arXiv subjects

Nancy K. Nichols

Publications and source records attributed to Nancy K. Nichols.

5 recordsLinked to original sources

arXiv2023

The Conditioning of Hybrid Variational Data Assimilation

In variational assimilation, the most probable state of a dynamical system under Gaussian assumptions for the prior and likelihood can be found by solving a least-squares minimization problem . In recent years, we have seen the popularity of hybrid variational data assimilation methods for Numerical Weather Prediction. In these methods, the prior error covariance matrix is a weighted sum of a climatological part and a flow-dependent ensemble part, the latter being rank deficient. The nonlinear least squares problem of variational data assimilation is solved using iterative numerical methods, and the condition number of the Hessian is a good proxy for the convergence behavior of such methods. In this paper, we study the conditioning of the least squares problem in a hybrid four-dimensional variational data assimilation (Hybrid 4D-Var) scheme by establishing bounds on the condition number of the Hessian. In particular, we consider the effect of the ensemble component of the prior covariance on the conditioning of the system. Numerical experiments show that the bounds obtained can be useful in predicting the behavior of the true condition number and the convergence speed of an iterative algorithm

math.NA↗

arXiv2021

Convergent least-squares optimisation methods for variational data assimilation

Data assimilation combines prior (or background) information with observations to estimate the initial state of a dynamical system over a given time-window. A common application is in numerical weather prediction where a previous forecast and atmospheric observations are used to obtain the initial conditions for a numerical weather forecast. In four-dimensional variational data assimilation (4D-Var), the problem is formulated as a nonlinear least-squares problem, usually solved using a variant of the classical Gauss-Newton (GN) method. However, we show that GN may not converge if poorly initialised. In particular, we show that this may occur when there is greater uncertainty in the background information compared to the observations, or when a long time-window is used in 4D-Var allowing more observations. The difficulties GN encounters may lead to inaccurate initial state conditions for subsequent forecasts. To overcome this, we apply two convergent GN variants (line search and regularisation) to the long time-window 4D-Var problem and investigate the cases where they locate a more accurate estimate compared to GN within a given budget of computational time and cost. We show that these methods are able to improve the estimate of the initial state, which may lead to a more accurate forecast.

math.OC↗

arXiv2021

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

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) centres 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 paper 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 e.g. 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 minimised 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.

math.NA↗

arXiv2019

Improving the condition number of estimated covariance matrices

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. 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. 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.

math.OC↗

arXiv2019

The impact of using reconditioned correlated observation error covariance matrices in the Met Office 1D-Var system

Recent developments in numerical weather prediction have led to the use of correlated observation error covariance (OEC) information in data assimilation and forecasting systems. However, diagnosed OEC matrices are often ill-conditioned and may cause convergence problems for variational data assimilation procedures. Reconditioning methods are used to improve the conditioning of covariance matrices while retaining correlation information. In this paper we study the impact of using the 'ridge regression' method of reconditioning to assimilate Infrared Atmospheric Sounding Interferometer (IASI) observations in the Met Office 1D-Var system. This is the first systematic investigation of how changing target condition numbers affects convergence of a 1D-Var routine. This procedure is used for quality control, and to estimate key variables (skin temperature, cloud top pressure, cloud fraction) that are not analysed by the main 4D-Var data assimilation system. Our new results show that the current (uncorrelated) OEC matrix requires more iterations to reach convergence than any choice of correlated OEC matrix studied. This suggests that using a correlated OEC matrix in the 1D-Var routine would have computational benefits for IASI observations. Using reconditioned correlated OEC matrices also increases the number of observations that pass quality control. However, the impact on skin temperature, cloud fraction and cloud top pressure is less clear. As the reconditioning parameter is increased, differences between retrieved variables for correlated OEC matrices and the operational diagonal OEC matrix reduce. As correlated choices of OEC matrix yield faster convergence, using stricter convergence criteria along with these matrices may increase efficiency and improve quality control.

physics.ao-ph↗