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Keywords:

  • data assimilation;
  • error correlations;
  • SSM/I;
  • AMSR-E

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method and data
  5. 3. Results
  6. 4. Conclusions
  7. Acknowledgements
  8. References

This article provides estimates of effective observation errors and their inter-channel and spatial correlations for microwave imager radiances currently used in the European Centre for Medium-Range Weather Forecasts (ECMWF) system. The estimates include the error contributions from the observation operator used in the assimilation system. We investigate how the estimates differ in clear and cloudy/rainy regions. The estimates are obtained using the Desroziers diagnostic.

The results suggest considerable inter-channel and spatial error correlations for current microwave imager radiances, with observation errors that are significantly higher than the measured instrument noise. Inter-channel error correlations are even stronger for cloudy/rainy situations, where channels with the same frequency but different polarizations show error correlations larger than 0.9. The findings suggest that a large proportion of the observation error originates from errors of representativeness and errors in the observation operator. The latter includes the errors from the forecast model, which can be significant in the case of humidity or cloud and rain.

Assimilation experiments with single SSM/I fields of view highlight how the filtering properties of a four-dimensional variational assimilation system are changed when inter-channel error correlations are taken into account in the assimilation. Depending on the first-guess (FG) departures in the channels used, increments can be larger as well as smaller in comparison with the use of diagonal observation errors. Copyright © 2011 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method and data
  5. 3. Results
  6. 4. Conclusions
  7. Acknowledgements
  8. References

This article provides estimates of observation errors and their inter-channel and spatial correlations for microwave imager radiances currently used in the European Centre for Medium-Range Weather Forecasts (ECMWF) system. In particular, we investigate how these estimates differ in clear and cloudy/rainy regions.

Microwave imager radiances in the ECMWF system are assimilated directly in a four-dimensional variational (4D-Var) assimilation framework in the ‘all-sky’ system (Bauer et al., 2010; Geer et al., 2010). That is, radiances in clear as well as cloudy or rainy conditions are directly assimilated, employing a radiative transfer model with scattering parametrization in the observation operator as required. The system is used operationally for the Special Sensor Microwave Imager (SSM/I: Hollinger et al., 1990) and the Advanced Microwave Scanning Radiometer for the Earth Observing System (AMSR-E: Kawanishi et al., 2003). As is currently common practice in radiance assimilation, the all-sky system treats the microwave imager radiance errors as independent and assumes a diagonal error correlation matrix. In the enhanced version used in the current article (described in Geer and Bauer, 2011), the observation errors (σo) are situation-dependent, employing a model that depends on the average cloudiness of the observations and the first guess (FG).

Observation-error covariances, together with background-error covariances, play an important role in determining the weight given to an observation in an assimilation system. For observation errors, assuming diagonal error covariances is currently common practice. However, for satellite radiances this assumption is questionable, especially as the observation-error covariance should include errors from the observation operator and errors of representativeness, as long as these have not been accounted for by the bias correction. The observation operator error includes errors from the radiative transfer and, in the case of strong-constraint 4D-Var, errors in the forecast model used to map from the analysis time to the observation time. The errors of representativeness include errors due to mismatches in the scales represented in the model fields and observations. Both of these errors are expected to be correlated, between channels or spatially. Indeed, recently several studies have provided evidence that observation errors for sounder radiances show inter-channel and spatial error correlations, especially in the case of water-vapour-sensitive radiances (Garand etal., 2007; Stewart et al., 2009; Bormann and Bauer, 2010; Bormann et al., 2010).

Estimation of observation-error covariances is not straightforward. Nevertheless, a number of methods have been devised that are based on FG or analysis departures taken from numerical weather prediction (NWP) systems (Rutherford, 1972; Hollingsworth and Lönnberg, 1986; Dee and da Silva, 1999; Desroziers and Ivanov, 2001; Desroziers et al., 2005). All of these methods rely on a number of assumptions (e.g. no biases and no error correlations between the observations and the background, assumptions on the spatial structure of the error correlations), and these have been summarized in more detail elsewhere (Dee and da Silva, 1999; Bormann and Bauer, 2010). Recently, results from three of the methods have been intercompared for all sounder radiances used in the ECMWF system (Bormann and Bauer, 2010; Bormann et al., 2010). The methods were the so-called Hollingsworth/Lönnberg method (Rutherford, 1972; Hollingsworth and Lönnberg, 1986), a method based on subtracting a scaled version of mapped assumed background errors from FG departure covariances and the Desroziers diagnostic (Desroziers et al., 2005). The ECMWF study showed considerable inter-channel and spatial error correlations for microwave and infrared water-vapour or window channel radiances. Overall, the results were consistent between the three methods applied, even though quantitatively some differences in the estimates could be noted, particularly for water-vapour channels. Given the findings about spatial error correlations for water-vapour radiances, the study found the Hollingsworth/Lönnberg method less reliable for these channels, as the method assumes that observation errors are spatially uncorrelated.

