Impact of AMSU-A Radiances in a Column Ensemble Kalman Filter

Herschel L. Mitchell Recherche en Prévision Numérique–Assimilation de Données, Environment and Climate Change Canada, Dorval, Québec, Canada

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P. L. Houtekamer Recherche en Prévision Numérique–Assimilation de Données, Environment and Climate Change Canada, Dorval, Québec, Canada

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Sylvain Heilliette Recherche en Prévision Numérique–Assimilation de Données, Environment and Climate Change Canada, Dorval, Québec, Canada

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Abstract

A column EnKF, based on the Canadian global EnKF and using the RTTOV radiative transfer (RT) model, is employed to investigate issues relating to the EnKF assimilation of Advanced Microwave Sounding Unit-A (AMSU-A) radiance measurements. Experiments are performed with large and small ensembles, with and without localization. Three different descriptions of background temperature error are considered: 1) using analytical vertical modes and hypothetical spectra, 2) using the vertical modes and spectrum of a covariance matrix obtained from the global EnKF after 2 weeks of cycling, and 3) using the vertical modes and spectrum of the static background error covariance matrix employed to initiate a global data assimilation cycle. It is found that the EnKF performs well in some of the experiments with background error description 1, and yields modest error reductions with background error description 3. However, the EnKF is virtually unable to reduce the background error (even when using a large ensemble) with background error description 2. To analyze these results, the different background error descriptions are viewed through the prism of the RT model by comparing the trace of the matrix , where is the RT model and is the background error covariance matrix. Indeed, this comparison is found to explain the difference in the results obtained, which relates to the degree to which deep modes are, or are not, present in the different background error covariances. The results suggest that, after 2 weeks of cycling, the global EnKF has virtually eliminated all background error structures that can be “seen” by the AMSU-A radiances.

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

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

Corresponding author: Herschel Mitchell, herschel.mitchell@canada.ca

Abstract

A column EnKF, based on the Canadian global EnKF and using the RTTOV radiative transfer (RT) model, is employed to investigate issues relating to the EnKF assimilation of Advanced Microwave Sounding Unit-A (AMSU-A) radiance measurements. Experiments are performed with large and small ensembles, with and without localization. Three different descriptions of background temperature error are considered: 1) using analytical vertical modes and hypothetical spectra, 2) using the vertical modes and spectrum of a covariance matrix obtained from the global EnKF after 2 weeks of cycling, and 3) using the vertical modes and spectrum of the static background error covariance matrix employed to initiate a global data assimilation cycle. It is found that the EnKF performs well in some of the experiments with background error description 1, and yields modest error reductions with background error description 3. However, the EnKF is virtually unable to reduce the background error (even when using a large ensemble) with background error description 2. To analyze these results, the different background error descriptions are viewed through the prism of the RT model by comparing the trace of the matrix , where is the RT model and is the background error covariance matrix. Indeed, this comparison is found to explain the difference in the results obtained, which relates to the degree to which deep modes are, or are not, present in the different background error covariances. The results suggest that, after 2 weeks of cycling, the global EnKF has virtually eliminated all background error structures that can be “seen” by the AMSU-A radiances.

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

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

Corresponding author: Herschel Mitchell, herschel.mitchell@canada.ca

1. Introduction

As the number of satellite observations available for assimilation has increased over the past several decades, radiances from the Advanced Microwave Sounding Unit-A (AMSU-A) instruments have been critically important over oceanic areas and in the stratosphere (Gelaro et al. 2010; Todling 2013; Joo et al. 2013). However, despite the global coverage they afford and their abundance, there are a number of reasons why assimilating AMSU-A radiance data is more difficult than assimilating in situ observations (such as radiosonde data). First, a radiative transfer (RT) model is needed. Second, unlike in situ point observations, AMSU-A radiances are sensitive to a relatively broad atmospheric layer and can “see” only large-scale vertical structures (Rodgers 2000, section 1.2.1). Third, AMSU-A observation errors may have significant horizontal, temporal, and interchannel correlations (Gorin and Tsyrulnikov 2011) [but see Bormann and Bauer (2010) for a different view]. If such correlations are indeed important, they need to be both estimated and accounted for (which is difficult) and they complicate the specification of observation errors for AMSU-A radiance data. [These correlations are thought to be due, at least in part, to bias correction procedures (e.g., Dee and Uppala 2009), which the radiance data must be subjected to.] While fast and convenient RT models are now available (e.g., RTTOV; Saunders et al. 1999) and are continually being improved, the latter two issues continue to be problematic. The result is that AMSU-A radiance profiles are more difficult to use and of more limited utility than profiles of conventional data.

Results from two EnKF–4D-Var intercomparison studies (Miyoshi et al. 2010; Bonavita et al. 2015) have led their authors to conclude that satellite radiance observations have a smaller impact in EnKF systems than they do in comparable variational systems. It is important to investigate this finding, as discussed by Houtekamer and Zhang (2016, section 7). It may explain the finding, in the intercomparison study of Buehner et al. (2010b), that when both EnKF and variational systems used the same EnKF-generated background fields to calculate background error covariances, better deterministic forecasts were produced from the variational assimilation system than from the EnKF assimilation system.

Further motivation for the current study was provided by the results of a project, undertaken several years ago by two of the current authors and M. Tsyrulnikov, aimed at improving the operational assimilation of AMSU-A radiances (Houtekamer et al. 2014a). That project attempted to replace the traditional diagonal specification of the observation error covariance matrix by a description that included horizontal, temporal, and interchannel correlations [based on Gorin and Tsyrulnikov (2011)]. However, this attempt to improve the assimilation of AMSU-A radiances in the Canadian EnKF (Houtekamer et al. 2014b) yielded disappointing results. In fact, this important change in the AMSU-A observation error specifications was found to have little impact.

This experience motivated a series of EnKF experiments aimed at improving the assimilation of AMSU-A radiances by modifying such aspects as the vertical localization, the horizontal thinning, and the observation error specification. The model error description was also experimented with, in an attempt to foster deeper vertical structures, which could be better resolved by the AMSU-A instrument. However, in general, it was found that the results obtained were either unsatisfying or inconclusive. It was concluded that there was a need for a study of AMSU-A radiance assimilation in a simpler, controlled experimental environment.

A vertical column EnKF environment was chosen for this purpose, following Campbell et al. (2010), as it can be used to investigate many of the aspects pertinent to the assimilation of AMSU-A radiance observations. This includes issues such as (i) the effect of background error vertical structure and, in particular, the effect of broad versus narrow background error structures; (ii) the impact of the observation error specification; and (iii) the effect of vertical localization and the impact of changing various aspects of its implementation.

The column EnKF is based on the Canadian operational EnKF. Like the operational EnKF, it uses the RT model RTTOV for the forward interpolation from model space to radiance space, that is, from a specified atmospheric profile to simulated brightness temperatures for each AMSU-A channel to be assimilated. A large number of experiments have been performed to examine issues (i)–(iii) above; however, this manuscript will focus on the background error vertical structure, which was found to be most important in determining the AMSU-A radiance impact. We begin by using an analytical approach to describe the background error perturbation structure; here, perturbation profiles are generated using vertical modes obtained analytically from the linearized meteorological equations. Subsequently, the analytical modes are replaced by the empirical modes of two covariance matrices, one obtained from, and the other used by, a data assimilation cycle with a research and development (R&D) version of the global EnKF.

The experimental setup and analytically defined background error perturbations are described in the next section. In section 3, some effects of the background error vertical structure and the impact of each of the AMSU-A channels are examined in an EnKF with a large ensemble. Results obtained using smaller ensembles with and without localization are presented in section 4. In section 5, we examine the empirical vertical modes and spectrum of a covariance matrix obtained from the global EnKF after 2 weeks of cycling and consider the impact of using them, in place of the analytical modes. In fact, this change is found to have a large impact, which motivates a synthesis of the results obtained with the two descriptions of background error perturbation structure in section 6 and an experiment with a static background error covariance matrix in section 7. The conclusions are discussed in section 8.

