Accounting for correlated error in the assimilation of high-resolution sounder data

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1 QuarterlyJournalof the RoyalMeteorologicalSociety Q. J. R. Meteorol. Soc. : , October 4 B DOI:.2/qj.236 Accounting for correlated error in the assimilation of high-resolution sounder data P. P. Weston,* W. Bell and J. R. Eyre Met Office, Exeter, UK *Correspondence to: P. P. Weston, Met Office, FitzRoy Road, Exeter EX 3PB, UK. peter.weston@metoffice.gov.uk This article is published with the permission of the Controller of HMSO and the Queen s Printer for Scotland. Until January 3, data from the high-resolution sounder IASI were assimilated with a diagonal observation-error covariance matrix within the Met Office 4D-Var assimilation scheme, assuming no correlation between channels. The errors were inflated to account indirectly for known inter-channel error correlations. This is sub-optimal as it artificially down-weights observations from these instruments. The true nature of these correlations for IASI are estimated here using data from the Met Office 4D-Var assimilation scheme and a posteriori diagnostics based on analysis and background departures. The diagnosed matrices are symmetrised and reconditioned, to make them suitable for use in the operational assimilation scheme. These matrices have been tested in full assimilation experiments. The results of these experiments show that using the new matrices improves forecast accuracy due to more weight in the assimilation being given to the IASI observations, particularly those from water-vapour-sensitive channels. Key Words: data assimilation; observation errors; error correlations; IASI Received 7 June 3; Revised 22 November 3; Accepted 25 November 3; Published online in Wiley Online Library 2 April 4. Introduction Numerical weather prediction (NWP) models require accurate initial conditions in order to produce accurate forecasts. The true state of the atmosphere at any given time is unknown, so data assimilation is used to blend observations and atmospheric state estimates from previous short-range forecasts (model background) to produce the best estimate of the initial state, which is called the analysis. This analysis is then used as an initial condition for the forecast runs. The Met Office currently employ an incremental four-dimensional variational (4D-Var) assimilation scheme (Rawlins et al., 7). Satellite data are among the most important sources of observations used in data assimilation for NWP due to their uniform global coverage and relatively high accuracy. In contrast, conventional observations are clustered over densely populated areas but are sparse elsewhere. A good understanding and accurate specification of the observation errors are vital so that the observations are suitably weighted against the model background. Errors in satellite data arise from a variety of sources including instrument noise, representativeness errors, forward model errors and other preprocessing errors. The statistics of errors from instrument noise are well known through in-orbit measurements of calibration targets. However, understanding of the characteristics of the other sources of error is limited. The total error can be estimated using statistical approaches such as that proposed by Desroziers et al. (5). Data from high spectral resolution sounders, such as the Infrared Atmospheric Sounding Instrument (IASI, on the EUMETSAT MetOp-A satellite), are assimilated into NWP models (e.g. Hilton et al., 9), and it has been shown that these instruments are amongst those which provide the largest benefit to NWP forecast accuracy (Joo et al., 3). The error covariance matrices used have usually been diagonal, assuming no correlations between different channels. However, previous work at Reading University and the Met Office (Stewart, 9; Stewart et al., 3) and at ECMWF (Bormann and Bauer, ; Bormann et al., ) demonstrated that correlations exist in IASI data, particularly for channels sensitive to water vapour. These correlations were suspected to exist when the data were initially assimilated, so the assumed error values were artificially inflated to indirectly account for this. Accounting for the correlations directly should allow more weight to be applied to IASI observations, particularly those from water-vapour-sensitive channels, thus improving the use of these instruments and the accuracy of the analysis. Section 2 outlines the theory and methods used to diagnose the observation-error covariances. Section 3 shows the results of performing a diagnostic technique described by Desroziers et al. (5) on IASI data to estimate the true structure of the observation-error covariance matrices. Section 4 shows the effects of accounting for correlations in a D-Var assimilation and how the diagnostic matrices are then modified to make them suitable for use in the Met Office 4D-Var assimilation scheme. Section 5 describes results from full assimilation experiments which account

