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    • Search Google Scholar
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  • View in gallery

    These plots show the time series of the influence of the perturbed runs on the diagnostics of the ECMWF 4DVar system. The runs were executed for the period 4 Apr through 31 May 2011. The run with the perturbed forward model for HIRS channel 12 is shown in red and the run with perturbed vertical diffusion physics is shown in green. (a) The first-guess departures for bias-corrected HIRS channel 12 observations, (b) the bias correction applied to HIRS channel 12 data, (c) the bias correction applied to AMSU-B channel 3 data, and (d) the bias correction applied to IASI channel 3645. All of these data types are most directly sensitive to water vapor in the upper troposphere. All are plotted with respect to the control run, are averages over the tropics, and have dimensions of brightness temperature (K). The perturbed bias corrections for AIRS channel 1783 (not shown) are nearly identical to those for IASI channel 3645.

  • View in gallery
  • View in gallery

    These plots show the (top) change in the mean state of specific humidity and the (bottom) analysis increments, zonally averaged, after the perturbed system relaxes. The analysis increments are relative to the control run and thus represent the response of the 4DVar system to the perturbed forward model for HIRS channel 12 or to the perturbed vertical diffusion physics. (left) When the forward model for the HIRS channel 12 is perturbed, there is little or no change to the analyzed state of specific humidity because the perturbation is correctly interpreted as a bias correction. (right) When the vertical diffusion physics is perturbed, however, the analyzed specific humidity changes, especially in the uppermost troposphere. The analysis increments act weakly to pull the system back toward reality, but mostly the system incorrectly interprets the perturbation as a bias correction to all data types that are sensitive to water vapor in the upper troposphere. The ordinate is latitude and the abscissa is pressure (hPa).

  • View in gallery

    This plot shows the first-guess departures for radio occultation bending angle at 12-km height. It is a zonal and time average beginning 20 days after the beginning of the runs of the ECMWF 4DVar system described in the text. The first-guess departures for perturbed HIRS channel 12 forward model are shown in red, and the departures for perturbed vertical diffusion are shown in green, and both are shown with respect to the control run. The units of the bending angles are microradians (μrad).

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Leveraging Highly Accurate Data in Diagnosing Errors in Atmospheric Models

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  • 1 Harvard School of Engineering and Applied Sciences, Cambridge, Massachusetts
  • 2 European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
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Highly accurate data can serve the numerical weather prediction, climate prediction, and atmospheric reanalysis communities by better enabling the diagnosis of model error through the careful examination of the diagnostics of data assimilation, especially the firstguess departures and the analysis increments. The highly accurate data require no bias correction for instrument error, leaving the possibility of confusion with error in forward models for observations as the lone hindrance to the diagnosis of model error. With this scenario in mind, we conducted numerical experiments to investigate the potential confusion using the data assimilation system at the European Centre for Medium-Range Weather Forecasts. We found that large-scale systematic model error can be misattributed to error in the forward models for observations, thereby reducing systematic firstguess departures and impeding the mitigation of model error. The same large-scale model error generated a 20% increase in analyzed specific humidity near the tropopause, suggesting that current observational data cannot constrain the upper tropospheric humidity in current models, which contributes substantially to greenhouse forcing of the climate. We expect that the confusion of model error for an error in the forward models for observations occurs regardless of the objective method used to diagnose model error.

Our findings underline the importance for continued improvement in radiative transfer calculations and highlight the value of multiple sources of accurate data that are redundant in their sensitivity to atmospheric variables yet orthogonal in their radiation physics.

CORRESPONDING AUTHOR: Stephen S. Leroy, Anderson Group, 12 Oxford St., Link Building, Cambridge, MA 02138 E-mail: leroy@huarp.harvard.edu

Highly accurate data can serve the numerical weather prediction, climate prediction, and atmospheric reanalysis communities by better enabling the diagnosis of model error through the careful examination of the diagnostics of data assimilation, especially the firstguess departures and the analysis increments. The highly accurate data require no bias correction for instrument error, leaving the possibility of confusion with error in forward models for observations as the lone hindrance to the diagnosis of model error. With this scenario in mind, we conducted numerical experiments to investigate the potential confusion using the data assimilation system at the European Centre for Medium-Range Weather Forecasts. We found that large-scale systematic model error can be misattributed to error in the forward models for observations, thereby reducing systematic firstguess departures and impeding the mitigation of model error. The same large-scale model error generated a 20% increase in analyzed specific humidity near the tropopause, suggesting that current observational data cannot constrain the upper tropospheric humidity in current models, which contributes substantially to greenhouse forcing of the climate. We expect that the confusion of model error for an error in the forward models for observations occurs regardless of the objective method used to diagnose model error.

