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  • View in gallery

    Jacobians with respect to temperature of the AIRS radiance observation operator for (from left to right) groups 1-a, 2, and 3-b.

  • View in gallery

    Observation and background error variances and innovation variances from the Environment and Climate Change Canada assimilation system. Those were taken as the true values in the experiments.

  • View in gallery

    (top) Estimated observation and (bottom) background error variance when both the full and are adjusted. Initially, is the identity, while is taken as the true value (). Results are obtained after a single iteration resulting in a perfect fit to the innovation covariances.

  • View in gallery

    Convergence of the iterations in the case where . The distance between each iterate and the true state is the dashed curve, while the thick line corresponds to the distance between two successive iterates.

  • View in gallery

    Observation error variance initial (dashed curve), after one (dotted curve), and after 10 (solid line) iterations when . The red curve corresponds to .

  • View in gallery

    Observation error variance when (overestimated case): initial (dashed curve), after one (dotted curve), and after 10 (solid line) iterations. The red curve corresponds to .

  • View in gallery

    As in Fig. 4, but when (overestimated case).

  • View in gallery

    Correlations of .

  • View in gallery

    Estimated when is diagonal and .

  • View in gallery

    True observation error correlations . The true variances are the same as in section 3.

  • View in gallery

    Observation error variance when : initial (dot–dashed curve), after one (solid curve), and after 10 (dashed line) iterations. The red curve corresponds to . True observation error is with the error correlations shown in Fig. 10.

  • View in gallery

    Convergence of the iterations in the case where . The distance between each iterate and the true state is the dashed curve, while the thick line corresponds to the distance between two successive iterates. This is as in Fig. 4, but when the true observation error is correlated.

  • View in gallery

    Estimated observation error correlations when after (a) one iteration and (b) 10 iterations. The true observation error is correlated.

  • View in gallery

    As in Fig. 4, but with diagonal with the true error variances.

  • View in gallery

    Observation error variance after one (dashed) and 10 (solid black) iterations when and is taken as diagonal with the true observation error variances. The red curve corresponds to and also to the initial .

  • View in gallery

    As in Fig. 11, but when : after one (solid blue curve), three (dashed black line), and 10 (solid black line) iterations, which, in this case, converged to . The red curve corresponds to . The initial is the identity.

  • View in gallery

    As in Fig. 12, but when .

  • View in gallery

    As in Fig. 13, but when .

  • View in gallery

    As in Fig. 16, but when .

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    As in Fig. 12, but when .

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    As in Fig. 13, but when .

  • View in gallery

    Observation error variance for AIRS EOS-2 as function of channel index: a priori prescribed (solid black), estimated from one iteration (blue), and solution at convergence (dashed black). The variance of the prescribed is shown in red.

  • View in gallery

    As in Fig. 22, but for IASI MetOp-1.

  • View in gallery

    Observation error variance for radiosonde temperatures as a function of height (in hPa). As in previous figures, the error variances prescribed (solid black), estimated with one iteration (blue), and the solution at convergence (dashed black) are shown. The variance of the prescribed is shown in red.

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Convergence Issues in the Estimation of Interchannel Correlated Observation Errors in Infrared Radiance Data

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  • 1 Department of Earth and Atmospheric Sciences, University of Québec in Montréal, Montréal, Québec, Canada
  • | 2 Meteorological Service of Canada, Dorval, Québec, Canada
  • | 3 Meteorological Research Division, Environment and Climate Change Canada, Dorval, Québec, Canada
Open access

Abstract

A posteriori consistency diagnostics have been used in recent years to estimate correlated observation error. These diagnostics provide an estimate of what the observation error covariances should be and could, in turn, be introduced in the assimilation to improve the statistical consistency between the error statistics used in the assimilation and those obtained from observation departures with respect to the background and the analysis. To estimate the observation error covariances, it is often assumed that the background error statistics are optimal, an assumption that is open to criticism. The consequence is that if the background error covariances are in error, then the estimated observation error statistics will adjust accordingly to fit the innovation error covariances. In this paper, the RTTOV radiative transfer model is used as the observation operator. Using controlled experiments, the background error is considered fixed, and it is shown that the iterative procedure to estimate the observation error may require more than one iteration. It is also shown that the underlying matrix equation being solved can be factorized, and the exact solution can be obtained. If the true background error covariances are used in the assimilation, the estimated observation error covariances are then obtained by subtracting the background error covariances from those of the innovations. This can be applied to the full set of assimilated observations. Using the Environment Canada assimilation system, the results for several types of observations indicate that the background error estimation would deserve additional attention.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Pierre Gauthier, gauthier.pierre@uqam.ca