2 Accounting for Correlated Error in Sounder Data Assimilation 242 for inter-channel error correlations in the assimilation of IASI observations. Finally section 6 draws conclusions from this study. 2. Theory and methods 2.. Data assimilation Data assimilation is the process of combining information from the model background and observations to produce an analysis. This is done by minimising a cost function of the form J(x) = ( x x b ) T B ( x x b) 2 + { } T y H(x) R { y H(x) } (), 2 where x is the model state vector, x b is the background state vector produced from a previous short-range forecast, y is the vector of observations, B is the background-error covariance matrix, R is the observation-error covariance matrix and H is the observation operator. The cost function displayed here is the 3D-Var cost function, but can easily be extended to four dimensions (Ide et al., 997). Also, additional terms can be added, e.g. to enable variational bias correction (Dee, 4). The observation- and background-error covariance matrices provide information on the error characteristics of the observations and background and therefore how much weight is given to each source of information. The second derivative of the cost function (Eq. ()) is the Hessian and is given by J = B + H T R H, (2) where H is the Jacobian matrix of the observation operator (Ide et al., 997). The conditioning of the Hessian matrix is linked to the speed at which the minimisation of the cost function converges (Haben et al., ). This is quantified by the condition number, which is defined as the ratio of the largest eigenvalue to the smallest eigenvalue of the matrix and effectively describes the inverse of the distance between the matrix and the set of singular or non-invertible matrices. Therefore, the larger the condition number of (2) is, the worse the conditioning and hence the slower the convergence of the cost function. However, at most NWP centres, including the Met Office, an incremental cost function is used, in which J is a function of δx = x x g,whereδx is the increment and x g is a known reference state (Bannister, 8). In addition, a control variable transform is applied before the cost function is minimised, and this takes the form δx = Uχ = U h U v U p χ, where U p is the parameter transform, U v is the vertical spatial transform and U h is the horizontal spatial transform (Lorenc et al., ). The control variable transform simplifies the cost function and the corresponding Hessian then becomes J = I + B T/2 H T R HB /2, (3) so that the smallest eigenvalue the Hessian can have is. The conditioning of the Hessian is then given by the conditioning of the second term B T/2 H T R HB /2 which is directly dependent on the conditioning of the R matrix. Therefore using an illconditioned R matrix in 4D-Var can result in convergence taking longer Observation errors Each observation assimilated into the model has a corresponding error. The two main types of observation error are systematic and random errors. Systematic errors are biases in the data. Observations can be successfully assimilated only if their bias is reduced to acceptable levels. All satellite data which are assimilated into the model are put through a bias correction procedure, which uses atmospheric predictors to remove any biases (Harris and Kelly, ). Deficiencies in the bias correction can lead to some systematic errors appearing in the assimilation, which can lead to some correlations in the residual errors, which are treated as random. Random errors are those which are represented in the R matrix. An observation error is defined as ɛ o = y o H(x t ), (4) where ɛ o is the observation error, y o is an observation, H is the observation operator and x t is the truth at model resolution. This can be split up into two parts using the fact that y t, the errorfree observation, is equal to H t (x t ), the error-free observation operator acting on the truth in model space, as follows: ɛ o = y o H(x t ) = y o H(x t ) y t + H t (x t ) = (y o y t ) + { H t (x t ) H(x t ) } = ɛ e + ɛ f, (5) where ɛ e is the error due to instrument noise and ɛ f is the error in the observation operator. Assuming no correlation between instrument noise and errors in the observation operator, the corresponding error covariance matrices are R = E [ ɛ o ɛ ot] = E [ ɛ e ɛ et] + E [ɛ f ɛ f T] [ + E ɛ e ɛ f T] + E [ ɛ f ɛ et] R = E + F. (6) For the majority of observations, the instrument noise is not correlated between channels, meaning that its corresponding covariance matrix, E, is diagonal. However, IASI measurements are apodised (Chamberlain, 979) which reduces the noise but introduces correlations between neighbouring channels, so in this case E is a band diagonal matrix with bands of non-zero covariances surrounding the diagonal. The channel selection used at the Met Office was chosen so that modelling the effect of the apodisation could be avoided by not choosing adjacent channels (Collard, 7). Errors in the observation operator can be caused by: forward model errors: errors in the radiative transfer model; representativeness errors: caused when there is a scale mismatch between the observations and the model; pre-processing errors: caused by an inaccurate or incomplete state vector, e.g. uncertainties in retrieved skin temperature, and all of these sources of error can contribute to correlations between different channels errors. This means that F can and probably will contain correlations represented by non-zero offdiagonal elements Estimation of observation errors To estimate the structure of the full R matrix, a diagnostic procedure introduced by Desroziers et al. (5) has been used. This uses observation minus background (O B) and observation minus analysis (O A) statistics to produce observation error variances and covariances, using the expression in Eq. (7), R = E [ {y H (x a ) }{ y H ( x b)} T ], (7) where the notation is as in sections 2. and 2.2.