The present article follows on from Bormann and Bauer (2010) and Bormann etal. (2010) by extending the study to microwave imager radiances in clear and cloudy conditions. We only apply one of the estimation methods, namely the Desroziers diagnostic (Desroziers et al., 2005). In line with the findings from our earlier studies, we do not apply the Hollingsworth/Lönnberg method, as we found it to produce less robust results for the water-vapour-sensitive window channel radiances of microwave imagers. We also do not apply our method using a scaled version of the assumed background-error covariances mapped into radiance space, primarily due to our current fairly poor understanding of background-error covariances in cloudy or rainy regions. The uncertainties in the observation-error estimates found for water-vapour radiances in Bormann and Bauer (2010) and Bormann etal. (2010) will give some indication of the reliability of our results from just one method.

The structure of the article is as follows. First we recall the estimation method employed and introduce the data and assimilation system used in this study. Next, we present the observation-error covariance estimates in terms of σo, inter-channel error correlations and spatial error correlations. We then present an illustration of the potential effect of inter-channel error correlations in a data assimilation system. Our results and conclusions are discussed in the last section.

2. Method and data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method and data
  5. 3. Results
  6. 4. Conclusions
  7. Acknowledgements
  8. References

2.1. Desroziers diagnostic

The observation-error estimates presented here are obtained with the Desroziers diagnostic (Desroziers et al., 2005). This diagnostic assumes that today's variational data assimilation schemes broadly follow linear estimation theory. It assumes that errors have zero bias and that there are no error correlations between the FG and the observations. In addition, a further assumption is that the weight given to the observations in the analysis is in approximate agreement with the true error covariances. Under these assumptions, the following relationships can be derived in observation space:

  • equation image(1)
  • equation image(2)

where R̃ is the diagnosed observation-error covariance matrix, B̃ 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 E[ ] is the expectation operator. Further details on the Desroziers diagnostic and applications of the method can be found in Desroziers et al. (2005) or Bormann and Bauer (2010). In the present article, we adopt the same methodology as used in Bormann and Bauer (2010) for the application of the Desroziers diagnostic.

It should be mentioned here that the applicability of the Desroziers diagnostic and its properties in realistic assimilation systems is an area of active research. Recently, Bormann and Bauer (2010) and Bormann et al. (2010) compared results from the Desroziers diagnostic with observation-error estimates from two different departure-based methods and the results were strikingly similar, especially for the estimates of the observation error (σo) and the inter-channel error correlations. Estimates for spatial error correlations also showed qualitatively good agreement. However, the study also showed some of the largest differences between methods for water-vapour channels, for which it is less clear to separate the FG departures into a spatially uncorrelated component, which can only be observation error, and a spatially correlated component, which may originate from observation or FG error. While the Desroziers diagnostic does not rely on assuming spatially uncorrelated observation error as is the case for the Hollingsworth/Lönnberg method, such different characteristics of the observation and background errors aid the separability of FG departures into observation and background-error contributions. For simple cases, Desroziers et al. (2009) show theoretically that the Desroziers 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. However, if the length-scales of the error correlations are too similar the method may give misleading results.

2.2. Instruments

We present estimates of observation errors for the SSM/I and AMSR-E instruments used in the ECMWF system. Both instruments are conically scanning microwave imagers with window channels sensitive to atmospheric total water vapour, clouds and rain. The channel specifications are given in Tables I and II. The SSM/I instrument scans with an incidence angle of 53.1°, with 128 scan positions for the 85 GHz channels and 64 scan positions for the other channels, sampled at every second scan. SSM/I is flown on the Defense Meteorological Satellite Program (DMSP) series of satellites, and in this article we consider the two instruments on the F-13 and F-15 satellites. Channel 3 on the F-15 SSMI is not used in the ECMWF system due to radio-frequency interference from a radar beacon on this satellite. AMSR-E is flown on the Aqua spacecraft and has a smaller field of view (FOV) compared with SSM/I and small changes in the centre frequency of some channels. The incidence angle for AMSR-E is 55° at the equator and 196 samples (392 for the 89 GHz channels) are obtained per scan.