Abstract

A posteriori consistency diagnostics have been used in recent years to estimate correlated observation error. These diagnostics provide an estimate of what the observation error covariances should be and could, in turn, be introduced in the assimilation to improve the statistical consistency between the error statistics used in the assimilation and those obtained from observation departures with respect to the background and the analysis. To estimate the observation error covariances, it is often assumed that the background error statistics are optimal, an assumption that is open to criticism. The consequence is that if the background error covariances are in error, then the estimated observation error statistics will adjust accordingly to fit the innovation error covariances. In this paper, the RTTOV radiative transfer model is used as the observation operator. Using controlled experiments, the background error is considered fixed, and it is shown that the iterative procedure to estimate the observation error may require more than one iteration. It is also shown that the underlying matrix equation being solved can be factorized, and the exact solution can be obtained. If the true background error covariances are used in the assimilation, the estimated observation error covariances are then obtained by subtracting the background error covariances from those of the innovations. This can be applied to the full set of assimilated observations. Using the Environment Canada assimilation system, the results for several types of observations indicate that the background error estimation would deserve additional attention.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Pierre Gauthier, gauthier.pierre@uqam.ca

1. Introduction

In the last decade, large volumes of satellite data were introduced in operational data assimilation systems. Advanced high-spectral-resolution infrared sounders like the Atmospheric Infrared Radiance Sounder (AIRS), the Infrared Atmospheric Sounding Interferometer (IASI), and the Cross-Track Infrared Sounder (CrIS) have become important components of the global observation system on which numerical weather prediction (NWP) centers rely. These sounders provide information about atmospheric temperature, humidity, and constituents based on measurements of radiance on 2378 (8461) wavelengths for AIRS (IASI) in the thermal infrared from 3.6 to 15.5 . The assimilation of these radiances presents challenges associated with, among others, channel selection, the identification of clear radiances, and the estimation of the observation error statistics. This paper is concerned with this last point.

It is understood that error correlations, if not accounted for, result in excessive weight given to the observations and suboptimal analyses and forecasts. In the case of spatiotemporal observation error correlations, short of considering these correlations explicitly, the problem is alleviated in practice by inflating the observation error and by thinning the density of observations (Liu and Rabier 2003; Collard and McNally 2009; Stewart et al. 2014). Some progress has been made recently on the estimation of interchannel error correlations and its explicit use in data assimilation (Garand et al. 2007; Heilliette and Garand 2015). It has been found that the interchannel observation error correlations (IOEC) are significant for the water vapor channels and channels sensitive to the surface. It has also been shown that introducing IOEC has a significant impact on the analysis. Bormann and Bauer (2010) and Bormann et al. (2010) used three different methods to estimate the spatial and interchannel observation error correlations of AIRS and IASI observations used in the European Centre for Medium-Range Weather Forecasts (ECMWF): the method of Hollingsworth and Lönnberg (1986), a method based on statistical consistency diagnostics introduced by Desroziers et al. (2005), and the so-called background error method. All three approaches concur to establish that the channels sensitive to the surface and the humidity-sounding channels are prone to interchannel error correlations. Moreover, Stewart et al. (2014) and Weston et al. (2014) also found noticeable correlations for IASI channels sensitive to water vapor and surface used in the 4DVar data assimilation process at the Met Office. Taking into account the observation error correlations in the assimilation requires first that we be able to estimate the statistical characteristics of the observation error (Weston et al. 2014). For NWP centers, it would be very useful to develop a reliable methodology to infer observation error statistics to better use satellite observations.

