The main purpose of this study is to estimate the IASI observation-error covariances and verify their impact on the quality of the ozone assimilation results. The setup of the experiment in terms of the period of the study, the model configuration, the choice of assimilated observations, and the background-error covariance matrix is reported in Table 1. The observation-error covariance matrix will be discussed in the results section (Sect. 3).

The model was initialized with a zonal climatology, and the spin-up time used is 1 month (June 2010). Then, our simulations were performed for the month of July 2010. The ozone forecast-error standard deviation was assumed to be proportional to the ozone concentration. In fact, have evaluated the standard deviation of the free model simulation against independent data (profiles from ozonesondes and MLS), and they found a small free forecast error in the stratosphere, larger error in the free troposphere, and the highest error close to the tropopause. This strategy was adopted previously by many studies . and have used a percentage of 15 % in the troposphere and 5 % in the stratosphere. In this study, we have adopted a detailed chemical scheme (discussed in Sect. 2.1.1). This scheme was shown to reduce the model bias compared to the scheme used in and (see Fig. 4 in ). Hence, we chose the same background error as in : 2 % of the O3 profile above 50 hPa and 10 % below. An important reason to keep the background errors similar to the setup of is also that we wanted to exclusively examine the impact of R, as mentioned in the introduction.

The ozone background-error covariance matrix is split into a diagonal matrix filled with the standard deviation and a correlation matrix modelled using a diffusion operator. The correlation, characterized by the length scale, spreads the assimilation increments in space. The configurations of horizontal and vertical length scales are described in Table 1.

The same preprocessing described in has been applied to our data before their use in the assimilation system. In order to avoid any contamination from clouds, data were filtered using a cloud mask, and only pixels with cloud fraction less than or equal to 1 % were kept. The cloud fraction values are obtained from the AVHRR measurements on board Metop-A. Since the spatial resolution of MOCAGE is coarser than the pixel size, the number of ground pixels was reduced by thinning the data using a grid of 1∘ × 1∘ resolution and only keeping the first pixel that falls in every two grid boxes. A dynamical rejection of observations – with a threshold of 12 % – based on the relative differences between simulated and measured values with respect to simulated values was considered. Some channels affected by H2O absorption (1008–1019, 1028–1030, 1064–1067, 1072–1076, 1089–1092 cm-1) were removed. Pixels affected by aerosols are detected and then removed using the index based on V-shaped sand signature as discussed in .

The observations used in the assimilation system could have a margin of error. We can identify two types of errors, systematic and random errors. The systematic error is ordinarily corrected before the data assimilation process. In NWP, these types of errors in satellite observations are in general corrected before assimilating the observations or within the data assimilation process by the VarBC scheme . The key assumption is that the background state provided by the NWP system is unbiased. This assumption is not valid in atmospheric chemistry applications, where models might have significant biases, which is the case in our study (see Fig. 4 in ). In such a case, VarBC requires some independent data (anchor) to prevent the drift of the analyses to unrealistic values that might be introduced by the model bias. In our case, we control tropospheric and stratospheric ozone. Identifying an anchor needs to be investigated carefully. Ozonesondes might be used as an anchor in the troposphere and low stratosphere, but the number of profiles provided is limited spatially and temporally. This might have an impact on the capacity of ozonesonde measurements to prevent the drift of the analyses due to the model bias. used IASI channel 1585 as an anchor in the assimilation of ozone for NWP. have used the same uncorrected channel as an anchor, and they showed that its impact was not sufficient to stabilize the bias correction process for a long period. This aspect needs to be explored carefully in a separate study. On the other side, a good understanding of sources of the measurement bias is a prerequisite to implement a bias correction scheme. VarBC in NWP applications, for instance, needs to define a linear model with some predictors . Before adapting this approach in an atmospheric chemistry framework, the possible sources of systematic errors in the IASI ozone window need to be assessed.