Table I. Channel characteristics of the SSM/I instrument (Hollinger et al., 1990). Note that the instrument noise values are measured estimates before the super-obbing employed in the data assimilation system. Also shown are the root-mean-square (RMS) values of the assumed observation error (σo) for the clear and cloudy classes (see main text for further details on the definition of these classes).
ChannelFrequencyField ofInstrumentUsed in theRMS ofRMS of
 [GHz] andviewnoise [K]ECMWFassumedassumed
 polarization[km × km] systemclear σo [K]cloudy σo [K]
1 (19V)19.35 V69 × 430.42yes2.15.5
2 (19V)19.35 H69 × 430.45yes4.410.2
3 (22V)22.23 V50 × 400.74F13 only2.94.1
4 (37V)37.0 V37 × 280.38yes3.56.7
5 (37H)37.0 H37 × 280.37no
6 (85V)85.5 V15 × 130.73yes4.16.5
7 (85H)85.5 H15 × 130.69no
Table II. As Table I, but for the AMSR-E instrument (Kawanishi et al., 2003).
ChannelFrequencyField ofInstrumentUsed in theRMS ofRMS of
 [GHz] andviewnoise [K]ECMWFassumedassumed
 polarization[km × km] systemclear σo [K]cloudy σo [K]
1 (6.9V)6.925 V75 × 430.32no
2 (6.9H)6.925 H75 × 430.34no
3 (10.6V)10.65 V51 × 290.49no
4 (10.6H)10.65 H51 × 290.57no
5 (19V)18.7 V27 × 160.55yes2.67.0
6 (19V)18.7 H27 × 160.47yes6.013.2
7 (24V)23.8 V32 × 180.56yes3.04.8
8 (24V)23.8 V32 × 180.54yes5.59.1
9 (37V)36.5 V14 × 8.20.51yes3.46.2
10 (37H)36.5 H14 × 8.20.41no
11 (89V)89.0 V5.9 × 3.51.18no
12 (89H)89.0 H5.9 × 3.50.91no

2.3. All-sky assimilation and data used

The assimilation choices for the all-sky system used are described in detail in Geer and Bauer (2010) and only the main points are highlighted here. The all-sky system treats clear and cloudy data in the same assimilation framework, calling a radiative transfer model as observation operator which can include a scattering parametrization if required. The channel selection for the two instruments is as given in Tables I and II. Only data over sea are assimilated, and only data within ±60° latitude. Observation biases are corrected using variational bias correction (Dee, 2004), employing a linear bias model that includes a global constant, skin temperature, total water vapour and surface wind speed from the FG as predictors. Scan-position-dependent biases are modelled through a third-order polynomial in the scan position. The bias correction model is the same for cloudy as well as clear conditions.

The radiance observations are ‘super-obbed’ to the Gaussian grid representation of T255 (≈80 km), as described in Geer and Bauer (2010). This results in at least 8 SSM/I FOVs or 50 AMSR-E FOVs being averaged together. The super-obbing is performed in order better to match the spatial scales and variability represented in the observations and the model fields. The chosen scale is coarser than the full model resolution, but representative of the effective resolution of cloud and rain structures in the FG. The super-obbing also accounts for different FOV sizes for the different instruments.

The current choice of observation-error model is described in Geer and Bauer (2011) in this issue. It uses situation-dependent observation errors, assigning observation errors based on the average cloudiness from the observations and the FG. Lower observation errors are used for clear cases, whereas larger ones are used for cloudy/rainy cases (e.g. Tables I and II) .

While the all-sky system treats clear as well as cloudy cases in the same way, it is useful for diagnostic purposes to separate the observations into clear and cloudy classes. This is done here on the basis of an estimate of the liquid water path, using a regression in the 22V (24V in the case of AMSR-E) and 37V brightness temperatures. A threshold of 0.05 kg m−2 has been selected to separate the data into roughly equal ‘clear’ and ‘cloudy/rainy’ classes. This classification can be done on the basis of the observations or on the basis of the FG, leading to four classes depending on whether the observations are clear or cloudy/rainy or the FG is clear or cloudy/rainy. Further details of the classification and its implications can be found in Geer etal. (2010).