Our findings underline the importance for continued improvement in radiative transfer calculations and highlight the value of multiple sources of accurate data that are redundant in their sensitivity to atmospheric variables yet orthogonal in their radiation physics.

CORRESPONDING AUTHOR: Stephen S. Leroy, Anderson Group, 12 Oxford St., Link Building, Cambridge, MA 02138 E-mail: leroy@huarp.harvard.edu

Highly accurate observations from satellites should facilitate the diagnosis of error in atmospheric models, but improved observation operators or data with independent calibration will be necessary.

Successful weather prediction relies on our ability to initialize forecast models with observations of the current state of the atmosphere. Although there have been significant improvements in models and observations over the years, model and observation errors will always exist. Automated approaches such as variational bias correction (Derber and Wu 1998) can reduce the impact of systematic error, but they leave the forecast system open to drift away from reality—a leading complication in atmospheric reanalysis. To prevent this drift, certain observations thought to be particularly accurate are assimilated without bias correction (Dee 2005). These observations “anchor” analyses of the atmosphere closer to the truth and are referred to as “anchor data.” First-guess departures for anchor data offer the opportunity to diagnose errors in the background model, but that opportunity is still hindered by the potential for confusion of error in the model and errors in the forward models for observations. (See the sidebar for an introduction to data assimilation and its diagnostics, including first-guess departures and forward models for observations.)

The diagnosis and consequent reduction in model error should yield great dividends for climate prediction in addition to numerical weather prediction and atmospheric reanalysis. The uncertainties in the model to be addressed by first-guess departures in anchor data are thought to be responsible for much of the uncertainty in the prediction of interdecadal climate change. One meteorological center has already established a “unified model” that is the basis for weather prediction and climate prediction to take advantage of this link between them (Brown et al. 2012).

Here we investigate the potential for highly accurate spectral infrared data, a data type rich in information content for numerical weather prediction [Anderson et al. 2004; Wielicki et al. 2013], to aid in the diagnosis of model error. The potential for the diagnosis of model error, through the examination of analysis increments, is increased when different observation types, with distinct corresponding forward models, are able to constrain the same atmospheric variables. We use the example of radio occultation data to examine this further possibility.

A NUMERICAL EXPERIMENT.

We investigate the possibility that model error can be wrongly interpreted as an error in the forward model for satellite radiance using the four-dimensional variational data assimilation (4DVar) of the European Centre for Medium-Range Weather Forecasts (ECMWF). In addition to an unperturbed control run, we simulate an error in a forward model by executing the ECMWF 4DVar system a second time after increasing the opacity of water vapor by 3% in the forward model for High Resolution Infrared Radiation Sounder (HIRS) channel 12 (6.52 μm). HIRS channel 12 sounds water vapor in the upper troposphere in the vicinity of 300 hPa, where water vapor is a strong radiative coolant. There were three meteorological satellites in that time period—National Oceanic and Atmospheric Administration-17 (NOAA-17), NOAA-19, and Meteorological Operational-A (MetOp-A)—that flew the same HIRS instrument type and delivered HIRS channel 12 data. We perturb the forward model for HIRS channel 12 for all of these instruments. We simulate a model error by executing the ECMWF 4DVar system after perturbing the physics associated with vertical diffusion in such a way that diffusion is reduced in regions of high static stability. All assimilations are executed for a 58-day period from 4 April through 31 May 2011. The perturbation to the forward model for HIRS channel 12 is consistent with the existing bias correction magnitudes in the operational forecast system, while the perturbation to the vertical diffusion physics falls within the bounds of realistic uncertainties.

