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Comparison of hybrid ensemble-4DVar with EnKFand 4DVar for regional-scale data assimilation
Jon Poterjoy and Fuqing Zhang
Department of MeteorologyThe Pennsylvania State University
Wednesday 18th December, 2013
Introduction
Zhang and Zhang(MWR, 2012)
Zhang, Zhang,and Poterjoy
(MWR, 2013)
A prototype hybrid ensemble-4DVar (E4DVar) systemoutperformed EnKF and 4DVar in a month-long experimentover the continental United States. Experiments used theWRF model with a 90-km grid spacing.
Introduction
The current study examines the performance of a newlydeveloped E4DVar system that is based on multi-incrementalWRF-4DVar.
EnKF, 4DVar, and E4DVar are applied for a case of tropicalcyclogenesis to compare the three methods at the mesoscale.
Coupled EnKF-Var data assimilation
x1,0f , x2,0
f ,... , xN ,0f
x0f
!x1,0f , !x2,0
f ,... , !xN ,0f
3D/4DVar - is used as first guess - are used in the cost function
x0f
!x1,0f , !x2,0
f ,... , !xN ,0f
EnKF - is used as first guess - are used to estimate the covariance
x0f
!x1,0f , !x2,0
f ,... , !xN ,0f
Ensemble Forecast
Separate ensemble into mean and perturbations
Run 3D/4DVar and EnKF separately
x0a
!x1,0a , !x2,0
a ,... , !xN ,0a
3D/4DVar analysis becomes posterior
mean for EnKF perturbations
x1,0a , x2,0
a ,... , xN ,0a
Ensemble Analysis
Zhang et al. (AAS, 2009)
Hybrid cost function
Composition of hybrid increments:
δx0 = δxc0 +
1√N − 1
N∑n=1
(an ◦ x′f n,0)
The N weighting vectors an are concatenated to form the αcontrol variables for the ensemble perturbations
αT = [aT1 , aT
2 , ..., aTN ].
Hybrid cost function
J(δx0) = βcJb(δxc0) + βeJe(α) + Jo(δx0)
= βc12δx
cT0 B−1δxc
0
+ βe12α
T A−1α
+12
τ∑t=0
[HtMtδx0 − dt ]T R−1
t [HtMtδx0 − dt ].
dt is calculated with respect to a model trajectory from x̄f0:
dt = yt − Ht [Mt(x̄f0)].
Lorenc (QJRMS, 2003), Wang et al. (MWR, 2008)
Tropical cyclogenesis case
100oW 90
oW 80
oW 70
oW 60
oW 50
oW
0o
10oN
20oN
30oN
00 UTC 9 Sept.00 UTC 18 Sept.
Track
− tropical wave
− tropical storm
− hurricane
4DVar, EnKF, and E4DVarare compared for 10 daysduring the genesis, rapidintensification and decay ofHurricane Karl (2010).
Karl developed from a tropical disturbance that initiated on 8Sept., and formed a tropical depression on 18 UTC 14 Sept.
Experiments are initialized from GFS/GDAS analyses on 18UTC 07 Sept.
Analyses are performed every six hours until 00 UTC 18 Sept.
ObservationsConventional observations
100oW 90
oW 80
oW 70
oW 60
oW 50
oW
0o
10oN
20oN
30oN
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xxx x
xxx
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x x xxx
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xxxxxxxx x xxxx x
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ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
+
+
+++
+
+
+
++
+
++
+
+
+++
+ +
+
+
+
+ ++ +
+++
+
+++++++
+++
+++
+
++
+
+++++++
++
+
+
+
++
++
+
++
+
+
++
+
+++++++
++
+
+ ++
00 UTC 9 Sept.00 UTC 18 Sept.
Track
− tropical wave
− tropical storm
− hurricane
Observations~ sat winds below 500 mb
~ sat winds above 500 mb
o soundings
+ buoys
x METARs
Field observations
100oW 90
oW 80
oW 70
oW 60
oW 50
oW
0o
10oN
20oN
30oN
o o o o o o
o
o
oooo
oo o o o o
ooo
o o o o o o
o
o
ooooo
o o o o
oooo
o
oo
o
o ooo
o o
o
o
ooooo
o
oo
o
o
oooooo
o
o o
o
o o
o
o o
o
o
o
o o
o
oooooo
o ooo
o ooo o
oo
oo o o
oo
o
oo
o
ooo
o
o
ooo
o
ooo
oo o
o
o
o
o
ooo
oo
oo
ooooo
oo
oo
oo
o
oo
oo
o
o o o o
oo
oo
oo
oo
o o
oo
oo
o
ooo
oo
oooo
ooooo
oooooo ooo
oooo o
oo
ooo
oooo
ooo
o oooooooo
ooooo oooooo
00 UTC 9 Sept.
00 UTC 18 Sept.
Track
− tropical wave
− tropical storm
− hurricane
PREDICT dropsondeso 10 Sept. flight missions (2)
o 11 Sept. flight mission
o 12 Sept. flight mission
o 13 Sept. flight mission
o 14 Sept. flight mission
GRIP dropsondeso 12 Sept. flight mission
o 13 Sept. flight mission
o 14 Sept. flight mission
o 16 Sept. flight mission
o 17 Sept. flight mission
The assimilated data includeconventional observations anddropsonde measurementscollected during the NSFPredepression Investigation ofCloud Systems in the Tropics(PREDICT) field campaign.