2. The experimental environment

The experimental setup is similar to that used in Houtekamer and Mitchell (1998) and Houtekamer and Mitchell (2001). As in the latter study, only a single data assimilation step is performed, so no forecast model is required. More specifically, the experimental procedure consists of using different realizations of the sets of simulated observations and of the ensembles of background profiles to produce many realizations of an analysis ensemble. This permits statistically meaningful results to be obtained, without requiring that data assimilation cycles be performed. The column EnKF will now be described.

a. The analysis levels and the reference and truth profiles

Since the AMSU-A channels that are assimilated in the global EnKF are primarily determined by the atmospheric temperature profile, the analysis variable is temperature. The number of analysis levels is 81. The locations of these levels are shown in Fig. 1 and are the same as those of the analysis levels for temperature of a grid column in the global EnKF having a surface pressure of 1013.25 hPa. In fact, these are the levels of the thermodynamic variables for such a column in the version of the Canadian GEM forecast model (Girard et al. 2014) used to drive the global EnKF. Note that although the top of the column (and global) EnKF is located at 0.1 hPa, the top temperature level is located at 0.1265 (not 0.1) hPa since, in the global system, the thermodynamic variables are staggered in the vertical with respect to the momentum variables. Also, although the bottom of the column EnKF domain is at 1013.25 hPa, the bottom temperature level of the column EnKF is located at 1013.06 (not 1013.25) hPa, since in the global model the lowest temperature level (a diagnostic level) is located 1.5 m above the surface.

Fig. 1.
Fig. 1.

The column of plus signs (+) plotted at 1 along the abscissa shows the 81 analysis levels used in the column EnKF. The column of circles with dots () plotted at 2 along the abscissa indicates the 47 RTTOV v12 coefficient levels that lie between 0.1 and 1013.25 hPa.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

The background temperature perturbations will be specified with respect to a reference profile. We choose the U.S. Standard Atmosphere 1976, obtained from Anderson et al. (1986, Table 1f) and interpolated to the 81 analysis levels, as the reference temperature profile (Table 1). This profile will also be taken to be the truth profile.

Table 1.

The altitude (km), pressure (hPa), temperature (K), and water mixing ratio (10−6 kg kg−1) of the reference profile at every second level of the 81 analysis levels, assuming a surface pressure of 1013.25 hPa. Also shown is the factor , which appears in (1) and (3).

Table 1.

For the assimilation of AMSU-A radiance measurements, RTTOV also requires a moisture profile. Again we use the U.S. Standard Atmosphere 1976 profile from Anderson et al. (1986, Table 1f) interpolated to the 81 analysis levels (Table 1). Surface parameters, also required by RTTOV, are set for an atmospheric column over ocean having a skin temperature equal to that of the air at the lowest level of the reference profile (i.e., 288.2 K) and surface zonal and meridional wind speeds of 5 m s−1. (For an ocean location, the surface wind speed is used by RTTOV to calculate the surface emissivity.) The satellite zenith angle is taken to be zero.

b. Interpolations associated with RTTOV

The RT model RTTOV uses coefficients that are defined at a specific set of prescribed pressure levels. For many years, RTTOV used coefficients defined on 43 pressure levels ranging from 0.1 to 1013.25 hPa. RTTOV version (v) 12.1 was released in February 2017 and is the version of RTTOV used in this study. It uses coefficients defined on 54 pressure levels ranging from 0.005 to 1050 hPa. Since the background profiles are defined on a different set of levels and are available to RTTOV only on those levels, the background profiles must be interpolated to the pressures of the RTTOV coefficients. In the global EnKF, which at this writing uses an older version of RTTOV (v10.2), this interpolation of the background profiles is performed for each satellite radiance profile in the EnKF code itself. Since RTTOV v12.1 has an enhanced set of interpolation routines (Hocking 2014), the column EnKF leaves it to RTTOV to interpolate the background profiles from the 81 input pressure levels to the RTTOV coefficient pressure levels. In both the global and column EnKFs, this interpolation is performed using the procedure described by Rochon et al. (2007). Figure 1 shows the analysis grid used by the column EnKF, as well as the 47 RTTOV v12.1 coefficient levels located between 0.1 and 1013.25 hPa.

c. Observations

At the time of this writing, the operational EnKF extends from the surface to 2 hPa. Over the ocean, it assimilates AMSU-A radiances for channels 4–12. [Following the Canadian ensemble–variational (EnVar) analysis, channels 4 and 5 are not assimilated over land or sea ice and, in general, the assimilation of channels 4–7 is subject to certain thresholds for the estimated cloud liquid water and precipitation intensity and the height of the model topography.] The R&D EnKF (a version of which was implemented at operations in September 2018) extends to 0.1 hPa and also assimilates channels 13 and 14. In this study, experiments will be performed assimilating 11 AMSU-A channels: 4–14. A set of observations, therefore, consists of a vertical profile of simulated AMSU-A radiances for channels 4–14. Perfect observations for these 11 channels are presented in Table 2 and were obtained from RTTOV using the truth profile as the input temperature profile. Characteristics and specifications, including frequencies, of the AMSU-A channels can be found in NOAA (1999, Table 3.3.2.1-1) and Weng et al. (2003, Table 1).

Table 2.

The brightness temperature of a perfect observation; the assigned observation error standard deviation (from the global EnKF); the innovation standard deviation (as monitored by the variational system during Jan 2015); and the assigned pressure [i.e., the maximum of the weighting function (used only when localization is employed)] for each of the AMSU-A channels considered in this study.

Table 2.

d. Specification of observation errors and generation of an ensemble of perturbed observations

Operational data assimilation systems often use a diagonal covariance matrix for the observation errors of the AMSU-A radiances (Liu and Rabier 2003). To “compensate” for the neglect of error correlations, the diagonal entries are usually inflated beyond realistic values. Inflating the diagonal entries has the effect of reducing the impact of individual radiance observations.

For the experiments below, the observation error covariance matrix is taken to be diagonal. Standard deviations of the observational errors for channels 4–12 have been obtained from the global EnKF. As channels 13 and 14 are not currently assimilated operationally, the values for these channels are from the R&D EnKF. All of these values, presented in Table 2, originate from the higher-resolution Canadian EnVar assimilation system. They had been obtained by monitoring O (observed) − P (predicted) statistics for each AMSU-A channel and inflating the resulting OP standard deviations. Inflation factors were 1.4 for channels 5–9, 1.6 for channels 4 and 10, and 1.8, 2.0, 2.4, and 3.7 for channels 11, 12, 13, and 14, respectively. [The larger factors for the highest peaking channels were motivated in part by a desire not to overly disturb the model climatology near the model top (L. Garand 2015, personal communication).] Consistent with the monitoring results, slightly different standard deviations are used in the global EnKF for the same AMSU-A channel on different satellites.

For each realization of an experiment, a profile of simulated radiance observations is obtained by adding random perturbations to the profile of perfect observations [as in Eq. (6) of Houtekamer and Mitchell (1998)]. The addition of different random perturbations to the profile of simulated radiance observations [as in Eq. (9) in Houtekamer and Mitchell (1998)] yields an ensemble of profiles of perturbed radiance observations. This ensemble will be assimilated into the ensemble of background profiles using the column EnKF.

e. Ensemble of background profiles

Assuming an isothermal basic state at rest and a constant Coriolis parameter, a set of vertical modes for temperature can be obtained from the linearized meteorological equations. Ensembles of background profiles are generated using these vertical modes.

First, we define the scale height , where R, , and are the gas constant, basic state temperature, and gravitational acceleration, respectively. Using an pressure vertical coordinate , where hPa, we define (i.e., Z at the top of the domain) to be the value of Z at p = 0.1 hPa. (This is 65.59 km for the reference profile.) Then, as discussed in appendix A, the vertical modes for temperature, , are given by
e1
where
e2
The modes correspond to the vertical modes for height and wind, that is,
e3
previously obtained by Simmons (1982) and utilized by Bartello and Mitchell (1992). Note that the exponential factor becomes quite large in the upper part of the domain (Table 1, last column).
The modes can be used to generate a realization of the background error as a random vector:
e4
where the are random numbers obtained from a normal probability distribution with mean zero and standard deviation one. The amplitude and the spectral coefficients will be specified below.

Using (4), background error ensembles are generated in the same way as in Houtekamer and Mitchell (1998, section 2) [see also Mitchell and Houtekamer (2009, section 2)].

f. Analysis algorithm

The EnKF used in this study is essentially a column version of the Canadian global EnKF (Houtekamer et al. 2014b,c). Thus, the column EnKF is stochastic and its ensemble members can be flexibly configured into subensembles with the gain for each subensemble being computed from the members in all of the other subensembles (Mitchell and Houtekamer 2009, section 5). This k-fold cross-validation technique (where k is the number of subensembles) prevents inbreeding (Houtekamer and Mitchell 1998) and results in analysis ensembles whose spread accurately reflects the RMS error of the ensemble mean, except when the ensemble is very small. No covariance inflation or covariance relaxation is employed in this study.

One difference between the column EnKF and its global parent relates to the use of an extended state vector (Houtekamer and Mitchell 2005, section 4e). This was introduced into the global EnKF to facilitate time interpolation, but is not used in the column EnKF, and could result in qualitatively different behavior in the case of sequential assimilation of radiance profiles.