First-guess departures and bias corrections have structure in longitude, latitude, and time and are different for each instrument whose data are assimilated. The departures and biases for the perturbed runs diverge from the control run only for those data types that are sensitive to upper tropospheric humidity. We examined the first-guess departures and bias corrections for upper tropospheric water vapor sounders HIRS channel 12, Advanced Microwave Sounding Unit B (AMSU-B) channel 3, Atmospheric Infrared Sounder (AIRS) channel 1783 (1556.1 cm−1), and Infrared Atmospheric Sounding Interferometer (IASI) channel 3645 (1556.0 cm−1) more closely, all of which are sounders of upper tropospheric humidity. The first two are actively assimilated and bias corrected; the last two are monitored and bias corrected but not assimilated. For these sounders the bias corrections in brightness temperature range from approximately –0.5 to 2.0 K. The departures and bias corrections for both perturbed runs relaxed to near constant offsets from the control run after approximately 15 days (see Fig. 1).

Fig. 1.
Fig. 1.

These plots show the time series of the influence of the perturbed runs on the diagnostics of the ECMWF 4DVar system. The runs were executed for the period 4 Apr through 31 May 2011. The run with the perturbed forward model for HIRS channel 12 is shown in red and the run with perturbed vertical diffusion physics is shown in green. (a) The first-guess departures for bias-corrected HIRS channel 12 observations, (b) the bias correction applied to HIRS channel 12 data, (c) the bias correction applied to AMSU-B channel 3 data, and (d) the bias correction applied to IASI channel 3645. All of these data types are most directly sensitive to water vapor in the upper troposphere. All are plotted with respect to the control run, are averages over the tropics, and have dimensions of brightness temperature (K). The perturbed bias corrections for AIRS channel 1783 (not shown) are nearly identical to those for IASI channel 3645.

Citation: Bulletin of the American Meteorological Society 95, 8; 10.1175/BAMS-D-12-00143.1

DATA ASSIMILATION AND DIAGNOSING ERROR

The first-guess departures and analysis increments, two objective diagnostics of data assimilation, should contain strong fingerprints of model error (Klinker and Sardeshmukh 1992; Rodwell and Palmer 2007). To understand what they are, how highly accurate data would benefit their utility, and why inaccuracy in the forward models for those accurate data remains a problem, it is necessary to present a brief qualitative description of data assimilation. For reference, the diagram above conceptually illustrates the steps in data assimilation for any particular state variable (e.g., temperature, pressure, and winds) and an observation that is sensitive to that state variable (e.g., radiance).

Data assimilation is the procedure by which a numerical model of the atmosphere is guided by observational data to best represent the atmosphere's true evolving state. It has been documented extensively elsewhere (e.g., Lorenc 1986; Talagrand 1997). To start, a best estimate of the true atmospheric state at a previous time (“original analysis”) serves as an initial state for the integration of the fluid dynamical equations of motion. In the course of that integration, the model's physics are evaluated to account for underresolved processes in the atmosphere that are important to its evolution such as radiation physics, diffusion, and gravity wave drag and clouds and convection. The result of the integration is a forecast of the atmospheric state at the time of a specific observation. The forecast atmospheric state is input to a forward model for the observation that predicts a first guess for that observation. The difference between the simulated observation and the bias-corrected observation is the first-guess departure. The new best estimate for the true state of the atmosphere is determined such that it is consistent with the first guess of the atmospheric state and the bias-corrected observation to within uncertainties associated with each. The update of the atmospheric state from the forecast state to the new analysis is the “analysis increment.” The analysis increments are best interpreted as the overall influence of the data on the analyzed atmospheric state.

The four contributors to a nonzero first-guess departure (Auligné et al. 2007) are as follows:

  1. error in the initial state vector,
  2. error in the observation,
  3. error in the forward model for the observation, and
  4. error in the model.

Averaging over a sufficient number of analysis cycles is expected to reduce the impact of the random components of these errors. Dynamic bias correction is intended to remove systematic error (bias) in observations and in forward models for observations (Derber and Wu 1998; Dee 2005). If done successfully, the remaining departures will reflect systematic error in the model and systematic error in the initial vector. The latter is itself due to the accumulation of model error over many analysis cycles. Diagnosing model error in this way has been applied to the physics of convective entrainment (Rodwell and Palmer 2007), the specification of atmospheric aerosol (Rodwell and Jung 2008), and to a gravity wave parameterization (Pulido et al. 2012).

