A judicious assessment of the observational data impact on a forecast aspect requires the development of observation sensitivity techniques that account for the specifics of the data assimilation system (Baker and Daley 2000, Langland and Baker 2004, Zhu and Gelaro 2007).

An optimal control perspective is used to provide forecast sensitivity to the observations and to the background field in 4D-Var data assimilation. The 4D-Var observation sensitivity framework accounts for time distributed data and it is shown to be a natural extension to the original work of Baker and Daley (2000). The impact of any 4D-Var assimilated data subset on the forecast skill may be thus estimated.

The model fit to each observational data set is weighted in the assimilation process by the inverse observation error covariance matrix and valuable information may be lost if the observation errors are misrepresented. A better understanding of the impact of the observation errors on the forecast may also point to the directions where improvements in the current observing system and data representation would be of most benefit. The equations of the forecast sensitivity to the observation error covariance matrix are presented and the relationship between the observation sensitivity and the sensitivity to the observation error is discussed.

In practical applications the estimation of the observation sensitivity in the full state space is hampered by the high computational requirements that involve the Hessian of the 4D-Var cost functional. To overcome this difficulty, a reduced-order observation sensitivity approach is formulated using a projection on a low-rank state subspace. An optimal basis to the reduced space is shown to be closely related to the Hessian singular vectors optimized at the observation time. A computationally feasible method is proposed to identify a projection operator on a low-dimensional control space that incorporates in a consistent fashion information pertinent to the 4D-Var data assimilation procedure and to the forecast sensitivity.

Idealized twin experiments are performed with a Lin-Rood finite volume global shallow-water model and initial conditions from Williamson et al. (1992) and from ECMWF ERA-40 data sets. A nonlinear 4D-Var assimilation scheme is implemented using first and second order adjoint modeling to provide gradient and Hessian information, respectively.

Numerical results and a comparative analysis with observation sensitivities evaluated in the full model space indicate that the reduced-order approach provides at a low computational cost accurate estimates of sensitivity to observations distributed in the time-space domain. Further simplifications that may result form the use of an incremental 4D-Var scheme and applications to targeted observations are discussed.