3 2422 P. P. Weston et al..5. Operational Standard Deviation.5 Diagnosed 6 8 Noise Channel Index Figure. Diagnostic IASI error standard deviations from 4D-Var output, IASI instrument noise and previously used operational values. N.B. The previously used operational error standard deviations for the water-vapour-sensitive channels indexed from 7 to 38 are 4 K Correlation Figure 2. Diagnostic IASI error correlation matrix from 4D-Var output. Two assumptions are made in the derivation of Eq. (7). The first is that observation errors and background errors are independent. The second is that the R and B matrices used to produce the analysis are exactly correct. In this application of the diagnostic, this assumption will be violated due to the artificial inflation of the previously used assumed observation errors. This violated assumption can lead to some unrealistic results which will be commented on further in section 4.2. Additionally it has been shown by Desroziers et al. (9) that the observations and background errors must have sufficiently different scales for the results of this diagnostic to be reliable. 3. Results of diagnostics The Desroziers diagnostic was performed on data from the 4D-Var data assimilation system at the Met Office. The model background and analysis fields in 4D-Var are specified on the model grid and the observation operator interpolates these values to observation times and locations before running the radiative transfer model to transform model variables to radiances. It is these background and analysis values in observation space, together with the bias-corrected observations, which are used to produce the diagnostics. The observations used have been run through operational quality control (QC) and so there are some cloudy profiles where only a subset of the channels out of the complete selection are used. Diagnostics using a set of such observations and those produced from only completely clear observations yielded very similar results. All of the diagnostics have been produced using days of data, which corresponds to approximately 75 profiles or equivalently approximately 53 million observations treating each channel separately. It appears that data obtained over one day provides a sufficiently large sample to produce robust diagnostics which are almost identical to those produced from more data. Also, there is very little daily or seasonal variability in the diagnostics. Figure shows that the diagnosed error standard deviations are much smaller than the values used operationally from the initial implementation of IASI data in November 7 until January 3 in 4D-Var. This is a result of the artificial inflation of the previously used operational standard deviations to account for the inter-channel error correlations that were not modelled directly. The inflation was larger for the water-vapour-sensitive channels since this was where the correlations were thought to be largest. There is also significant inflation in the temperaturesounding channels which was implemented to be consistent with the assumed observation errors of similar channels on other sounders such as the Advanced Microwave Sounding Unit and the Atmospheric Infrared Sounder (AMSU-A and AIRS; Hilton et al., 9), and to compensate for inaccuracies in the assumed B. Prior to the development of objective methods for the estimation of R,such ad hoc error inflation was commonly used to optimise the influence of new observations. The instrument noise in Figure has been derived from principal component residuals by EUMETSAT (). For the temperature-sounding channels (selected channel indexes 86) in the spectral range cm,where the diagnosed standard deviations are close to the instrument noise, other sources of error are small compared to instrument noise. However, for the surface-sensitive channels (indexed 87 2) in the spectral range cm, there is a slightly larger difference showing that the other contributions to the errors are slightly larger for these channels. Finally for the water-vapour-sensitive channels (indexed 3 38) in the spectral range cm, the diagnosed standard deviations are much larger than the instrument noise, showing that the other sources of error are large for these channels. Figure 2 shows that the strongest positive error correlations are between water vapour and surface-sensitive channels. These are shown by the red blocks surrounding the diagonal towards the bottom right corner of the matrix. Also apparent are the weaker error correlations between temperature-sounding channels shown by the paler colours in the top left corner of the matrix. The correlations are caused by a combination of forward model error, representativeness error, apodisation and other pre-processing errors. However, it is difficult to split these up to see which source of error is the dominant cause. One way to attempt to isolate the horizontal representativeness errors is to compare the results of the diagnostic from the D-Var and 4D-Var, as done by Stewart et al. (3). However, there are some other differences in addition to the representativeness errors between the D-Var and 4D-Var, such as different state vector variables which could contribute to the differences in the diagnostics.