Some of the experiments below employ covariance localization. For EnKFs using small ensembles, this is the standard technique for avoiding the noisy covariance estimates associated with distant observations (Houtekamer and Mitchell 1998, 2001; Hamill et al. 2001). In this study, as in the operational EnKF, vertical localization is implemented in radiance (i.e., observation) space using a Schur (element wise) product of the covariances calculated from the background ensemble and a fifth-order piecewise rational function [Gaspari and Cohn 1999, Eq. (4.10)] with the natural logarithm of pressure as the vertical coordinate. As shown in Fig. 6 of Gaspari and Cohn’s paper, the form of the fifth-order piecewise rational function is very similar to that of a Gaussian function. The severity of the localization can be controlled by an adjustable parameter that determines the function half-width. Focusing on satellite radiance assimilation, Campbell et al. (2010) compared model-space and radiance-space localization [see their Eqs. (1) and (2), respectively, or equivalently, Eqs. (5) and (6) in Houtekamer and Mitchell (2001)] and found that radiance-space localization yields larger errors than model-space localization. However, recently Lei and Whitaker (2015) determined that the opposite can be true when there are negative background error covariances. We note that, unlike the EnKF, variational solvers typically use model-space localization (e.g., Buehner et al. 2010a, section 5).

As discussed by Houtekamer et al. (2005, p. 608), vertical localization is more problematic than horizontal localization. For example, while a surface pressure observation is valid at the surface, it reflects the entire atmospheric column above it. In addition, using vertical localization to assimilate AMSU-A radiances requires that a specific pressure be assigned to each AMSU-A channel, as is done for in situ point observations. In the current study, we follow the global EnKF and define this pressure as the maximum of the weighting function, that is, the maximum of the derivative with respect to of the transmission function between the top of the atmosphere and the current pressure level p. The assigned pressures obtained using this approach for the reference profile are shown in Table 2.

g. The experimental and evaluation procedure

Each result below is based on 20 000 realizations of the analysis procedure. The performance of the EnKF in each experiment will be evaluated, as in Houtekamer and Mitchell (1998) and Houtekamer and Mitchell (2001), by examining (i) the RMS difference between the ensemble mean and the (known) true state (i.e., the RMS error of the ensemble mean) and (ii) the RMS spread in the ensemble (which will sometimes simply be referred to as the ensemble spread). When showing vertical profiles of these quantities in sections 3 and 4, we scale them by , as in, for example, Matsuno (1970) and Kirkwood and Derome (1977). Since this scaling factor is the reciprocal of the exponential term in (1), the dominating exponential growth with height exhibited by vertical profiles of, for example, background error generated using the modes given by (1), is not evident after this scaling. Also, as noted in the Matsuno and Kirkwood–Derome papers [see also Holton (1975, p. 56)], this scaling yields a measure that is proportional to the square root of the error kinetic energy density.

3. Basic experiments with a large ensemble

For the experiments in this section, the ensemble has 4 subensembles with 96 members per subensemble, for a total of 4 × 96 = 384 members. In view of the large ensemble size, no vertical localization is employed in this section.

a. Effect of background error vertical structure

We perform a series of four experiments varying the vertical structure of the background; in each case, a full AMSU-A profile consisting of channels 4–14 is assimilated. Each panel in Fig. 2 shows the vertical profiles of RMS error for the background and for the resulting analysis. In the top-left panel, the background for each realization is defined using (4) with , , and . In the other panels, the background is defined in the same way but with changed lower and upper limits of summation: these are set to 5 and 8 for the top-right panel, to 9 and 12 for the bottom-left panel, and to 13 and 16 for the bottom-right panel. Thus for all four panels, the background consists of only four vertical modes: broad modes for the top-left panel, intermediate modes for the top-right panel, narrow modes for the bottom-left panel, and very narrow modes for the bottom-right panel.

Fig. 2.
Fig. 2.

Profiles of RMS error of the ensemble mean for the background (solid line with circles) and for the analysis (dashed line with squares). The EnKF uses an ensemble having 4 × 96 members and does not use localization. The four panels differ in how the background is specified. In all four cases the background is defined using four vertical modes: modes (top left) 1–4, (top right) 5–8, (bottom left) 9–12, and (bottom right) 13–16.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

We see that the results in the four panels differ markedly with respect to the effectiveness of the analysis in reducing the error. When the background consists of broad modes (top-left panel), it can be seen that the analysis substantially reduces the error. When the background consists of narrower modes (top-right panel), there is a significant reduction of error due to the analysis, but the reduction is not nearly as large as was the case in the top-left panel. When the background consists of even narrower modes (bottom panels), we see that the analysis is very ineffective at reducing the error. This is due to the fact that AMSU-A radiances see only large-scale vertical structure (e.g., Rodgers 2000, section 1.2.1).

b. Reference experiment with a broader spectrum for the background error

Here, for each realization, the background is defined using (4) with (about the mode limit that our 81 vertical levels can resolve), (as before), and
e5
where . With this definition, assumes a maximum value of 1 for and and decreases as increases. This yields the vertical profiles of the background RMS ensemble mean error and spread given by the rightmost (virtually superposed) pair of curves in Fig. 3. [As in section 3a, yields a profile of background RMS ensemble mean error that is commensurate with the profile of the observation error standard deviation (Table 2).]
Fig. 3.
Fig. 3.

Two pairs of vertical error profiles, virtually superposed in each case. The rightmost pair of profiles are the background RMS ensemble mean error and ensemble spread, specified (as discussed in the text) using (4) and (5). The other pair of profiles are the analysis RMS ensemble mean error and ensemble spread obtained by assimilating AMSU-A channels 4–14 with an EnKF having 4 × 96 ensemble members. Symbols are plotted along the curves only at every second analysis level.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

Another pair of virtually superposed vertical profiles is presented in Fig. 3. These profiles are the analysis RMS ensemble mean error and spread obtained by assimilating an AMSU-A radiance profile (i.e., channels 4–14). It can be seen that the assimilation results in a substantial reduction of the error from the surface up to about 2 hPa. Above this level, and especially above 1 hPa, the analysis does not result in a substantial reduction in the background error. This is consistent with the fact that the AMSU-A instrument is designed to retrieve vertical temperature profiles from the Earth’s surface to about 45 km (NOAA 1999, section 3.3). From Table 1, 45 km lies between the sixth and seventh levels of the 81 analysis levels.

c. Impact of each of the AMSU-A channels

The assimilation of each AMSU-A channel should result in a reduction of the background error. To examine the magnitude and vertical structure of this reduction, we repeated the experiment of the previous subsection, but now assimilating each AMSU-A channel in turn. For each channel, we then computed the analysis effectiveness or impact relative to the background as , where A and B are the analysis and background RMS errors as defined in (B2). The results for each channel are shown in Fig. 4.

Fig. 4.
Fig. 4.

The analysis impact (or effectiveness) of AMSU-A channels 4–14 when the background realizations are defined using (4) and (5) with , , and . The top-left and bottom-right panels each pertain to two AMSU-A channels as indicated in those figure panels while each of the other panels pertains to a single AMSU-A channel. The analysis impact at every second analysis level is indicated by a plus sign (+) except that for the lowest- and highest-peaking channels (i.e., channel 4 in the top-left panel and channel 14 in the bottom-right panel) where an ex (×) is used.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

With regard to the structure of the analysis impact, it can be seen from Fig. 4 that for each channel this exhibits a primary peak, whose altitude increases as the channel number increases, and a number of secondary peaks. For most channels, the analysis impact exhibits four peaks. With regard to the magnitude of the analysis impact, it can be seen that this varies substantially; on the one hand, for channels 4 and 14, it never exceeds 0.06, while on the other hand, the peak analysis impacts for channels 9 and 10 exceed 0.30.

Consideration of how the magnitude of the analysis impact varies between the different channels suggests an important dependence on the assigned observation errors (Table 2, column 3). To examine this hypothesis, the RTTOV level at which each AMSU-A channel has its maximum impact was determined and noted in Table 3 (column 2) and the background error standard deviation at that level was also noted (column 4). Then, at each of these levels, the observation and background errors, and , respectively, were considered to apply to two independent unbiased point measurements. Combining these two measurements, following, for example, Ghil and Malanotte-Rizzoli (1991, 157–158) as in appendix B, the optimal linear unbiased estimates of the relative analysis error, , and the analysis impact relative to the background, , were calculated for each channel, using (B5) and (B6). While this simple model cannot be expected to simulate the assimilation of AMSU-A radiance measurements, our objective here is limited to how the assigned observation errors might affect the manner in which the analysis impact relative to the background varies from channel to channel. Consequently, we introduce a scaling factor for specified so that, for channel 11, the modeled analysis impact matches the actual analysis impact. (The value of this scaling factor is 0.416.) The modeled and actual analysis impacts are presented in Table 3, columns 5 and 6, respectively. As can be seen in Table 3, the modeled analysis impact simulates the channel-to-channel variation in the actual analysis impact quite well, thus verifying our hypothesis.