Dropsondes from the NASAGenesis and RapidIntensification Processes(GRIP) experiment are used forverification.
Experiment configuration
The data assimilation is performed on a 13.5-km grid spacingdomain with 34 vertical levels.
Deterministic forecasts use a 4.5-km storm-following nesteddomain.
Inner-loop iterations (4DVar/E4DVar) use a 40.5-km gridspacing.
Ensemble experiments use 60 members with a 900-km ROIfor localization and 80% relaxation to prior perturbations
The hybrid cost function uses 80% of the ensemble incrementand 20% of the climatological increment.
Deterministic forecast results
0 h 24 h 48 h 72 h1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
a) u (m s−1)
RM
SD
(observ
ations)
0 h 24 h 48 h 72 h1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
b) v (m s−1)
0 h 24 h 48 h 72 h0.6
0.7
0.8
0.9
1
1.1
c) T (K)
0 h 24 h 48 h 72 h1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
d) q (g kg−1)
0 h 24 h 48 h 72 h1.8
2
2.2
2.4
2.6
2.8
3
e) u (m s−1)
RM
SD
(G
DA
S)
Forecast lead time0 h 24 h 48 h 72 h
1.6
1.8
2
2.2
2.4
2.6
2.8
3
f) v (m s−1)
Forecast lead time0 h 24 h 48 h 72 h
0.45
0.5
0.55
0.6
0.65
0.7
g) T (K)
Forecast lead time0 h 24 h 48 h 72 h
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
h) q (g kg−1)
Forecast lead time
EnKF
4DVar
E4DVar
RMSD to routinesoundings and fieldobservations within 800km of storm center
RMSD to GDAS datawithin 2500 km of stormcenter
Deterministic track and intensity forecasts
100oW
95oW 90
oW 85
oW 80
oW 75
oW 70
oW 65oW
10oN
12oN
14oN
16oN
18oN
20oN
22oN
24oN
xx
xx
xxxxxxxxx
EnKF
a)
12 Sept. 14 Sept. 16 Sept. 18 Sept.
10
20
30
40
50
x x
x
x
x x
x
x
x
x
x x
x
Ma
x w
ind
sp
ee
d (
m s
−1)
d)
Analysis time
00 UTC 12 Sept.
06 UTC 12 Sept.
12 UTC 12 Sept.18 UTC 12 Sept.
00 UTC 13 Sept.
06 UTC 13 Sept.12 UTC 13 Sept.
18 UTC 13 Sept.
00 UTC 14 Sept.06 UTC 14 Sept.
12 UTC 14 Sept.
18 UTC 14 Sept.
00 UTC 15 Sept.
100oW
95oW 90
oW 85
oW 80
oW 75
oW 70
oW 65oW
10oN
12oN
14oN
16oN
18oN
20oN
22oN
24oN
xxx
x
xx
xx
xxxxx
4DVar
b)
12 Sept. 14 Sept. 16 Sept. 18 Sept.
10
20
30
40
50
xx
x
xx
x x
x
x
xx
x x
e)
100oW
95oW 90
oW 85
oW 80
oW 75
oW 70
oW 65oW
10oN
12oN
14oN
16oN
18oN
20oN
22oN
24oN
xx
xxx
x
xxxxx
xx
E4DVar
c)
12 Sept. 14 Sept. 16 Sept. 18 Sept.
10
20
30
40
50
x xx
x
xx
x x
x
x
x x x
f)
Forecasts to 00 UTC 18 Sept. using analysis times leading upto genesis.
Deterministic track and intensity forecasts
100oW
95oW 90
oW 85
oW 80
oW 75
oW 70
oW 65oW
10oN
12oN
14oN
16oN
18oN
20oN
22oN
24oN
xxxx
EnKF
a)
12 Sept. 14 Sept. 16 Sept. 18 Sept.
10
20
30
40
50
x
x
x x
Ma
x w
ind
sp
ee
d (
m s
−1)
d)
Analysis time
00 UTC 14 Sept.
06 UTC 14 Sept.
12 UTC 14 Sept.
18 UTC 14 Sept.
100oW
95oW 90
oW 85
oW 80
oW 75
oW 70
oW 65oW
10oN
12oN
14oN
16oN
18oN
20oN
22oN
24oN
xxxx
4DVar
b)
12 Sept. 14 Sept. 16 Sept. 18 Sept.
10
20
30
40
50
x
xx
x
e)
100oW
95oW 90
oW 85
oW 80
oW 75
oW 70
oW 65oW
10oN
12oN
14oN
16oN
18oN
20oN
22oN
24oN
xxx
x
E4DVar
c)
12 Sept. 14 Sept. 16 Sept. 18 Sept.
10
20
30
40
50
x
x
x x
f)
Forecasts to 00 UTC 18 Sept. using analysis times leading upto genesis.
