Variational data assimilation for very large environmental problems

Lawless, Amos S.

doi:10.1515/9783110282269.55

Cited by 17 publications

(11 citation statements)

References 118 publications

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“…A property which these applications all share is the vast dimensionality of the state vectors involved. In numerical weather prediction the systems have variables of order 10 8 [24]. In addition to the requirement that these computations to be solved quickly, the storage requirement presents an obstacle.…”

Section: Introductionmentioning

confidence: 99%

A low-rank approach to the solution of weak constraint variational data assimilation problems

Freitag

Green

2018

Journal of Computational Physics

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Weak constraint four-dimensional variational data assimilation is an important method for incorporating data (typically observations) into a model. The linearised system arising within the minimisation process can be formulated as a saddle point problem. A disadvantage of this formulation is the large storage requirements involved in the linear system. In this paper, we present a low-rank approach which exploits the structure of the saddle point system using techniques and theory from solving large scale matrix equations. Numerical experiments with the linear advection-diffusion equation, and the non-linear Lorenz-95 model demonstrate the effectiveness of a low-rank Krylov subspace solver when compared to a traditional solver.

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Section: Introductionmentioning

confidence: 99%

A low-rank approach to the solution of weak constraint variational data assimilation problems

Freitag

Green

2018

Journal of Computational Physics

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“…This involves linearising H l and M l+1,l about the current iterative state of the numerical model x (k) l , where k ∈ N 0 is the iteration number. The linearised numerical model and its transpose are referred to as the tangent linear model (TLM) and the adjoint model [1,22]. The resulting cost function to be minimised is then constructed in part from these linearised models.…”

Section: Discussionmentioning

confidence: 99%

“…Here N, m l ∈ N for all l. More details on these variables and 4D-Var can be found in [2][3][4]21,22].…”

Section: Problem Formulationmentioning

confidence: 99%

“…Despite looking deceptively simple, this system provides a challenge numerically and has historically been an essential test equation for the development and analysis of many numerical methods, for example see [23]. Such linear problems are also directly relevant for adjoint methods and to tangent linear models used in incremental 4D-Var [22]. The schemes introduce numerical model error through the approximation of derivatives [7].…”

Section: Problem Formulationmentioning

confidence: 99%

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The effect of numerical model error on data assimilation

Jenkins

Budd

Freitag

et al. 2015

Journal of Computational and Applied Mathematics

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a b s t r a c tStrong constraint 4D-Variational data assimilation (4D-Var) is a method used to create an initialisation for a numerical model, that best replicates subsequent observations of the system it aims to recreate. The method does not take into account the presence of errors in the model, using the model equations as a strong constraint. This paper gives a rigorous and quantitative analysis of the errors introduced into the initialisation through the use of finite difference schemes to numerically solve the model equations. The 1D linear advection equation together with circulant boundary conditions, are chosen as the model equations of interest as they are representative of the advective processes relevant to numerical weather prediction, where 4D-Var is widely used. We consider the deterministic error introduced by finite difference approximations in the form of numerical dissipation and numerical dispersion and identify the relationship between these properties and the error in the 4D-Var initialisation. In particular, we find that a solely numerically dispersive scheme has the potential to introduce destructive interference resulting in the loss of some wavenumber components in the initialisation. Bounds for the error in the initialisation due to finite difference approximations are determined with and without observation errors. The bounds are found to depend on the smoothness of the true initial condition we wish to recover and the numerically dissipative and dispersive properties of the scheme. Numerical results are presented to demonstrate the effectiveness of the bounds. These lead to the conclusion that there exists a critical number of discretisation points when considering full sets of observations, where the effects of both the considered numerical model error and observational errors on the initialisation are minimised. The numerically dissipative and dispersive properties of the finite difference schemes also have the potential to alter the properties of the noise found in observations. Correlated noise structures may be introduced into the 4D-Var initialisation as a result. We determine when this occurs for observational errors in the form of additive white noise and find that the effect is reduced through the use of numerically non-dissipative finite difference schemes.

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“…However, it is a huge undertaking to develop adjoint models for ocean models (Lawless 2012). In this paper, 4Dvar will be used to an existing storm surge model for the German Bight (Bruss et al 2011).…”

Section: Introductionmentioning

confidence: 99%

Adjoint free four-dimensional variational data assimilation for a storm surge model of the German North Sea

et al. 2016

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In this paper, a data assimilation scheme based on the adjoint free Four-Dimensional Variational(4DVar) method is applied to an existing storm surge model of the German North Sea. To avoid the need of an adjoint model, an ensemble-like method to explicitly represent the linear tangent equation is adopted. Results of twin experiments have shown that the method is able to recover the contaminated low dimension model parameters to their true values. The data assimilation scheme was applied to a severe storm surge event which occurred in the North Sea in December 5, 2013. By adjusting wind drag coefficient, the predictive ability of the model increased significantly. Preliminary experiments have shown that an increase in the predictive ability is attained by narrowing the data assimilation time window.

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Variational data assimilation for very large environmental problems

Cited by 17 publications

References 118 publications

A low-rank approach to the solution of weak constraint variational data assimilation problems

A low-rank approach to the solution of weak constraint variational data assimilation problems

The effect of numerical model error on data assimilation

Adjoint free four-dimensional variational data assimilation for a storm surge model of the German North Sea

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