In the present article, we only consider samples in two of the four classes, that is the class in which the observation and the FG both indicate that the situation is clear and the class for which observation and FG both indicate cloud or rain. The reason for this is that the other two classes inherently suffer from sampling biases, which lead to biases in the FG departures and therefore make an estimation of the (random) observation error less meaningful. For simplicity, we will refer to the two classes just as ‘clear’ and ‘cloudy’ classes.

The departure statistics required for the Desroziers diagnostic were taken from an assimilation experiment that covered June and July 2009, also used in Geer and Bauer (2010). It was performed with the ECMWF system (Rabier et al., 2000), using a T799 (≈25 km) model resolution, an incremental analysis resolution of T255 (≈80 km) and 12 h 4D-Var. The other observations were as used operationally at the time. All statistics presented here are based on data for the whole of July 2009, using the effective observation departures used in the assimilation system, based on super-obbed observations and after bias correction.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method and data
  5. 3. Results
  6. 4. Conclusions
  7. Acknowledgements
  8. References

In the following, we present the estimates for observation-error covariances for the three instruments considered here. They are shown in terms of the observation errors (σo), inter-channel error correlations and isotropic spatial error correlations.

3.1. Observation errors (σo)

Estimates of observation errors from the Desroziers diagnostic for the three instruments are shown in Figures 1–3. The results for equivalent channels are consistent overall between the three instruments. Estimates for the clear sample are typically between 1 and 2 K, whereas the cloudy class shows larger error estimates with values typically between 2 and 5 K. Observation-error estimates for the cloudy class are expected to be larger due to larger observation-operator and representativeness errors (see also Geer and Bauer, 2011), and it is reassuring that the results are consistent with this. The estimates are considerably larger than the instrument noise (cf. Tables I and II), in particular after the averaging employed in the super-obbing process used in the assimilation, which will act to reduce further the random part of the instrument noise. The finding suggests that most of the observation error is due to errors in the observation operator or errors of representativeness. The latter should be reduced through the super-obbing applied, but a precise matching of represented scales is nevertheless unlikely.

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Figure 1. Observation-error estimates (solid lines) and standard deviations of FG departures (dashed lines) for the F13 SSM/I. Estimates for the clear sample are shown in black; estimates for the cloudy sample are shown in grey.

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Figure 2. As 1, but for the F15 SSM/I.

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Figure 3. As 1, but for AMSR-E on Aqua.

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The observation-error estimates can be compared with the standard deviations of FG departures for the respective channels. The standard deviations of FG departures are typically around 1.5 times the observation-error estimates. As the covariance of FG departures is the sum of the true background-error covariances and the true observation-error covariance (assuming no correlation between background and observation errors), the Desroziers diagnostic hence suggests that the size of the background errors and observation errors is roughly comparable in both classes.

The estimates of the observation errors are considerably smaller than the assumed observation error used in the assimilation system (cf. Tables I and II). The definition of the observation errors for the system used is described in detail in Geer and Bauer (2011) and is dependent on the cloudiness of the particular FOV. The observation errors are therefore situation-dependent. The observation-error model has been derived on the basis of FG departures only, assuming that the background error is small (1 K). This results in relatively large observation errors, reflecting a cautious approach to the assimilation of the microwave imager data. An inflation of observation errors is a common approach aimed at counteracting effects of neglected error correlations.

3.2. Inter-channel error correlations

Inter-channel observation-error correlations as estimated by the Desroziers diagnostic are shown in Figure 4. Two aspects are striking: firstly, for the clear sample, all channels exhibit significant error correlations that are generally above 0.5 and frequently much higher than that. The finding is consistent with the significant error correlations found for humidity-sounding instruments in Bormann and Bauer (2010). Secondly, for the 19–37 GHz channels the cloudy class exhibits even stronger inter-channel error correlations, which are generally above 0.7. In particular, the vertically and horizontally polarized channels of the same frequency have error correlations above 0.9 for this class, reflecting the depolarization effect of clouds and rain. In contrast, the 85 GHz channel on SSM/I is slightly less correlated with the other channels in the cloudy class than in the clear class, probably reflecting the different sensitivity to ice hydrometeors between these frequencies.

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Figure 4. (a) Estimates of inter-channel error correlations for the F13 SSM/I in clear conditions. (b) As (a), but for the cloudy sample. (c) and (d) As (a) and (b), respectively, but for the F15 SSM/I. (e) and (f) As (a) and (b), respectively, but for AMSR-E on Aqua.