Table 3.

The modeled and actual analysis impact relative to the background for each of the AMSU-A channels considered in this study. Also shown for each channel are the analysis-level index, the pressure, and the background error standard deviation at the analysis level where the analysis impact relative to the background is a maximum.

Table 3.

To investigate the origin of the secondary peaks observed in Fig. 4, we changed the background error specification, in particular, the value of in (5). Recall that determines which vertical modes dominate the background error spectrum and that modes 4 and 5 are dominant when . The three panels in Fig. 5 can be compared with the central panel in Fig. 4 and show the analysis impact relative to the background for channel 9 when the background error spectrum is shifted so that modes 2 and 3, 6 and 7, and 8 and 9 dominate. It can be seen from Fig. 5 that such a change affects both the magnitude and vertical structure of the analysis impact relative to the background. With respect to the magnitude of the analysis impact, we see that, consistent with the results in section 3a, this increases when the background is dominated by broader modes (Fig. 5, top panel) and decreases as the modes dominating the background become narrower (middle and bottom panels). (Note that the large analysis impact in the top panel necessitated a change in scale in Fig. 5 with respect to Fig. 4.) With respect to the vertical structure of the analysis impact, we see that the number of peaks in the analysis impact increases and decreases with the dominant mode numbers in the background error spectrum. That is, the analysis increment reflects the background error spectrum, as expected (Daley and Ménard 1993, Fig. 2; Bannister 2008, section 3).

Fig. 5.
Fig. 5.

As in Fig. 4, but all panels pertain to channel 9 and it is the specification of that differs from panel to panel: (top) , (middle) , and (bottom) . Note that the scale for the analysis impact is different from that used in Fig. 4.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

The results obtained up to this point indicate that, when the ensemble is large, the EnKF behaves in a reasonable manner. To the extent that problems have been noted, for example, when the background error consists of narrow modes or with observation errors that have been severely inflated, variational solvers could be expected to behave in a similar manner (since no localization has been used in this section).

4. Smaller ensembles

Reducing the ensemble size can be expected to degrade EnKF performance. We begin with the case where a single radiance profile is assimilated, as in section 3. Then, since the operational EnKF assimilates batches of observations sequentially (Houtekamer and Mitchell 2001), the experimental setup is extended to the situation where a number of radiance profiles are assimilated sequentially. We expect that problems due to the use of small ensembles and/or severe localization will be compounded by the sequential assimilation of radiance profiles. For conciseness, we here present results only with two ensemble sizes at opposite ends of the ensemble-size spectrum.

a. Results with a small ensemble and no localization

The reference experiment of section 3b, in which an ensemble having 4 × 96 members is used to assimilate AMSU-A radiances from channels 4–14, is now repeated but with fewer ensemble members. The results of two analysis experiments are presented in Fig. 6. In the first experiment, the ensemble size (with 4 × 24 = 96 members) is still large and comparison with the corresponding curves in Fig. 3 shows little change in analysis quality. In fact, comparing the RMS analysis error from Fig. 3 with the RMS analysis error from the current experiment yields differences that do not exceed 0.002 K at any analysis level.

Fig. 6.
Fig. 6.

Profiles of RMS error of the ensemble mean for the background (solid line with circles) and for the analysis for ensembles having 4 × 24 (solid line with squares) and 3 × 2 (solid line with triangles) ensemble members. In each case, the corresponding RMS spread in the ensemble [dashed line with exes (×)] is also plotted. No localization is used. Symbols are plotted along the curves at every second analysis level.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

The impact of reducing the ensemble size much more drastically can be seen in the other two curves in Fig. 6. These curves show the analysis RMS ensemble mean error and spread when the ensemble has three subensembles with two members per subensemble for a total of only 3 × 2 = 6 members. Even with such a small ensemble, it can be seen that the analysis is generally effective in reducing the RMS error, although not as effective as in the case of a large ensemble. In addition, there are now two indications that, with this small ensemble, the algorithm is under stress. First, we observe that the RMS spread in the analysis ensemble is now larger than the RMS difference between the mean of the analysis ensemble and the true state [consistent with previous results using k-fold cross validation and very small ensembles (Mitchell and Houtekamer 2009, Fig. 3)], whereas with larger ensembles we observed almost perfect agreement between these two measures of error. Second, we see that at the uppermost levels, the analysis error is larger than the background error. As noted earlier, these levels are above the region that the AMSU-A instrument was designed to observe.

b. Sequential assimilation with a small ensemble and no localization

As in section 4a, we begin with a large ensemble (4 × 24 members). Figure 7a shows the profiles of analysis impact relative to the background after the assimilation of a single radiance profile (as in Fig. 6) and after the sequential assimilation of two, three, and four radiance profiles. It can be seen that with the assimilation of each additional radiance profile, there is an increase in the analysis impact and, as could be expected, the incremental increase in the analysis impact decreases as the number of assimilated profiles increases. We note that in the region above 1 hPa, the analysis impact is small but certainly remains positive as successive radiance profiles are assimilated.

Fig. 7.
Fig. 7.

(a) The impact of assimilating a single radiance profile and of sequentially assimilating two, three, and four radiance profiles. An EnKF with 4 × 24 members has been used. (b) The ratio of ensemble spread to RMS ensemble mean error for the same experiment. In both (a) and (b), symbols are plotted along the curves at every second analysis level. (c),(d) As in (a),(b), but the ensemble has only 3 × 2 members and results after the sequential assimilation of four radiance profiles are not shown. Note the difference in scale between the ratio of ensemble spread to RMS ensemble mean error in (b) and (d).

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

Figure 7b shows the corresponding profiles of the ratio of ensemble spread to RMS ensemble mean error. Ideally, this quantity should equal 1. In fact, we see that this quantity is close to 1, mostly taking values between 1 and 1.01 after the assimilation of a single radiance profile and increasing slightly as more radiance profiles are assimilated to assume a value of about 1.02 after the assimilation of four radiance profiles.

This same experiment was repeated with the very much smaller (3 × 2 member) ensemble, yielding the results shown in the bottom panels in Fig. 7. As presaged by Fig. 6, this results in two important changes. First, the overestimation of the RMS analysis error by the ensemble spread is now substantially larger than before, ranging from about 12% after the assimilation of a single radiance profile to about 4% after the assimilation of three radiance profiles. Second, in the region that is not observed by the AMSU-A radiances (above ≈1.2 hPa), the analysis impact is negative; that is, the RMS analysis error exceeds the RMS background error. A similar phenomenon occurs at the lowest analysis levels. These problems are symptomatic of generally poor performance in this case. In fact, unlike the situation with the 4 × 24 member ensemble, in this case (i) there are regions where the analysis impact actually decreases as more radiance profiles are assimilated and (ii) above about 5 hPa and near the surface the analysis impact actually becomes increasingly negative with the assimilation of each additional radiance profile. Furthermore, the attempt to sequentially assimilate a fourth radiance profile resulted in a program abort when RTTOV detected a temperature profile in the trial ensemble with a value exceeding 400 K. This temperature, which RTTOV deemed to be invalid (i.e., physically unrealizable), was detected at level 2 (0.202 hPa) during the assimilation of the fourth radiance profile of the 15 992nd realization of the projected 20 000 realizations of this experiment.

The possibility that the problems we have encountered with a small ensemble can be addressed by vertical localization will now be investigated.

c. Sequential assimilation with localization

We repeat the experiments of the previous subsection but now with localization. Figure 8a shows the analysis impact with respect to the background after the sequential assimilation of five radiance profiles with an EnKF having 4 × 24 members. Results are shown for three different localization half-widths, that is, 1, 2, and 3 units of . [We note that the global EnKF used a half-width of one unit of for many years (Houtekamer and Mitchell 2005, section 3d).] Corresponding profiles of the ratio of ensemble spread to RMS error of the ensemble mean are shown in Fig. 8b. In the present circumstances where the ensemble is large for the problem under consideration, localization is unnecessary and, to the extent that it has an impact, can actually be expected to have a detrimental effect. For this reason, results without any localization are included in the figure and serve as a standard of comparison in these experiments.

Fig. 8.
Fig. 8.

(a) The impact of sequentially assimilating five radiance profiles using localization having half-widths of 1, 2, and 3 units of . As a standard of comparison (denoted std), the corresponding result with no localization is also shown. The EnKF has 4 × 24 members in all cases. (b) Ratio of ensemble spread to RMS ensemble mean error for the same experiment. In both panels, symbols are plotted along the curves at every second analysis level. (c),(d) As in (a),(b) showing the result from a 4 × 24-member EnKF with no localization, but, for the results with localization, the EnKF has only 3 × 2 members. Note the difference in scale between (b) and (d).