This approach to diagnosing model error succeeds if the model error is isolated from the other contributors to error. If the bias in any observation is incorrectly modeled and not completely removed from first-guess departures, though, some residual observation bias would instead be confused for model error. It is also possible that model error could be misattributed to observation bias because of the simplicity of the linear models used in dynamic bias correction (Dee and Uppala 2009), thus obscuring the signature of model error. The potential for confusion is expected to be especially acute for passive nadir sounders of the atmosphere and the atmospheric state variables to which they are most sensitive. With highly accurate data instead of inaccurate data, however, there would be no need for bias correction that accounts for observation bias, and the potential for confusion of observational error for model error (and vice versa) would be eliminated. We anticipate that numerical weather prediction centers will assimilate highly accurate satellite data without bias correction if the forward model for the observations is also highly accurate, but with bias correction that accounts only for error in the forward model for the observations otherwise.

Perturbing the forward model for HIRS channel 12 (red in Fig. 1) has the initial effect of simulating brightness temperatures for channel 12 that are lower than the observations, and the signature is positive, systematic first-guess departures in HIRS channel 12 (see Fig. 1a). Within one day, the data assimilation interprets the large first-guess departures in HIRS channel 12 as forecast errors and then immediately as bias corrections in all of the passive operational sounders of upper tropospheric water vapor. After approximately 15 days, the system skillfully recognizes the perturbation as an error in the forward model for HIRS channel 12 alone (see Fig. 1b), with the bias corrections of the non-HIRS sounders of upper tropospheric water vapor converged to the bias corrections of the unperturbed run (see Figs. 1c,d). Overall, this is a successful demonstration of dynamic bias correction: an error in the forward model for a data type is interpreted as such by the bias correction algorithm, and the system adjusts and obtains the same analyses and forecasts as if there were no error in the forward model for the data.

Perturbing vertical diffusion (green in Fig. 1) impacts the system differently: what should be unbiased data inherit a bias because of model error, thus removing the signature of model error from the first-guess departures. Initially there is no difference between the perturbed run and the control run, but after approximately 15 days, the perturbed vertical diffusion physics redistributes water vapor in the vertical and relaxes to a new climatological state for water vapor in the upper troposphere. Decreased diffusion at the tropopause, where static stability is high, reduces the ventilation of tropospheric water vapor into the stratosphere, and water vapor is enhanced immediately beneath the tropopause as a result. The relaxation is slow enough so that the first-guess departures are never large. Instead, the differences between predicted observations and actual observations are always interpreted as bias corrections for the aforementioned water vapor sounders (not shown). Once the departures are largely interpreted as bias corrections for the passive radiance sounders, it becomes impossible to discover model error in the first-guess departures. Moreover, it is unlikely that model error can be discerned in the bias corrections because the pattern of model error in bias corrections to be discerned against a complex background of real bias is the same as the pattern of forward model error.

The repercussions of confusing an error in vertical diffusion for an error in forward models for water vapor sounders are substantial. See the upper panels of Fig. 2, which show the change in the zonal average mean state of specific humidity in the analyses beginning 20 days after the start of the runs. Perturbing the forward model for HIRS channel 12, which was correctly interpreted as a bias and subtracted from the data before assimilation, has no effect on the mean state of specific humidity. Perturbing the vertical diffusion physics, however, effectively increases the specific humidity in the analyses in the layer immediately below the tropopause worldwide by up to 20%.

Fig. 2.
Fig. 2.

These plots show the (top) change in the mean state of specific humidity and the (bottom) analysis increments, zonally averaged, after the perturbed system relaxes. The analysis increments are relative to the control run and thus represent the response of the 4DVar system to the perturbed forward model for HIRS channel 12 or to the perturbed vertical diffusion physics. (left) When the forward model for the HIRS channel 12 is perturbed, there is little or no change to the analyzed state of specific humidity because the perturbation is correctly interpreted as a bias correction. (right) When the vertical diffusion physics is perturbed, however, the analyzed specific humidity changes, especially in the uppermost troposphere. The analysis increments act weakly to pull the system back toward reality, but mostly the system incorrectly interprets the perturbation as a bias correction to all data types that are sensitive to water vapor in the upper troposphere. The ordinate is latitude and the abscissa is pressure (hPa).