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The inter-channel error correlations are fairly similar for the three instruments (note that the comparison is not entirely straightforward due to the different channel selection for the three instruments). AMSR-E exhibits slightly larger error correlations between the 24V and 19V channels in the clear class compared with the error correlations between the 22V and the 19V channels for SSM/I. However, this may merely reflect that the 24 GHz channel is slightly less sensitive to water vapour than the 22 GHz channel, making it more similar to the 19 GHz channel.

3.3. Spatial error correlations

Next we will investigate spatial observation-error correlations. To estimate these, we generated a database of pairs of FOVs for the respective instruments. All observations from the same orbit were matched up with each other, making sure that each observation pair is represented only once. The observation pairs were binned by separation distance in order to calculate isotropic error correlations with the Desroziers diagnostic (1). The binning interval used was 25 km.

Estimates of spatial error correlations of the F13 SSM/I are shown in Figure 5. The results for equivalent channels for the other instruments are similar and are therefore not shown here. All channels show observation-error correlations of around 0.2 or higher for separations of less than 100 km. For the 85V channel, the spatial observation-error correlations for the cloudy class are broadest and considerably broader than their clear counterparts, reaching 0.2 only at around 400 km. For the 19–37 GHz channels, the clear class tends to exhibit slightly larger spatial observation-error correlations than the cloudy class.

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Figure 5. Estimates of spatial error correlations for the F13 SSM/I from the Desroziers diagnostic. Estimates of the observation-error correlations are shown as solid lines, whereas those for the background-error correlations are shown as dashed lines, with the clear sample shown in black and the cloudy sample shown in grey. The last panel gives the number of observation pairs (in thousands) that the estimates are based on. The highly variable number of observation pairs is due to the thinning/super-obbing applied to the data prior to assimilation, which averages observations to the T255 Gaussian grid.

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The reasons for the different behaviour for the spatial observation-error correlations in clear and cloudy classes are not clear. The larger spatial observation-error correlations for the clear class compared with the cloudy class for the 19–37 GHz channels may reflect a larger sensitivity to surface emissivity errors or shorter spatial error correlations present in the model cloud parametrizations. The opposite behaviour for the 85V channel may be due to the smaller dynamic range for this channel for the cloudy class, favouring broader error correlations. Alternatively, it may also point to larger-scale deficiencies in the forecast model's parametrization of ice clouds. The latter were previously noted by Geer and Bauer (2010). However, these results may also be due to shortcomings in the Desroziers diagnostic. In simulation studies, Desroziers et al. (2009) show that the Desroziers diagnostic requires several iterative applications in cases where observation error and background error show spatial error correlations, and that it may fail if the correlation length-scales in both are too similar. As a consistency check, we also used the Desroziers diagnostic to estimate spatial background-error correlations in observation space, and provide these as well in Figure 5. As can be seen, for the 85 GHz cloudy class the spatial background and observation-error correlations are the most similar, suggesting that the estimates for this channel probably need to be treated with some caution. For other channels, correlation length-scales for the background error still appear significantly broader than the estimates for the observation-error correlations, such that similarity of the spatial correlation length-scales is less of a problem.

3.4. Single-FOV assimilation experiments

Given the strong inter-channel error correlations, particularly in the cloudy class, we will now investigate how the filtering properties of the assimilation system are affected by neglecting these or accounting for them. We will do this for the example of the cloudy class from the F-13 SSM/I; similar results would be obtained for the other instruments.

