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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– 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
• ICs for hurricane forecasts often involve some form of bogussing
• A simple, empirical approach to intializing hurricane vortex– 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) scheme1. Remove existing vortex from ICs2. Spin up vortex in an axisymmetric model, constraining low-level
winds to match those from specified bogus vortex3. Add axisymmetric vortex to ICs at observed location
A Simple 2D “Hurricane” Experiment
• 2D (barotropic) vorticity dynamics• (2400 km)2, doubly periodic domain• Strong vortex (80-km radius) embedded
in large-scale 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
10-4 s-1
A Simple 2D Experiment (cont)
t=24h 91 km
127 km
Position Errors in Hurricane Data Assimilation
• 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
• 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
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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 viaxa = 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
• 2D (barotropic) vorticity dynamics• (2400 km)2, doubly periodic domain• Strong vortex (80-km radius) embedded
in large-scale 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
• Simulate obs of vortex position with random error
• Assimilate 1-hourly obs with EnKF
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
t=24h 91 km
127 km
Without assimilation With assimilation
Have also explored assimilation of intensity and shape of vortex
Experiments with WRF/DART
• WRF -- Weather Research and Forecasting model • DART -- Data Assimilation Research Testbed:
http://www.image.ucar.edu/DAReS/DART/• 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: – 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
– 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
– 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
– 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
– Analysis: • 36-km horizontal resolution, 26 ensemble members• Assimilate position, intensity, and satellite winds every hour for a total of 12 hours
– Forecasts:• Compare forecasts initialized from the EnKF analysis, from the GFS/AVN forecasts and from GFDL analysis at 36-km and 12-km
resolutions
36-km 12-km
Hurricane Rita and Ophelia 2005
– 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
– 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
• EnKF analysis• WRF 3DVAR analysis (3DVAR, cycling for 2.5 days)• GFS analysis (3DVAR-non)
Forecast time (day)
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 2004Surface 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 observations3. Repeat
EnKF initializes its own ensemble and provides short-range ensemble forecast; unifies DA and
EF
D1
D2D3
.
.obs
obs
WRF/DART for Doppler Radar
Analysis reflectivity (color), obs. (20 dBZ, black contour)
21:10 UTC 21:30 UTC
21:50 UTC 22:10 UTC
km km
kmkm
KOUN
WRF/DART for Doppler Radar
Background minus observation statistics(av’d over 3 analysis cycles/3 elevation angles)
Analysis time (UTC) Analysis time (UTC)
Vel
ocity
(m
/s)
Ref
lect
ivity
(dB
Z)