Citation: Bulletin of the American Meteorological Society 95, 8; 10.1175/BAMS-D-12-00143.1

While the passive radiance sounders do not seem able to effectively correct biases in the mean state of upper tropospheric water vapor due to model error (perturbed vertical diffusion physics), the data assimilation system nevertheless contains some observational information that tends to correct for the model error, albeit not strongly. The two lower panels of Fig. 2 show the systematic analysis increments (see the sidebar) in specific humidity after perturbing the forward model for HIRS channel 12 and after perturbing the vertical diffusion physics, each with respect to the analysis increments of the unperturbed run. In the case of the perturbed forward model for HIRS channel 12, the change to the analysis increments in specific humidity from the unperturbed run to the perturbed run is spatially incoherent and hence not significantly different than zero. In the case of the perturbed vertical diffusion physics, however, the analysis increments are significantly more negative in the region of the mid- and high-latitude tropopause, where they tend to oppose systematic errors in the analyses. The only information that can induce these analysis increments must not be subject to a bias correction that can be confused for physical error. This suggests that the anchor data or data types that do not confuse model error for bias correction act to oppose the influence of model error, but do so too weakly to correct the model forecast. Moreover, the assimilated data collectively do not exert a noticeable influence in specific humidity above 200 hPa and thus have no discernible effect in specific humidity at the tropopause in the tropics.

It is not difficult to understand how an error in vertical diffusion physics can be confused for an error in the forward models for passive radiance sounders. Both perturbing the opacity of water vapor and redistributing water vapor in the vertical by perturbing vertical diffusion have the effect of altering the weighting function associated with the modeling of any passive radiance observation of water vapor. Not only would a perturbation to vertical diffusion physics result in such confusion, but a perturbation to any model error that redistributes water vapor in the upper troposphere would result in the same confusion.

There are some data types sensitive to upper tropospheric water vapor that must play a role in inducing the analysis increments in upper tropospheric water vapor. One such satellite data type is radio occultation using the Global Positioning System (GPS; Kursinski et al. 1997)—data that are already being assimilated at ECMWF (Healy and Thépault 2006; Poli et al. 2010) and many other centers. GPS radio occultation differs fundamentally from the passive radiance sounders in that its remote sensing physics is based in optical refraction rather than in radiative absorption and also because it is not subjected to bias correction. At ECMWF, the radio occultation bending angle as a function of height is assimilated. We show the zonal average systematic first-guess departures of radio occultation bending angle at 12-km height in Fig. 3. The fingerprint of perturbed vertical diffusion exists in the first-guess departures of radio occultation (RO) data in both hemispheres between 30° and 45° latitude, which is where the elevated climatology of water vapor intersects the 12-km height surface. For the purpose of searching out model error in the diagnostics of data assimilation, it should be helpful to examine the first-guess departures of data types that obey different remote sensing physics.

Fig. 3.
Fig. 3.

This plot shows the first-guess departures for radio occultation bending angle at 12-km height. It is a zonal and time average beginning 20 days after the beginning of the runs of the ECMWF 4DVar system described in the text. The first-guess departures for perturbed HIRS channel 12 forward model are shown in red, and the departures for perturbed vertical diffusion are shown in green, and both are shown with respect to the control run. The units of the bending angles are microradians (μrad).

Citation: Bulletin of the American Meteorological Society 95, 8; 10.1175/BAMS-D-12-00143.1

DISCUSSION.

New, highly accurate data should have the capability to better enable the diagnosis of model error by examination of the analysis increments and first-guess departures of data assimilation; however, we have found that highly accurate satellite radiance data are not sufficient for the diagnosis of model error. It will also be necessary to improve the accuracy of their corresponding forward models. With sufficient accuracy in the forward model for high accuracy observations, it is more likely that the signature of model error can be identified in the diagnostics of data assimilation. The forward models associated with the anchor data should be made at least as accurate as the data themselves to take full advantage of the data's accuracy. Should improving accuracy of radiative transfer be an insurmountable problem, though, it may still be possible to discern model error when considering different types of highly accurate data that obey different remote sounding physics, including in situ data types, but are sensitive to the same atmospheric state variables. In this study we found this to be the case. GPS radio occultation reveals the signature of perturbed vertical diffusion physics while passive, bias-corrected remote sounders of upper tropospheric water vapor do not.