In atmospheric chemistry, we used to assimilate level 2 products of ozone . Only recently has the direct assimilation of IASI radiances been introduced in our chemistry transport model . Implementing a bias correction scheme requires careful diagnosis of the bias from observation monitoring. On the other hand, choosing an anchor demands particular care, and the choice depends on the full set of assimilated instruments. In this work, which is not based on a preexisting operational setup, we do not assimilate other ozone instruments. Thus, we had to assume that our observations are unbiased and we did not perform any bias correction. This assumption has been adopted in many chemical analysis studies (e.g. ).

Random errors can arise from the measurements (e.g. instrumental error), forward model, representativeness error (e.g. difference between point measurements and model representation), or quality control error (e.g. error due to the cloud detection scheme missing some clouds within clear-sky-only assimilation). These types of errors should be accounted for by the observation-error covariance matrix R. According to , the instrument noise could be assumed to be uncorrelated. However, the IASI measurements are apodized, which may introduce correlations between neighbouring channels, particularly in our case where we are assimilating a subset of adjacent channels. The radiative transfer model may also introduce correlations between channels. The error statistics from the instrument noise are known, while the characteristics of other sources of error are not yet well understood.

In this paper, we estimate the total error using the statistical approach introduced by . 1R=E[(y-H(xa))(y-H(xb))T] Here xa is the analysis state vector, xb is the background state vector, y is the vector of observations, and H is the observation operator that computes model counterpart in the observation space.

This method has been used to estimate observation errors and inter-channel error correlations . It can potentially provide information on imperfectly known observation and background-error statistics with a nearly cost-free computation . However, this approach assumes that the R and B matrices used to produce the analysis are exactly correct, which is almost never the case in practice. Furthermore, Desroziers diagnostics compute the total covariances, but more efforts are needed to understand and distinguish the sources of the error.

The Desroziers method was computed on the output of a 3D-Var experiment using a diagonal matrix R (with a standard deviation of 0.7 mW m-2 sr-1 cm as in ). The diagnosed R could not be used directly in the assimilation system. In fact, the estimated matrix was asymmetric and not positive definite. Similar unrealistic features in the diagnosed covariance matrices were encountered in and , where an artificial inflation of observation errors was applied. R needs to be a valid covariance matrix before being used in the 3D-Var assimilation system. Therefore, we first symmetrize the estimated matrix by taking the mean of the original matrix and its transpose. Then we impose the negative eigenvalues to be equal to the smallest positive eigenvalue as in and . Another method which consists of increasing all eigenvalues of R by the same amount was tested here to recondition the estimated matrix. We favoured the first method since the standard deviation and the correlation values remain closer to the initially estimated quantities.

Using outputs (analyses and forecasts) derived from a 3D-Var experiment that used a diagonal R matrix (hereafter called the first 3D-Var experiment) in the estimation process might have an impact on the diagnosed R matrix. The matrix derived using these outputs is hereafter called the first estimation. We performed another 3D-Var experiment (second 3D-Var experiment) using the first estimation. The outputs (analyses and forecasts) of this experiment (second 3D-Var experiment) were used to estimate another R matrix called the second estimation. The standard deviation of the second estimation is larger than that of the first estimation (not shown). The same goes for correlations (not shown). It should be noted that the second estimation was positive definite, unlike the first estimation where some unrealistic features were encountered. We have followed the same process to further estimate two other matrices (third and fourth estimations). The differences of the estimations in terms of standard deviation and correlations became smaller as we reestimated the matrices, suggesting a sort of convergence of the estimation. We have adopted the second estimation for the results shown in this work. The reason for this choice will be discussed later (Sect. 5.2).

Figure 1 presents the standard deviation diagnosed using the Desroziers approach (solid black line) and that used in (dotted blue line). The latter was set equal to 0.7 mW m-2 sr-1 cm for all channels, which is a common setting for most IASI O3 retrievals . At first glance, we note that the standard deviation used in previous studies is highly underestimated for the SST-sensitive channels and overestimated for some ozone-sensitive channels (around 1040 and 1050 cm-1). The diagnosed standard deviation increases to reach 2 mW m-2 sr-1 cm for SST-sensitive channels (the first and the last 20 channels of the band (980–1000 cm-1 and 1080–1100 cm-1) and the channels between 1040 and 1045 cm-1) and varies from 0.2 to 1.4 mW m-2 sr-1 cm for the ozone-sensitive channels. The radiance values for the observations are greater for the SST channels than those of the ozone. The same goes for the corresponding background and the analysis values. Since these diagnostics are based on observation, background and analysis residuals, a larger standard deviation for the SST channels than for ozone channels might be expected. We have plotted the R standard deviation, the average of observations, and the average of the background in the observation space on the same figure (not shown). We have noticed that the estimated standard deviation has a very similar shape to that of the observed radiances or the equivalent of the background in the observation space. This may suggest that the larger absolute error in the SST band compared to the ozone channels might be explained by the large values of the observation and the background for the SST channels in comparison with respect to the ozone channels. It could also be attributed to greater sensitivity to emissivity and representativity error.

