On the Sensitivity Equations of Four-Dimensional Variational (4D-Var) Data Assimilation

Abstract

The equations of the forecast sensitivity to observations and to the background estimate in a fourdimensional variational data assimilation system (4D-Var DAS) are derived from the first-order optimality condition in unconstrained minimization. Estimation of the impact of uncertainties in the specification of the error statistics is considered by evaluating the sensitivity to the observation and background error covariance matrices. The information provided by the error covariance sensitivity analysis is used to identify the input components for which improved estimates of the statistical properties of the errors are of most benefit to the analysis and forecast. A close relationship is established between the sensitivities within each input pair data/error covariance such that once the observation and background sensitivities are available the evaluation of the sensitivity to the specification of the corresponding error statistics requires little additional computational effort. The relevance of the 4D-Var sensitivity equations to assess the data impact in practical applications is discussed. Computational issues are addressed and idealized 4D-Var experiments are set up with a finite-volume shallow-water model to illustrate the theoretical concepts. Time-dependent observation sensitivity and potential applications to improve the model forecast are presented. Guidance provided by the sensitivity fields is used to adjust a 4D-Var DAS to achieve forecast error reduction through assimilation of supplementary data and through an accurate specification of a few of the background error variances.

A Second Order Adjoint Method to Targeted Observations

“…It is also noticed that insertion of targeted observations at t = 6h is of significant benefit to the forecast, indicating a larger forecast impact from these observations. This is consistent to the observation sensitivity study in [6] where is was found that the forecast sensitivity to observations increases for observations near the end of the assimilation window, and thus closer to the verification time. In addition, accounting for data interaction is essential when multiple targeting instants are considered [15], and this is an area where further research is much needed.…”

“…Adjoint modeling has been an essential tool for the development of targeting strategies in the context of variational data assimilation methods. The adjoint of the tangent linear model associated to an atmospheric model is a key ingredient to implementing various targeting strategies, such as gradient sensitivity, dominant singular vectors, and sensitivity to observations ( [1], [2], [3], [4], [5], [6]). …”

Adjoint sensitivity of the model forecast to data assimilation system error covariance parameters

“…where H = ∂h/∂x ∈ R p×n is the Jacobian matrix of the observation operator at x a , allows close relationships to be established between the forecast sensitivities to observations/background and to the associated error covariances (Daescu, 2008):…”

“…In Daescu (2008), the error covariance sensitivity equations (14) and (16) were expressed in column vector format. The matrix format adopted here emphasizes the rank-one property of the B-and R-forecast sensitivity matrices, and it is more suitable for the purpose of this study.…”

“…The derivation of the sensitivity equations and applications of the adjoint-DAS approach to error covariance sensitivity analysis in VDA are presented by Daescu (2008). The current work investigates novel adjoint-DAS applications to parametric error covariance sensitivity and forecast impact assessment and as a tool to provide DAS diagnostics.…”