The method proposed by Desroziers et al. (2005), mentioned above, uses diagnostics of statistical consistency of the observation departures from the analysis and the background state to estimate the observation error statistics. Recently, this approach has been used by many to estimate the error statistics from information contained in the by-products of the assimilation. Significant benefits have been obtained in terms of forecast impact through explicitly accounting for interchannel observation error correlations in global NWP (Bormann and Bauer 2010; Bormann et al. 2010; Stewart et al. 2014; Weston et al. 2014; Bormann et al. 2016; Heilliette and Garand 2015; Waller et al. 2016; Ménard 2016; Campbell et al. 2017). However, due to the complexity of operational data assimilation systems, it is difficult to iterate the algorithm to convergence, and the inferred correlations correspond to a single iteration of the algorithm. Li et al. (2009) and Miyoshi et al. (2013) have also studied this problem in the context of the ensemble Kalman filter.

The objective of the present paper is to use a one-dimensional (1D) assimilation based on the Radiative Transfer for TOVS (RTTOV) model (Matricardi and Saunders 1999) as the observation operator to study the convergence properties of the Desroziers approach to estimate the true observation error covariances. Many studies have used, for practical reasons, a single application of the consistency diagnostics of Desroziers et al. (2005) to estimate the observation error. The question we want to address is whether more than one iteration is required, and if so, if it converges to reliable error statistics. The problem is cast in a framework similar to observing system simulation experiments (OSSEs) in which the true observation and background error statistics are known. Innovations alone are not enough to discriminate what part can be attributed to observation or background error, so additional assumptions are needed (Talagrand 1999, 2003). It is practical to assume that the background error statistics are correct, knowing, however, that they can be incorrect. This question relates to the background error method of Bormann et al. (2010). Since the background error statistics are assumed to be known, the observation error covariances can be immediately obtained by subtracting them from the innovation error covariances.

The paper is organized as follows. Section 2 recalls the method of Desroziers and its main properties. It is shown that in the case where the background error covariance matrix is known, the iterative method seeks to solve a matrix equation having an exact solution to which the iterative method should be converging. Section 3 introduces the 1D system used to estimate the observation errors and their correlations associated with the RTTOV observation operator used for the assimilation of AIRS observations. A discussion is given of some properties of the matrix equation that raise some problems for solving it iteratively. Section 4 shows the results of the experiments on the estimation of observation error covariances when the true ones include error correlations. The results show that in most cases, more than one iteration is required to reach convergence, and furthermore, it may not converge to the exact solution of the matrix equation. In section 5, innovations obtained from experiments done with the assimilation system of Environment and Climate Change Canada are used to compute the exact solution, assuming the background error covariances to be true. The resulting error covariance matrices are examined for several assimilated observation types. A summary and conclusions are presented in section 6.