4 Accounting for Correlated Error in Sounder Data Assimilation Percentage Channel Index Figure 3. Percentage difference in diagnostic IASI error standard deviations from 4D-Var output run at N26 and N48 resolutions Correlation Figure 4. The correlation matrix of the difference in diagnostic IASI error covariance matrices from 4D-Var output run at N26 and N48 resolutions. Another way to partially isolate the contribution from horizontal representativeness errors is to evaluate the Desroziers diagnostic on first-guess and analysis departures from 4D-Var run at different resolutions. The Met Office 4D-Var system was run at N48 ( 27 km) resolution using an N48 background and N26 ( 6 km) resolution using an N26 background with the same QC and thinning, and hence the same set of observations. The only differences between the two runs are the forecast and data assimilation model resolutions. Performing the Desroziers diagnostic on output from these runs and inspecting differences will illustrate to what extent different IASI channels are affected by errors of representativeness. It will also show which of the correlations are caused by representativeness errors. The diagnosed errors from the N48 analysis would be expected to be larger than those from the N26 analysis. The larger the differences, the larger the representativeness error in that channel. Figure 3 shows that there is a negligible increase in representativeness error between the two resolutions in the highest-peaking temperature-sounding channels (indexed 5). The lower-peaking temperature-sounding channels (indexed 6 86) show some evidence of increased representativeness error at the lower resolution, with the increases generally larger for the lowest peaking of these channels (higher indices). The surface-sensitive channels (indexed 87 2) have relatively large changes in representativeness errors. The mid-tropospheric peaking water-vapour-sensitive channels (indexed 9 2 and 28 38) have the largest changes in representativeness errors, whereas the lowest- and highest-peaking water-vapour-sensitive channels (indexed 3 8 and respectively) have smaller changes in representativeness errors. These results are what were expected with small representativeness errors in temperature-sounding channels and larger representativeness errors in surface-sensitive and water-vapoursensitive channels, although there are a couple of anomalies to this general pattern with the small representativeness errors for the highest- and lowest-peaking water-vapour-sensitive channels. Figure 4 shows that the representativeness errors are responsible for correlations between the lowest-peaking temperaturesounding channels, surface-sensitive channels and the watervapour-sensitive channels. The unrealistic results such as the negative values on the diagonal (which cause the blue lines running the length and breadth of the matrix) for some high-peaking channels and some correlation values being larger than are caused by the violated assumptions in the Desroziers method and are not significant results. What should be stressed is that the differences between representativeness errors at N48 and N26 resolutions do not reveal the full extent of the representativeness errors. They merely suggest which channels are affected by these errors the most and which are affected by them the least. Also, the effective spatial resolution given by the footprint size of IASI varies from 2 2 km at nadir to 39 km at the edge of the scan. The diagnostics calculated to produce Figures 3 and 4 were averaged over all scan positions, meaning that quantifying representativeness error using the above method will not be entirely accurate. An alternative way to isolate horizontal representativeness error would be to keep the model resolution fixed and compare diagnostics using observations from different scan positions; this will be the subject of future investigations. 4. Matrix modifications 4.. Effect of accounting for the correlations Modelling the correlations will have an effect on the weights given to the observations in the assimilation. However, will this effect be the same for all observations or will it vary depending on the relationship between the O B departures for different channels? A pair of D-Var experiments was run using the same observations and background, but one run used a diagonal R matrix and one used a correlated R matrix (with the same diagonal values as diagnosed by Stewart et al., 3). This means that comparisons between the two experiments will isolate the effect that modelling the correlations has on the retrieval. Figure 5(a) shows that the O Bs fluctuate about zero for all channels including the surface- and water-vapour-sensitive channels (indexed 87 38) which have the most correlated observation errors. Figure 5(b) shows the normalised departures which are the original departures pre-multiplied by the inverse of the R matrix for both the diagonal and correlated matrices. This forms part of the increment which is added to the background to form the analysis. In this case, these normalised departures are larger in magnitude when using the correlated matrix (particularly for the most correlated channels). This results in larger retrieval increments for both temperature and

