1. Data assimilation, short-term forecast, and forecasting error
at convective scales
Kao-Shen Chung
鍾高陞
Department of Atmospheric and Oceanic Sciences
McGill University
April

2. Outline
1. Introduction
2. McGill radar assimilation system
* impact of the background term
3. Initialization and the short term forecast
* case study on 12 July 2004
4. Sensitivity test of the short term forecasts
5. Summary / conclusion
6. Future work

3. 1.What is Data Assimilation ?
Talagrand (1997):
Assimilation of meteorological or oceanographical observations
can be described as the process through which all the available
information is used in order to estimate as accurately as possible
the state of atmospheric or oceanic flow.
Satellite
Radiosonde
Radar
Observations
The physical laws:
Govern the evolution of the flow
e.g. Equation of Motion,
Thermodynamic Equation,
Mass and Water Continuity
……etc
Information coming from previous
model forecast

4. Purpose of Data Assimilation
NWP
Data Assimilation
Best estimate of the
initial conditions
At large scale
Main observing system : radiosonde network 
only W (vertical velocity) is unknown for initialization of
a forecast model
Main objective: optimal interpolation

5. Main observing system at convective scales
Doppler Radar provides:
- high spatial resolution ( 1km )
- high temporal resolution ( ~ 5 min )
Capable of sampling the structure of individual
Convective cells in a convective system.

6. (Doppler wind, reflectivity) Numerical Weather Prediction
Data assimilation at convective scale:
Challenge :
High temporal and space variability
No simple balances can be used
Observing system: measurements (e.g. radar network)
are not direct model variables ( U,V,W,P,T)
Radial component
Real wind
Radar observations
(Doppler wind, reflectivity)
To Initialize
Numerical Weather Prediction
(U,V,P,T…)

7. History of the McGill radar Assimilation system
Laroche and Zawadzki (1994): retrieved 3D wind within precipitation area
Horizontal Wind Vector
Vertical Velocity (w)
Pressure perturbation
Temperature perturbation
McGill Radar observations network
Protat and Zawadzki (1999, 2000)

8. Initialize the numerical model
Montmerle et. al (2001)

9. Current McGill radar observations
McGill S-Band Radar:
Reflectivity
Doppler Velocity

10. Challenge and main objective of the research
Observations: single Doppler radar
(radial velocity, reflectivity )
Information other than radar:
a prior forecast of a high-resolution model
The predictability / uncertainty of the short term
forecast at convective scale

11. state of the atmosphere
2. McGill Radar Data Assimilation System
Based on Caya (2001): Variational algorithm
Background
A Cloud model:
weak constraint
Best estimate the
state of the atmosphere
Cost Function (J)
Observations
Consider model is not perfect
No Adjoint model, reduce the computational time

12. Present state of the Model Governing Equations
Momentum equations
(u,v,w)
Mass continuity equation
Thermodynamic equation
Kessler microphysics
(rain and cloud)

13. Impact of including model term in the cost function
Single observation test
Horizontal Wind
Temperature

14. Impact of the background term
Background field, xb
Fill the non-precipitation
area in the domain
Role of the background term:
Background error matrix, B:
determine the filtering and propagation
of the observed information

15. Former assimilation system
Diagonal Part:
Variance
Assume:
Uncorrelated
Too simplified
need smoothness constraint in the algorithm
Smoothness Term

16. Without penalty term: With penalty term: Horizontal Wind
Vertical Velocity
With penalty term:

17. In the current assimilation system
B is modeled by a recursive filter [followed Purser et al. (2003)]
* Assume the error correlation of the control variables is
isotropic and homogeneous
* Applied the filters to control variables
A prior high-resolution model forecast is used as the background field
* MC2 (Mesoscale Compressible Community)
Non-hydrostatic
Horizontal resolution: 1km
Stretched vertical levels
Explicit scheme in microphysics process

18. Modified assimilation system
Comparison of the two assimilation systems
Former assimilation system
Modified assimilation system
(Recursive filters)
No penalty term (smoothness constraint)

19. McGill assimilation system
3. Initialization and the short term forecast
McGill assimilation system

20. Case Study (12th July, 2004 ): Radar site 1810 UTC 1840 UTC 1910 UTC
Deep convective and
long lasting
storm system
We start from the early
stage of the storm !!
1940 UTC
2010 UTC

21. Impact of using a previous numerical weather prediction
CAPE value Convective potential
Stable
Marginally Unstable
Moderately Unstable
Very Unstable
Extremely Unstable

22. Impact of assimilating radar observations
Before assimilation
After Assimilation UU UU VV VV

23. Results of short term forecast at 30 min
Vertical velocity
Radar observation
Model simulation
1840 UTC

24. Results of short term forecast at 60 min
Radar observation
Model simulation
1910 UTC
1 hour forecast

25. Forecast results with a cycling strategy
Model simulation (60 min)
Radar observation
1940 UTC
Radar observation
Model simulation (90 min)
2010 UTC

26. Verification: Radial component : Observed radial velocity
1910Z H = 2.5km
RMSE of Doppler wind
The errors are larger
in the upper levels
Simulated radial velocity
1910Z H = 2.5km

27. Observation Reflectivity Simulated Reflectivity
At what kind of scales do we have more predictability?
Followed Turner et al. (2004)
Wavelet Transform Analysis:
Similar to the Fourier transform, but the wavelet transform are more
effective in representing localized, intermittent fields.