4DVar and E4DVar structure functions
Balance
0 2 4 6 8 10 1210
−6
10−5
10−4
10−3
Domain root mean square ∂2p
s/∂t
2
Me
an
(h
Pa
s−
2)
a)
0 2 4 6 8 10 1210
−7
10−6
10−5
10−4
Sta
nd
ard
de
via
tio
n (
hP
a s
−2)
Forecast lead time (h)
b)
EnKF4DVarE4DVar
Domain-averaged ∂2Ps∂t2 is
used to quantify gravitywave activity afterinitialization. Value areaveraged over alldeterministic forecasts.
Standard deviations in mean∂2Ps∂t2 show amount of
variability between cycles.
Cost
Approximating a minimum to the 4DVar/E4DVar costfunction requires many iterations of the tangent linear andadjoint model (can be expensive).
Inner-loop iterations are averaged over the 40 dataassimilation cycles:
4DVar E4DVarMean iterations 37.5 25.6
Mean analysis time (256 cores) 2236 s 1566 s
Conclusions
A two-way coupled hybrid method (E4DVar) is found tooutperform the benchmark EnKF and 4DVar systems for a10-day tropical cyclogenesis and rapid intensification event.
E4DVar does not require a long assimilation window todevelop flow-dependent information.
It may also use observations at the beginning of the timewindow more affectively than 4DVar.
The ensemble information improves initial condition balanceover 4DVar and EnKF and reduces the number of iterationsrequired to minimize the 4-D cost function.
Future research will focus on optimal window length,conditioning, and comparison with 4D-ens-Var.
Particle filtering
Approximate moments of the posterior error distribution using
f (xt) =∫
f (xt)p(xt |yt)dxt ,
≈N∑
n=1wn
t f (xnt ).
For the simplest particle filter, the weights are given by
wnt =
p(yt |xnt )∑N
n=1(yt |xnt ).
Local likelihood particle filter (LLPF)
Problem: unless the ensemble is relatively large, it willeventually lose track of the signal, in which case, the weightsbecome concentrated on a small number of particles. Thisproblem occurs faster when the dimensions of x and y arelarge (Snyder et al. 2008, MWR).
Possible solution: use a likelihood function that decaysexponentially away from observation locations. Filterdegenerecy is then restricted to local regions of the domain.
Local likelihood particle filter (LLPF)The Lorenze-96 model is used to test this idea and comparewith EnKF:
dxt+1,i
dt = (xt,i+1 − xt,i−1)xt,i−1 − xt,i + 8.
Experiment details:
100 variables
Observations are taken from a “truth” run at every third gridpoint, with added Gaussian noise (σ = 1).
These observations are assimilated every 24 h (dt = 0.2) for1000 cycles.
Relaxation coefficient and localization radius for EnKF aretuned for each ensemble size.
Local likelihood particle filter (LLPF)
Ne = 50
0 5 10 15 20 25 300.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
30−
cycle
mean R
MS
E
30−day averaging period
EnKF
LLPF
RMSEs are averaged every 30 cycles.
Local likelihood particle filter (LLPF)
Ne = 100
0 5 10 15 20 25 300.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
30−
cycle
mean R
MS
E
30−day averaging period
EnKF
LLPF
RMSEs are averaged every 30 cycles.
Local likelihood particle filter (LLPF)
Ne = 200
0 5 10 15 20 25 300.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
30−
cycle
mean R
MS
E
30−day averaging period
EnKF
LLPF
RMSEs are averaged every 30 cycles.
Local likelihood particle filter (LLPF)
Ne = 300
0 5 10 15 20 25 300.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
30−
cycle
mean R
MS
E
30−day averaging period
EnKF
LLPF
RMSEs are averaged every 30 cycles.
Conclusions
A localized likelihood approach to particle filtering is beinginvestigated for systems that contain a large spatialdimension.
This method makes use of a distance-dependent likelihoodfunction that can be applied to prevent filter degeneracy(much like localization in EnKF).
Much more work needs to be done; e.g., resampling, inflation,better choices for likelihood function, etc.
Extra
A simple localization example is to use a function that decaysexponentially away from the location of an observation, sothat the likelihood of the the i th variable given the j th
observation is written
p(yt,j |xnt,i) = [p(yt,j |xn
t )−1
NeNy]exp(−di ,j
R ) +1
NeNy,
where di ,j is the physical distance between the observation andmodel grid point, and R is a tunable localization radius.
Extra
EnKF(N = 60)
4DVar E4DVar(N = 60)
Mean number of observations 26140 26140 26140Mean inner-loop iterations N/A 37.4 25.6
Number of NLM runs 60 2 62Mean TLM time (s) N/A 17.3 17.3Mean ADM time (s) N/A 39.5 39.5Mean NLM time (s) 13.4 13.4 13.4
Mean IO (s) N/A 2.2 2.2Mean analysis time (s) 840 2236 1566
TOTAL TIME (s): 1644 2263 2397
Computing time for 40.5 km grid spacing, 34 vertical levelsand 256 nodes.