5 2424 P. P. Weston et al. (a) O-B (K) (a) O-B (K) (c) Pressure (hpa) (b) Normalised value (K - ) Correlated matrix Diagonal matrix Retrieval - Background (K) (d) Pressure (hpa) Correlated matrix Diagonal matrix Retrieval - Background (g/kg) Figure 5. (a) Raw and (b) normalised first-guess (O B) departures for all assimilated channels, and (c) temperature and (d) specific humidity retrieval increments, for an IASI profile located in the Atlantic Ocean. Solid blue and dotted red lines represent results when using the correlated matrix and diagonal matrix respectively. specific humidity (illustrated by Figure 5(c) and (d)) when accounting for the correlations, which implies that the retrieval is pulling closer to the observations. Thus accounting for the correlations increases the weight given to the observations, in this case. Figure 6 shows the opposite case, where accounting for the correlations results in smaller retrieval increments thus decreasing the weight given to the observations. The O Bs are all negative for the most correlated channels and the normalised departures are smaller in magnitude when using the correlated matrix. This results in the retrievals staying closer to the background for both temperature and specific humidity. These two examples show that accounting for correlations can both down- and up-weight the observations in individual cases thus supporting the findings of Bormann and Collard (2). However, it is also interesting to look at the effect which accounting for the correlations has on average over a large number of observations. Figure 7 shows that the standard deviations of the retrieval departures (O As) for most channels are increased when accounting for correlations. The effect is largest in the water vapour channels where the correlations are strongest. This means that, in general, accounting for the correlations results in a small down-weighting of the observations. Hence, directly accounting for correlations and inflating the errors both result in a downweighting of the observations overall, highlighting why error inflation is commonly found to be a pragmatic solution to counteract error correlations. However, the examples above show that error inflation will never fully compensate for neglecting error correlations. (b) (c) Pressure (hpa) Normalised value (K - ) (d) Correlated matrix Diagonal matrix Retrieval - Background (K) Pressure (hpa) Correlated matrix Diagonal matrix Retrieval - Background (g/kg) Figure 6. As Figure 5, but for a different profile located in the Atlantic Ocean Using the matrices in the assimilation The matrices diagnosed from 4D-Var output were then tested in the Met Office s assimilation scheme. All of the diagnosed matrices were asymmetric and some were not positive definite. These are examples of unrealistic features in the diagnostic covariance matrices as a result of violated assumptions in the Desroziers diagnostic mentioned in section 2.3. The matrices have to be valid covariance matrices when used in the 4D-Var assimilation scheme and hence must be both symmetric and positive definite. So, before the matrices were used in the assimilation, they were symmetrised (by taking the mean of the original matrix and its transpose) and any negative eigenvalues of the matrix were modified so that they were positive. A potential problem with accounting for correlated observation errors in the assimilation is the computational cost of inverting the full R matrix. This process can be done by the very efficient Cholesky decomposition method (Golub and Van Loan, 996), but has to be done separately for each observation and at each minimisation iteration. It has to be done for each observation because the channel selection can vary for each observation depending on the height and amount of retrieved cloud. In locations where there is detected cloud in the field of view, those channels whose temperature Jacobian matrices have % of their integrated value below the cloud top are rejected, as explained by Pavelin et al. (8). Put simply, only those channels which have most of their sensitivity above the cloud are used. In these cases, when using the full R matrix, it is only the sub-matrix formed by the relevant channel selection which will need to be inverted. The matrix has to be inverted for each minimisation iteration because the only alternative is to store the inverses of all possible

6 Accounting for Correlated Error in Sounder Data Assimilation 2425 Percentage difference Figure 7. Percentage difference in standard deviations of retrieval departures for each channel averaged over all observations in a 6 h window when using the correlated matrix compared to using the diagonal matrix. Table. The number of iterations, timings and cost function values resulting from using diagonaland full R matrices in 4D-Var. Number of Overall Final value R matrix used iterations time (s) of J Old operational diagonal version Diagnosed version with correlations channel selections ( 5); this requires too much memory to be a viable option. In addition to timings, another indicator of the health of the data assimilation is the number of iterations the minimisation takes to converge. The more iterations, the slower is the convergence of the minimisation. This can be caused by the observations and background being too different or the assumed errors being too small. It can also be caused by ill-conditioning in the Hessian (Eq. (3)). One final indicator is the value of the cost function (Eq. ()) which is calculated from the inverses of both the backgrounderror covariance matrix (B) and R. This means that the smaller the errors used, the larger the cost function will be at the start of the minimisation. The value of the cost function at the end of the minimisation is determined by how close the analysis fits to both the observations and the background. If the observation errors used in the assimilation are correctly specified, then the final cost function value should be equal to the total number of observations divided by 2, as derived by Desroziers and Ivanov () and Talagrand (999). In a typical operational data assimilation run, there are.9 million separate observations assimilated, of which.5 million are from IASI. Therefore, if the observation errors are correctly specified, then the total cost function value should be 95 with a contribution of 25 from IASI alone. Table shows that introducing the full R matrix results in a large increase (+ 28%) in the number of iterations required for the minimisation to converge. This in turn results in a large increase (+ 27%) in the cost of the analysis. What is interesting is that the percentage increase in time is almost exactly matched by the increase in iterations, suggesting that the extra time to invert the full matrices is negligible. Further tests using a fixed number of iterations