28. Wavelet analysis
The simulation has more predictability
at the longer scale( > 30 km) beyond
20 minutes.

29. Definition of the forecasting error
4. Sensitivity test of the short term forecasts
Definition of the forecasting error
What is the Forecasting (background) errors?
The best estimate of atmosphere
Forecasting errors
Truth
Short-range forecast
Due to the uncertainty in the initial conditions
(assumption: model is perfect)

30. Motivation Ensemble forecasts : characterize the forecasting errors
We want to know how sensitive of the short-term forecasts
relative to the uncertainty of the initial conditions
What is the structure of the forecasting/background errors at
convective scale?
In data assimilation:
The “optimal” analysis fields can be obtained only if the statistics
of the background and observations errors can be accurately
described.
Hamill et al and Anderson 2001:
the underestimate of the forecasting error covariances
 Due to the small size of ensemble members
 Any other factor could cause the underestimate of the
forecasting error?
Ensemble forecasts :
characterize the forecasting errors

31. Ensemble scheme Model simulation 1h forecast Calculate the
Background at
1500, 1600 and 1700 UTC
McGill radar
Data assimilation
Model simulation
1h forecast
Calculate the
Statistics of
forecasting errors.
Perturb observations
(error of observation)

32. About the observation errors
By given the standard deviation of the observational variables
e.g. Reflectivity: 2~5dB ; Radial velocity: 1m/s
Uncorrelated observations errors
However…..

33. Errors have correlation in space / time !
Berenguer and Zawadzki (2008)

34. Prescribed observation errors
Reflectivity: error correlation length: 10km (Hori),
vertical correlation = 0.85 (250m)
standard deviation: 2.5 dB
Radial velocity: error correlation length: 5km (Hori),
vertical correlation = 0.75 (250m)
standard dev: 1m/s
Correlated observation errors

35. Characterize the forecasting error
Variance
(Ensemble spread)
Correlation
Prescribed errors
correlation length

36. Convective Storm: 12 Jul 2004 1800UTC to 1900 UTC
McGill Radar
1 hour forecast

37. The impact of the observation errors
(Ensemble mean)
Unperturbed reference
Correlated noise
Uncorrelated noise

38. The impact of the observation errors
(Ensemble spread)
Uncorrelated noise
Correlated noise
Uncorrelated noise
Correlated noise

39. The impact of the observation errors:
(Forecasting error of ACF in space )
Correlated noise
Forecasting error of ACF - U
Forecasting error of ACF - U
Uncorrelated noise
Forecasting error of ACF - T
lag [km]
Correlated noise
Forecasting error of ACF - T
lag [km]
Uncorrelated noise

40. The impact of the observation and background errors (Ensemble spread)
Uncorrelated noise
Correlated noise
Perturbed both
observation and background

41. Forecasting error of ACF : Entire domain v.s. precipitation area
Forecasting error of ACF - U
The error structure could be different within and outside of precipitation area

42. Cross correlations errors:
Vertical velocity and cloud water
Strong connection between dynamics and microphysics processes

43. Verification of the ensemble forecasts
30 min simulation
40 min simulation

44. Error growth of the forecasting error:
It takes about min to double the error growth
The limit of predictability depends on the rate of error growth
by one assimilation window, the one hour forecast may reach
the limit of the predictability at this scale

45. 5. Summary / Conclusion
The McGill radar assimilation system successfully triggered
the convective storms at the right time and place based on single
Doppler radar observations.
The cycling process helps to capture the evolution of the storm in
intensity and location. However, after 1.5- hour forecast the result
indicates an error in the position of the convective cells.
The verification of the radial wind in time reveals that the errors are
larger in the high levels. In addition, the simulation has more
predictability at the longer scale( > 30 km) beyond 20 minutes.
larger in the high levels. This may explain the position errors of the
simulation.

46. Show how sensitive forecasting errors are to the representation of
the initial perturbation (from observation error correlation).
The correlation of the errors greatly increases the spread of the
ensemble as well as its correlation in space.
The result of error ACF indicates the need to discriminate background
error covariances within and outside the precipitating areas
Cross-correlation errors reveal the strong coupling between
dynamics and microphysics.

47. 6. Future work
Semi-Operational mode: Meso-Analysis System (MAS)
Convective system
Stratiform system

48. Extend the radar observation to the radar network.
Investigate the impact of flow-dependent background
errors in the short term forecast. (cycling process)
Study and apply the observation error covariance
into the system.
Investigate the model error at the very short term
forecast. ( model is not perfect )
background + observation + model