A non-linear least squares enhanced POD-4DVar algorithm for data
                        assimilation

Tian, Xiangjun; Feng, Xin

doi:10.3402/tellusa.v67.25340

Cited by 42 publications

(77 citation statements)

References 63 publications

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“…As an advanced 4DEnVar method, NLS‐4DVar (Tian & Feng, ; Tian et al, ) determines the analysis increment of the initial condition (IC), δ x (at initial time t 0 ), by minimizing the following cost function:

J ((), δ boldx) = \frac{1}{2} {(δ x)}^{T} B^{- 1} ((), δ boldx) + \frac{1}{2} \sum_{k = 0}^{S} {[L_{k}^{'} (δ x) - {boldd}_{k}]}^{T} R_{k}^{- 1} ([], L_{k}^{'} ((), δ boldx) - d_{k}),

where δ x = x − x b is the perturbation of the background field x b at t 0 :

L_{k}^{'} ((), δ boldx) = L_{k} ((), x^{b} + δ boldx) - L_{k} ((), {boldx}^{b}),

d_{k} = y_{k}^{0} - L_{k} ((), {boldx}^{b}),

and

L_{k} = H_{k} M ((),, t_{k}, t_{0}) .

…”

Section: Methodsmentioning

confidence: 99%

“…Here

P_{y, k} = ((),,, {boldy}_{k, 1}^{'} {boldy}_{k, 2}^{'} \dots {boldy}_{k, N}^{'})

are the simulated observation perturbations (OPs) and

y_{k}^{'} = L_{k} ((), x^{b} + δ boldx) - L_{k} ((), {boldx}^{b})

. Following Tian and Feng (),

β^{l} = β^{l - 1} + \sum_{normalk = 0}^{S} P_{y, k}^{*} L_{k}^{'} ((), P_{x} β^{l ‐ 1}) + \sum_{normalk = 0}^{S} P_{y, k}^{#} ([], d_{k} - L_{k}^{'} ((), P_{x} β^{l ‐ 1})),

P_{y, k}^{*} = - ((), N - 1) {[{false\sum}_{k = 0}^{S} {boldP}_{y, k}^{normalT} {boldR}_{k}^{- 1} {boldP}_{y, k} + (N - 1) I]}^{- 1} {[{false\sum}_{k = 0}^{S} {boldP}_{y, k}^{normalT} {boldP}_{y, k}]}^{- 1} P_{y, k}^{T},

P_{y, k...}

…”

Section: Methodsmentioning

confidence: 99%

“…The time of the observation was the analysis time. The single‐observation temperature was 27.20 K with an error standard deviation of 0.1 K. The simulated temperature from the background field at the single‐observation location was 27.80 K. The MG‐NLS‐4DVar scheme based on the ensemble framework adapted a 4‐D moving sampling strategy (Tian & Feng, ) to produce the MPs ( P x ) and OPs ( P y ) from 78‐hr forecasts. More specifically, two 78‐hr forecasts were initialized with 1° × 1° NCEP/FNL global analysis data, from 0000 UTC on 7 June 2010 and 0000 UTC on 6 June 2010, respectively.…”

Section: Evaluation Of the Mg‐nls‐4dvar Scheme Using The Arw‐wrf Modelmentioning

confidence: 99%

“…The establishment of the adjoint model is difficult, and its costs in programming and computing time are high, especially for complex and strongly nonlinear models (Tian & Feng, ). Several existing four‐dimensional ensemble‐variational (4DEnVar) methods could be possible alternatives by using approximations of both methods by combining their strengths (Tian & Feng, ). The 4DEnVar methods, for example, singular value decomposition 4DVar (Qiu et al, ), dimension‐reduced projection 4DVar (Wang et al, ), and nonlinear least squares 4DVar (NLS‐4DVar) (Tian et al, , ; Tian & Feng, ), are based on an ensemble framework, such that the adjoint model is no longer needed, thus significantly simplifying the assimilation process.…”

