## 1. Introduction

This is the first part of a two-part article in which we estimate observation errors and their correlations for clear-sky radiances used in the European Centre for Medium-Range Weather Forecasts (ECMWF) system. In the first part we describe the methods used and summarize the results for instruments of the Advanced TIROS Operational Vertical Sounder (ATOVS) suite. The second part gives the results for advanced high-spectral-resolution infrared sounders. The assumed observation-error covariances, together with assumed background-error covariances, play an important role in determining the weight of a given observation in data-assimilation systems. For technical or computational reasons, observation-error covariance matrices used in data-assimilation systems are mostly assumed to be diagonal.

Satellite radiances currently provide the largest input to today's data-assimilation systems for numerical weather prediction (NWP), in terms of both numbers and forecast impact, but the assumption of uncorrelated error is questionable for these observations. This is true for spatial as well as interchannel error correlations. Observation errors used in data assimilation include errors from the observation operator, and radiative-transfer models are likely to exhibit correlated errors, for instance due to errors in the spectroscopy or in the assumed gas concentrations (Sherlock, 2000). Other aspects are also expected to lead to correlated observation errors, such as aspects of the instrument design or calibration, errors arising from the different representativeness of the radiances and the model fields, or some common practices of quality control used in the assimilation system (e.g. cloud screening). Neglecting spatial-error correlations in the assimilation can lead to sub-optimal analyses if the observations are used too densely (Liu and Rabier, 2003).

While there is general agreement that radiances potentially have correlated observation errors, relatively few estimates of such error correlations are available, especially in the case of spatial-error correlations. This is partly due to difficulties with the methods commonly used for such error estimation. A number of methods exist that are based on first-guess (FG) or analysis departures. Without further input, any such method can only be successful in separating FG errors and observation errors if the FG error and the observation error show sufficiently different characteristics (Dee and da Silva, 1999). A commonly made assumption is that FG errors are spatially correlated, whereas observation errors are not. This is the basis of the so-called Hollingsworth/Lönnberg method (Rutherford, 1972; Hollingsworth and Lönnberg, 1986). This method has been applied by Garand *et al.* (2007), who found considerable interchannel radiance-error correlations for data from the Atmospheric Infrared Sounder (AIRS). The assumption of spatially uncorrelated observation error of course rules out any estimates for such errors, and it is questionable in the case of satellite radiances, as outlined above. Another method has recently been used to estimate observation-error characteristics based on a consistency diagnostic summarized by Desroziers *et al.* (2005). The diagnostic can recover aspects of observation errors as long as the correlation scales of background and observation error are sufficiently different (Desroziers *et al.*, 2009). The method has been applied to estimate variances of observation errors, as well as interchannel error correlations (Ménard *et al.*, 2009; Stewart *et al.*, 2009). As long as the length-scales of FG and observation errors are sufficiently different, the diagnostic should be able to provide some estimates of spatial observation-error correlations, but no attempts at calculating these are known to the authors.

Related methods have been developed for observation-error tuning based on FG or analysis departures, based on fitting an assumed model for the observation-error correlations. Desroziers and Ivanov (2001) proposed a method to tune scaling coefficients for the observation-error covariance based on an optimality criterion for the cost function at the minimum, assuming the correlations are accurately represented in the initially assumed observation-error covariance. This has been applied by Chapnick *et al.* (2006) and others for diagonal observation-error covariance matrices. Related to this method is the maximum-likelihood estimation (Dee and da Silva, 1999), which directly fits free parameters of covariance models to FG-departure statistics. While these methods provide useful tools, the drawback in our case is that little is known about which covariance models would be appropriate for interchannel or spatial-covariance models for satellite radiances. Using incorrect covariance methods (for instance, assuming uncorrelated observation error when error correlations are present in the real data) can lead to undesired results in these estimates (Liu and Rabier, 2003; Chapnick *et al.*, 2006). A less constrained characterization of error correlations for radiance data is needed first.

Common methods to counteract spatial or interchannel error correlations are spatial thinning or error inflation. Both are applied widely in data-assimilation systems (Dando *et al.*, 2007; Collard and McNally, 2009). Guidance for selecting optimal thinning scales can be taken from Liu and Rabier (2003), who found that thinning scales of a threshold-error correlation value of around 0.2 produced the smallest analysis error when error correlations are neglected and the diagonal observation errors are not inflated. However, since reliable estimates of spatial-error correlations are lacking for radiance data, thinning scales currently used are mostly ad hoc estimates. Given the ever-increasing horizontal resolution of today's operational global NWP models, it is desirable to optimize the thinning or to take horizontal error correlations explicitly into account in the assimilation to enhance the spatial information extracted from satellite radiances.

In the present article we introduce the methods employed to estimate observation errors and their interchannel and spatial correlations. Given the difficulties of estimating observation errors, we employ three methods to better characterize the uncertainty inherent in these estimates. The aim is to provide guidance for the specification of observation-error covariances and thinning scales in data assimilation. The data used for this study are described in the next section, followed by an overview of the methods employed. Next, the estimation methods are applied to data from the ATOVS instruments, including the Advanced Microwave Sounding Unit (AMSU)-A, the Microwave Humidity Sounder (MHS) and the High-Resolution Infrared Radiation Sounder (HIRS). A discussion of the results and conclusions are provided in the last section. A companion article summarizes the results for the advanced high-spectral resolution sounders, that is the Atmospheric Infrared Sounder (AIRS) and the Infrared Atmospheric Sounding Interferometer (IASI) (see Bormann *et al.*, 2010).