Hybrid 4D EnVar for the NCEP GFS
(with comments on initialization)
Daryl Kleist1,3, Jeff Whitaker2, Kayo Ide3, and John Derber1
Lili Lei2,4, Rahul Mahajan1,5, Catherine Thomas1,3,5
1NOAA/NWS/NCEP/EMC
2NOAA/OAR/ESRL/PSD
3University
of Maryland
4CIRES
5IMSG
With acknowledgements to Dave Parrish (EMC), Ricardo Todling (NASA/GMAO), Xuguang
Wang (OU), Ting Lei (OU) and many others
6th EnKF Workshop
1
Ensemble-Var methods: nomenclature
• En-4DVar: Propagate ensemble Pb from one
assimilation window to the next (updated using EnKF
for example), replace static Pb with ensemble estimate
of Pb at start of 4DVar window, Pb propagated with
tangent linear model within window.
• 4D-EnVar: Pb at every time in the assim. window
comes from ensemble estimate (TLM no longer used).
• As above, with hybrid in name: Pb is a linear
combination of static and ensemble components.
• 3D-EnVar: same as 4D ensemble Var, but Pb is
assumed to be constant through the assim. window
(current NCEP implementation).
2
member 1
forecast
Generate new ensemble
perturbations given the
latest set of observations
and first-guess ensemble
member 2
forecast
EnKF
member update
recenter analysis ensemble
T254L64
Dual-Res Coupled Hybrid
Var/EnKF Cycling
member 3
forecast
T574L64
Ensemble contribution to
background error
covariance
high res
forecast
Previous Cycle
GSI
Hybrid Ens/Var
member 1
analysis
member 2
analysis
member 3
analysis
Replace the EnKF
ensemble mean analysis
and inflate
high res
analysis
Current Update Cycle
3
4D EnVar: The Way Forward ?
• Natural extension to operational 3D EnVar
– Uses variational approach with already available 4D ensemble perts
• No need for development of maintenance of TLM and ADJ models
– Makes use of 4D ensemble to perform 4D analysis
– Modular, usable across a wide variety of models
• Highly scalable
– Aligns with technological/computing advances
• Computationally inexpensive relative to 4DVAR (with TL/AD)
– Estimates of improved efficiency by 10x or more, e.g. at Env. Canada (6x
faster than 4DVAR on half as many cpus)
• Compromises to gain best aspects of (4D) variational and ensemble
DA algorithms
• Other centers exploring similar path forward for deterministic NWP
– Canada (potentially replace 4DVAR), UKMO (potentially replace En4DVar)
Single Observation (-3h) Example
for 4D Variants
4DVAR
H-4DVAR_AD
bf-1=0.25
TLMADJ
4DENSV
ENSONLY
TLMADJ
H-4DENSV
bf-1=0.25
ENSONLY
5
Observing System Simulation
Experiments (OSSE)
• Joint OSSE
– International, collaborative effort between ECMWF, NASA/GMAO,
NOAA (NCEP/EMC, NESDIS, JCSDA), NOAA/ESRL, others
– ECMWF-generated nature run (c31r1)
• T511L91, 13 month free run, prescribed SST, snow, ice
• Observations from (operational) 2005/2006 observing system
developed
– NCEP: ‘conventional’, sbuv ozone retrievals, GOES sounder radiances
– NASA/GMAO: all other radiances (AMSUA/B, HIRS, AIRS, MSU)
• Older version of the CRTM
• Simulated observation errors developed by Ron Errico
(GMAO/UMBC)
– Horizontally correlated errors for radiances
– Vertically correlated errors for conventional soundings
• Synthetic observations used in this study were calibrated by
Nikki Prive (GMAO)
– Attempt to match impact of various observation types with results
from data denial experiments
6
3D to 4D hybrid
• Model/Assimilation
• NCEP Global Forecast System
(GFS) (T382L64; post May 2011
version – v9.0.1)
• 80 member EnKF, 25% static
hybrid, additive/mult. inflation
• Test Period
• 01 July 2005-31 August 2005 (3
weeks ignored for spin-up)
• Analysis Error Difference (August)
• H-4DEnVar (With TLNMC)
minus 3D hybrid
7
500 hPa AC (against EC NR)
8
Constraint Options
• Tangent Linear Normal Mode Constraint
– Based on past experience and tests with 3D hybrid, default configuration
includes TLNMC over all time levels (quite expensive)


