Estimates of spatial and interchannel observation-error characteristics for current sounder radiances for numerical weather prediction. I: Methods and application to ATOVS data

Authors


Abstract

This is the first part of a two-part article that uses three methods to estimate observation errors and their correlations for clear-sky sounder radiances used in the European Centre for Medium-Range Weather Forecasts (ECMWF) assimilation system. The analysis is based on covariances derived from pairs of first-guess and analysis departures. The methods used are the so-called Hollingsworth/Lönnberg method, a method based on subtracting a scaled version of mapped assumed background errors from first-guess departure covariances and the Desroziers diagnostic. The present article reports the results for the three Advanced TIROS Operational Vertical Sounder (ATOVS) instruments: the Advanced Microwave Sounding Unit (AMSU)-A, High-Resolution Infrared Radiation Sounder (HIRS) and Microwave Humidity Sounder (MHS).

The findings suggest that all AMSU-A sounding channels show little or no interchannel or spatial observation-error correlations, except for surface-sensitive channels over land. Estimates for the observation error are mostly close to the instrument noise. In contrast, HIRS temperature-sounding channels exhibit some interchannel error correlations, and these are stronger for surface-sensitive channels. There are also indications for stronger spatial-error correlations for the HIRS short-wave channels. There is good agreement between the estimates from the three methods for temperature-sounding channels.

Estimating observation errors for humidity-sounding channels of MHS and HIRS appears more difficult. A considerable proportion of the observation error for humidity-sounding channels appears correlated spatially for short separation distances, as well as between channels. Observation error estimates for humidity channels are generally considerably larger than the instrument noise.

Observation error estimates from this study are consistently lower than those assumed in the ECMWF assimilation system. As error correlations are small for AMSU-A, the study suggests that the current use of AMSU-A data in the ECMWF system in terms of observation-error or thinning-scale choices is fairly conservative. Copyright © 2010 Royal Meteorological Society

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).

2. Data

The statistics presented here are based on FG and analysis departures for pairs of observations for the respective instruments. The observations in each pair are required to be less than 1 hour apart and originate from the same instrument on the same satellite. All possible pairs were collected over the study period, and the pairs of observations were binned by separation distance, using a binning interval of 25 km unless indicated otherwise. This allows us to calculate isotropic spatial covariance statistics as a function of separation distance. Note that for longer separation distances the sampling for these statistics will be biased towards the north– south direction, as separations in the east– west direction are limited by the instrument's swath width. Departures from isotropy were also investigated, and they are presented where noteworthy effects have been found.

The departures were taken from the ECMWF system, using only data that were actively assimilated. The use of assimilated observations is a requirement of the Desroziers diagnostic (Desroziers et al., 2005; see below). We restrict ourselves to fields of view (FOVs) for which all active channels of an instrument are diagnosed as cloud-free and pass further quality control checks. This is in contrast to the practice employed in the assimilation where the aim is to identify clear-sky channels rather than clear-sky FOVs. The reason for restricting ourselves to clear-sky FOVs is primarily to simplify the statistics and to harmonize the sampling for different channels in order to estimate interchannel error correlations. The methods employed for cloud-screening are summarized in the results section for each instrument.

During the course of this study, we analysed departures taken from assimilation experiments performed for different periods, at different model resolution (ranging from T255 to T799) and with different thinning intervals (operational thinning and halved thinning interval). While some seasonal variations in the statistics exist, the overall results for the observation-error covariance estimates were comparable for these variations, and differences were usually within the range of results from the three methods considered in this study. The results presented here are based on data for the 21-day period 22 August– 11 September 2008. The FG and analysis departures were taken from an assimilation experiment that used four-dimensional variational data assimilation (4DVAR) with a 12 hour observation window, a model resolution of T799 (≈25 km), an incremental analysis resolution of T255 (≈80 km) and 91 levels in the vertical up to 0.01 hPa. The version of the assimilation system and the data selection was the same as that used in operations in April 2009, except that the thinning scale was approximately halved for all radiance data in our experiment. For the results presented here, one radiance datum is selected per 60 km × 60 km box for each instrument where data is available. This increases the sample size for our purposes, especially for small separation distances. The reduction of the thinning scale did not negatively affect the forecast accuracy. Note that in the chosen approach to thinning (i.e. selecting one radiance datum per 60 km × 60 km box), observations can still be less than 60 km apart, for instance if the two selected observations are close to the common border of neighbouring thinning boxes.

We use departure statistics after bias correction, as only these influence the atmospheric analysis. The bias correction is performed within the analysis using variational bias correction (Dee, 2004). Unless indicated otherwise, the bias correction uses a linear model for the airmass bias, with a constant component and four layer thicknesses calculated from the FG as predictors (1000– 300 hPa, 200– 50 hPa, 50– 5 hPa, 10– 1 hPa). Scan biases are modelled through a third-order polynomial in the scan angle. The bias correction will partly correct for errors in the radiative transfer, as these tend to introduce large-scale air-mass-dependent biases (Bormann et al., 2009). The radiative-transfer model used in this study is RTTOV, version 9 (Bormann et al., 2009; Saunders et al., 1999).