The IASI instrumental error is provided by the CNES (Centre National d'Etudes Spatiales), taking into account different known effects such as flight homogeneity and apodization effect (Le Barbier Laura, personal communication). The instrumental error covariance matrix is computed as described in . This error remains smaller (about 0.2 mW m-2 sr-1 cm) than that used in the previous studies (0.7 mW m-2 sr-1 cm). Then, the large estimated standard deviation noticed in our estimation might be due to the radiative transfer input error.

To investigate the off-diagonal part of R, we present the diagnosed correlation matrix in Fig. 2. The results show high correlations between the majority of the channels (larger than 0.4). In particular, a very high correlation is observed among SST-sensitive channels (around 0.9 to 1). The regions of, relatively, lower correlation (around 0.4 to 0.7) represent the ozone channel correlations and cross correlation between ozone- and SST-sensitive channels.

The high correlation found here was expected since previous studies have highlighted the same behaviour in this spectral region . In fact, the use of the same radiative transfer model for all channels may introduce similar errors among these channels.

The diagnostic discussed above is based on a global estimation, without any distinction between the type of the surface (land or sea) or the time of the observation (day or night). Since the emissivity varies according to the type of the surface, and the skin temperature is strongly driven by the sun radiation, we evaluated R taking these differences into account. In terms of standard deviation, the error over land reveals large values for the SST-sensitive channels in comparison with that estimated over the sea which, in turn, reproduces a slightly different error in comparison with the global estimation (not shown). The two surfaces also introduce a slightly different error regarding the ozone band. The same behaviour as the global estimation is reproduced when the statistics were performed from the data measured separately from the day and from the night. The variability in terms of correlations is more pronounced when the surface type is considered than in the case of the observation time. The difference between the correlations estimated using all observations and pixels over the sea surface varies between 0 % and 40 % for the majority of the channels with values that can reach 60 %. These differences are located around 1035 and 1060 cm-1, which correspond to the regions of low correlations (not shown).

The separate treatment of land–sea covariance matrices did not yield significant differences in terms of assimilation results compared with the use of global estimation. Hence, we have adopted the global estimation in our study. The rationale for this choice will be given during the discussion of the validation results (Sect. 5.2).

In this section, we discuss the impact of the observation-error covariances estimated previously on the ozone analysis. To this end, three experiments for the month of July 2010 were carried out:

(i.) model run without data assimilation hereafter called the free run (or Control), and denoted in the rest of this paper as ControlExp;

The assimilated spectra include channels sensitive to both ozone and surface skin temperature. The IFS skin temperature was taken as a background in the assimilation process. We have computed the difference between the SST analysis and the background at the end of each assimilation experiment (RdiagExp and RfullExp). The skin temperature is physically linked to the ozone measured. In fact, the skin temperature interacts with the ambient atmosphere. An increase in SST can for example create a convective movement impacting the transport of the ozone. However, the skin temperature is given only at the observation location in this study, and it is specified with values interpolated from NWP forecasts (IFS), whereas ozone is a 3D field issued from the chemistry transport model. Hence, the estimation and potential account of error correlations between the two variables seem challenging in our system. In this work, we did not consider the background-error correlation that might exist between O3 and SST.