2. Estimation of the observation error using statistical consistency diagnostics

Desroziers et al. (2005) showed that if the error statistics used in the assimilation were statistically consistent, the results from the assimilation should satisfy some simple relationships. Considering a background state represented by the vector with error covariance , and observations with error covariance , the departure of the background and the resulting analysis from observations are represented, respectively, as
eq1
eq2
with , , and being the observation, background, and analysis errors, respectively. H is the nonlinear observation operator, while is its Jacobian evaluated at . The analysis error is then related to by
eq3
where is the gain matrix, and . The assimilation links the analysis error with the innovations based on the a priori estimates of the background error covariances and the observation error covariances through the gain matrix . This assumes that the data assimilation system can be reasonably approximated by linearized observation operators and Gaussian error statistics. Assuming the observation and background errors and to be uncorrelated, it follows that , where the tilde stands for estimated quantities, while stands for the statistical average estimated from an ensemble of innovations assumed to have the same error statistics using an ergodicity assumption. This is usually what is done when collecting all innovations associated with observations deemed to be of the same types, even though they may be taken by the same instrument but at different locations and times.
Relationships (1)–(3) of Desroziers et al. (2005) hold if there is statistical consistency. If this is not the case, then
e1
e2
What (1) and (2) state is that the estimated (respectively, ) are not measured directly but derived from through the a priori covariances and (respectively, ). Furthermore, the estimated and are not even symmetric matrices in general. If the a priori error statistics and used by the assimilation are such that the estimated innovation covariance matrix is consistent with those of the a priori error statistics, then
eq4
and, from (1) and (2), and , which does not imply that either corresponds to its true value. It is important to state that this requires only that the observed background and observation error statistics be consistent (Talagrand 1999, 2003).
This result is useful in itself as it permits to evaluate if the error statistics in the assimilation are properly calibrated. In practice, they are not, and it is necessary to devise a method to estimate these errors. The next step is to use (1) and (2) iteratively as successive estimates of and converging, hopefully, to meet the consistency diagnostic. The iterative procedure is then
e3
e4
starting with and associated with the error statistics prescribed in the assimilation system. It is important to state that when the full matrices are available, (3) and (4) do not require redoing an assimilation to estimate and . Given that is estimated and the link to the analysis is established through the a priori and , the approach is to modify those covariances to improve the fit between , the resulting innovations’ covariances, and the measured ones .
At convergence, the exact solutions of (3) and (4) are either that and both vanish or that they are and , such that
eq5
If one is to tune both the full matrices and , then adding (3) and (4) yields immediately that
eq6
At the very first iterate, and adjust to match the estimated innovations. As discussed in Ménard (2016), an additional condition is needed to be able to estimate and . In Weston et al. (2014), the additional condition is that is kept constant. When an explicit form of the innovations is not available, it is more appropriate to estimate them by doing an analysis using as the a priori in the assimilation and obtain
eq7
which is what Weston et al. did, keeping constant. It was then concluded that one iteration was enough to obtain an estimate deemed reasonable.
Tuning should then be made for considering to be given, but then, having estimated and being assumed to be true, it follows that
e5
and is the exact solution that should come out when (3) is solved iteratively. This is related to what Bormann and Bauer (2010) considered as the third approach. It is then possible to rewrite (1) as
eq8
This means that when is taken to be fixed, the problem can be stated as solving the matrix equation
e6
with as in (5). There are two exact solutions to this equation: , which is unphysical, and . When solving (3) iteratively, we are interested to know whether convergence can be reached, and if so, if it converges to the exact value. This will be looked at in a context where it will be possible to solve the full matrix problem while being relevant for the estimation of interchannel correlations for infrared radiance measurements.

3. Application of diagnostics to AIRS observation data in a simplified 1D case

The estimation of interchannel cross correlations in observation error is studied here using an appropriate observation operator based on radiative transfer. It will be simplified to a 1D assimilation problem for which the background error covariances will be modulated by a linearized but representative radiative transfer operator.

a. A 1D linear radiative transfer model

RTTOV, described in Matricardi and Saunders (1999), was used for the assimilation of 85 AIRS channels presented in Table 1, which have been regrouped according to common properties. Group 2 is referred to as the water vapor channels, as they are sensitive to water vapor but also to temperature, while groups 1 and 3 are sensitive to the lower-tropospheric temperature and surface temperature . Figure 1 shows the Jacobians for temperature for groups 1-a, 2, and 3-b, showing that group 2 is sampling at different heights, which provides more information about the vertical structure. Groups 1-a and 3-b are sensitive to surface temperature and temperature in similar layers. In the present case, the Jacobian is a linearization of RTTOV around a profile representative of the midlatitudes.

Table 1.

Channel groups for the 85 AIRS channels, excluding channels 1–4.

Table 1.
Fig. 1.
Fig. 1.

Jacobians with respect to temperature of the AIRS radiance observation operator for (from left to right) groups 1-a, 2, and 3-b.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

A simple model has been constructed that uses this Jacobian with respect to temperature T, the logarithm of specific humidity , surface pressure , and surface temperature , the vertical discretization for T and being on 80 levels up to 0.1 hPa. This has been used to define the observation operator as an explicit 85 × 162 matrix. The background error covariances for the state variables (T, , , ) were derived from those used in the static component of the ensemble variational system (EnVar) of Environment and Climate Change Canada (Buehner et al. 2015), neglecting, however, the cross covariances between temperature and humidity. Figure 2 shows the observation and background error variances (in observation space) for these 85 channels altogether with the resulting innovation variances, which here is simply the sum of the observation and background error variances.

Fig. 2.
Fig. 2.

