## 1. Introduction

Atmospheric data assimilation systems (DASs) rely on the state estimation theory to ingest the information content of observations into numerical weather prediction (NWP) models (Jazwinski, 1970; Lahoz *et al.*, 2010). The observation performance in reducing the analysis and forecast errors is closely determined by the representation in the DAS of the statistical properties of the errors in the prior state estimate, model, and observations. Consistency diagnostics for the covariances of the observation and background errors may be obtained *a* *posteriori* from the statistical analysis of the DAS products (Talagrand, 1999; Desroziers *et al.*, 2005). Diagnosis studies indicate that both spatial and inter-channel error correlations are present in the satellite radiances provided by the atmospheric sounders (Garand *et al.*, 2007; Bormann and Bauer, 2010; Bormann *et al.*, 2010, 2011; Gorin and Tsyrulnikov, 2011) and additional error correlations are introduced when order reduction techniques are used to extract the information content of high-resolution datasets (Collard *et al.*, 2010).

A common approach taken at NWP centres is to assign a diagonal weight matrix to the observational component, and error variance inflation is performed to compensate for unrepresented error correlations. Quantification of the loss of information as a result of suboptimal weighting, and implementation of efficient procedures to adjust the error covariance parameters to a configuration that improves the forecasts' skill, are areas of active research. Synergistic efforts include the development of error covariance models for NWP applications (Derber and Bouttier, 1999; Gaspari and Cohn, 1999; Lorenc, 2003; Fisher, 2003; Bannister, 2008a, 2008b; Frehlich, 2011; Bishop *et al.*, 2011; Raynaud *et al.*, 2011) and of computationally feasible techniques for diagnosis, estimation, and tuning of the error covariance parameters (Wahba *et al.*, 1995; Dee, 1995; Dee and Da Silva, 1999; Andersson *et al.*, 2000; Desroziers and Ivanov, 2001; Cardinali *et al.*, 2004; Buehner *et al.*, 2005; Chapnik *et al.*, 2006; Zupanski and Zupanski, 2006; Trémolet, 2007; Anderson, 2007; Liu and Kalnay, 2008; Desroziers *et al.*, 2009; Li *et al.*, 2009).

Observing system experiments (OSEs) are the traditional tool to assess the observation value (Atlas, 1997; Kelly *et al.*, 2007) and, to date, studies on the forecast impact as a result of variations in the specification of the observation- and background-error covariance parameters have been only performed through additional data assimilation experiments (Zhang and Anderson, 2003; Joiner *et al.*, 2007). OSEs allow the investigation of only a few parameters in the DAS since a new experiment is required for each parameter input. The assimilation of radiances from hyperspectral instruments such as the Atmospheric Infrared Sounder (AIRS) and the Infrared Atmospheric Sounding Interferometer (IASI) has increased the number of ‘tunable’ parameters to a dimension where an analysis by trial-and-error is not feasible. Computationally efficient techniques are necessary to identify those error covariance parameters of potentially large forecast impact and to obtain insight on the performance of a new error covariance model *prior to* its actual implementation in the DAS i.e. without performing an additional assimilation experiment. The adjoint approach to parameter sensitivity and impact estimation provides a basis to advance research in this area.

Baker and Daley (2000) have shown that an *all-at-once* evaluation of the forecast sensitivity to observations may be performed by developing the adjoint of the data assimilation system (adjoint-DAS). The analysis of the information content of observations and the observation impact assessment through observation sensitivity and adjoint-DAS techniques are routine activities at NWP centres to monitor the observing system performance on reducing the short-range forecast errors (Langland and Baker, 2004; Trémolet, 2008; Baker and Langland, 2009; Cardinali, 2009; Daescu and Todling, 2009; Gelaro and Zhu, 2009; Gelaro *et* *al.*, 2010; Cardinali and Prates, 2011; Lupu *et al.*, 2011). The adjoint-DAS applications may be extended to incorporate the sensitivity analysis with respect to error covariance parameters and the estimation of the forecast impact from adjusting the error covariance models. Daescu (2008) derived the equations of the forecast sensitivity with respect to the observation-error covariance model (**R**-sensitivity) and to the background-error covariance model (**B**-sensitivity) in a nonlinear four-dimensional variational (4D-Var) DAS. The practical ability to estimate the forecast sensitivity to **R**- and **B**-weight parameters in a 3D-Var DAS was shown by Daescu and Todling (2010).

The current work presents novel theoretical aspects of the adjoint-DAS error covariance sensitivity analysis and first applications in an operational 4D-Var data assimilation and forecast system, the Naval Research Laboratory Atmospheric Variational Data Assimilation System–Accelerated Representer (NAVDAS-AR; Daley and Barker, 2001; Xu *et al.*, 2005; Rosmond and Xu, 2006) and the Navy Operational Global Atmospheric Prediction System (NOGAPS; Hogan and Rosmond, 1991).

The article is organized as follows. A review of the analysis equation in a strong-constraint 4D-Var implementing a single outer-loop iteration is put forward in section 2 to elucidate the notational convenience and the four-dimensional structure of the operators involved in the error covariance sensitivity estimation. Section 3 provides the derivation of the forecast **R**- and **B**-sensitivity equations consistent with the analysis scheme. Special care is taken to account for the dependence of the linearized model and the linearized observation operator on the background forecast, and it is shown that both **R**- and **B**-sensitivities may be evaluated without the need of a second-order adjoint model. Section 4 includes additional sensitivity properties and applications of the adjoint-DAS approach as a guidance tool to error covariance parameter tuning and impact assessment. A computationally efficient procedure to evaluate the first-order forecast impact of a low-rank **B**-covariance update is presented. In section 5, numerical results obtained with NAVDAS-AR/NOGAPS are used to emphasize the complementary information derived from various techniques to analyze the DAS performance: observation impact, sensitivity to error covariance weighting, and error covariance diagnosis. First applications are also presented of the adjoint-DAS observation-error covariance sensitivity to provide guidance on the forecast impact of error variance estimates derived from an *a posteriori* diagnosis. A summary and further research perspectives are in section 6.