Figure 4a shows the difference between the analysis of the SST given by RdiagExp and the IFS SST forecast, whereas Fig. 4b shows the difference between the analysis of the SST given by RfullExp and the IFS SST forecast. In terms of geographical distribution, we notice that the differences are smaller through the tropics and mid-latitudes, especially over sea, when the estimated R was adopted. Looking at the average values, RdiagExp decreases the surface skin temperature by about 0.55 ∘C with respect to the background. The introduction of the estimated R decreases the difference between the SST analysis and that of IFS to almost -0.18 ∘C instead of -0.55 ∘C. The standard deviation was also reduced from 1.39 to 1.05 ∘C. Thus, the use of the estimated R lets the SST analysis stay closer to the IFS forecasts. However, there is an increase in difference on land using RdiagExp, mainly in Africa and South America. This increase in difference over the land seems related to the dependence of observation errors on the surface. In fact, the number of observations over the sea represents almost 70 % of the total observations we have used in this study. Consequently, our SST analysis stays closer to background values (IFS forecasts) over the sea than over the land.

In our assimilation setup, the cost function is minimized hourly. For each window, the minimizer needs to converge after a certain number of iterations. The cost of each iteration is dominated by the cost of the radiative transfer operators (tangent linear, the adjoint model) and of the background-error covariance operators. When the observation error was assumed to be uncorrelated (RdiagExp), the number of iterations needed for each hourly cycle is significantly higher than when the estimated observation-error covariance matrix is used. In fact, the introduction of the estimated R reduces the number of iterations from 150 (a fixed value to stop iterations if the convergence criteria were not attained to save computational time) to 89 iterations on average. This means that the CPU time is reduced by more than 150 % for each assimilation cycle. The convergence criteria of the BFGS algorithm are based on either the reduction of the cost function or the norm of its gradient below some given small thresholds. For the RfullExp, the convergence is achieved due to the stationarity of the cost function (first criterion). The widespread correlations (high condition number) and larger variance of the estimated R matrix lead to a downweight of the observations and are likely the reason for the improved convergence in RfullExp. This increase in the convergence speed was encountered in the Met Office 1D-Var system where a correlated observation matrix was introduced in the system. Moreover, in the matrix R and the observation-error variance appear in the expression of the condition number of the Hessian of the variational assimilation problem, indicating that these terms are important for convergence of the minimization function.

In an attempt to distinguish the impact of the variance on the convergence speed from that of the correlations, we have performed three assimilation experiments using different R matrices. The first experiment (first experiment) employed R that was estimated from the analysis computed using a diagonal R matrix. The minimizer takes 149 iterations on average to converge (average computed for all the assimilation cycles of the month). We used the analysis given by the first experiment to estimate another R matrix. We have used this estimation to run another assimilation cycle (second experiment). We have noticed that the minimizer needs about 89 iterations on average. We have modified the R matrix of the first experiment by keeping its correlations and replacing its standard deviation with that of R used in the second experiment. The resulting matrix was used to run a third assimilation experiment. The minimizer needs about 90 iterations to converge. The results of the third experiment seem to suggest that updating the variance has a larger impact on the convergence speed.

Figure 5 shows the difference of the ozone total column (in Dobson units (DU)) provided by OMI and that of RdiagExp (a) and that of RfullExp (b). At first sight, we note smaller differences over the tropics between the OMI total column and the total column given by RfullExp in comparison with that given by RdiagExp. This behaviour was expected since a large reduction of the amount of ozone was observed in these regions (see Fig. 3). In the high northern latitudes, the differences were slightly increased when the estimated matrix was adopted. This is a consequence of the increase in the amount of ozone encountered in these regions in the stratosphere, compared to the amount reduced in the same region in the troposphere (Fig. 3). On the other hand, the global mean and the standard deviation of these differences are lower in the case of using the new estimated matrix (10.1 DU as a mean and 6.3 as a standard deviation when the new estimated matrix was used instead of 10.6 DU as a mean and 7.3 as a standard deviation when a diagonal matrix was used). Hence, we conclude that the estimated matrix R has slightly improved the results in terms of ozone total columns.

In this section, we validate the two simulations against radiosoundings and MLS data. We use the root-mean-square error (RMSE) as the main statistical indicator to quantify the accuracy of the experiments.