confirmed that the increase in processing time associated with inverting the full matrices is less than %. The final column of the table shows that the final value of the total cost function increases significantly. Looking at the cost function values in more detail shows that the contribution from IASI increases from to 244. This value is much closer to the predicted optimum value of 25. This suggests that the cause of the increase in iterations is not that the error values used are too small. The other potential reason for a large increase in iterations is the ill-conditioning of the Hessian (Eq. (3)). Given that the Hessian s conditioning is directly proportional to the conditioning of the R matrix, this could be the cause of the observed increase in iterations. To investigate this further, the condition numbers of the full matrices used are examined. The condition number of the full matrix for IASI is and is significantly larger than that of the diagonal matrix, which is 64.. Generally a condition number of would not be considered to be large for a covariance matrix. However, the minimisation is very sensitive to the conditioning of the Hessian and hence the R matrix. This suggests that reducing the condition number of the R matrix could be a way of reducing the number of iterations required for convergence and hence improving the stability of the minimisation. This can be done by modifying the eigenvalues of the matrix Methods of reconditioning Reducing the condition number of a matrix by modifying the eigenvalues can be done in various ways. The main idea is to move the smallest and largest eigenvalues so that they are relatively closer together. The larger eigenvalues are dominant and modifying these will result in significant changes to the structure of the matrix. Therefore the modifications should be concentrated on the smaller eigenvalues. The first method tested is to set a minimum eigenvalue threshold as λ thresh = λ max κ req, (8) where λ thresh is the minimum eigenvalue threshold value, λ max is the existing maximum eigenvalue and κ req is the required condition number. Then all eigenvalues smaller than this threshold are set to the threshold and the matrix is reconstructed using a reverse eigendecomposition using the original eigenvectors and new eigenvalues (Golub and Van Loan, 996). The advantage of this method is that it keeps the largest eigenvalues constant, and so should change the overall structure of the matrix minimally. The disadvantage is that the effect of setting the smallest eigenvalues to a constant threshold results in many of the small diagonal values of the matrix getting set to an almost constant value. This effect seems to be most pronounced for the errors of channels which have the weakest inter-channel error correlations (the temperature-sounding channels). This is intuitive given that the eigenvalues of a diagonal matrix are just the diagonal values themselves and thus increasing the eigenvalues will increase the diagonal elements by the same amount. This unwanted effect means that many channels will have the same error and be given the same weight when they may really have quite different errors and should be given different weights. The second method tested is to increment the diagonal of the matrix (which has the effect of incrementing every eigenvalue by the same amount) by a quantity calculated to give the required condition number: λ inc = λ max λ min κ req, (9) κ req where λ inc is the increment and λ min is the existing minimum eigenvalue. The advantage of this method is that the relationship between the errors of different channels is unchanged, i.e. they are all incremented rather than being set to an almost constant value. The disadvantage is that the largest eigenvalues are changed albeit by a relatively small amount. This has the effect of weakening the correlations between channels slightly more than in the first method.

7 2426 P. P. Weston et al. Initial tests showed that matrices reconditioned using the first method resulted in the minimisation taking either the same or more iterations to converge than using matrices reconditioned using the second method. Also, reducing the condition number significantly using the first method resulted in most of the temperature-sounding channels using the same or very similar error values to each other. However, Figure shows that the diagnosed errors for the temperature-sounding channels vary quite significantly between.4 K for the highest-peaking channels to.25 K for the lowest-peaking channels. Therefore using the first method results in more unrealistic errors than using the second method. For these reasons the second method, using the increment, was chosen to produce the matrices which would be tested more thoroughly. The choice of matrix to use was a compromise between preserving the stability of the minimisation and using the most accurate errors. The lower the condition number chosen, the more stable the minimisation but the less accurate the errors. By testing matrices with different condition numbers and analysing the convergence of the cost function value in each case, it was found empirically that a good compromise was to use a matrix with a condition number of 67 for subsequent testing. This is very similar to the condition number (64) of the previously used diagonal matrix for IASI. Using this matrix in the original minimisation configuration resulted in 69 iterations, which took 548 s and with a final penalty value of Comparing these values with Table shows that, although these values are smaller than when using the ill-conditioned diagnostic matrices, they are still significantly larger than the control using the diagonal matrices. However, using this matrix in the operational minimisation, which has a fixed number of 6 iterations, 3 at lower resolution ( km) and 3 at higher resolution( 6 km), and uses pre-conditioning of the full Hessian, results in good convergence. It is unclear whether the need for the reconditioning of the R matrix is specific to the Met Office data