Section: Introductionmentioning

confidence: 99%

“…Several existing four‐dimensional ensemble‐variational (4DEnVar) methods could be possible alternatives by using approximations of both methods by combining their strengths (Tian & Feng, ). The 4DEnVar methods, for example, singular value decomposition 4DVar (Qiu et al, ), dimension‐reduced projection 4DVar (Wang et al, ), and nonlinear least squares 4DVar (NLS‐4DVar) (Tian et al, , ; Tian & Feng, ), are based on an ensemble framework, such that the adjoint model is no longer needed, thus significantly simplifying the assimilation process. When finite differentiation is used to approximate the gradient of the cost function, using the approximation method (i.e., 4DEnVar) requires that the perturbation be sufficiently small.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Multigrid Nonlinear Least Squares Four‐Dimensional Variational Data Assimilation Scheme With the Advanced Research Weather Research and Forecasting Model

Zhang

Tian

2018

JGR Atmospheres

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The motions of the atmosphere have multiscale properties in space and/or time, and the background error covariance matrix (Β) should thus contain error information at different correlation scales. To obtain an optimal analysis, the multigrid three‐dimensional variational data assimilation scheme is used widely when sequentially correcting errors from large to small scales. However, introduction of the multigrid technique into four‐dimensional variational data assimilation is not easy due to its strong dependence on the adjoint model, which has high computational costs in data coding, maintenance, and updating, especially for large‐scale, complex problems. In this study, the multigrid technique was introduced into the nonlinear least squares four‐dimensional variational assimilation (NLS‐4DVar) method, which is an advanced four‐dimensional ensemble‐variational method that can be applied without invoking the adjoint models. The multigrid NLS‐4DVar (MG‐NLS‐4DVar) scheme uses the number of grid points to control the scale, with doubling of this number when moving from coarser to finer grid levels. Furthermore, the MG‐NLS‐4DVar scheme not only retains the advantages of NLS‐4DVar but also sufficiently corrects multiscale errors to achieve a highly accurate analysis. The effectiveness and efficiency of the proposed MG‐NLS‐4DVar scheme were evaluated by one group of single‐observation experiments and one group of comprehensive evaluation experiments using the Advanced Research Weather Research and Forecasting Model. MG‐NLS‐4DVar outperformed NLS‐4DVar, with a lower computational cost.

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J ((), δ boldx) = \frac{1}{2} {(δ x)}^{T} B^{- 1} ((), δ boldx) + \frac{1}{2} \sum_{k = 0}^{S} {[L_{k}^{'} (δ x) - {boldd}_{k}]}^{T} R_{k}^{- 1} ([], L_{k}^{'} ((), δ boldx) - d_{k}),

where δ x = x − x b is the perturbation of the background field x b at t 0 :

L_{k}^{'} ((), δ boldx) = L_{k} ((), x^{b} + δ boldx) - L_{k} ((), {boldx}^{b}),

d_{k} = y_{k}^{0} - L_{k} ((), {boldx}^{b}),

and

L_{k} = H_{k} M ((),, t_{k}, t_{0}) .

…”

Section: Methodsmentioning

confidence: 99%

“…Here

P_{y, k} = ((),,, {boldy}_{k, 1}^{'} {boldy}_{k, 2}^{'} \dots {boldy}_{k, N}^{'})

are the simulated observation perturbations (OPs) and

y_{k}^{'} = L_{k} ((), x^{b} + δ boldx) - L_{k} ((), {boldx}^{b})

. Following Tian and Feng (),

β^{l} = β^{l - 1} + \sum_{normalk = 0}^{S} P_{y, k}^{*} L_{k}^{'} ((), P_{x} β^{l ‐ 1}) + \sum_{normalk = 0}^{S} P_{y, k}^{#} ([], d_{k} - L_{k}^{'} ((), P_{x} β^{l ‐ 1})),