3. Methods

In the following, we describe the methods used here and in the companion article to estimate the observation errors and their correlations. They are all based on the FG or analysis departure statistics from the database of pairs of observations introduced above. The observation-error covariances are intended to be the sum of all errors relevant to the interpretation of the radiances in data assimilation. This includes instrument and calibration errors, errors of representation (from the representation of different scales in the horizontal or vertical in the radiances and the model data) and errors in the observation operator (i.e. errors in the radiative transfer used to assimilate the radiances, such as errors in spectroscopy), as long as they have not been addressed by the bias correction. We will use the term ‘observation error’ to refer to the diagonal of the observation-error covariance matrix.

3.1. Hollingsworth/Lönnberg

This method is based on the assumption that true background errors are spatially correlated, whereas observation errors are spatially uncorrelated. As a result, observation errors can be estimated by calculating FG-departure covariances from pairs of FG departures as a function of separation distance (see, for instance, the black lines in Figure 1). Observation errors are estimated by extrapolating the covariance/separation relationship from non-zero separations to zero separation, so that the FG-departure variance at zero separation is split into a spatially correlated part and a spatially uncorrelated component. The latter is assumed to give the observation error. The method also assumes that observation and background errors are uncorrelated. The method has been applied numerous times to estimate background errors from FG departures of radiosonde networks (Rutherford, 1972; Hollingsworth and Lönnberg, 1986), or, more recently, errors and interchannel error correlations for AIRS radiances (Garand et al., 2007). A variant of the method has been used to estimate spatial-error correlations in atmospheric motion vectors (AMVs) from differences between AMVs and radiosondes (Bormann et al., 2003). This variant is not well-suited to radiance data, as standard radiosonde observations do not reach high enough in the atmosphere to perform the necessary radiative-transfer calculations. More details on the Hollingsworth/Lönnberg method can be found in the above references. The method will be used here to estimate the spatially uncorrelated component of the observation errors and their interchannel error correlations.

Figure 1.

First-guess departure covariances (black), Desroziers background-error diagnostic (solid grey) and mapped background-error covariance (dashed grey) as a function of separation distance for the NOAA-18 AMSU-A channels used in the ECMWF system. The number of collocations as a function of separation distance is shown in the last panel; the spikes in the number of pairs for certain separation distances are due to Moiré effects from the binning interval and the AMSU-A sampling grid. Separation bins with fewer than 5000 observation pairs are not shown.

To perform the extrapolation to zero distance from non-zero separations, a correlation function is frequently fitted to the covariance statistics as a function of separation distance for non-zero separations. For most data considered in this article, the shortest separation distances are fairly small compared with the length-scales of background-error correlations. Instead of employing such a correlation function, we therefore subtract the FG-departure covariance at the first sufficiently populated non-zero separation bin from the one for zero separation. We found this to produce more robust results, as the use of a correlation function gives results that are highly dependent on the choice of correlation function. The first sufficiently populated separation bin is typically in the range of 12.5– 50 km, and the actual choice is stated by instrument in the results section. As FG-departure covariances tend to increase with decreasing separation, neglecting the use of a correlation function introduces an underestimation of the spatially correlated part of the FG-departure variances, and hence an overestimation of the observation error. On the other hand, the presence of any spatially correlated observation error will lead to an underestimation of the observation error, as such spatial correlations are neglected.

The assumption that observation errors are spatially uncorrelated is questionable in the case of satellite radiances, as outlined earlier, for instance due to spatially correlated radiative-transfer or screening errors. However, Garand et al. (2007) argue that such radiative-transfer or screening errors are reduced through the bias correction. Nevertheless, it should be kept in mind that the Hollingsworth/Lönnberg method will only be able to estimate the spatially uncorrelated part of the observation error, and the results will be misleading if there are significant spatial observation-error correlations.

3.2. Background-error method

This method uses covariances of FG departures and subtracts from these the assumed background errors, mapped into radiance space, and possibly scaled as described below. The background-error estimates are taken from the assimilation system, and they have been derived using the ensemble method (Fisher, 2003). The background-error-subtraction method assumes that observation and background errors are uncorrelated, and that the assumed background errors provide good estimates of the true background errors. The method is applied to derive spatial and interchannel observation-error characteristics.

The spatial and interchannel background-error characteristics in radiance space were calculated from an ensemble of 50 random perturbations to short-term forecast fields, with perturbations consistent with the assumed background-error characteristics. These global perturbations were calculated at a horizontal truncation of T255, consistent with the incremental analysis resolution of the assimilation system configuration used in this study. The perturbations were calculated for the 12 hour analysis cycle for 1 September 2008 0000 Z, in the middle of our study period. The perturbations were mapped from the analysis variables into radiance space using the tangent linear of the radiative-transfer model, assuming nadir viewing conditions. The resulting radiance perturbations were sampled at the respective observation locations for the instruments used in this study. Interchannel and spatial background-error covariances were derived from these perturbations in the same way as the FG-departure covariances introduced earlier.