assimilation system at these resolutions or whether this type of adjustment is compensating for other sub-optimalities (e.g. neglected spatial error correlations), and will also be necessary in other assimilation systems. Another potential solution to the problem of slow convergence could be to further pre-condition the Hessian, and this may become a more viable option if correlated observation errors are to be used for more observation types Effect of reconditioning on the matrices Figure 8 shows the eigenvalues of the originally diagnosed matrix and the newly reconditioned matrix. Despite the method of reconditioning incrementing all of the eigenvalues, this has a much larger relative effect on the smallest original eigenvalues than on the largest original eigenvalues. Reconditioning the matrix in this way increases the error variances of all channels by a constant amount (. K 2 ). Figure 9 shows that the corresponding error standard deviations change by differing amounts depending on the original value for each channel (.33 and.5 K when the initial error standard deviations are close to and K respectively). However, this method does preserve the structure of the standard deviations between channels. The equivalent plot (not shown) for the minimum eigenvalue method was examined and the standard deviations of all of the channels which originally had the smallest errors were all very similar after reconditioning this way. Also, the standard deviations of the water-vapour-sensitive channels, which had larger standard deviations originally, are changed by a much smaller amount. Although reconditioning using the chosen method does not affect the off-diagonal elements of the R matrix, comparing Figures 2 and shows that it does affect the correlations markedly. In general the correlations are weakened due to the larger diagonal values in the error covariance matrix. In the Eigenvalue Eigenvector number Figure 8. Eigenvalues of the diagnosed (red) and reconditioned (black) IASI error covariance matrix. Standard Deviation (K) Operational Reconditioned Diagnosed Figure 9. Operational, diagnosed and reconditioned IASI error standard deviations. channels with the strongest correlations, they are reduced by as much as Assimilation trial results The use of correlated observation errors for IASI was tested in two assimilation trials. Both controls were set up using a configuration which was operational in July with a forecast model at

8 Accounting for Correlated Error in Sounder Data Assimilation 2427 (a) 6 8 Percentage change - -2 (b) Correlation Figure. Reconditioned IASI error correlation matrix. N3 ( km) resolution and a two-stage data assimilation run at model resolutions of N8 ( km) followed by N26 ( 6 km). Control and experiment ran for 3 days between December and 3 December, and control and experiment 2 ran for days between 4 June and 4 July. Experiment will be referred to as the winter trial and experiment 2 as the summer trial. Figure shows this change generally improves the forecast accuracy, with reductions in the forecast error of most of the forecast verification metrics against both observations and analyses. The most significant improvements are in the pressure at mean sea level (PMSL) and 5 hpa geopotential height in the Southern Hemisphere with reductions in forecast RMSE of between.3 and 2.5% against observations and between.5 and.5% against analyses for these metrics. There are also smaller reductions in forecast RMSE in most of these metrics in the Northern Hemisphere. There is not a significant change in forecast RMSE of tropical winds against observations but against analyses there is a slight increase in forecast RMSE. This may be because of the correlated errors changing the fine-scale humidity structure in the analysis and this not being carried forward into the forecasts. For this reason, these apparent degradations in the Tropics are not true indicators of a reduction in forecast accuracy. Significant changes to the humidity field in the analysis have been previously found to suffer from similar problems when verifying against own analyses (Geer and Bauer, ). Figure 2 shows that the analysis fit to IASI changes significantly when using the correlated errors. Slightly more weight is being given to the high-peaking temperature-sounding channels, as shown by the reductions in standard deviation of analysis departures of 2%. This is to be expected because the error standard deviations for these channels are smaller in the full matrix than in the diagonal matrix and there are negligible correlations between these channels. The new errors for these channels are smaller by approximately a factor of 2 than the corresponding channels on the AIRS and AMSU-A instruments. This may have an effect on the minimisation convergence, but the difference in the assumed errors is consistent with the difference in instrument noise between these instruments. Much more weight is being given to the water-vapour-sensitive channels, as shown by the reductions in standard deviation of analysis departures of 5 %. This is due to the much smaller error standard deviations and despite the significant correlations in the full matrix for these channels. Conversely, less weight is Percentage change Winter trial Summer trial NH T+24 PMSL NH T+48 PMSL NH T+72 PMSL NH T+96 PMSL NH T+ PMSL NH T+24 H5 NH T+48 H5 NH T+72 H5 NH T+24 W25 TR T+24 W85 TR T+48 W85 TR T+72 W85 TR T+24 W25 SH T+24 PMSL SH T+48 PMSL SH T+72 PMSL SH T+96 PMSL SH T+ PMSL SH T+24 H5 SH T+48 H5 SH T+72 H5 SH T+24 W25 Verification metric Figure. Change in mean percentage forecast root mean squared error (RMSE) and weighted skill against (a) observations and (b) analysis for winter and summer trials of accounting for correlated errors for IASI. The verification metrics used are pressure at mean sea level (PMSL), geopotential height at 5 hpa (H5) and winds at 25 and 85 hpa (W25 and W85) for the Tropics (TR) and Northern and Southern Hemispheres (NH, SH). being given to the lower-peaking temperature-sounding channels and some of the