We compute the relative (to the control simulation) difference of RMSE with respect to radiosoundings and MLS averages globally and for five different latitude bands. The difference is computed by subtracting the RMSE of each experiment from that of the control simulation. Negative values indicate an improvement of the O3 profiles. It should be noted that the representativeness of the statistics given by the MLS in the stratosphere is better than that of the radiosoundings because the number of profiles provided by MLS is much higher compared to the radiosounding ones. Consequently, higher confidence is given to the validation using the MLS data in the stratosphere.

Figure 6 reports the RMSE differences with respect to the radiosoundings. Considering the global RMSE (ALL), we notice that the experiment with the estimated matrix improves the results above 150 hPa, around 400 hPa, and near the surface. However, it also reduces the improvement from 30 % (the case of using a diagonal matrix) to 15 % in the vicinity of the upper troposphere–lower stratosphere (UTLS, 100–300 hPa).

The issue of increasing the ozone analysis errors compared to the control simulation encountered in is especially severe in the tropics (30∘ S–30∘ N). The use of the estimated R has substantially enhanced the results in this latitude band, bringing the error from +15 % to -2 %. Apart from the vicinity of 50 and 400 hPa, the results in the tropics were improved over the entire vertical profile. Regarding other latitude bands, almost the same feature of the global validation is found in the Northern Hemisphere. The two experiments show almost the same behaviour in the southern latitudes, with a slight improvement for RfullExp in the southern high latitudes (60–90∘ S).

The MLS validation in Fig. 7 shows almost the same behaviour reported by radiosounding validation in the tropical stratosphere, where the reduction of error is remarkable. In the other latitude bands, MLS reports a similar behaviour of the two experiments, with some small differences in the Northern Hemisphere.

To evaluate the significance of the differences between the analyses of the two experiments with respect to MLS and ozonesounding measurements, we have performed the t test of the differences between analyses and observations (ozonesondes then MLS). We have noticed that for the ozonesoundings, the significance differs among vertical levels. The reduction of the error between 20 and 50 hPa and between 300 and 400 hPa reported in Fig. 6 is statistically significant. For the low troposphere the differences are not significant. Unlike the ozonesounding results, the differences with respect to the MLS measurements are statistically significant for all levels discussed in MLS validation.

All things considered, the introduction of the estimated R has globally improved the O3 profiles in the stratosphere and in the free troposphere, especially in the tropics. In spite of its degradation in the vicinity of the UTLS, the improvement always remains advantageous with respect to the control run.

The matrix used for this study (see Sect. 3.2) will now be discussed in this section since the decision was also based on the outcome of the assimilation experiments presented in this section. We sequentially performed three assimilation experiments using the first, second, and third estimations of R (Sect. 3.2). The results of validation against radiosoundings and MLS showed small differences (not shown). Therefore, to avoid the initial impact of using a diagonal matrix, we have adopted the second estimation (which uses the analyses derived from the experiment using the first estimation). In an operational framework, we may estimate the matrix daily (weekly or monthly if the period of the study is considerably long) using the analyses of the previous day (using the analysis of the previous week or month respectively). In other words, we may use a diagonal matrix to produce analyses for the first day or spin-up period, these analyses will be used to estimate the matrix that will be used for the second day, and so on throughout the period of the study.

We have also discussed the type (sea or land) and the time (day or night) of the observations while estimating the matrices. To check the impact of these differences on the assimilation results, we ran an additional assimilation experiment using the matrix estimated considering the type of the surface of each observation (since the differences were more important than if the time of the observation was considered). Only slight differences among the results have been noticed (not shown). This behaviour might be explained by the number of observations over the sea and over the land. In fact, the observations over the sea represent more than 70 % of the total observations. The differences, in terms of standard deviation, of the global estimation and that using pixels over the sea is very small in comparison with that using pixels over the land (not shown). The differences are also small in terms of correlations in the case of the sea surface in comparison with the land surface (not shown). Hence, we consider that the predominance of observations over sea averages out the potential differences caused by a separate land–sea specification of R. Thus, for simplicity, it seems reasonable to adopt the global estimation of the matrix and neglect the effect of the time and the type of the surface of the observations.