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

Examination of Fig. 8a shows that localization results in a decrease in the analysis impact and that decrease, relatively small when the localization half-width is three units of , becomes more pronounced as the localization becomes more severe, as expected. Examination of Fig. 8b shows that, for this large ensemble, the ratio of ensemble spread to RMS error of the ensemble mean is close to 1, generally taking values between 1 and 1.02. Between 0.1 and 1 hPa, where the effect of the localization is to severely curtail any impact of the radiance observations, the ratio of ensemble spread to RMS error of the ensemble mean is very close to 1.

When the ensemble size is small, as in the experiments summarized in the bottom panels of Fig. 8, we do expect localization to be beneficial. Indeed, even with mild localization—having a half-width of three units of —it is now possible to sequentially assimilate five radiance profiles without the negative values of analysis impact that we saw in Fig. 7. Moreover the analysis impact with respect to the background is substantial, although there is a sizeable gap compared to the large-ensemble no-localization standard of comparison. A comparison of the analysis impacts with the three localization half-widths indicates that the analysis impact is not greatly affected by the severity of the localization and that, despite the small ensemble size, the most severe localization [half-width of one unit of ] does not yield the largest analysis impact. Moreover, it seems to result in a somewhat noisy analysis impact profile. The bottom panels in Fig. 8, like the top panels, indicate that the localization acts to isolate the region above about 2 hPa from the effects of the assimilation.

Looking at Fig. 8d, we see that below about 5 hPa the ensemble spread tends to overestimate the RMS ensemble mean error by a substantial margin (i.e., more than 10%). The margin of overestimation decreases as the severity of the localization is relaxed. Curiously, however, this is not the case after the assimilation of, for example, a single radiance profile (not shown).

In section 4, we have seen that the use of a small ensemble negatively impacts EnKF performance, as expected. Nevertheless, with only mild localization [localization half-width of three units of ], the EnKF is able to substantially reduce the background error. In fact, the EnKF performs surprising well even with as few as six members. This may be due to the small number of vertical modes that are playing an important role here—a subject that will be investigated further in section 6.

First, however, to more closely approach an operational context, we will use a modified procedure to generate background error profiles for the column EnKF.

5. Background error profiles from the global EnKF

Given the crucial importance of background error vertical structure in determining the impact of AMSU-A radiance assimilation (section 3), it would be desirable to perform experiments using background error profiles similar to those in operational systems. This would help us to understand how the results obtained in sections 3 and 4 above relate to an operational environment.

In this section, a background error covariance matrix obtained from the global EnKF after 14 days of cycling, will be used to generate background error perturbation profiles for use in the column EnKF. That is, we use the same experimental setup as before, but with the following change in the implementation of (4), we set , , and and equal to the nth eigenvector and square root of the nth eigenvalue, respectively, of the background error covariance matrix . The resulting background error profiles will be similar to those in the global EnKF.

a. The matrix and its eigenvalues and eigenvectors

The background error covariance matrix was calculated from a 256-member ensemble of global background temperature fields by taking the (surface-area weighted) average of the sample covariance matrices computed at every analysis grid column. The ensemble was obtained from a data assimilation cycle with an R&D version of the global EnKF and is valid at 1800 UTC 10 January 2015. The R&D global EnKF uses 256 ensemble members and a 50-km horizontal grid and is a further-developed version of the EnKF described in Houtekamer et al. (2014b). As described in the first paragraph of section 2a above, the vertical domain and the placement of the 81 vertical levels are the same in the column EnKF and the global EnKF. The R&D EnKF assimilated the following observation types over the 14-day assimilation period: conventional observations (i.e., from sondes, surface stations, ships, buoys, and aircraft), AMSU-A and -B radiances, Advanced Technology Microwave Sounder (ATMS) radiances, satellite winds, and GPS radio occultation (RO) and scatterometer observations, as well as some infrared radiance observations. At every analysis time (i.e., every 6 h), ≈328 000 AMSU-A radiances and ≈67 000 AMSU-A-like ATMS radiances were assimilated.

An eigen decomposition of the 81 × 81 matrix yields the set of eigenvalues (i.e., spectrum) plotted in Fig. 9. A selection of the corresponding eigenmodes is plotted in Fig. 10. It can be seen that the eigenmodes exhibit a multitude of different structures and in many cases are confined to the upper or lower atmosphere or some intermediate region. Thus, eigenmodes 1–5 are essentially stratospheric modes, but bear some resemblance to the first five analytic modes given by (1). Eigenmode 6 is essentially confined to the troposphere and has its largest amplitude at the surface. Eigenmodes 7 and 8 are largely confined to the region around 100 hPa. Eigenmodes 9–11 and 21 are oscillatory, with the first three of these being confined to the region above 100 hPa. Interestingly, eigenmode 31 and to a lesser extent eigenmode 45 (not shown) exhibit deep vertical structure. However, generally the higher modes exhibit increasingly narrow vertical structure, often changing sign from one vertical level to the next. In view of their narrow vertical structure and small eigenvalues, these higher modes are not expected to play an important role here.

Fig. 9.
Fig. 9.

The spectrum of the background error covariance matrix where the latter was obtained from the global EnKF.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

Fig. 10.
Fig. 10.

A selection of eigenmodes of the background error covariance matrix . The first 10 eigenmodes and the 8 eigenmodes 11, 21, …, 81 are shown. The first eigenmode noted in the top-right-hand corner of each panel is plotted using an open circle ; an ex (×) is used for the second eigenmode. Note the difference in scale for the eigenmode amplitude between the three right-hand panels and the other panels.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

b. Assimilation results

To judge the effectiveness of the analysis with these more realistic background error profiles, we perform an experiment similar to the one in section 3b. That is, using an ensemble having 4 × 96 (= 384) members and no localization, we assimilate a radiance profile consisting of AMSU-A channels 4–14. The resulting vertical profiles of spread and RMS error for the background ensemble and for the subsequent analysis ensemble are shown in Fig. 11a. Note that since the eigenmodes in Fig. 10 do not generally exhibit exponential growth with height, these vertical profiles (unlike those shown in sections 3 and 4) have not been scaled by . A complementary view of these results, in terms of analysis impact, is presented in Fig. 11b. The failure of the analysis to have any substantial impact on the error is very striking.

Fig. 11.
Fig. 11.

(a) The RMS error of the ensemble mean and the ensemble spread for the background and analysis when background error perturbation profiles are generated from a covariance matrix estimated from a global EnKF data assimilation cycle. The EnKF has 4 × 96 members. (b) The corresponding analysis impact. (c),(d) As in (a),(b), but uninflated observation errors (i.e., equal to the OP values as monitored during January 2015) are used. Symbols are plotted along the curves at every second analysis level. The label in the bottom-right-hand corner of each panel relates to the observation errors used [i.e., standard (Std) or OP (OmP) values].

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

To investigate possible regional variability in , the latitude circles at 30°N and 30°S are used to partition the globe into three regions (i.e., the tropics and the northern and southern extratropics). Considering each of these regions separately, a background error covariance matrix is calculated for each region from the 256-member ensemble of background temperature fields. Then, the analysis experiment is repeated using the eigenmodes and spectrum calculated for each region. The resulting figures for each of the three regions (not shown) are similar to those shown for the whole globe in Figs. 11a and 11b, although above 100 hPa background errors are roughly 20% larger in the tropics than in the extratropical regions. Analysis impacts are generally very small: generally less than 1% in the tropics, marginally larger but below 2% in the southern extratropics, and a bit larger still in the northern extratropical stratosphere, but exceeding 2% only from 9 to 29 hPa.

In an attempt to increase the analysis impact, we repeat the global experiment but with reduced observation errors for the AMSU-A radiances. More precisely, instead of using the standard observation errors from the R&D global EnKF, we use the uninflated OP values, as monitored by the variational system in January 2015 (see Table 2 and the related discussion in section 2d). The results of this experiment are shown in Figs. 11c and 11d. It can be seen that while reducing the observation errors in this way does have a very small positive impact, the analysis is still very ineffective at reducing the background error.

In the next section, we will attempt to reconcile these results with the results obtained previously in sections 3 and 4.

