Abstract
SummaryIn variational data assimilation a least‐squares objective function is minimised to obtain the most likely state of a dynamical system. This objective function combines observation and prior (or background) data weighted by their respective error statistics. In numerical weather prediction, data assimilation is used to estimate the current atmospheric state, which then serves as an initial condition for a forecast. New developments in the treatment of observation uncertainties have recently been shown to cause convergence problems for this least‐squares minimisation. This is important for operational numerical weather prediction centres due to the time constraints of producing regular forecasts. The condition number of the Hessian of the objective function can be used as a proxy to investigate the speed of convergence of the least‐squares minimisation. In this paper we develop novel theoretical bounds on the condition number of the Hessian. These new bounds depend on the minimum eigenvalue of the observation error covariance matrix and the ratio of background error variance to observation error variance. Numerical tests in a linear setting show that the location of observation measurements has an important effect on the condition number of the Hessian. We identify that the conditioning of the problem is related to the complex interactions between observation error covariance and background error covariance matrices. Increased understanding of the role of each constituent matrix in the conditioning of the Hessian will prove useful for informing the choice of correlated observation error covariance matrix and observation location, particularly for practical applications.
Highlights
One of the most well-known applications of data assimilation is to numerical weather prediction (NWP), where observations of the atmosphere and ocean are combined with a prior model state of the atmosphere in order to produce the initial conditions for a weather forecast
We develop a new theory for bounding the condition number of the Hessian of the least-squares objective function
We present new bounds on the condition number of the Hessian given by (3)
Summary
Data assimilation combines the output from a numerical model of a dynamical system, the background or prior, with observations of the system to yield an accurate description of the current dynamical state (analysis). This work provides motivation to investigate further the role of the minimum eigenvalue of the observation error covariance matrix on the conditioning of the variational data assimilation problem; in turn, developing this crucial understanding will permit the optimal use of correlated observation errors in data assimilation systems. We develop a new theory for bounding the condition number of the Hessian of the least-squares objective function This theory applies to both uncorrelated and correlated choices of observation error. We see that the minimum eigenvalue of the observation error covariance matrix and the ratio of background variance to observation variance are important terms for controlling the conditioning of the variational problem for both the bounds in Section 3 and the numerical results from Section 5. The primary motivation for the investigation of the impact of correlated observation errors arises from their application in meteorology, the theory and conclusions presented here are very general and apply to any other application of variational data assimilation such as in neuroscience[33,34] and ecology.[35,36]
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