N

n 
n
xk  Ck xf  α  xe k 
n 1


• Weak Constraint “Digital Filter”
– Construct filtered/initialized state as weighted sum of 4D states
J dfi   x m x im , x m x im
K
x   h k  m x uk
i
m
k 1
• Combination of the two
– Apply TLNMC to center of assimilation window only in combination with JcDFI
(Cost effective alternative?)
9
Constraint impact (single case)
• Impact on tendencies
• Dashed: Total tendencies
• Solid: Gravity mode tendencies
• All constraints reduce
incremental tendencies
• Impact on ratio of gravity
mode/total tendencies
• JcDFI increases ratio of gravity
mode to total tendencies
• TLNMC most effective (but
most expensive)
• Combined constraint potential
(cost effective alternative)
Analysis Error (cycled OSSE)
• Time mean (August) change in
analysis error (total energy) relative
to 4D hybrid EnVar experiment that
utilized no constraints at all
• TLNMC universally better
• Combined constraint mixed
• JcDFI increases analysis error
Low Resolution GFS/GDAS
Experiments with real observations
• Basic configuration
– T254L64 GFS, opnl obs, GFS/GDAS cycles 20120701-20121001
• PR3LOEX0
– 3DVAR
• PRHLOEX1
– Hybrid 3D EnVar, 80 member T126L64 ensemble with fully
coupled (two-way) EnKF update, slightly re-tuned localization
and inflation for lower resolution, TLNMC on total increment,
75% ensemble & 25% static
• PRH4DEX1
– Hybrid 4D EnVar, TLNMC on all time levels, only 1x150
iterations
– Hourly TC relocation, O-G, binning of observations
12
500 hPa Die Off Curves
Northern Hemisphere
Southern Hemisphere
4DHYB ---3DHYB ---3DVAR ----
4DHYB ---3DHYB ---3DVAR ----
4DHYB-3DHYB
3DVAR-3DHYB
Move from 3D Hybrid (current operations) to Hybrid 4DEnVar yields improvement that is about 75% in amplitude
in comparison from going to 3D Hybrid from 3DVAR.
13
14
(Preliminary) Results and Comments
• 4D extension has positive impact in OSSE and real
observation (low resolution) framework
• 4D EnVar does have slower convergence
• As configured, 4D EnVar was 40% more expensive
than 3D hybrid (caveats being different iteration
count, low resolution and machine variability)
– TLNMC (balance constraint) over all time levels
quite expensive
– I/O potential issue, optimization will be necessary
prior to implementation.
• Option to post-process ensemble prior to use in
assimilation in subsequent cycle
15
(4D) Incremental Analysis Update
• In addition to TLNMC, we currently utilize full-field digital filter
initialization in the GFS
– Can have undesirable impacts being a full field filter
• IAU (Bloom et al.) has been used at GMAO and elsewhere as alternative to
DFI
– Increment is passed to model throughout the window as a forcing term
– However, this has typically been done using a 3D increment (rescaled)
through the window
• 4D EnVar lends itself to a minor modification of the IAU procedure to use
the 4D increment
– This also provides a mechanism for passing the 4D solution to the model
– Perhaps a way to help spin up/spin down of clouds, precipitation, etc.
• Promising preliminary results in EnKF context, work underway in 4D
hybrid EnVar context
16
EnKF-IAU and EnKF-DFI
EnKF-IAU noticeably better
Courtesy of Lili Lei
17
Compare 6-h forecasts to EC analysis
EnKF-IAU forecasts closer to EC analysis, esp for winds
Temperature
Uwind (Vwind is similar to Uwind)
Courtesy of Lili Lei
18
Next Steps/Future Development
• Configure update and outer loop
– Continue to investigate use of IAU to force 4D increment into model
• Replace all other initialization options? Get rid of TLNMC as well?
– Quasi-outer loop (rerun of nonlinear model as in 4D Var)
– Lengthen assimilation window (backward) in catch-up cycle
• May require temporal localization (code in place courtesy of OU collaboration)
• Test role of stochastic physics as replacement for additive inflation in 4D
EnVar context
• Improved (or at least tuned) localization
• Optimize static B for 4D hybrid
–
–
–
–
Computationally (current static B is bottleneck in code)
Add temporal information (FOTO)
Weights, including scale-dependence
Increase role of ensemble (to 100%?)
• 4D EnVar tentatively scheduled to be part of GDAS/GFS upgrade in 2015
19
Backup Slides
20
Time Evolution of Increment
TLMADJ
ENSONLY
t=-3h
Solution at beginning of window
same to within round-off (because
observation is taken at that time,
and same weighting parameters
used)
t=0h
Evolution of increment qualitatively
similar between dynamic and
ensemble specification
** Current linear and adjoint
models in GSI are computationally
impractical for use in 4DVAR other
than simple single observation
testing at low resolution
t=+3h
H-4DVAR_AD
H-4DENSV
21
Accounting for under-represented
source of uncertainty
• Current operations
– Multiplicative inflation – Helps compensate for finite-sized ensemble
– Additive perturbations – Helps compensate for lack of consideration
of model uncertainty
• Proposed configuration for T1534 SL (3072 x 1536) GFS
– Keep multiplicative inflation
– Reduce additive perturbations from 32 to 5%
– Add stochastic physics in model
•
•
•
•
SPPT -- Stochastic Physics Perturbation Tendency
SKEB – Stochastic Energy Backscatter
VC – Vorticity Confinement
SHUM – Boundary Layer Humidity perturbations
• See talk by Jeff Whitaker in Session 9
22
Better spread behavior (2014042400)
• Current operations
– Spread too large
– Spread decays and
recovers
3HR
6HR
9HR
• Stochastic Physics
– Spread decreased
overall (consistent
with error estimates)
– Spread grows
through assimilation
window
23
Hybrid ensemble-4DVAR
[H-4DVAR_AD]
Incremental 4DVAR: bin observations throughout window and solve for
increment at beginning of window (x0’). Requires linear (M) and adjoint
(MT) models
1
1 K
T 1
T
J x0   x0  B f x0    H k M k x0  d k  R k1 H k M k x0  d k 
2
2 k 1
ADJ
TLM
Can be expanded to include hybrid just as in the 3DHYB case
 