6. A synthesis of the results

In Fig. 11 we saw that when background error (perturbation) profiles were obtained from the global EnKF, the assimilation of AMSU-A radiances with the column EnKF did not result in an appreciable reduction of background errors. This was the case even though the EnKF had the benefit of a large ensemble with no need for localization. This result is consistent with our previous experience, mentioned in section 1, that changing various parameters in the global EnKF (e.g., relating to vertical localization, horizontal thinning, and observation error specification) did not seem to yield a consistent benefit or, in fact, have an important impact on the assimilation of AMSU-A radiances. However, one could wonder how this result is consistent with the results obtained in sections 3 and 4, where we saw that background errors could be reduced substantially by the column EnKF.

As we saw already in Fig. 2, the ability of the EnKF to effectively utilize AMSU-A radiances depends crucially on the background error vertical structure. Moreover, almost every occurrence of the background in the EnKF data assimilation algorithm (e.g., Houtekamer and Mitchell 1998, section 2f), involves it first being acted upon by the forward interpolation operator , which in the present case is the RT model. Consequently, to gain a better understanding of our current results, it would seem to be useful to consider how the different backgrounds that we have used appear through the prism of the RT model. To do this, we consider the matrix [defined as in Houtekamer and Mitchell (2001, Eq. 3)] and focus on its trace as a summary diagnostic. In section 6a, we consider the contribution of each vertical mode n to . Then, in section 6b, we consider the contribution of each AMSU-A channel to .

a. Decomposition of by vertical mode

For both the background perturbation profiles used in sections 3 and 4 (defined in terms of analytical vertical modes) and those used in section 5 (defined in terms of empirically obtained vertical modes), we know both the modes and the spectral amplitudes . So we can write
e6
Consider a linear forward operator to the AMSU-A radiances, such as the tangent linear to . Then, we can write
e7
Therefore, can be obtained by calculating for each mode n. For our 81-level vertical grid and our 11 AMSU-A channels, each is a vector of length 81 and is a vector of length 11. Consequently,
e8
is an 11 × 11 matrix.
Here, we use the nonlinear operator to approximate the operator ; that is,
e9
where is the reference temperature profile and δ is a perturbation. This is a good approximation because AMSU-A radiances are largely linear with respect to temperature. Therefore, we can write
e10
Consequently,
e11
which enables us to calculate the contribution of vertical mode n to the trace of . Equation (11), like Eq. (3) in Houtekamer and Mitchell (2001), motivates our use of the notation . We first consider the spectrum of in the case that all analytical vertical modes of have the same amplitude and then consider the corresponding spectra for the different backgrounds that we have used in sections 35.

Figure 12a shows the spectrum of in the case of a background error covariance matrix similar to those used in the set of experiments shown in Fig. 2. Here, however, instead of setting the amplitudes of only four modes at a time to be nonzero and equal to 25, the amplitudes of modes 1–24 are all set equal to 25. It can be seen that this yields a fairly monotonic spectrum of , with amplitudes that drop by four orders of magnitude as the modal index goes from 1 to 16. These results are consistent with those shown in Fig. 2 and seem to indicate that the magnitude of for a given background mode is indeed indicative of the ability of the AMSU-A radiances to perceive that mode. These results (together with those of Fig. 2) would seem to indicate that once the magnitude of for a given background mode has dropped below about 0.1 K2, AMSU-A radiance observations will be ineffective in reducing that error variance.

Fig. 12.
Fig. 12.

Spectra of for two background error covariance matrices . In both cases, the eigenmodes of are given by (1). (a) All modes have the same amplitude; (b) the modal amplitudes are given by (5).

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

Figure 12b is similar to Fig. 12a but now the background error spectrum is as given in (5), with , which we used in many experiments (e.g., the experiments the results of which are shown in Fig. 3). Comparing Fig. 12b with Fig. 12a, we see that the values of for modes 4 and 5 are the same in the two panels, while values are smaller for all of the other modes. [This is due to the form of , given by (5).] The result is that the spectrum drops more quickly in Fig. 12b and only for modes 1–6 are the values of larger than 0.1. This confirms that in sections 3 and 4 only six modes are actually playing a significant role in the experiments with the background error spectrum given by (5).

Having considered the background error profiles that were used in sections 3 and 4, we now turn to the profiles used in section 5, which were generated using the background error covariance matrix obtained from the global EnKF. Corresponding results for the background error profiles generated from the global covariance matrix , for all 81 vertical eigenmodes, are presented in Fig. 13. It can be seen that only beyond mode 45 does the spectrum drop off in a fairly consistent manner. Interestingly, it is mode 31 (see Fig. 10) that yields the largest value of and, significantly, this largest value is only 0.098 (i.e., just below our postulated threshold of 0.1). Thus, it seems that the spectrum shown in Fig. 13 offers an explanation for the ineffectiveness of the analysis in section 5.

Fig. 13.
Fig. 13.

Spectrum of where the background error covariance matrix was obtained from the global EnKF after 2 weeks of cycling.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

In the same way, the spectra of were calculated for the covariance matrices, , for the three regions considered in section 5b. The spectra themselves (not shown) are generally consistent with the assimilation results discussed in that section, as can be seen in the brief summary presented in Table 4. That is, the magnitudes of the trace and of the largest eigenvalues in the three regions are similar to those for the entire globe, with the northern extratropics exhibiting the largest magnitudes [as was the case for the analysis impact (section 5b)].

Table 4.

Trace of and its three largest modal contributions for five different covariance matrices . The first four covariance matrices originate from the global EnKF after 14 days of cycling; the fifth is the static background error covariance matrix . NExt and SExt denote the northern and southern extratropics, respectively.

Table 4.

b. Decomposition of by channel

In (11), for a fixed vertical mode n, we summed over AMSU-A channels 4–14 to calculate the contribution of mode n to . In a similar manner, for a fixed AMSU-A channel, , we can sum the terms over all vertical modes to calculate , the contribution of that particular channel to . This calculation has been performed (i) in the case where the background error is specified as in the reference experiment in section 3b and (ii) in the case where the background error covariance matrix is obtained from the global EnKF. The results are presented in the second and third columns in Table 5, respectively.

Table 5.

(columns from left to right) The channel number, the value of when the background error is specified as in section 3b, the value of when the background error covariance matrix is obtained from the global EnKF, the value on the diagonal of (i.e., the square of the assigned observation error; from the global EnKF), and the value on the diagonal of if the assigned observation errors were taken to be the innovation amplitudes (as monitored by the variational system during Jan 2015). All values in columns 2–5 are in units of K2.

Table 5.

Comparing columns 2 and 3, we see that (with the exception of channel 4) is smaller, and sometimes substantially so, when calculated from the global EnKF profiles than when calculated using the analytically defined modes as in the reference experiment in section 3b. In fact, there is almost an order of magnitude difference for channels such as 9, 10, 11, and 12 that we expect (from Fig. 4) will have the largest impact.

The expression , where is the observation error covariance matrix, arises in the EnKF (and Kalman filter) equations. Moreover, as mentioned in section 2d above, in the present study we use the traditional assumption that is diagonal. This suggests that, for each channel, chan, the value of be compared to the corresponding value from the diagonal of . These latter variances for the standard observation errors from the global EnKF are shown in the fourth column in Table 5. The corresponding uninflated OP variances are shown in the fifth column in Table 5.

We first compare the values that pertain to the experiments in sections 3 and 4 (i.e., columns 2 and 4 in Table 5). Focusing on channels 9–12, we see that and the corresponding observation error variances are comparable in magnitude, with the former values being somewhat larger than the latter values for channels 9–11. This is consistent with the generally substantial reduction in the error that assimilating AMSU-A radiances was found to have in sections 3 and 4. Comparing column 4 with column 3 (i.e., with when is obtained from the global EnKF), we see that the observation error variances are now consistently much larger than the corresponding values of . Again, this is consistent with the results shown in the top panels in Fig. 11. Finally, a comparison of columns 3 and 5, instead of 3 and 4, indicates that while using uninflated innovation amplitudes as the observation errors substantially reduces the observation error variances, the latter are still substantially larger than when is obtained from the global EnKF. This is consistent with the results in the bottom panels in Fig. 11.

We have succeeded in reconciling the results of section 5 with those of sections 3 and 4. In particular, we have shown that when is estimated from the global EnKF, not only are the mode-by-mode and channel-by-channel contributions to the trace of the important matrix substantially smaller than are the corresponding quantities in the reference experiment of section 3b, but the channel-by-channel contributions are also substantially smaller than the corresponding diagonal entries (i.e., variances) of the observation error covariance matrix .

7. Background error profiles from a static background error covariance matrix

We have seen, in sections 5 and 6 above, that there is a paucity of deep modes in the background error profiles generated using the vertical modes and spectrum of the covariance matrix obtained from the global EnKF after 2 weeks of cycling. We hypothesize that, having assimilated AMSU-A radiances over a 2-week period, the global EnKF has virtually eliminated all of the background error structures that can be “seen” by the AMSU-A radiances. To test this hypothesis, it is of interest to see to what extent the situation differs at the beginning of a data assimilation cycle.