Note that the mapped background errors will be hampered by an incomplete modelling of skin-temperature errors for channels with strong sensitivity to the surface. Skin temperature is not an analysis variable in the ECMWF system. Over sea, skin temperatures are prescribed through a sea-surface-temperature analysis (Stark et al., 2007). For our background-error computations we assumed an error of 0.4 K for the skin-temperature error over sea, based on in situ validation of the sea-surface-temperature analysis (Stark et al., 2007). This error is assumed to be spatially uncorrelated. This assumption is probably unrealistic, so the mapped background errors will show unrealistic spatial characteristics for channels with strong sensitivity to the surface. For the same reason, the method is not applied for such channels over land, where skin-temperature errors are expected to be larger. More details are provided for the channels in question.

During the course of this work, spatial characteristics of the mapped assumed background errors indicated that the assumed background errors are too large or too small when compared with spatial FG-departure covariances for some channels. In cases where the spatial correlation structures nevertheless appeared consistent with FG-departure covariances, but only the magnitude appeared off, a channel-specific scaling factor was introduced to make FG-departure covariances and the mapped background errors more consistent for larger separation distances. The scaling factor was calculated from data over separation distances between 200 and 1200 km. The scaling was only performed when the mapped background-error covariances were larger than the FG-departure covariances at larger separation distances. This is to avoid a situation in which the scaling masks indications of spatial observation-error correlations.

Used with scaling, the method becomes an extension of the Hollingsworth/Lönnberg method: we assume that background errors dominate FG-departure covariances at larger separation distances, and use as correlation function the empirical relationship between mapped assumed background errors and separation distance. We therefore allow for some spatial observation-error correlations at shorter distances by assuming that the spatial background-error correlations follow this empirical relationship. In the following we will refer to this method simply as the background-error method.

3.3. Desroziers diagnostic

Assuming that variational data-assimilation schemes broadly follow linear estimation theory, consistency diagnostics can be derived for observation, background and analysis errors in observation space from FG and analysis departures. These diagnostics have been derived and summarized by Desroziers et al. (2005), and here we make use of the following relationships:

equation image(1)
equation image(2)

where is the diagnosed observation-error covariance matrix, is the diagnosed background-error covariance matrix, H is the linearized observation operator, db are the background departures of the observations, da are the analysis departures of the observations and equation image is the expectation operator. Apart from the usual assumptions on Gaussian errors and no error correlations between FG and observation, etc., the diagnostic expressions also assume that the weight given to the observations in the analysis is in agreement with the true error covariances.

While primarily introduced as a consistency diagnostic, Desroziers et al. (2005) argue that the diagnostic equations may be used to estimate improved versions of the background- or observation-error covariances. They point out that the diagnostic equations formulate a fixed-point problem, and the solution may be derived iteratively by using the diagnosed values in a subsequent assimilation, which is then used again to calculate the diagnostics. The method has been used to estimate observation errors and interchannel error correlations (Ménard et al., 2009; Stewart et al., 2009). For a simple case, Desroziers et al. (2005) show that the method has the capability of retrieving spatial correlation structures of observation errors, even if the initial assumed observation error is uncorrelated. This is possible as long as the true background errors and the true observation errors have sufficiently different correlation structures. The applicability of the method and its properties in the case of estimating spatial observation-error correlations in realistic assimilation systems is an area of active research.

In the current article, we refer to the results of Eqns (1) and (2) as the Desroziers diagnostics, and we use the results as further estimates of improved observation or background errors. The method is used to obtain spatial as well as interchannel error correlations. For the current article, we do not use the diagnostic observation-error characteristics in subsequent assimilations, i.e. we show results only after one iteration of the tuning method suggested by Desroziers et al. (2005).

It should be noted here that all three methods assume that errors in the FG and observation errors are uncorrelated. The assumption, however, is not strictly true. Quality control based on FG departures is likely to introduce apparent correlations between FG errors and observation errors. Representativeness errors in the observations are also likely to be correlated with FG errors. Nevertheless, such error correlations are assumed to be small.

For all three methods, biases have been removed for the statistics presented here, either through variational bias correction or by subtracting a global mean of residual biases. While variational bias correction will ensure that biases between the observations and analyses are close to zero, residual biases can occur between observations and the FG in cases where there is considerable bias in the forecast model. This is the case, for instance, for some stratospheric channels or some water-vapour channels. This aspect needs to be kept in mind when the results are considered for use in data-assimilation systems, as such biases may warrant adjustments to the observation errors that can be used in data assimilation.

4. Results

In the following, we present the results for the three ATOVS instruments considered here. We present the results for data from one satellite only. Instruments from other satellites have been investigated, and the findings for error correlations were comparable. Small differences in the estimates for observation errors were nevertheless found, reflecting small variations in instrument performance, as apparent from routine data monitoring. Unless indicated otherwise, statistics are shown for data over sea. Results for data over land are also shown for surface-sensitive channels if any are used for the particular instrument.

