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Dacian N. Daescu

Abstract

The equations of the forecast sensitivity to observations and to the background estimate in a four-dimensional 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.

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Dacian N. Daescu

Abstract

An optimal use of the atmospheric data in numerical weather prediction requires an objective assessment of the value added by observations to improve the analyses and forecasts of a specific data assimilation system (DAS). This research brings forward the issue of uncertainties in the assessment of observation values based on deterministic observation impact (OBSI) estimations using observing system experiments (OSEs) and the adjoint-DAS framework. The state-to-observation space uncertainty propagation as a result of the errors in the verification state is investigated. For a quadratic forecast error measure, a geometrical perspective is used to provide insight and to convey some of the key aspects of this research. The study is specialized to a DAS implementing a linear analysis scheme and numerical experiments are presented using the Lorenz 40-variable model.

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Dacian N. Daescu and Ricardo Todling

Abstract

A parametric approach to the adjoint estimation of the variation in model functional output due to the assimilation of data is considered as a tool to analyze and develop observation impact measures. The parametric approach is specialized to a linear analysis scheme and it is used to derive various high-order approximation equations. This framework includes the Kalman filter and incremental three-and four-dimensional variational data assimilation schemes implementing a single outer loop iteration. Distinction is made between Taylor series methods and numerical quadrature methods. The novel quadrature approximations require minimal additional software development and are suitable for testing and implementation at operational numerical weather prediction centers where a data assimilation system (DAS) and the associated adjoint DAS are in place. Their potential use as tools for observation impact estimates needs to be further investigated. Preliminary numerical experiments are provided using the fifth-generation NASA Goddard Earth Observing System (GEOS-5) atmospheric DAS.

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Dacian N. Daescu and Gregory R. Carmichael

Abstract

The spatiotemporal distribution of observations plays an essential role in the data assimilation process. An adjoint sensitivity method is applied to the problem of adaptive location of the observational system for a nonlinear transport-chemistry model in the context of 4D variational data assimilation. The method is presented in a general framework and it is shown that in addition to the initial state of the model, sensitivity with respect to emission and deposition rates and certain types of boundary values may be obtained at a minimal additional cost. The adjoint modeling is used to evaluate the influence function and to identify the domain of influence associated with the observations. These essential tools are further used to develop a novel algorithm for targeting observations that takes into account the interaction among observations at different instants in time and spatial locations. Issues related to the case of multiple observations are addressed and it is shown that by using the adjoint modeling an efficient implementation may be achieved. Computational and practical aspects are discussed and this analysis indicates that it is feasible to implement the proposed method for comprehensive air quality models. Numerical experiments performed with a two-dimensional test model show promising results.

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Francois-Xavier Le Dimet, I. M. Navon, and Dacian N. Daescu

Abstract

In variational data assimilation (VDA) for meteorological and/or oceanic models, the assimilated fields are deduced by combining the model and the gradient of a cost functional measuring discrepancy between model solution and observation, via a first-order optimality system. However, existence and uniqueness of the VDA problem along with convergence of the algorithms for its implementation depend on the convexity of the cost function. Properties of local convexity can be deduced by studying the Hessian of the cost function in the vicinity of the optimum. This shows the necessity of second-order information to ensure a unique solution to the VDA problem.

In this paper a comprehensive review of issues related to second-order analysis of the problem of VDA is presented along with many important issues closely connected to it. In particular issues of existence, uniqueness, and regularization through second-order properties are examined. The focus then shifts to second-order information related to statistical properties and to issues related to preconditioning and optimization methods and second-order VDA analysis. Predictability and its relation to the structure of the Hessian of the cost functional is then discussed along with issues of sensitivity analysis in the presence of data being assimilated. Computational complexity issues are also addressed and discussed.

Automatic differentiation issues related to second-order information are also discussed along with the computational complexity of deriving the second-order adjoint.

Finally an application aimed at illustrating the use of automatic differentiation for deriving the second-order adjoint as well as the Hessian/vector product applied to minimizing a cost functional of a meteorological problem using the truncated-Newton method is presented. Results verifying numerically the computational cost of deriving the second-order adjoint as well as results related to the spectrum of the Hessian of the cost functional are displayed and discussed.

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