Observation and background error variances and innovation variances from the Environment and Climate Change Canada assimilation system. Those were taken as the true values in the experiments.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

b. Tuning the observation error covariance matrix: Evaluation of the iterative method

As in an OSSE, the true values , , and are defined as those of the operational system (see Fig. 2). In all experiments, it then follows that . Setting then the a priori statistics as and , one has to solve the matricial equations (3) and (4). The low dimension of and makes it possible in this case to solve the problem using these full matrices.

As discussed in section 2, the exact solution is known to be from (5), which differs from the true value . Two questions are asked: Does the iterative procedure defined by (3) converge to , and is it a reasonable approximation of the true value? First, it should be said that since for any two symmetric matrices and ,
eq9
in general, (3) yields a matrix that is not necessarily symmetric or positive definite. Therefore, at each iteration, only the symmetric part must be retained by filtering . Thus,
e7
is used to constrain the result to have this essential property of an error covariance matrix. However, the solution may not be positive definite.
To measure the convergence of the iterates, the Frobenius norm for any matrix
eq10
will be used to evaluate , while the departure from the true value is .

c. Two simple experiments with diagonal

In a first experiment with diagonal, was initially set to its true value and to the identity. This is the best of cases, and both and were tuned using (3) and (4). As expected from the discussion in section 2, exact convergence was reached in a single iteration. However, Fig. 3 shows that in particular for channels 40–55, the variances of neither nor , the values at convergence, are correct, being underestimated for the former and overestimated for the latter. Even though was initially correct, there is no information to indicate that the misfit to the innovations can be attributed to either the observation or the background error. Hence, , but and . Although the full matrices and were estimated, the retrieved remained nearly diagonal (not shown).

Fig. 3.
Fig. 3.

(top) Estimated observation and (bottom) background error variance when both the full and are adjusted. Initially, is the identity, while is taken as the true value (). Results are obtained after a single iteration resulting in a perfect fit to the innovation covariances.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Unless mentioned otherwise, from now on, is kept constant in all experiments and will be such that , α being a scaling factor. Therefore, it keeps the structure of , but its amplitude is altered to make it perfectly known, underestimated (), or overestimated ().

The starting point was taken to be the identity, and (3) was iterated 10 times. In an experiment in which the background error was set to the true value (), Fig. 4 shows that convergence is reached to what it should be in fewer than five iterations. Figure 5 presents the resulting variances, which show differences between the first and last iterations, mostly in the water vapor channels. In this case, is the true value, and this is what is recovered since is kept constant.

Fig. 4.
Fig. 4.

Convergence of the iterations in the case where . The distance between each iterate and the true state is the dashed curve, while the thick line corresponds to the distance between two successive iterates.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 5.
Fig. 5.

Observation error variance initial (dashed curve), after one (dotted curve), and after 10 (solid line) iterations when . The red curve corresponds to .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

The exact solution in this case is [(5)], and negative variances in can be obtained when α is too large. This occurs when setting . Figure 6 shows that has slightly negative variances for channels index 16–28. Figure 7 shows that even though convergence is reached, it does not converge to the solution , which means that
eq11
The estimate so obtained is not consistent with the information provided by the innovations. What is surprising, however, is that convergence is reached quickly to values that are closer to the true solution than the solution to which it should have converged. Such a behavior occurs only if the solution is “unphysical” (negative variances). These examples show that in some cases, solving iteratively (6) does not lead to its true solution because solving iteratively a nonlinear matrix equation is quite different than solving a similar nonlinear scalar equation (Higham 2008).
Fig. 6.
Fig. 6.

Observation error variance when (overestimated case): initial (dashed curve), after one (dotted curve), and after 10 (solid line) iterations. The red curve corresponds to .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 7.
Fig. 7.

As in Fig. 4, but when (overestimated case).

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

4. Estimation of observation error covariances

Interchannel correlations are to be expected in observation error of infrared measurements sensitive to water vapor (group 2 in Table 1) and to the surface (all other groups in Table 1). When estimating the structure of correlations from innovations, keeping constant, the estimated observation error covariance matrix is
e8
e9
and these correlations will include those from , which, as shown in Fig. 8, induce interchannel correlations. If is indeed correct, then the retrieved observation error would be
eq12
but if , then
eq13
Fig. 8.
Fig. 8.