4.1. AMSU-A

AMSU-A is a 15 channel cross-track scanning microwave radiometer, primarily designed to sound atmospheric temperature in the 50 GHz oxygen band (Goodrum et al., 2009). It provides data sampled at 48 km across-track and 52.7 km along-track resolution at nadir. We discuss statistics for the NOAA-18 AMSU-A, as it is considered the best AMSU-A in orbit at the moment, with all channels functioning nominally. Channels 5– 14 are considered for assimilation at ECMWF, with channels 5 and 6 rejected over land over higher orography. Lower tropospheric channels are rejected over sea when the FG departure for channel 3 exceeds 3 K to avoid regions with a strong cloud or rain signal (a FG-departure threshold of 0.7 K on channel 4 departures is used over land). Additional checks for scattering signatures are also performed. The outermost three scan positions of each scan line are rejected (out of 30 scan positions). Channel 14 is used without bias correction to anchor the stratospheric temperature analysis; interaction between model bias and variational bias correction otherwise leads to an undesired drift in the applied bias correction.

4.1.1. General results

Figure 1 shows covariance statistics for the FG departures for the NOAA-18 AMSU-A data used (diagonal only) as a function of separation distance. They show sizeable differences between the covariance values at zero separation and those at non-zero separation, and the expected reduction of covariance values with separation distance. Given the size of the difference between zero separation and non-zero separation and the fact that errors in the FG will be spatially correlated, it is already apparent that the spatially uncorrelated part of the observation error for AMSU-A dominates.

The FG-departure statistics are shown together with characteristics calculated from the assumed background-error statistics and the diagnostic for the background error given by Desroziers (Eq. (2)). For channels 5– 8, 11 and 13– 14, the sampled background-error statistics are somewhat larger than the covariances calculated from the FG departures for separations greater than 200– 300 km. This suggests that either the assumed background errors are larger than the true background errors or the average spatial characteristics of the assumed background errors in radiance space are not consistent with observations. The Desroziers-estimated background errors are, by definition, smaller than the covariances from the FG departures. While this means they are also considerably smaller than the assumed background errors, the shape of the reduction with separation distance is actually fairly similar. It is therefore likely that the assumed background errors are in fact inflated for the channels in question, whereas the spatial characteristics are consistent with observations. For channels 9, 10 and 12, the Desroziers-estimated background errors are very close to the sampled background errors.

Figure 2 shows estimates of the spatial-error correlations for AMSU-A observations. One estimate is based on the Desroziers diagnostic (Eq. (1)), whereas the other is based on the background-error method. Both estimates are fairly consistent and give relatively small spatial-error correlations for separations larger than the thinning scales currently used at ECMWF. For the current operational thinning scale of 125 km, the correlations are at or below 0.2 for all channels. Channels 5 and 6 have slightly higher correlations at the 50 km separation bin, but they are still relatively small (less than 0.3). Channels 5 and 6 have some sensitivity to the surface and to thick clouds and rain, and these aspects may lead to higher spatial-error correlations, for instance through the surface emission, undetected cloud or rain or the quality control applied.

Figure 2.

Estimates of spatial-error correlations as a function of separation distance for NOAA-18 AMSU-A for the channels used at ECMWF. The estimates are based on the Desroziers diagnostic (black) and the background-error method (grey). The number of collocations as a function of separation distance is shown in the last panel.

Estimates for the total observation errors are summarized in Figure 3. Also shown are the standard deviations of FG departures by channel, which should provide an upper limit for the observation error, and the measured mean instrument noise, which should provide a lower limit of the observation error. The estimates for the three methods employed here are in good agreement, with values of less than 0.2 K for channels 5– 10. For these channels, the estimates of the observation error are at or below the mean measured instrument noise. This is most likely due to sampling and quality control, which will act to reduce the standard deviations of the FG departures that are the basis of the observation-error estimates. The finding nevertheless suggests that the radiative-transfer error for these channels is relatively small, at least after applying the bias correction used in the ECMWF system. While intriguing, the latter result is consistent with the finding that spatial-error correlations appear to be small, as radiative-transfer errors are expected to be spatially correlated. The three estimates of observation error are much smaller than currently assumed in the ECMWF assimilation system, typically by about 40%.

Figure 3.

Estimates of observation errors for NOAA-18 AMSU-A channels used in the ECMWF system. The estimates are based on the measured in-flight instrument error (solid grey), the observation error assumed in ECMWF's assimilation system (dotted grey), the Hollingsworth/Lönnberg method (dotted black, calculated from the difference in FG-departure covariances at 0 km and 50 km separation), the background-error method (dashed black) and the Desroziers diagnostic (solid black). Also shown are the standard deviations of FG departures (dash– dotted grey).

There is little evidence of interchannel error correlations for AMSU-A (Figure 4). The three methods employed here consistently give correlations of less than 0.2 between any channels. The Desroziers diagnostic gives the highest correlations between any channels, with around 0.13 between channels 5 and 6 and channels 6 and 7.

Figure 4.

