{\partial y} and v = - f^{- 1} g \frac{\partial h}{\partial x} at t = - 60 hr

for 60 hr. The background state x b was produced in a similar manner, but using the imperfect model (i.e., h 0 = 0 m).…”

Section: Osses Using the Shallow‐water Equation Modelmentioning

confidence: 99%

“…Among the many 4DVar methods equipped with ensemble‐estimated error covariances B e , the nonlinear least squares 4‐D variational data assimilation method (NLS‐4DVar; Tian et al, ; Tian & Feng, ) is distinctive as an advanced 4DEnVar algorithm that utilizes a Gauss‐Newton iterative scheme (Dennis & Schnabel, ) to handle the nonlinearity of the 4DVar cost function, thereby enhancing 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, ), which has superior performance compared to the standard NLS‐4DVar, without increasing computational costs. Remarkably, BD‐NLS4DVar suitably adjusts and steadily maintains the ensemble spreads by including a historical big data ensemble.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Tian

Zhang

2019

Earth and Space Science

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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.

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“…The cost function (equations –) in terms of the new control β can be further transformed into the following format:

J ((), β) = \frac{1}{2} Q {(boldβ)}^{T} Q ((), β)

based on the approximation of

L_{k}^{'} ((), {boldx}^{'}) \approx H_{k}^{'} M_{k}^{'} ((), {boldx}^{'})

, where

H_{k}^{'}

and

M_{k}^{'}

are the tangent linear models of H k and

M_{t_{0} ⟶ t_{k}}

, respectively, after further calculations (see Tian & Zhang, , for details), where

Q ((), β) = ((), \begin{array}{c} \sqrt{N - 1} boldβ \\ R_{+, 0}^{- 1 / 2} ([], P_{y}^{0} boldβ - y_{obs, 0}^{'}) \\ ⋮ \\ R_{+, S}^{- 1 / 2} ([], P_{y}^{S} boldβ - y_{obs, S}^{'}) \end{array}),

…”

Section: Methodsmentioning

confidence: 99%

“…Thus, the Gauss‐Newton iteration scheme for the nonlinear least squares problem (equations and ) is defined by (Dennis & Schnabel, )

\begin{matrix} lefttrue \\ {boldβ}^{l} = {boldβ}^{l - 1} - {([], ((), N - 1) I_{N \times N} + \sum_{k = 0}^{S} {(P_{y}^{k})}^{T} R_{k}^{- 1} ((), {boldP}_{y}^{k}))}^{- 1} \\ \times \{(N - 1) {boldβ}^{l - 1} + {false\sum}_{k = 0}^{S} {((), {boldP}_{y}^{k})}^{normalT} {boldR}_{k}^{- 1} [L_{k}^{'} ({boldP}_{x} {boldβ}^{l - 1}) - {boldy}_{obs, k}^{'}]\} \end{matrix}

\begin{matrix} lefttrue \\ {boldβ}_{ρ}^{l} = {boldβ}_{ρ}^{l - 1} + {false\sum}_{k = 0}^{S} {((), ρ_{y} < e > P_{y,}^{k *})}^{normalT} L_{k}^{'} (x^{', l - 1}) \\ + {false\sum}_{k = 0}^{S} {((), ρ_{y} < e > P_{y}^{k #})}^{normalT} {boldR}_{k}^{- 1} [{boldy}_{obs, k}^{'} ...] \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

“…Similar to Tian and Zhang (), we utilized the following 2‐D shallow‐water equations, with the f‐ plane formulation, as the forecast model for the OSSEs: