Putting a Vortex in Its Place
Chris Snyder
National Center for Atmospheric Research
Introduction
Data assimilation spanning a range of scales is difficult---a central
unsolved problem in assimilation/state estimation
Hurricanes are an obvious example
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Large-scale “steering” flow
Axisymmetric vortex
Asymmetric structure; rain bands
Convective elements, eye-wall details, …
Introduction
Importance of remotely sensed observations
– Indirect; instrument does not measure model variables
– Patchy in time and space
Also special, in-situ observations
– Reconaissance flights provide position and intensity of vortex
Themes
1. Initializing forecast/simulation model with vortex in correct
location
– Two scales: “environment” and vortex
2. Monte-Carlo (ensemble) methods for DA
Bogussing
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ICs for hurricane forecasts often involve some form of bogussing
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A simple, empirical approach to intializing hurricane vortex
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Obs of intensity, size of vortex (e.g. from reconnaisance flights)
Use these to determine parameters in analytic, axisymmetric model of vortex … a
“bogus” vortex
Information from bogus vortex inserted into ICs at observed location of vortex
Operational (NHC/GFDL) scheme
1. Remove existing vortex from ICs
2. Spin up vortex in an axisymmetric model, constraining low-level winds to match those
from specified bogus vortex
3. Add axisymmetric vortex to ICs at observed location
A Simple 2D “Hurricane” Experiment
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10-4 s-1
2D (barotropic) vorticity dynamics
(2400 km)2, doubly periodic domain
Strong vortex (80-km radius) embedded in largescale turbulent flow
Construct 31 ICs with small disp.s of vortex and
small diff.s in large scale
 30 ensemble members + 1 “true”/reference state
A Simple 2D Experiment (cont)
t=24h 91 km
127 km
Position Errors in Hurricane Data Assimilation
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Errors in large scales produce wind errors local to vortex, and
thus position/track errors
Vortex intensity and structure also influence track and can lead
to position errors
Resulting difficulties for data assimilation:
• Influence of obs depends strongly on presence, location of
vortex
• Even small displacements of vortex imply non-Gaussian pdfs
Most practical DA schemes assume Gaussian prior with stationary,
isotropic covariances
PARTICLE FILTER 1
PARTICLE FILTER 2
PARTICLE FILTER 3
PARTICLE FILTER 3
Other Non-Gaussian Assimilation Schemes
• 4D variational methods
– Assume Gaussian prior and observation errors
– Compute maximum likelihood estimate given obs in time interval
– Nonlinear minimization in many variables
• Methods based on “alignment” or “distortion”
– Assume prior is known function of uncertain spatial coords
– E.g. suppose  = (x + x, y + y), with x, y Gaussian
– Lawson and Hansen (2004), Ravela et al. (2007)
Assimilation of Position Observations
http://cimss.ssec.wisc.edu/tropic/archive/montage/atlantic/2004/IVAN-track.gif
• Wish to avoid difficulties associated with large position errors
• Geostationary satellites provide vortex position almost
continuously in time
• Assimilating such obs should limit position errors in analysis
Details of Position Assimilation
• Need operator that returns vortex position given model fields,
e.g., location of minimum surface pressure
• For small, Gaussian displacements, errors are Gaussian with
covariances related to gradient of original field
(x + x, y + y) - (x, y)    (x, y)
• If position obs are accurate and frequent, can assimilate with a
linear scheme
Ensemble Kalman Filter (EnKF)
• Estimates/models of forecast and obs. pdfs are crucial to DA.
• EnKF uses sample (ensemble) estimates
• EnKF considers only 1st, 2nd moments---linear scheme
EnKF Analysis Equations
• Assimilate obs serially (one at a time)
• Given single obs y, any state variable x is updated via
xa = xf + k ( y - yf ),
where
yf = Hxf,
k = cov( xf, yf ) / ( var(yf ) + 2 ) .
Both cov( xf, yf ) and var(yf ) are sample (ensemble) estimates
• Loop over state variables, loop over observations
• For large ensembles, converges to KF (or BLUE)
• No adjoint or minimization algorithm required.
2D Experiment Revisited
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2D (barotropic) vorticity dynamics
(2400 km)2, doubly periodic domain
Strong vortex (80-km radius) embedded in largescale turbulent flow
Construct 31 ICs with small disp.s of vortex and
small diff.s in large scale
 30 ensemble members + 1 “true”/reference state
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Simulate obs of vortex position with random error
Assimilate 1-hourly obs with EnKF
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Chen, Y. and C. Snyder, 2007: Assimilating vortex
position with an ensemble Kalman filter. Mon. Wea.
Rev., in press.
10-4 s-1
2D Experiment Revisited
Without assimilation
With assimilation
t=24h 91 km
127 km
Have also explored assimilation of intensity and shape of vortex
Experiments with WRF/DART
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WRF -- Weather Research and Forecasting model
DART -- Data Assimilation Research Testbed:
http://www.image.ucar.edu/DAReS/DART/
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36 km horizontal resolution, 35 vertical levels
26/28 ensemble members
Ensemble initial and boundary conditions are generated by perturbing
GFS(AVN) analysis/forecast with WRF-VAR error statistics
Assimilated observations:
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hurricane track (center position and minimum sea level pressure from NHC advisories)
Satellite winds (3% available observations)
Compare forecasts initialized from the EnKF mean analysis and from the
GFS analysis
Hurricane Ivan 2004
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36-km horizontal resolution, 28 ensemble members
Assimilate position, intensity and satellite winds every 3h for a total of 24h
Compare forecasts initialized from the EnKF analysis and from the GFS analysis
Hurricane Ivan 2004
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36-km horizontal resolution, 28 ensemble members
Assimilate position, intensity and satellite winds every 3h for a total of 24h
Compare forecasts initialized from the EnKF analysis and from the GFS analysis
Hurricane Ivan 2004
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36-km horizontal resolution, 28 ensemble members
Assimilate position, intensity and satellite winds every 3h for a total of 24h
Compare forecasts initialized from the EnKF analysis and from the GFS analysis
Hurricane Katrina 2005
36-km
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Analysis:
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12-km
36-km horizontal resolution, 26 ensemble members
Assimilate position, intensity, and satellite winds every hour for a total of 12 hours
Forecasts:
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Compare forecasts initialized from the EnKF analysis, from the GFS/AVN forecasts and from GFDL analysis at 36-km and 12-km resolutions
Hurricane Rita and Ophelia 2005
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36-km horizontal resolution, 26 ensemble members
Assimilate position, intensity, and satellite winds every hour for a total of 12 hours
Compare forecasts initialized from the EnKF analysis and from the GFS/AVN forecasts.
Typhoon Dujuan 2003
Forecast time (day)
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45-km horizontal resolution, 28 ensemble members
Assimilate position, intensity, satellite winds, and GPS refractivity for 1 day or 2.5 days
Compare forecasts initialized from
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EnKF analysis
WRF 3DVAR analysis (3DVAR, cycling for 2.5 days)
GFS analysis (3DVAR-non)
Increments to Vortex Structure
RITA at 2005-09-20-23Z
RITA 2005-09-20-23Z
Center Lat. (oN)
Center Lon. (oW)
Mini. SLP(mb)
Observation
(error)
24.00 (0.3)
-82.20 (0.3)
973.0 (5.0)
Prior mean
(spread)
23.85 (0.24)
-82.33 (0.23)
988.6 (2.0)
Posterior mean (spread)
23.89 (0.15)
-82.29 (0.18)
986.5 (2.0)
Vortex Spin-up
6-h Accumulated Precipitation
Katrina 2005, 12-km
Vortex Spin-up
Hurricane Ivan 2004
Surface Pressure Tendency
GFS0913
EnKF
2006 Real-time Forecast
2-way nested domain: 36km (183x133x35) – 12km (103x103x35)
Assimilation window: 12Z – 00Z, every hour
Observations: vortex position, intensity; MADIS satellite wind;
dropsondes
Helene (2006)
forecast hour
forecast hour
Summary
Hurricane track observations can be easily assimilated with an EnKF--effectiveness depends on frequent, accurate observations.
When position errors are larger, non-Gaussian effects important. General
purpose ensemble filters (esp. PF) are not feasible solutions.
Track forecasts initialized from the EnKF analysis are significantly
improved in retrospective tests.
EnKF analysis produces dynamically consistent increments, and reduces
spurious transient evolution of initial vortex.
EnKF Forecast/Analysis Cycle
1. Forecast: integrate ensemble members to time of next
available observations
2. Update members given new observations
3. Repeat
EnKF initializes its own ensemble and provides short-range
ensemble forecast; unifies DA and EF
D1
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D3
obs
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obs
D2
WRF/DART for Doppler Radar
Analysis reflectivity (color), obs. (20 dBZ, black contour)
21:30 UTC
km
21:10 UTC
KOUN
22:10 UTC
km
21:50 UTC
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WRF/DART for Doppler Radar
Velocity (m/s)
Reflectivity (dBZ)
Background minus observation statistics
(av’d over 3 analysis cycles/3 elevation angles)
Analysis time (UTC)
Analysis time (UTC)