Correlations of .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Thus, the estimated interchannel correlations can, in fact, come from the background error covariances embedded within the innovation covariances. On the other hand, if the observation error of a certain type of observation is known not to be correlated (e.g., radiosondes), then if the corresponding does show error correlations, this would be an indication that differs from its true value. In the preceding section, experiments in which was diagonal, the estimated observation error correlations when did induce such erroneous correlations. These are shown in Fig. 9 for a similar experiment in which the total background error variance was underestimated (). It shows that correlations are present in , while has no correlations in this case.

Fig. 9.
Fig. 9.

Estimated when is diagonal and .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

In a previous study by Heilliette and Garand (2015), the Desroziers diagnostics were used to estimate these correlations. In our study, covariances were constructed using the operational observation error variances and these correlations. From the point of view of its impact on the analysis and forecasts, this was a reasonable representation of a true . In our experiments, this estimate has been taken as the true observation error covariances (see Fig. 10), while the true is the same as the one used before.

Fig. 10.
Fig. 10.

True observation error correlations . The true variances are the same as in section 3.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

When the background error is assumed to be perfect (), Fig. 11 shows the resulting observation error variance, while Fig. 12, similar to Fig. 4, shows that more than one iteration is needed and that convergence is not monotonic. At times, this can lead to very bad solutions (e.g., after three iterations). Figure 13 shows the estimated observation error correlations obtained after one (Fig. 13a) and 10 iterations (Fig. 13b). Comparing with the true observation error correlations (Fig. 10), there are significant improvements brought in by iterating, indicating that not only the variance but also the correlations could be estimated.

Fig. 11.
Fig. 11.

Observation error variance when : initial (dot–dashed curve), after one (solid curve), and after 10 (dashed line) iterations. The red curve corresponds to . True observation error is with the error correlations shown in Fig. 10.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 12.
Fig. 12.

Convergence of the iterations in the case where . The distance between each iterate and the true state is the dashed curve, while the thick line corresponds to the distance between two successive iterates. This is as in Fig. 4, but when the true observation error is correlated.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 13.
Fig. 13.

Estimated observation error correlations when after (a) one iteration and (b) 10 iterations. The true observation error is correlated.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

This small difference between the first and last iterations may be because the initial is the identity, which is unrealistic. Using instead to be diagonal with the true error variances is closer to the situation at hand when estimating the observation error correlations. Figure 14 shows that convergence seems to be reached after three iterations, while a significant difference remains with respect to the true value. This figure shows that convergence is not monotonic but finally converges to the true value after 20 iterations. Figure 15 shows the variances after one and 10 iterations when the true state is finally recovered. The initial variances of correspond to the true values in this case. One or 10 iterations are not enough to converge to , which is the true solution in this case. This shows that the road to convergence may depend on the initial , but the conclusion that more than one iteration is needed remains.

Fig. 14.
Fig. 14.

As in Fig. 4, but with diagonal with the true error variances.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 15.
Fig. 15.

Observation error variance after one (dashed) and 10 (solid black) iterations when and is taken as diagonal with the true observation error variances. The red curve corresponds to and also to the initial .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

In the following experiments, the starting point is again taken as the identity to test whether the iterative procedure converges or not to what it should. In the next experiment, the total background error variance is underestimated (). Figure 16 shows the resulting observation error variance after one and 10 iterations when convergence to has been reached. This, as discussed above, implies that since . This figure also shows that there are substantial changes in the solution between the first and last iterations. Figure 17 presents the error () and the departure () as a function of iteration. The decrease is nonmonotonic, and it takes 10 iterations to converge to . Figure 18 shows also significant differences in the correlations obtained after one and 10 iterations.

Fig. 16.
Fig. 16.

As in Fig. 11, but when : after one (solid blue curve), three (dashed black line), and 10 (solid black line) iterations, which, in this case, converged to . The red curve corresponds to . The initial is the identity.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 17.
Fig. 17.

As in Fig. 12, but when .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 18.
Fig. 18.