Estimates of interchannel error correlations for NOAA-18 AMSU-A. (a) Based on the Hollingsworth/Lönnberg method (calculated from the difference in FG-departure covariances at 0 km and 50 km separation). (b) Based on the background error. (c) Based on the Desroziers diagnostic. Note that the Desroziers diagnostic does not necessarily result in symmetric matrices (although deviations from symmetry were found to be fairly small); the matrices presented here have been made symmetric by using equation image.

4.1.2. Land

The statistics presented so far were calculated for data over sea only; the same analysis has been repeated for data over land. Due to poor knowledge of the background errors and their correlations for skin temperature, the method of subtracting the scaled mapped background error gives poor results, so only the Desroziers diagnostic is available to estimate spatial-error correlations. As expected, only channels 5 and 6 show appreciable differences in the observation-error estimates, as other channels show little or no sensitivity to the surface characteristics. The Desroziers diagnostic suggests larger observation errors with stronger spatial-error correlations, particularly for channel 5 (Figure 5). The Hollingsworth/Lönnberg method also estimates larger observation errors (not shown), but the considerable size of the spatial-error correlations suggested by the Desroziers method make the applicability of this method more questionable. Interchannel error correlations between channels 5 and 6 are also increased slightly over land (by ≈ 0.05). The larger observation-error estimates for the surface-sensitive channels over land are likely due to larger radiative-transfer errors, as a result of a more difficult specification of the surface emission.

Figure 5.

(a) Comparisons of spatial observation-error correlations as a function of separation distance over land (grey) and sea (black) from the Desroziers diagnostics for NOAA-18 AMSU-A channel 5. (b) As (a), but for channel 6. (c) As (a), but for channel 7. (d) Estimates of the observation errors from the Desroziers diagnostic over land (grey) and sea (black).

4.1.3. Anisotropy

The error statistics have also been compiled as a function of the difference in scan position and scan line between the observation pairs, thus allowing for anisotropy. While for most channels FG-departure characteristics or observation-error correlation estimates appear primarily isotropic, some channels show more complicated patterns. One example is channel 6, which shows some stripiness in the cross-track FG-departure covariances, in particular larger values for scan-position differences of 21 or 22 (Figure 6(a)). The pattern is satellite- and channel-specific; channel 6 on METOP-A, for instance, does not show the same pattern (not shown). Nevertheless, broadly similar characteristics have been found for channels 5– 8 for some satellites. The reason for the features is unknown, but the finding that it is satellite-specific suggests an instrument-related scan characteristic that is not taken into account by the scan-bias correction. Consistent with this interpretation, the Desroziers diagnostics give a roughly isotropic estimate for the background error for channel 6 on NOAA-18, and attribute the stripes to a correlated observation error (Figure 6(b) and (c)). As shown in our isotropic analysis, observation-error correlations remain small, even with anisotropic features taken into account.

Figure 6.

Diagnostics for NOAA-18 AMSU-A channel 6 over sea. (a) FG-departure covariance statistics [K2] as a function of scan position and scan-line difference. The grey-scale has been adjusted to emphasize values for non-zero differences; the FG-departure variance for zero separation is 0.024 K2 and appears as white. (b) Background-error covariance estimates from the Desroziers diagnostic [K2] as a function of scan position and scan-line difference. (c) As (b), but for the observation-error correlations. (d) Number of observation pairs used in thousands. Note that only one quadrant of each figure has been calculated; the rest is derived from symmetry considerations.

4.2. MHS

We will now analyze data from the MHS instrument on METOP-A. MHS is a five-channel cross-track scanning microwave radiometer, with the primary aim of sounding atmospheric water vapour around the 183 GHz water-vapour band (Goodrum et al., 2009). It provides very dense sampling, 16 km across-track and 17.6 km along-track at nadir. The quality control for MHS in the ECMWF system is as follows: channels 3 and 4 are used over sea and low orography, whereas the use of channel 5 is restricted to data over sea only. No data are used over sea ice. Cloud- or rain-affected data are rejected when FG departures for channel 2 exceed 5 K. The outermost nine scan positions on either side are also not considered for assimilation (out of 90 scan positions).

4.2.1. General results

Figure 7 shows covariances for FG departures as a function of separation distance, compared with the estimates for the background-error covariances from the Desroziers diagnostic and the background error assumed in the assimilation. The behaviour of the FG-departure covariances is quite different from that observed for AMSU-A: the covariances are much sharper with separation distance, reflecting the smaller correlation scales in the FG errors for humidity. Also, there is no clear separation between the FG-departure covariance at zero separation and that at non-zero separation. This reflects that the relative contribution from the instrument error to the FG departures is much smaller than was the case for AMSU-A (instrument errors for MHS are around 0.5 K, compared with standard deviations at zero separation of 1.3– 1.7 K). Both aspects make it difficult to separate the FG-departure covariances clearly into a spatially correlated part (which is expected to be primarily due to FG errors) and a spatially uncorrelated part (observation error). To account for this, the finer spatial binning of 12.5 km has been used for the calculation of the FG-departure covariances.