Failure to implement these suggestions may well lead to climatologically significant errors in atmospheric analyses, especially in upper tropospheric humidity. While it is possible that future developments in data assimilation may enable the discernment of model error from observational error in bias corrections, there is little doubt that highly accurate data will make any such discernment more tractable. Thus, we emphasize that information-rich climate benchmarking systems—space-based observing systems that obtain empirically demonstrable highly accurate data—should be deployed to better enable the diagnosis of model error.

Finally, analysis of the diagnostics of fourdimensional variational assimilation is not the only approach to diagnosing model error, but by virtue of its ability to disentangle the different physical processes at work in the context of a system that considers both the full dynamical equations of motion and a large abundance of data, it is likely that any problem encountered in our numerical experiment will also be encountered in less direct and less objective approaches. Hence, our findings should be applicable to any method of atmospheric (re)analysis.

ACKNOWLEDGMENTS

We thank Dick Dee, Tony McNally, Peter Bauer, Marco Matricardi, Irina Sandu, Anton Beljaars, Erland Källén (ECMWF), and James Anderson (Harvard) for advice and assistance in this work. Funding was provided in part by Grants NNX11AE74G of the U.S. National Aeronautics and Space Administration and ATM-0755099 of the U.S. National Science Foundation.

REFERENCES

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    • Search Google Scholar
    • Export Citation
  • Auligné, T., , A. P. McNally, , and D. P. Dee, 2007: Adaptive bias correction for satellite data in a numerical weather prediction system. Quart. J. Roy. Meteor. Soc., 133, 631642, doi:10.1002/qj.56.

    • Search Google Scholar
    • Export Citation
  • Brown, A., , S. Milton, , M. Cullen, , B. Golding, , J. Mitchell, , and A. Shelly, 2012: Unified modeling and prediction of weather and climate. Bull. Amer. Meteor. Soc., 93, 18651877, doi:10.1175/BAMS-D-12-00018.1.

    • Search Google Scholar
    • Export Citation
  • Dee, D. P., 2005: Bias and data assimilation. Quart. J. Roy. Meteor. Soc., 131, 33233343, doi:10.1256/qj.05.137.

  • 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, doi:10.1002/qj.493.

    • Search Google Scholar
    • Export Citation
  • Derber, J. C., , and W.-S. Wu, 1998: The use of TOVS cloud-cleared radiances in the NCEP SSI analysis system. Mon. Wea. Rev., 126, 22872299, doi:10.1175/1520-0493(1998)1262.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Healy, S. B., , and J.-N. Thépault, 2006: Assimilation experiments with CHAMP GPS radio occultation measurements. Quart. J. Roy. Meteor. Soc., 132, 605623, doi:10.1256/qj.04.182.

    • Search Google Scholar
    • Export Citation
  • Klinker, E., , and P. D. Sardeshmukh, 1992: The diagnostics of mechanical dissipation in the atmosphere from large-scale balance requirements. J. Atmos. Sci., 49, 608627, doi:10.1175/1520-0469(1992)0492.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kursinski, E. R., , G. A. Hajj, , J. T. Schofield, , R. P. Linfield, , and K. R. Hardy, 1997: Observing Earth's atmosphere with radio occultation measurements using the global positioning system. J. Geophys. Res., 102 (D10), 23 42923 465, doi:10.1029/97JD01569.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Poli, P., , S. B. Healy, , and D. P. Dee, 2010: Assimilation of global positioning system radio occultation data in the ECMWF ERA-Interim reanalysis. Quart. J. Roy. Meteor. Soc., 136, 19721990, doi:10.1002/qj.722.

    • Search Google Scholar
    • Export Citation
  • Pulido, M., , S. Polavarapu, , T. G. Shepherd, , and J. Thuburn, 2012: Estimation of optimal gravity wave parameters for climate models using data assimilation. Quart. J. Roy. Meteor. Soc., 138, 298309, doi:10.1002/qj.932.

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