As in Fig. 13, but when .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Finally, if the total background error variance is overestimated (), this results in negative variances in (see Fig. 19). These results also show that iterating does not converge to , but to a solution that is closer to . This can also be seen in Fig. 20, and this implies that the innovation covariances expected from the estimated observation and background error covariances differ from the true ones since here, . The surprising and unexplained fact is that by not fitting the innovations, the estimated correlations get closer to the true ones . Furthermore, Figs. 21a and 21b show that the resulting correlations compare relatively well to the true ones. This can only be an artifact of the iterative procedure employed to solve (3). As was mentioned earlier, each iteration yields a matrix , which is, in general, not symmetric, and it is a necessity to filter out the antisymmetric component [see (7)].

Fig. 19.
Fig. 19.

As in Fig. 16, but when .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 20.
Fig. 20.

As in Fig. 12, but when .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 21.
Fig. 21.

As in Fig. 13, but when .

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

In summary, when the incorrect background error covariances are used, fitting the innovations yields an exact solution to which an iterative approach should be converging. This is what has been observed, but more than one iteration is needed. There can also be cases where there is a fundamental disagreement between the assumed background error covariances and what is measured by the innovation covariances: this can translate as negative variances in , which is unphysical. Solving iteratively then converges to a solution that, when added to the assumed , results in innovation covariances that do not fit the measured ones . The surprising result is that this solution seems to be closer to the true observation error covariance matrix . The fact that the estimate for does not lead, with the assumed , to innovation covariances that agree with the true ones suggests that this estimate is incorrect.

As was done in other studies (Desroziers et al. 2005; Bormann et al. 2010), one can alter the background error covariance, tuning then both and . The results from the first experiment, presented in section 3c, show that when tuning both the observation and background error covariance matrices, a perfect fit is obtained at the first iteration, but the estimated and may both differ from the true ones (see Fig. 3). Since the same influences the fit to all observations, such changes to this matrix would have to be validated with respect to all assimilated observations.

5. Estimation based on innovations obtained from the Environment and Climate Change Canada data assimilation system

The results presented up to now have shown that assuming the background error covariances to be correct, the results of one iteration of the Desroziers diagnostics differed from what the iterative process should be converging to. Using innovations and the a priori error statistics of the Environment and Climate Change Canada (ECCC) assimilation system (Buehner et al. 2015; Heilliette and Garand 2015), the results obtained after one iteration were compared to the exact solution for several types of data. The associated background errors in observation space were obtained using a randomization method to obtain for all observation types. The variances obtained after one iteration were compared to that solution and the prescribed observation error statistics.

The results for AIRS radiances (Fig. 22) show significant differences between the estimated ones after one iteration and . The latter even shows negative variances, which indicates that the corresponding background error variances () are too large. The a priori observation error variances are significantly larger than both. Similar results were obtained for IASI on MetOp-1 (Fig. 23).

Fig. 22.
Fig. 22.

Observation error variance for AIRS EOS-2 as function of channel index: a priori prescribed (solid black), estimated from one iteration (blue), and solution at convergence (dashed black). The variance of the prescribed is shown in red.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

Fig. 23.
Fig. 23.

As in Fig. 22, but for IASI MetOp-1.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

For radiosonde measurements, the estimated observation error variances (Fig. 24) after one iteration and at convergence are closer to one another and are both positive. Other sets of observations were also looked at that also showed that the variances of could be negative and quite different from the result obtained after one iteration. This suggests that the background error variances are overestimated. This sheds some concern on the estimated observation error obtained from those diagnostics.

Fig. 24.
Fig. 24.

Observation error variance for radiosonde temperatures as a function of height (in hPa). As in previous figures, the error variances prescribed (solid black), estimated with one iteration (blue), and the solution at convergence (dashed black) are shown. The variance of the prescribed is shown in red.

Citation: Monthly Weather Review 146, 10; 10.1175/MWR-D-17-0273.1

The main conclusion from this experiment is that since negative variances for are observed, the background error variances are overestimated and inconsistent with those of the innovations. One could then use this to make changes to and recalculate and verify if it still has negative variances. One should keep in mind that some of the assumptions made in this approach may not hold. For instance, as pointed out by Janjić et al. (2018), departure-based quality control may introduce some correlation between observation and background error. When is assumed to be statistically consistent, the results presented here indicate that iterating would eventually converge to , but this does not mean that they correspond necessarily to the true . Using this approach makes it simpler and provides a comparison against all observations to ensure statistically consistent results.