As a first step, it is useful to inspect the eigenvectors and eigenvalues of the error correlation matrix presented in Figure 4(b) (Daley, 1993). These are shown in Figure 6. Normalized with the σo values and projected onto these eigenvectors, the errors in the SSM/I observations will all be independent. For each eigenvector, the square root of the eigenvalues gives a measure of how the errors associated with that eigenvector are inflated or deflated relative to a diagonal matrix (see Bormann etal., 2003, for a discussion of the spatial equivalent). This can be seen as follows: for a diagonal error correlation matrix, the set of vectors shown in Figure 6 is still a set of eigenvectors, but with eigenvalues of 1. So if we normalize the errors in the SSM/I observations using σo and project them onto a given eigenvector, the error will be 1 in the case of a diagonal error correlation matrix, but the square root of the associated eigenvalue in the case with error correlations. We can see that in the case of correlated observation error, mean-like structures associated with the leading eigenvector will show a larger error compared with the case when the error covariance is diagonal (square root of the eigenvalue of 1.962 > 1). In contrast, structures associated with higherorder eigenvectors instead show a smaller error, with eigenvalues of less than 1. These structures broadly represent weighted differences between groups of channels. As a result, if inter-channel observation-error correlations are taken into account in an assimilation system then we can expect to see a situation-dependent down- or upweighting of the observations compared with using a diagonal matrix, depending on whether the σo-normalized observation departures primarily project onto the leading or the higher eigenvectors (compare, for instance, the analysis of the filtering properties of a simplified and idealized assimilation system in Daley, 1993, p 127). The effect for the clear case is similar, with different eigenvectors (not shown) and square roots of eigenvalues of 1.931, 0.803, 0.559, 0.487 and 0.279.

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Figure 6. Eigenvectors of the error correlation matrix shown in Figure 4(b). Also given are the square roots of the associated eigenvalues above each panel.

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The behaviour of up- or downweighting the observations can be demonstrated in assimilation experiments. To highlight this, we perform assimilation experiments in which only a single selected SSM/I FOV is assimilated; all other observations are excluded. In the control experiment we use a diagonal observation-error covariance matrix, whereas in the error-correlation experiment we explicitly take the inter-channel error correlations into account. The observation error (σo) is the same for both experiments and it is as defined in Geer and Bauer (2011). The σo estimates presented in the present article are smaller than these assumed ones and would therefore lead to larger increments, but the effect of the error correlations is expected to be similar. In all other aspects, the set-up of the assimilation system is as for the assimilation experiment used earlier. We study two cases here, both of them diagnosed as cloudy in the observations and the FG. The locations are shown in Figure 7; these are F13 SSM/I cases taken on 11 and 14 July 2009, respectively, with observation times close to the beginning of the 12Z 4D-Var assimilation window.

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Figure 7. Locations of the two cases used for single-FOV experiments. The precise locations are 8.8N, 46.4W for case 1 and 29.1S, 44.2W for case 2.

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FG departures for the two cases are shown in Figures 8 and 9, respectively. Case 1 shows FG departures with differing signs for different channels; once normalized by σo these project primarily onto the second and fourth eigenvectors of the error correlation matrix, associated with eigenvalues of less than 1. In contrast, case 2 shows FG departures consistently smaller than −10 K for all channels, projecting well onto the leading eigenvector after normalization with σo.

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Figure 8. FG departures (observation minus FG) [K] for the F13 SSM/I channels used in the ECMWF system for case 1.

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Figure 9. As Figure 8, but for case 2.

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As a result, taking the error correlations explicitly into account in the assimilation leads to larger increments for case 1 (Figure 10), as the observations are receiving more weight. The increments are the difference between the analysis and the FG and therefore a measure of how strongly the observation affects the analysis. In contrast, case 2 exhibits smaller increments in the error correlation experiments (Figure 11), as the inter-channel error correlations act to reduce the weight of the observations. In both cases, the increment profiles shown have been extracted at the observation locations; as a result of the observations being close to the beginning of the observation window, the increments are approximately isotropic spatially around the observation locations (not shown), with the scales determined by the assumed background errors. The findings illustrate how taking observation-error correlations into account in the assimilation can act to increase as well as decrease the weight of the observations.

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Figure 10. Vertical profile of the increments (analysis minus FG) for case 1 of the single-FOV experiments in terms of temperature (left) and humidity (right), valid at the beginning of the assimilation window. The increment profile has been extracted at the observation location. Grey indicates the results for a diagonal observation-error covariance matrix, whereas black gives the results with inter-channel error correlations taken into account.

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Figure 11. As Figure 10, but for case 2.

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Longer assimilation trials with and without taking the error correlations into account are beyond the scope of the present article. However, the above results highlight that taking these error correlations into account will indeed introduce a situation-dependent change of the filtering properties of the assimilation system. As the error correlations increase as well as decrease the weight of the observations, care has to be taken that the error correlations are correctly specified. The error correlations themselves may also be situation-dependent, and further work into the use of inter-channel or spatial error correlations is clearly needed.

4. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method and data
  5. 3. Results
  6. 4. Conclusions
  7. Acknowledgements
  8. References

In the present study we have estimated observation errors and their spatial and inter-channel error correlations for microwave imager radiances in the ECMWF system, and highlighted how inter-channel error correlations can alter the filtering properties of a 4D-Var assimilation system. The main findings are as follows.