As described by Houtekamer et al. (2005, 606 and 609) and Houtekamer et al. (2009), the global EnKF uses a 3D covariance matrix (which originates from our center’s former 3D-Var analysis; Gauthier et al. 1999) for two purposes. At the beginning of a cycle, is used to generate the initial ensemble and, at every data assimilation time, (with reduced amplitude) serves as the largest component of the model (or system) error. In addition, serves as the static background error component in the Canadian EnVar system, which produces the analysis for the high-resolution deterministic forecast. Therefore, to test our hypothesis, we repeated the calculations of section 5 using the ensemble of temperature fields generated from . We now present the main results.

Figures 14a and 14b are the same as Figs. 11a and 11b but for the ensemble of background error temperature fields with the statistics of . A comparison of Figs. 14a and 14b and Figs. 11a and 11b reveals several important differences. First, with respect to vertical structure in the upper atmosphere, we see that whereas the error increases virtually monotonically with height above 10 hPa in Fig. 11a, the error decreases substantially above 2 hPa in Fig. 14a. This decrease is due to the fact that in the construction of the covariances were tapered toward zero at the top of the analysis domain. Second, Figs. 14a and 14b indicate a modest reduction of error due to the analysis, which is markedly larger than the minimal error reduction seen in Figs. 11a and 11b.

Fig. 14.
Fig. 14.

(a) The RMS error of the ensemble mean and the ensemble spread for the background and analysis when background error perturbation profiles are generated from the static covariance matrix . The EnKF has members. (b) The corresponding analysis impact. Symbols are plotted along the curves at every second analysis level.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

Proceeding as in section 6, we then computed the spectrum of . This is presented in Fig. 15, which is directly comparable to Fig. 13. It can be seen that for , there are eight modes for which the values of are larger than 0.1, but with no modes having values as large as those seen in the two panels in Fig. 12. This is consistent with the analysis results (Fig. 14), which showed an analysis impact appreciably larger than that indicated by Figs. 11a and 11b, but substantially smaller than that seen in the top panels of Fig. 2 and in Fig. 3. Thus, again the magnitude of has proven to be indicative of the magnitude of the analysis impact. The same conclusions result from the consideration of Table 4, where some pertinent aspects of the spectrum of when are compared with the corresponding quantities when was obtained from the global EnKF after 2 weeks of cycling.

Fig. 15.
Fig. 15.

As in Fig. 13, but for the static covariance matrix .

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0093.1

Given that the same covariance matrix, , is used both in the system error description of the global EnKF and as the static background error component in the EnVar, receives more weight in the EnVar system than in the EnKF (Houtekamer and Zhang 2016, section 9a); in fact, it receives more than 3 times as much weight, given the weights assigned to the flow-dependent and static covariances in the Canadian EnVar system. As a consequence of the additional weight accorded to , background error profiles from the Canadian EnVar system will tend to suffer less from a paucity of deep structures than profiles from the EnKF.

8. Summary and concluding discussion

As discussed in section 1, this study was undertaken to uncover and investigate possible problems relating to the EnKF assimilation of AMSU-A radiances. Three different descriptions of background temperature error were considered: (i) using analytical vertical modes and hypothetical spectra (sections 3 and 4), (ii) using the vertical modes and spectrum of a covariance matrix obtained from an R&D version of the Canadian global EnKF after 2 weeks of cycling (section 5), and (iii) using the vertical modes and spectrum of the static background error covariance matrix used to initiate a global data assimilation cycle (section 7).

Results with analytical vertical modes, hypothetical spectra, and a large ensemble (section 3) indicate that the column EnKF is able to significantly reduce error levels in the region below about 2 hPa, if the background error contains broad vertical structures. Note that this is the region for which the AMSU-A instrument was designed to retrieve atmospheric temperature profiles. For small ensembles (section 4), the magnitude of the error reduction in this region is smaller, while the assimilation actually causes the error to increase above this region. The sequential assimilation of several radiance profiles results in a further reduction of the error in most of the domain but, when the ensemble size is small, there is a cumulative error increase in the region above about 2 hPa and near the surface (Fig. 7). We found that this eventually led to unphysical temperatures near the top of the domain. Vertical localization, implemented as in the global EnKF using a fifth-order piecewise rational function [Gaspari and Cohn 1999, Eq. (4.10)] and with the natural logarithm of pressure as the vertical coordinate, was introduced to overcome this problem.

A factor that limits the impact of AMSU-A radiance assimilation is that, in the Canadian data assimilation systems, the observation errors assigned to AMSU-A radiances have been inflated beyond the values indicated by OP monitoring. (This is commonly done in an attempt to compensate for the neglect of observation error correlations.) As discussed in section 2d, in the Canadian EnKF and EnVar analysis systems, the inflation factors for the channels 4–12 observation error standard deviation currently vary from 1.4 to 2.0, with even larger values for channels 13 and 14. These inflation factors severely reduce the observation impact since it is largely the observation error variances that determine the observation impact of the different AMSU-A channels, as is the case for point observations (cf. the modeled and actual analysis impacts in Table 3). As a result, although radiances from 11 AMSU-A channels are assimilated, the analysis impact is dominated by four channels: 9–12 (Fig. 4). Thus, for the other seven channels, with the background error perturbation spectrum used in the reference experiment (section 3b), there is not even a single pressure level at which the reduction of the error standard deviation due to the analysis exceeds 20% (Fig. 4 and Table 3).

With regard to the use of vertical localization and small ensembles, the results in section 4 indicate that reducing the ensemble size reduces the impact of the AMSU-A radiances. In addition, localization is detrimental for large ensembles and, while it can be beneficial for small ensembles, our results do not indicate a benefit of using severe localization. Thus, setting the localization half-width equal to one unit of yields results that are somewhat worse than using a localization half-width of two or three units of with an ensemble having 3 × 2 (i.e., 6) members (Fig. 8c). These results would imply that EnKF assimilation should be performed with ensembles that are as large as possible, so that the vertical localization can be relaxed as much as possible. This is consistent with results obtained using a larger ensemble and broader vertical covariance localization in the ECMWF global research EnKF (Hamrud et al. 2015, section 5a) and is the same as the conclusion arrived at by Lei and Whitaker (2015) in their comparison of model-space and radiance-space localization for the assimilation of satellite radiances. However, the good performance with small ensemble sizes and the insensitivity to the localization half-width that we observed in section 4 is likely due to the small number of vertical modes that are actually playing an important role here (Figs. 2 and 12).

Use of the vertical modes and spectrum of a covariance matrix obtained from the global EnKF after 2 weeks of cycling (Figs. 10 and 9, respectively) to generate background error perturbation profiles (section 5) yielded rather different, and very striking, results: the EnKF was virtually unable to reduce the background error even when using a large ensemble with no localization (Fig. 11). It would seem then that the generally unsatisfying impact of increasing the number of assimilated AMSU-A radiances and the general unresponsiveness of the global EnKF to what are considered to be important changes in the method of assimilating AMSU-A radiances (mentioned in section 1) have been successfully reproduced by the column EnKF. Moreover, the problem is not associated with the use of small ensembles or the manner in which localization has been implemented. Rather, it stems from the vertical structure of the background error profiles. Ensemble-variational (EnVar) systems that use the same background ensembles would likely also see limited impact due to AMSU-A radiance assimilation. However, operational implementations use a hybrid formulation with a static background error covariance that can mitigate the problem, as seen in section 7.

An examination of for the various background error profiles used in this study was performed in sections 6 and 7. The results (e.g., Figs. 12, 13, and 15) indicate that the magnitude of the contribution of a given background mode to this quantity is a good indicator of the extent to which the error in that mode can effectively be reduced by the assimilation of AMSU-A radiances. It was found that the background error profiles most often used in sections 3 and 4 are dominated by about six vertical modes sufficiently broad that they are susceptible to error reduction by the assimilation of AMSU-A radiances. In contrast, there are virtually no such modes in the background error profiles generated using the vertical modes and spectrum of the covariance matrix obtained from the global EnKF after 2 weeks of cycling.

These last results are consistent with the hypothesis that, having assimilated AMSU-A radiances over a 2-week period, the global EnKF has virtually eliminated all of the background error structures that can be “seen” by the AMSU-A radiances. This raises questions about the utility of assimilating ever-larger volumes of AMSU-A or AMSU-A-like (e.g., ATMS) radiance profiles within the current operational context. The absence of deep modes in the evolved background error of the global EnKF likely indicates that there are error sources, relating to deep modes, that are currently not being sampled and points to the need for a more comprehensive description of system error. Perhaps, for example, we should try to explicitly sample the uncertainty relating to the bias correction for AMSU-A radiances. Progress on AMSU-A assimilation in all-sky conditions (i.e., also in the presence of thick clouds or significant precipitation; e.g., Zhu et al. 2016 and references therein) would further increase the impact of AMSU-A radiances.