P_{y, k}^{*} = - ((), N - 1) {[{false\sum}_{k = 0}^{S} {boldP}_{y, k}^{normalT} {boldR}_{k}^{- 1} {boldP}_{y, k} + (N - 1) I]}^{- 1} {[{false\sum}_{k = 0}^{S} {boldP}_{y, k}^{normalT} {boldP}_{y, k}]}^{- 1} P_{y, k}^{T},

P_{y, k...}

…”

Section: Methodsmentioning

confidence: 99%

Section: Evaluation Of the Mg‐nls‐4dvar Scheme Using The Arw‐wrf Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Multigrid Nonlinear Least Squares Four‐Dimensional Variational Data Assimilation Scheme With the Advanced Research Weather Research and Forecasting Model

Zhang

Tian

2018

JGR Atmospheres

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An Efficient Local Correlation Matrix Decomposition Approach for the Localization Implementation of Ensemble‐Based Assimilation Methods

Zhang

Tian

2018

JGR Atmospheres

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Ensemble‐based data assimilation methods often use the so‐called localization scheme to improve the representation of the ensemble background error covariance (Be). Extensive research has been undertaken to reduce the computational cost of these methods by using the localized ensemble samples to localize Be by means of a direct decomposition of the local correlation matrix C. However, the computational costs of the direct decomposition of the local correlation matrix C are still extremely high due to its high dimension. In this paper, we propose an efficient local correlation matrix decomposition approach based on the concept of alternating directions. This approach is intended to avoid direct decomposition of the correlation matrix. Instead, we first decompose the correlation matrix into 1‐D correlation matrices in the three coordinate directions, then construct their empirical orthogonal function decomposition at low resolution. This procedure is followed by the 1‐D spline interpolation process to transform the above decompositions to the high‐resolution grid. Finally, an efficient correlation matrix decomposition is achieved by computing the very similar Kronecker product. We conducted a series of comparison experiments to illustrate the validity and accuracy of the proposed local correlation matrix decomposition approach. The effectiveness of the proposed correlation matrix decomposition approach and its efficient localization implementation of the nonlinear least‐squares four‐dimensional variational assimilation are further demonstrated by several groups of numerical experiments based on the Advanced Research Weather Research and Forecasting model.

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Numerical simulation of the temperature field of the stadium building foundation in frozen areas based on the finite element method and proper orthogonal decomposition technique

Song

2021

Math Methods in App Sciences

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This paper is mainly concerned with the development and design of a reduced order model in numerical simulation of the unsteady temperature field of the stadium building foundation in frozen areas, based the finite element (FE) method and proper orthogonal decomposition (POD) with the snapshot technique. We first derive the standard FE formulation of the unsteady temperature field and compute its FE full solutions, from which we choose a few spatio‐temporal solutions as snapshots. Based on POD technique, we then build a set of optimal POD bases maximizing the energy content in the original ensemble data, and in the new space spanned by the POD basis, we establish a low‐order numerical model of reduced‐order finite element (ROFE) formulation of the unsteady temperature field. And we prove the error estimates of the ROFE solutions of the unsteady temperature field. Finally, numerical tests are provided to simulate the temperature field of crushed stone clay under the natural pavement near the foundation of a stadium building in a frozen soil area of the northwest of Sichuan province in China.

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A non-linear least squares enhanced POD-4DVar algorithm for data assimilation

Cited by 42 publications

References 63 publications

A Multigrid Nonlinear Least Squares Four‐Dimensional Variational Data Assimilation Scheme With the Advanced Research Weather Research and Forecasting Model

A Multigrid Nonlinear Least Squares Four‐Dimensional Variational Data Assimilation Scheme With the Advanced Research Weather Research and Forecasting Model

An Efficient Local Correlation Matrix Decomposition Approach for the Localization Implementation of Ensemble‐Based Assimilation Methods

Numerical simulation of the temperature field of the stadium building foundation in frozen areas based on the finite element method and proper orthogonal decomposition technique

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