Figure 7.

As Figure 1, but for MHS on METOP-A with a binning interval of 12.5 km. Note also the smaller range of separation distances shown compared with AMSU-A.

Figure 8.

As Figure 2, but for MHS on METOP-A with a binning interval of 12.5 km. Estimates for the observation-error correlations from subtracting values for the mapped background-error covariances from the FG-departure covariances are based on unscaled values. Note also the smaller range of separation distances shown compared with AMSU-A.

The mapped assumed background errors and Desroziers-diagnostic background errors show relatively good agreement in terms of the length-scales of the shorter spatial background-error correlations. The assumed background errors are considerably smaller than the covariances computed from FG departures or Desroziers-diagnostic background-error estimates. The findings suggest that the assumed background errors are underestimated, or that some of the observation error for these MHS channels is spatially correlated.

For short separations (<200 km), the estimates of spatial observation-error correlations for MHS are significantly larger than those obtained for AMSU-A. Figure 9 shows the estimates for spatial-error correlations obtained from the Desroziers diagnostic and by subtracting the unscaled assumed background-error covariance from the FG-departure covariances. Even though there are considerable differences between the two estimates, both indicate correlations close to or above 0.2 for some channels for separations of less than 140 km, the thinning scale currently used for MHS in the operational ECMWF system.

Figure 9.

As Figure 3, but for MHS on METOP-A. Estimates for the Hollingsworth/Lönnberg method are based on subtracting the FG-departure covariances from the 12.5 km bin (covering 6.25– 18.75 km separations, with an average distance of 17.5 km) from the FG-departure variances at zero separation. Estimates for the observation errors from subtracting values for the mapped assumed background-error covariances from the FG-departure covariances are based on unscaled values.

The estimates of spatial observation-error correlations partly reflect aspects of representation. The analysis increments in the incremental assimilation system used here and the mapped background-error estimates are calculated at a resolution of T255 (≈80 km), much coarser than the MHS FOV of 16 km (at nadir) and coarser than the model resolution of T799 (≈25 km). The mismatch in representation between the MHS FOV size and the resolution of the analysis increments will lead to errors of representation that are likely to be spatially correlated. Also, the FG for the FG departures is used at full model resolution; any spatial FG-error correlations on finer scales than allowed by the coarser analysis increments will therefore be interpreted as spatially correlated observation error. Both aspects are much more prominent for observations sensitive to humidity, with its small-scale variations and FG errors, than for temperature-sensitive observations.

Figure 9 gives estimates for the observation error for MHS. As expected, there is considerable variation between these estimates. The estimates from the Hollingsworth/Lönnberg method give the lowest observation errors, as they explicitly neglect any spatial correlations in the observation error. The values estimated with the Hollingsworth/Lönnberg method are therefore also closest to the instrument error. The other two methods provide estimates that are considerably larger than the instrument noise. This is partly a result of the representation issues discussed earlier, but may also be due to errors in the observation operator. The three methods again provide lower values for the observation error than currently assumed in the ECMWF system. Given that the representation issues discussed earlier should be reflected in the choice of observation errors, our estimates from the Hollingsworth/Lönnberg method are, however, not providing useful guidance for observation-error specification for data-assimilation systems for MHS.

Estimates for interchannel error correlations are shown in Figure 10. Again, there is some spread in the estimates for the error correlations, but all three methods employed here show significant interchannel error correlations, in the range of 0.4– 0.8 for neighbouring channels. Hollingsworth/Lönnberg gives the smallest values for the reasons discussed earlier.

Figure 10.

Estimates of interchannel error correlations for METOP-A MHS. (a) Based on the Hollingsworth/Lönnberg method (calculated from the difference in FG-departure covariances for the 0 km and 12.5 km separation bins). (b) Based on subtracting the unscaled mapped background-error covariance from the FG-departure covariance. (c) Based on the Desroziers diagnostic.

4.2.2. Land

Estimates for observation-error covariances over land show only slight differences compared with the findings over sea. Only channels 3 and 4 are used over land. The Desroziers diagnostic suggests slightly larger observation errors over land than over sea for these channels, but spatial or interchannel error correlations are similar (e.g. Figure 11).

Figure 11.

(a) Comparisons of spatial observation-error correlations as a function of separation distance over land (grey) and sea (black) from the Desroziers diagnostics for METOP-A MHS channel 3. (b) As (a), but for channel 4. (d) Estimates of the observation errors from the Desroziers diagnostic over land (grey) and sea (black).

4.3. HIRS

Next, we present results for the HIRS-4 instrument on METOP-A. HIRS is a 20-channel radiometer, with channels in the infrared and visible part of the spectrum (Goodrum et al., 2009). Sampling is 26 km across-scan and 42.2 km along-scan at nadir. The quality control for HIRS in the ECMWF system is as follows: channels 4– 7, 11, 14 and 15 are assimilated over sea, whereas channel 12 is used over sea and land areas with low orography. Cloud screening is based on checks of the FG departures and interchannel gradients to identify clear channels (Krzeminski et al., 2009). The three outermost scan positions on either side in each scan are excluded (out of 56). The standard model used in the bias correction for HIRS is modified for channels 14 and 15 to include a predictor that is zero during night-time and the cosine of the solar zenith angle during daytime (Bormann et al., 2008).