  • The Desroziers diagnostic indicates larger observation errors for microwave imager data in cloudy regions, as would be expected due to larger observation-operator and representativeness errors. Observation errors and background errors are estimated to be similar in size.

  • Estimates for inter-channel error correlations are rather large for all three instruments, particularly for the 19–37 GHz channels in the cloudy class, for which error correlations are generally above 0.7. For the cloudy class, channels with the same frequency but different polarizations show particularly strong correlations exceeding 0.9, suggesting that errors arising from the cloud parametrization (either in the moist physics or in the radiative transfer) dominate in these cases.

  • Considerable spatial error correlations can be found for separations of less than 100 km for all channels. The 85 GHz channel in cloudy conditions shows the broadest spatial error correlations, reaching 0.2 at around 400 km. Note that in the experiments used here the data are super-obbed to T255 (≈80 km) resolution.

  • Taking the inter-channel error correlations into account in the assimilation system can increase as well as decrease the weight given to the observations relative to assuming a diagonal error correlation matrix.

The present results are consistent with the findings for water-vapour and window radiances in our earlier studies (Bormann and Bauer, 2010; Bormann et al., 2010). In both studies, we find considerable spatial and inter-channel error correlations for these channels and a σo that is significantly larger than the measured instrument noise. This suggests that a large proportion of the observation error originates from the observation operator or from errors of representativeness. It is worth noting here that in strong-constraint 4D-Var the observation operator effectively includes the integration of the forecast model up to the observing time. These errors will inherently lead to inter-channel as well as spatial error correlations. Geostationary water-vapour radiances with their frequent sampling of the same airmasses show very clearly that such errors can be significant for water-vapour sensitive radiances over an assimilation window. For instance, routine monitoring performed at ECMWF for the Meteosat-9 water-vapour channels shows that standard deviations of FG departures in clear-sky conditions are typically 1.1–1.2 K at the end of a 12 h assimilation window, compared with around 0.7 K at the beginning of the assimilation window (cf. Munro et al., 2004). As the observations and the radiative transfer model are unaffected by the position of the observation within the assimilation window, the increase in the FG departures must come from the forecast model. This behaviour is more difficult to demonstrate for SSM/I or AMSR-E due to the variable geographical sampling inherent in a polar-orbiting instrument. It is, however, expected that a similar situation occurs for the humidity-sensitive microwave imager radiances investigated here and that the situation may be more severe in the cloudy case, where the observation operator includes the forecast model's cloud parametrization.

The conclusion that a large proportion of the observation error characterized here is likely due to errors in the forecast model raises another issue. As a consequence, it is likely that background errors and forward model errors are correlated. Such error correlations are neglected in the estimation method used in our present study and this would imply that observation errors are likely to be underestimated. Correlations between background and forward model errors are also neglected in today's variational data assimilation systems by assuming that the cost function contributions from observations and background are separable. Further research is required to evaluate to what extent this impacts the assimilation of observations sensitive to humidity and clouds, and how this assumption could be relaxed.

The finding of significant error correlations for the microwave imager radiances, especially in cloudy conditions, prompts the question as to what implications these have for the assimilation of data within the current assumptions of variational data assimilation systems. One way to address the error correlations could be to revise the spatial thinning or the channel selection, while continuing to use a diagonal observation-error covariance matrix. This is the most cautious option and is likely to also reduce the effect of limitations due to the assumption of uncorrelated background and forward model error made in today's assimilation systems. Alternatively, observation error correlations could be included explicitly in the assimilation system. Our single-FOV assimilation experiments with inter-channel error correlations demonstrate that accounting for these can increase as well as decrease the weight of the observations, suggesting that an accurate specification of these inter-channel error correlations is likely to be important. Extending the error model presented in Geer and Bauer (2011) to include inter-channel error correlations or to investigate data thinning strategies will be the subject of future work.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method and data
  5. 3. Results
  6. 4. Conclusions
  7. Acknowledgements
  8. References

The assimilation experiments with inter-channel error correlations used software originally developed by Andrew Collard. Comments from two anonymous reviewers helped to improve the manuscript. Alan Geer was funded by the EUMETSAT fellowship programme.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method and data
  5. 3. Results
  6. 4. Conclusions
  7. Acknowledgements
  8. References