The conditioning of least‐squares problems in variational data assimilation

Tabeart, Jemima M.; Dance, Sarah L.; Haben, Stephen; Lawless, Amos S.; Nichols, Nancy K.; Waller, John C.

doi:10.1002/nla.2165

Cited by 28 publications

(124 citation statements)

References 66 publications

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“…Again, this was due to interacting eigenstructures between observation and background terms, which could not be measured by the condition number alone. Additionally, Haben [2011], Tabeart et al [2018] find that the ratio of background to observation error variance is important for the convergence of a conjugate gradient problem. In the case where observation errors are small, poor performance of conjugate gradient methods is therefore likely.…”

Section: Discussionmentioning

confidence: 99%

“…However, the improved understanding of alterations to correlations and standard deviations for each method of reconditioning provided here may allow users to anticipate changes to the analysis for a particular system of interest using the results from previous idealised and operational studies (e.g. Tabeart et al [2018], Fowler et al [2018], Simonin et al [2019], Weston et al [2014], Bormann et al [2016]).…”

Section: Discussionmentioning

confidence: 99%

“…The first covariance matrix is constructed using a second-order auto-regressive (SOAR) correlation function [Yaglom, 1987] with lengthscale 0.2 on a unit circle. This correlation function is used in NWP systems [Fowler et al, 2018, Stewart et al, 2013, Tabeart et al, 2018, Waller et al, 2016b, Thiebaux, 1976 where its long tails approximate the estimated horizontal spatial correlation structure well. In order to construct a SOAR error correlation matrix, S, on the finite domain, we follow the method described in Haben [2011], Tabeart et al [2018].…”

Section: Numerical Frameworkmentioning

confidence: 99%

“…Both methods reduce the correlation lengthscale of the error covariance matrix. In Tabeart et al [2018], it was shown that reducing the lengthscale of the observation error covariance matrix decreases the condition number of the Hessian of the 3D-Var objective function and results in improved convergence of the minimisation problem. Hence the application of reconditioning methods to the observation error covariance matrix is likely to improve convergence of the overall data assimilation problem.…”

Section: Changes To the Covariance Matrixmentioning

confidence: 99%

“…This is true in both the unpreconditioned and preconditioned variational data assimilation problem using the control variable transform, as the inverse of the observation error covariance matrix appears in both formulations. The convergence of the data assimilation problem can be poor if either the background or observation variance is small; however, the condition number and eigenvalues of background and observation error covariance matrices have also been shown to be important for convergence in both the unpreconditioned and preconditioned case in Haben et al [2011b,a], Haben [2011], Tabeart et al [2018]. Furthermore, the conditioning and solution of the data assimilation system can be affected by complex interactions between the background and observation error covariance matrices and the observation operator [Tabeart et al, 2018, Johnson et al, 2005.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Improving the condition number of estimated covariance matrices

Tabeart¹,

Dance²,

Lawless³

et al. 2020

Tellus A: Dynamic Meteorology and Oceanography

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High dimensional error covariance matrices and their inverses are used to weight the contribution of observation and background information in data assimilation procedures. As observation error covariance matrices are often obtained by sampling methods, estimates are often degenerate or ill-conditioned, making it impossible to invert an observation error covariance matrix without the use of techniques to reduce its condition number. In this paper we present new theory for two existing methods that can be used to 'recondition' any covariance matrix: ridge regression, and the minimum eigenvalue method. We compare these methods with multiplicative variance inflation, which cannot alter the condition number of a matrix, but is often used to account for neglected correlation information. We investigate the impact of reconditioning on variances and correlations of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users regarding method selection, and choice of target condition number. The new theory shows that, for the same target condition number, both methods increase variances compared to the original matrix, with larger increases for ridge regression than the minimum eigenvalue method. We prove that the ridge regression method strictly decreases the absolute value of off-diagonal correlations. Theoretical comparison of the impact of reconditioning and multiplicative variance inflation on the data assimilation objective function shows that variance inflation alters information across all scales uniformly, whereas reconditioning has a larger effect on scales corresponding to smaller eigenvalues. We then consider two examples: a general correlation function, and an observation error covariance matrix arising from interchannel correlations. The minimum eigenvalue method results in smaller overall changes to the correlation matrix than ridge regression, but can increase off-diagonal correlations. Data assimilation experiments reveal that reconditioning corrects spurious noise in the analysis but underestimates the true signal compared to multiplicative variance inflation.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Numerical Frameworkmentioning

confidence: 99%

Section: Changes To the Covariance Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Improving the condition number of estimated covariance matrices

Tabeart¹,

Dance²,

Lawless³

et al. 2020

Tellus A: Dynamic Meteorology and Oceanography

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show abstract

Numerical linear algebra in data assimilation

Freitag

2020

GAMM-Mitteilungen

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Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

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New bounds on the condition number of the Hessian of the preconditioned variational data assimilation problem

Tabeart

Dance

Lawless

et al. 2021

Numerical Linear Algebra App

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Data assimilation algorithms combine prior and observational information, weighted by their respective uncertainties, to obtain the most likely posterior of a dynamical system. In variational data assimilation the posterior is computed by solving a nonlinear least squares problem. Many numerical weather prediction (NWP) centers use full observation error covariance (OEC) weighting matrices, which can slow convergence of the data assimilation procedure. Previous work revealed the importance of the minimum eigenvalue of the OEC matrix for conditioning and convergence of the unpreconditioned data assimilation problem. In this article we examine the use of correlated OEC matrices in the preconditioned data assimilation problem for the first time. We consider the case where there are more state variables than observations, which is typical for applications with sparse measurements, for example, NWP and remote sensing. We find that similarly to the unpreconditioned problem, the minimum eigenvalue of the OEC matrix appears in new bounds on the condition number of the Hessian of the preconditioned objective function. Numerical experiments reveal that the condition number of the Hessian is minimized when the background and observation lengthscales are equal. This contrasts with the unpreconditioned case, where decreasing the observation error lengthscale always improves conditioning. Conjugate gradient experiments show that in this framework the condition number of the Hessian is a good proxy for convergence. Eigenvalue clustering explains cases where convergence is faster than expected.

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The conditioning of least‐squares problems in variational data assimilation

Cited by 28 publications

References 66 publications

Improving the condition number of estimated covariance matrices

Improving the condition number of estimated covariance matrices

Numerical linear algebra in data assimilation

New bounds on the condition number of the Hessian of the preconditioned variational data assimilation problem

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