6. Summary and conclusions

Diagnostics of statistical consistency are very useful to assess if error statistics used in an assimilation are correct. However, this only tests if the sum of the a priori observation () and background error () covariances agrees with those embedded within the measured innovation covariances () obtained through comparison with real observations. As pointed out by Talagrand (1999), additional information is needed to correctly evaluate the observation error from innovations. An assumption that can be made is to consider to be correct and subtract it from the innovation covariances to obtain .

Based on departures among the observations, the analysis, and the background, these diagnostics can be used to correct and/or . This leads to an iterative procedure that has been used in several studies, mostly with a single iteration. When observation error correlations are estimated, this leads to matrix equations that are resolved iteratively to obtain and . If both are tuned, then their sum fits the innovation error covariances but does not necessarily correspond to the true observation and background (in observation space) error covariances. Assuming to be fixed, the solution to which the iterates should converge is known, and this has been used here to analyze the convergence.

The results presented here showed that in this context, one iteration is not enough and that the iterative procedure may not be monotonic. Moreover, if is assumed to be fixed, then the solution is The preferred approach should be to use instead of trying to solve the problem iteratively. In cases where has negative variances, the iterative procedure did not converge to as it should have, and this implies that the resulting innovation covariances do not agree with . Negative variances indicate that cannot be assumed to be true. The question is then to find a way to approximate . Using innovations obtained from the ECCC assimilation system, was evaluated for all observations. For several types of observations, the estimated observation error variance obtained from one iteration differs from the solution it should be converging to. Furthermore, reveals negative variances for nearly all types of satellite observations.

These results indicate that recovering observation error correlations from covariances requires a good confidence in the background error covariances used in the assimilation. As discussed by Bormann et al. (2010) and Chun et al. (2015), a physically based approach would be a good alternative, in which measurement and representation error (Janjić et al. 2018) must be estimated.

It is important to recall that the assimilation uses a linear observation operator, with Gaussian observation and background error with no biases assumed in either the background or the observation. But, as pointed out by many (e.g., Desroziers et al. 2005), the problem is much more complex in reality, as these assumptions have limitations. In practice, many studies (Bormann and Bauer 2010; Bormann et al. 2010; Stewart et al. 2014; Weston et al. 2014; Heilliette and Garand 2015; Waller et al. 2016; Campbell et al. 2017) reported a positive impact on analyses and forecasts by introducing observation error correlations estimated with the Desroziers method. Many have found it necessary to also alter the background error covariances.

The diagnostic offered by can be used to validate changes to the background error covariances obtained, for instance, from ensemble prediction systems. A comparison of the background state and the analysis against all available observations provides insight on the impact of on the estimation of observation error covariances (Buehner et al. 2005; Waller et al. 2017). By computing for a proposed , the presence of negative variances for any observation type would be an indication that does not lead to statistically consistent results. Given the assumptions mentioned above, it would be necessary to reevaluate innovations with an assimilation using a new estimate of . Innovation statistics from different observations reveal different aspects of the background error. A comparison against all observations provides information on different facets of the background error in terms of variables (), regions, or atmospheric conditions. Calibration of ensemble statistics would benefit from a comparison of innovations with respect to the larger body of observations used in the assimilation. This will be the object of future work.

Acknowledgments

The first author would like to thank Prof. Peter J. van Leeuwen and Prof. Nancy Nichols, who hosted him for a sabbatical in the Department of Meteorology at the University of Reading from October 2015 to April 2016. Part of this work was done during this visit and benefited from discussions with many at the University of Reading, the Met Office, and ECMWF. Special thanks to Dr. Sarah Dance, associate professor, Prof. Nancy Nichols, and Dr. Joanne Waller for discussions and comments on an early version of the manuscript that were very much appreciated. The authors thank also the three anonymous reviewers who did a careful and detailed review of the manuscript. The paper was significantly improved by their comments. This research has been funded in part by the grants and Contribution program of Environment and Climate Change Canada (ECCC) and a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant program.

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