4.3.1. General results

Spatial FG-departure covariances for HIRS are shown in Figure 12. Channels 4, 5 and 15 show a clear separation into spatially correlated and spatially uncorrelated parts. However, for channels 4 and 5 the FG-departure covariances for the 25 km separation bin (the bin with the shortest separations) are already considerably larger than the 50 km bin, possibly due to very small-scale correlations for the observation error. For the other temperature channels (5, 6, 7 and 14) and the water-vapour channels (11 and 12), the separation between the spatially correlated and spatially uncorrelated part in the FG-departure covariances is less clear: the covariances increase smoothly with decreasing separation distance. This makes the application of the Hollingsworth/Lönnberg method questionable, especially in the case of the water-vapour channels, which suffer from the same issues as outlined in the case of MHS. The covariances of the mapped assumed background errors show a similar behaviour compared with the FG-departure covariances as found for AMSU-A and MHS: for the lower temperature-channels, assumed background errors appear slightly too large, similarly to what was seen for the lower-peaking AMSU-A channels, and for the water-vapour channels the mapped assumed background errors are at or below the FG-departure covariances.

Figure 12.

As Figure 1, but for HIRS-4 on METOP-A.

Estimates for spatial observation-error correlations are small for the long-wave temperature-sounding channels, except for very short separation distances of less than 125 km, where they can exceed 0.2 (Figure 13). For channel 7, the background-error method gives slightly negative correlations around 200 km separation– most likely an artefact of insufficient scaling or poor representation of the spatial characteristics of skin-temperature errors in our mapped background errors. Otherwise, estimates for spatial observation-error correlations are similar for the two methods for the long-wave channels. For the short-wave temperature channels 14 and 15, there is some disagreement between the Desroziers diagnostic and the result from the background-error method for the spatial observation-error correlation. The Desroziers diagnostic gives broad correlations for both channels, reaching 0.2 at around 300 and 400 km, respectively. For channel 14, the background-error method instead gives much smaller estimates. For channel 15, the background-error method also indicates sizeable error correlations, albeit smaller than suggested by Desroziers. For this channel, the estimated error correlations for the background-error subtraction method do not appear to converge to zero with increasing separations. This is likely due to insufficient scaling of the assumed background error, as the scaling is calculated with the assumption that observation-error correlations are small for separation distances larger than 200 km. A smaller scaling factor would give more appropriate convergence to zero, leading to larger estimates of observation-error correlation, in better agreement with the Desroziers diagnostic. It appears that the short-wave channels are more prone to spatially correlated errors or that our bias correction model is less capable of correcting radiative-transfer errors for these channels. Reasons for this are probably poorer spectroscopy, solar effects or contributions of other atmospheric gases that are held constant in our radiative-transfer calculations (e.g. CO). For the water-vapour channels (11 and 12), the Desroziers diagnostic gives similar spatial-error correlations to those found for MHS, with considerable error correlations for short distances. For channel 11, the background-error method again gives some artefacts, due either to insufficient scaling of the mapped background-error characteristics or to a misrepresentation of spatial scales in the mapped assumed background error.

Figure 13.

As Figure 2, but for HIRS-4 on METOP-A.

Estimates for the size of the observation errors for HIRS-4 show reasonable agreement between the four methods used (Figure 14). For channels with the clearest separation of FG-departure covariances into spatially correlated and spatially uncorrelated part (4, 5, 15), the estimates are close to or even below the instrument noise. Again, values below the instrument noise are likely a result of sampling and quality control. The finding that the estimates are close to the instrument noise suggests that the radiative-transfer error is small after bias correction. For channels more sensitive to the surface or the water-vapour channels, the estimates for the observation error are larger than the instrument noise, most likely due to contributions from radiative-transfer or representation error. The exception is the estimate from the Hollingsworth/Lönnberg method for channel 12. Here, the estimate is below the instrument noise; however, the estimates for the water-vapour channels from the Hollingsworth/Lönnberg method are considered less reliable due to the problems outlined for MHS. All estimates for observation errors are significantly smaller than the observation errors currently assumed, especially for temperature-sounding channels where the estimates are about a quarter of the currently assigned error.

Figure 14.

As Figure 3, but for HIRS-4 on METOP-A. Estimates for the Hollingsworth/Lönnberg method are based on subtracting the FG-departure covariances for the 25 km separation bin from the FG-departure variances at zero separation. Estimates for the instrument noise have been converted to brightness temperature errors using the channel-specific mean observed temperature.

Figure 15.

As Figure 4, but for estimates of interchannel error correlations for METOP-A HIRS-4. Estimates for the Hollingsworth/Lönnberg method are based on subtracting the FG-departure covariances for the 25 km separation bin from the FG-departure variances at zero separation.