surface-sensitive channels, as shown by the increases in standard deviation of analysis departures of 2%. This is an interesting result because the error standard deviations for these channels are similar in both the full matrix and the diagonal matrix. However, there are significant correlations between these channels in the full matrix, so this supports the discussion in section 4., which suggests that accounting for correlations down-weights observations overall. We now look at the change in background fit to observation types whose treatment is unchanged in the control and the trial; this will give an idea of how accounting for correlated errors affects the short-range forecast of those variables to which those observation types are sensitive. Figure 3 shows that the background fit to AIRS is improved for almost all channels. The improvement is largest ( 2%) for the water-vapoursensitive channels (wavenumbers greater than 25 cm )but also significant (.5.5%) for the surface-sensitive channels (wavenumbers cm ). There is a very small change (.%) in the fit to the temperature-sounding channels (wavenumbers less than 75 cm ). These results show that accounting for correlated errors in IASI data improves the shortrange forecasts of temperature near the surface and of humidity in the troposphere. There is a reduction in the standard deviation of Microwave Humidity Sounder (MHS) first-guess departures of.% for channel 3 (83 ± GHz),.6% for channel 4 (83 ± 3GHz)

9 2428 P. P. Weston et al. Percentage Change Wavenumber cm - Figure 2. Percentage change in analysis fit to IASI observations for each IASI channel averaged over the entire winter trial period. Percentage Change Wavenumber cm - Figure 3. Percentage change in background fit to AIRS observations averaged over the entire winter trial period. and.7% for channel 5 (83 ± 7 GHz). These channels peak in the lower to mid-troposphere, with the most improved fit in channel 3 which peaks in the mid-troposphere. This shows that the extra weight being given to water-vapour-sensitive IASI observations consistently improves the short-range forecasts of humidity in the mid-troposphere. 6. Conclusions Inter-channel error correlations in IASI data have been diagnosed using the Desroziers diagnostic. The results show that there are very small correlations between high-peaking temperaturesounding channels, weak correlations between lower-peaking temperature-sounding channels, stronger correlations between surface-sensitive channels and very strong correlations between water-vapour-sensitive channels. These results are consistent with previous estimates of error correlations for IASI. It has been shown that correlations between most of the channels are caused by horizontal representativeness errors, although forward model errors, vertical representativeness errors, apodisation and other pre-processing errors could also contribute. Comparisons between estimates of the diagonal of the full error covariance matrix and the previously used operational values highlight the artificial inflation of the latter to account indirectly for the neglected correlations. Further investigations into the direct effects of accounting for correlations show that this inflation of errors was justified since both accounting for correlations and inflating the diagonal errors down-weight the observations on average. It has also been shown that the instrument noise is the main source of error in the temperature-sounding channels, while it is a minor source of error in both the surface-sensitive and water-vapour-sensitive channels. Using the symmetrised versions of the diagnostic matrices in the Met Office 4D-Var assimilation scheme results in slow convergence of the minimisation. Improving the conditioning of the matrices by adding an artificial error term to the diagonals of the diagnosed matrices and using these modified matrices helps to alleviate this problem. It remains to be seen whether there are alternative solutions to this problem, for example by performing further pre-conditioning to the Hessian. The use of these reconditioned matrices has been tested in two assimilation trials. The treatment of the correlated errors for IASI in 4D-Var leads to a significant improvement in forecast accuracy, as shown by the reductions in forecast RMSE in different variables and at different forecast lead times. In addition, background fits to most AIRS and MHS channels are improved, indicating general improvements to the model temperature and humidity fields. The reason behind these improvements is that more weight is given to IASI observations, particularly those from water-vapour-sensitive channels, in the 4D-Var assimilation scheme. For this reason the use of correlated errors for IASI was implemented into the Met Office operational system in January 3. Future work will concentrate on implementing the operational use of correlated observation errors for the other hyperspectral IR sounders assimilated at the Met Office: AIRS and CrIS (the Crosstrack Infrared Sounder). This method could also be applied to improve the assimilation of data from microwave sounders such as AMSU-A, MHS and the Advanced Technology Microwave Sounder (ATMS). An extension to the work presented at the end of section 3 would be to completely isolate the representativeness errors from other sources of error. To do this, the Desroziers diagnostic could be run on output from a 4D-Var run where the observations resolution is degraded to the model resolution. This would eliminate any contribution from horizontal representativeness error and, by comparing these diagnostics to Figures and 2, the full characteristics of representativeness error could be estimated. Acknowledgements The authors acknowledge the technical and scientific guidance provided by Laura Stewart, James Cameron, Brett Candy, Fiona Smith, Ed Pavelin, Nigel Atkinson and Gordon Inverarity. The authors are also grateful to the two anonymous reviewers whose comments helped to improve the manuscript. 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