## 1. Introduction

Over the last decade, data assimilation schemes have evolved towards very sophisticated systems, such as the four-dimensional variational system (4D-Var) (Rabier *et al.*, 2000) that operates at the European Centre for Medium-Range Weather Forecasts (ECMWF). The scheme handles a large variety of both space and surface-based meteorological observations. It combines the observations with prior (or background) information on the atmospheric state and uses a comprehensive (linearized) forecast model to ensure that the observations are given a dynamically realistic, as well as statistically likely, response in the analysis. Effective performance monitoring of such a complex system, with an order of 10^{8} degrees of freedom and more than 10^{7} observations per 12 h assimilation cycle, has become an absolute necessity.

The assessment of the contribution of each observation to the analysis is among one of the most challenging diagnostics in data assimilation and numerical weather prediction. Methods have been derived to measure the observational influence in data assimilation schemes (Purser and Huang, 1993; Fisher, 2003; Cardinali *et al.*, 2004; Chapnick *et al.*, 2004). These techniques therefore show how the influence is assigned during the assimilation procedure, which partition is given to the observation and which is given to the background or pseudo-observation. They therefore provide an indication of the robustness of the fit between model and observations and allow some tuning of the weights assigned in the assimilation system. Measures of the observational influence are useful for understanding the data assimilation (DA) scheme itself. How large is the influence of the latest data on the analysis and how much influence is due to the background? How much would the analysis change if one single influential observation were removed? How much information is extracted from the available data?

To answer these questions, it was necessary to consider the diagnostic methods that have been developed for monitoring statistical multiple regression analyses; 4D-Var is, in fact, a special case of the generalized least square (GLS) problem (Talagrand, 1997) for weighted regression, thoroughly investigated in the statistical literature.

Recently, adjoint-based observation sensitivity techniques have been used (Baker and Daley, 2000; Cardinali and Buizza, 2004; Langland and Baker, 2004; Morneau *et**al.*, 2006; Zhu and Gelaro, 2008; Cardinali, 2009) to measure the observation contribution to the forecast, where the observation impact is evaluated with respect to a scalar function representing the short-range forecast error. In general, the adjoint methodology can be used to estimate the sensitivity measure with respect to any parameter of importance of the assimilation system. Very recently, Daescu (2008) and Daescu and Todling (2010) derived a sensitivity equation of an unconstrained variational DA system from the first-order necessary condition with respect to the main input parameters: observation, background and their error covariance matrices. The paper provides the theoretical framework for further diagnostic tool development not only to evaluate the observation impact on the forecast but also the impact of the other analysis parameters. Sensitivity to background covariance matrix can help in evaluating the correct specification of the background weight and their correlation. Limitations and weaknesses of the covariance matrices are well known, and several assumptions and simplifications are made to derive them. Desroziers and Ivanov (2001) and Chapnik *et al.* (2006) discussed the importance of diagnosing and tuning the error variances in a data assimilation scheme.

The adjoint-based observation sensitivity technique measures the impact of observations when the entire observation dataset is present in the assimilation system. It also measures the response of a single forecast metric to all perturbations of the observing system. It provides the impact of all observations assimilated at a single analysis time.

The adjoint-based technique is limited by the tangent linear assumption, valid up to 3 days. Furthermore, a simplified adjoint model is usually used to carry the forecast error information backwards, which further limits the validity of the linear assumption, and therefore restricts the use of the diagnostic to a typical forecast range of 24–48 h.

In this paper, a comprehensive assessment of the impact of all-sky microwave imager radiances in the assimilation and forecast system is obtained by applying the diagnostic tools introduced above. All-sky assimilation of microwave imager radiances became operational in March 2009 (Bauer *et al.*, 2010; Geer *et al.*, 2010). In the all-sky approach, radiance observations from the Special Sensor Microwave/Imager (SSM/I; Hollinger *et al.*, 1990) and Advanced Microwave Scanning Radiometer for the Earth Observing System (AMSR-E; Kawanishi *et al.*, 2003) are assimilated under all atmospheric conditions, whether clear, cloudy or rainy. At the frequencies used by microwave imagers, the atmosphere is semi-transparent except in heavy cloud and precipitation conditions. Observations are only assimilated over oceans, where the observations are sensitive to ocean surface properties (e.g. surface temperature and wind speed), atmospheric water vapour, cloud, hydrometeor profiles and precipitation.

Recently, a complete revision of observation error variance, quality control, thinning and resolution matching of the all-sky assimilation of microwave imagers was implemented in the operational ECMWF 4D-Var system (Bauer *et al.*, 2010; Geer *et al.*, 2010).

As defined, observation error variance and bias correction make use of information from both model forecast and observations. Because cloud and precipitation structures are often misplaced in the model forecast, a variational bias correction that is only based on model information would often misrepresent cases where model and observations disagree on the sky representation. Therefore, the observation error variances and the observation bias correction are, in the new approach, a function of the mean cloud amount as indicated by the model and the observations (Geer and Bauer, 2010). The spatial scale of the observations has also been examined. Instead of taking the nearest single all-sky observation to a grid point, an average or ‘superob’ of all observations falling into a grid box is calculated prior to the assimilation. Also, the new approach screens out observations where the model shows very cold and heavy snowfall biases. These biases are in fact too difficult to deal with using a predictor-based bias correction scheme.

The paper is organized as follows: sections 2 and 3 describe the mathematical framework of the adjoint tools used in the investigation. Section 4 presents the results, and conclusions are provided in section 5.