The three methods consistently suggest sizeable interchannel error correlations for HIRS-4 (Figure 14). The lowest-peaking temperature channels (5, 6, and 7; 7 and 14) and the short-wave channels (14, 15) in particular show correlations of 0.6 or higher for neighbouring channels for at least two methods. Cloud contamination, cloud screening or errors in the surface emission are likely to contribute to this. The water-vapour channels also show interchannel error correlations, but only of about 0.3– 0.5.

4.3.2. Anisotropy

The Desroziers diagnostic has also been used to investigate error aspects as a function of difference in scan position and scan line, in order to check for anisotropic characteristics. Again, most channels show primarily isotropic features. However, channels 14 and 15 show additional structures. These are already apparent in FG-departure covariance statistics, which show a clear striping, alternating between slightly higher and slightly lower values with the difference in scan position (Figure 16(a)). In addition, channel 15 exhibits higher FG-departure covariances between scan positions located towards the edges of the swath. NOAA-17 HIRS data do not show the striping features to the same extent, but also shows higher FG-departure covariances between scan positions located towards the edges of the swath for observation pairs with small scan-line differences (not shown). The Desroziers diagnostic attributes the additional structure primarily to correlated observation errors, although it also shows some anisotropic behaviour for the estimates of the background error (Figure 16(b) and (c); note that the cross-track sampling for HIRS is finer than the along-track sampling, so the x- and y-axis scales correspond to different spatial scales). The latter possibly reflects that the weighting function for scan positions located towards the edges of the swath will be shifted in the vertical due to the higher viewing angles. The reason for the pattern in the observation-error correlations is unknown, but the characteristics suggest that they originate from the instrument design or the integration of the instrument on the satellite.

Figure 16.

As Figure 6, but for channel 15 of the METOP-A HIRS. The FG-departure variance for zero separation is 0.024 K2.

5. Conclusions

In the present study we have estimated observation errors and their spatial and interchannel error correlations for clear-sky radiances from the ATOVS instruments assimilated at ECMWF. The main findings are as follows.

  • AMSU-A shows little spatial or interchannel observation-error correlation for all channels used in the assimilation, except for surface-sensitive channels over land. Estimates of the observation error are close to the instrument noise, again except for channel 5 over land. The current use of thinning scales and observation errors in the ECMWF assimilation system appears very conservative for AMSU-A.

  • The finding that observation errors for AMSU-A are comparable to the instrument noise suggests that the radiative-transfer error is small after bias correction for these channels.

  • Long-wave temperature-sounding channels from HIRS show considerable interchannel error correlations, but relatively small spatial-error correlations, especially for the thinning scales currently employed. Short-wave temperature-sounding channels exhibit considerable spatial as well as interchannel error correlations. Estimates for the observation error are around the instrument noise for channels 4– 6 and 15, and larger for other channels.

  • Channels with stronger sensitivity to the surface show larger observation errors compared with the instrument noise for AMSU-A and HIRS, and interchannel error correlations also tend to be larger for these channels. This is particularly true for the HIRS instrument but also, to a lesser extent, for AMSU-A. Residual cloud contamination or problems in the specification of the surface emission may be a contributing factor to this.

  • Estimating observation errors for humidity-sounding channels of HIRS or MHS from FG or analysis departures is more difficult, primarily due to the combination of smaller-scale and larger errors in the FG for humidity. A considerable proportion of the observation error for humidity-sounding channels appears correlated spatially for short separation distances, as well as between channels. Representativeness appears to be an important contributor in this respect. Observation-error estimates for humidity channels are generally considerably larger than those provided by the instrument noise.

  • All observation-error estimates are lower than the observation errors currently assigned in the assimilation, by a factor ranging from 2 for AMSU-A to about 4 for HIRS. Given the interchannel error correlations and some spatial-error correlations, some error inflation appears justified for HIRS and MHS, but less so for AMSU-A.

The use of three methods to estimate observation errors gives an indication about the reliability of the estimates presented. For most temperature-sounding channels, the methods show good agreement, suggesting good reliability of the results, at least within the assumptions that are common to the three methods. In contrast, the three methods show the worst agreement for the humidity-sounding channels. Here, the results from the Desroziers diagnostic and the background-error method contradict the assumption of the Hollingsworth/Lönnberg method of spatially uncorrelated observation errors. Also, smaller length-scales in the FG errors and larger FG errors in radiance space make contributions from observation errors less identifiable for any FG-departure-based method.

The present study suggests that there is scope for an improved assimilation of some of the instruments investigated here, even with diagonal observation errors. For instance, our statistics suggest that AMSU-A may be used more densely or with smaller observation errors, given that spatial and interchannel observation-error correlations appear small and the current assimilation choices appear rather conservative.

For further discussions of the results, and observation-error covariance estimates for AIRS and IASI, the reader is referred to Part II of this article (Bormann et al., 2010).

Acknowledgments

Mike Fisher, Andrew Collard and Gabor Radnoti provided assistance and feedback on various aspects of the study. Comments from two anonymous reviewers were also greatly appreciated.

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