Acknowledgments

The authors are grateful to their many colleagues who engaged in helpful discussions and provided advice about various mathematical issues and computer utilities. We particularly want to thank Seung-Jong Baek, Jeffrey Blezius, Djamel Bouhemhem, Jean Côté, Stephen Macpherson, Richard Ménard, Kristjan Onu, André Plante, and Michel Valin. Section 7 was added in response to comments from Massimo Bonavita and Michael Tsyrulnikov. The many helpful comments from these two reviewers and a third anonymous reviewer resulted in numerous clarifications and improvements to the manuscript.

APPENDIX A

Vertical Modes for Temperature

Bartello and Mitchell’s (1992, hereafter BM92) derivation of vertical modes is here extended to temperature. We assume an isothermal basic state at rest and a constant Coriolis parameter and consider the linearized equations of motion, continuity, and thermodynamics [Holton 2004, section 8.4.1; BM92, Eq. (1)].

Using an pressure vertical coordinate Z, the boundary conditions at the top and bottom (i.e., at and , respectively) are expressed in terms of and given by BM92 [Eq. (2)]. Choosing the boundary condition at the top and bottom (i.e., setting the parameter δ of BM92 to zero) implies
ea1
As in BM92 [Eq. (3)], we seek separable solutions of the form
ea2
where u and are the wind components and ϕ is the geopotential.
Using the hydrostatic equation, it can be shown that
ea3
Substituting (A3) into the linearized thermodynamic equation [e.g., BM92, Eq. (1d)] yields
ea4
where , the buoyancy frequency squared, is constant for an isothermal basic state.
Equation (A4) implies that a separable solution for T of the same form as the separable solutions proposed in (A2) for u, υ, ϕ, and W will have the same vertical structure as does W; that is,
ea5
As shown in BM92, the equations obtained by substituting the separable solutions (A2) into the linearized meteorological equations can be separated into a set of equations governing , , and (i.e., the linearized shallow water equations) and a vertical structure equation for [by eliminating ]. So too can these equations be used (Andrews et al. 1987, 152–153) to yield the linearized shallow water equations and a vertical structure equation for . This is done by eliminating from these equations using the relation
ea6
which is equivalent to Andrews et al.’s (1987) Eq. (4.2.6a). The resulting vertical structure equation is identical to the vertical structure equation for , given by BM92 [Eq. (5)].
Boundary conditions, from (A1), are
ea7

The vertical structure equation together with boundary conditions (A7) is a regular Sturm–Liouville problem whose solutions, , are given by (1). The vertical modes, , satisfy the same orthonormality property [i.e., BM92, Eq. (7)] as the appropriately scaled modes .

APPENDIX B

Analysis Effectiveness or Impact

For each realization i of an analysis experiment, we have (i) a background ensemble, with mean , and (ii) an analysis ensemble, with mean . Let t denote the truth and define
eb1
Let
eb2

Then, is a measure of the relative analysis error and is a measure of the analysis effectiveness or analysis impact relative to the background.

Consider the classical case of two (unbiased) estimates or measurements (say and ) of an observable x. Assume that the errors of these estimates or measurements are uncorrelated and that their variances and are known. Then [see, e.g., Ghil and Malanotte-Rizzoli (1991, 157–158)] the optimal linear unbiased estimate of x is given by
eb3
and , the variance of the optimal estimate, satisfies
eb4
It follows from (B4) that
eb5
Taking square roots yields a measure of the relative analysis error, , and the following measure of analysis effectiveness or impact relative to the background:
eb6
The modeled analysis impact for each AMSU-A channel shown in Table 3 is calculated using (B6).

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Save
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    • Search Google Scholar
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  • Bartello, P., and H. L. Mitchell, 1992: A continuous three-dimensional model of short-range forecast error covariances. Tellus, 44A, 217235, https://doi.org/10.3402/tellusa.v44i3.14955.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bonavita, M., M. Hamrud, and L. Isaksen, 2015: EnKF and hybrid gain ensemble data assimilation. Part II: EnKF and hybrid gain results. Mon. Wea. Rev., 143, 48654882, https://doi.org/10.1175/MWR-D-15-0071.1.

    • Crossref
    • Search Google Scholar
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  • Bormann, N., and P. Bauer, 2010: Estimates of spatial and interchannel observation-error characteristics for current sounder radiances for numerical weather prediction. I: Methods and application to ATOVS data. Quart. J. Roy. Meteor. Soc., 136, 10361050, https://doi.org/10.1002/qj.616.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buehner, M., P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, 2010a: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part I: Description of single-observation experiments. Mon. Wea. Rev., 138, 15501566, https://doi.org/10.1175/2009MWR3157.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buehner, M., P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, 2010b: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part II: One-month experiments with real observations. Mon. Wea. Rev., 138, 15671586, https://doi.org/10.1175/2009MWR3158.1.

    • Crossref
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  • Campbell, W. F., C. H. Bishop, and D. Hodyss, 2010: Vertical covariance localization for satellite radiances in ensemble Kalman filters. Mon. Wea. Rev., 138, 282290, https://doi.org/10.1175/2009MWR3017.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
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  • Dee, D. P., and S. Uppala, 2009: Variational bias correction of satellite radiance data in the ERA-Interim reanalysis. Quart. J. Roy. Meteor. Soc., 135, 18301841, https://doi.org/10.1002/qj.493.

    • Crossref
    • Search Google Scholar
    • Export Citation
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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gauthier, P., M. Buehner, and L. Fillion, 1999: Background-error statistics modelling in a 3D variational data assimilation scheme: Estimation and impact on the analyses. Proc. Workshop on Diagnosis of Data Assimilation Systems, Reading, United Kingdom, ECMWF, 131–145.

  • Gelaro, R., R. H. Langland, S. Pellerin, and R. Todling, 2010: The THORPEX observation impact intercomparison experiment. Mon. Wea. Rev., 138, 40094025, https://doi.org/10.1175/2010MWR3393.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ghil, M., and P. Malanotte-Rizzoli, 1991: Data assimilation in meteorology and oceanography. Advances in Geophysics, Vol. 33, Academic Press, 141–266, https://doi.org/10.1016/S0065-2687(08)60442-2.

    • Crossref
    • Export Citation
  • Girard, C., and Coauthors, 2014: Staggered vertical discretization of the Canadian Environmental Multiscale (GEM) model using a coordinate of the log-hydrostatic-pressure type. Mon. Wea. Rev., 142, 11831196, https://doi.org/10.1175/MWR-D-13-00255.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gorin, V. E., and M. D. Tsyrulnikov, 2011: Estimation of multivariate observation-error statistics for AMSU-A data. Mon. Wea. Rev., 139, 37653780, https://doi.org/10.1175/2011MWR3554.1.

    • Crossref
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  • Fig. 1.

    The column of plus signs (+) plotted at 1 along the abscissa shows the 81 analysis levels used in the column EnKF. The column of circles with dots () plotted at 2 along the abscissa indicates the 47 RTTOV v12 coefficient levels that lie between 0.1 and 1013.25 hPa.

  • Fig. 2.

    Profiles of RMS error of the ensemble mean for the background (solid line with circles) and for the analysis (dashed line with squares). The EnKF uses an ensemble having 4 × 96 members and does not use localization. The four panels differ in how the background is specified. In all four cases the background is defined using four vertical modes: modes (top left) 1–4, (top right) 5–8, (bottom left) 9–12, and (bottom right) 13–16.

  • Fig. 3.

    Two pairs of vertical error profiles, virtually superposed in each case. The rightmost pair of profiles are the background RMS ensemble mean error and ensemble spread, specified (as discussed in the text) using (4) and (5). The other pair of profiles are the analysis RMS ensemble mean error and ensemble spread obtained by assimilating AMSU-A channels 4–14 with an EnKF having 4 × 96 ensemble members. Symbols are plotted along the curves only at every second analysis level.

  • Fig. 4.

    The analysis impact (or effectiveness) of AMSU-A channels 4–14 when the background realizations are defined using (4) and (5) with , , and . The top-left and bottom-right panels each pertain to two AMSU-A channels as indicated in those figure panels while each of the other panels pertains to a single AMSU-A channel. The analysis impact at every second analysis level is indicated by a plus sign (+) except that for the lowest- and highest-peaking channels (i.e., channel 4 in the top-left panel and channel 14 in the bottom-right panel) where an ex (×) is used.