 
1
1 N n T 1 n
T 1
J xf ,α   b f xf  B f xf   b e  α L α 
2
2 n 1
1 K
T
1




H
M
x

d
R

k
k 0
k
k H k M k x 0  d k 
2 k 1
ADJ
TLM
With a static and ensemble contribution to the increment at the beginning of
the window
N

n
x0  xf  T α n x e
n 1

24
4D-Ensemble-Var
[4DENSV]
As in Buehner (2010), the H-4DVAR_AD cost function can be modified to solve
for the ensemble control variable (without static contribution)
 
 
1 N n T 1 n 1 K
T
J α    α L α   H k xk  d k  R k1 H k xk  d k 
2 n 1
2 k 1
Where the 4D increment is prescribed exclusively through linear
combinations of the 4D ensemble perturbations
N

n
xk  T α n x e  k
n 1

Here, the control variables (ensemble weights) are assumed to be valid
throughout the assimilation window (analogous to the 4D-LETKF without
temporal localization). Need for computationally expensive linear and
adjoint models in the minimization is conveniently avoided.
25
Hybrid 4D-Ensemble-Var
[H-4DENSV]
The 4DENSV cost function can be easily expanded to include a static
contribution
 
 
1
1 N n T 1 n
T 1
J xf ,α   b f xf  B f xf   b e  α L α 
2
2 n 1
1 K
T
1




H
x

d
R

k k
k
k H k x k  d k 
2 k 1
Where the 4D increment is prescribed exclusively through linear
combinations of the 4D ensemble perturbations plus static contribution
N

n
xk  xf  T α n x e  k
n 1

Here, the static contribution is considered time-invariant (i.e. from 3DVARFGAT). Weighting parameters exist just as in the other hybrid variants.
Again, no TLM or ADJ.
26