lecture 4 data assimilation...university of california irvine, cee & ees university of...
TRANSCRIPT
![Page 1: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/1.jpg)
Jasper A. Vrugt
University of California Irvine, CEE & EES
University of Amsterdam, CGE Email: [email protected]
LECTURE 4
DATA ASSIMILATION
![Page 2: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/2.jpg)
true input
true response
observed input
simulated response
measurement
outp
ut
time f
parameters prior info
observed response
optimize parameters
TUNING THE PARAMETERS SO THAT CLOSEST FIT TO THE OBSERVED SYSTEM RESPONSE IS OBTAINED
ENVIRONMENTAL MODELING FRAMEWORK
![Page 3: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/3.jpg)
MATHEMATICAL FORMULATION
LETS USE A STATE SPACE FORMULATION
THE MEASUREMENT OPERATOR
THE ERROR RESIDUAL
![Page 4: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/4.jpg)
SINGLE LAYER CANOPY INTERCEPTION MODEL
Rainfall, P Evaporation, E
Drainage, D
Storage, S
MAIN MODEL EQUATIONS
Vrugt et al., WRR, (2003)
![Page 5: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/5.jpg)
INTERCEPTION MODELING
True rainfall
Rainfall data
Time [hours]
Sto
rage
[m
m]
Inte
rcept
ion
Dra
inag
e
Eva
pora
tion
Max
imum
S
tora
ge
![Page 6: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/6.jpg)
INTERCEPTION MODELING (continued)
True rainfall
Rainfall data
Time [hours]
Sto
rage
[m
m]
Simulation
Unable to fit
![Page 7: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/7.jpg)
DATA ASSIMILATION
True rainfall
Rainfall data
Time [hours]
Sto
rage
[m
m]
Simulation
REMEMBER
Blow up at time t
X
?
t +1
![Page 8: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/8.jpg)
DATA ASSIMILATION (continued)
True rainfall
Rainfall data
Time [hours]
Sto
rage
[m
m]
Data assimilation
REMEMBER
DATA ASSIMILATION REMOVES PERSISTENT BIAS BY UPDATING STATE VARIABLES
Simulation
![Page 9: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/9.jpg)
HOW TO DETERMINE SIZE STATE UPDATES?
?
t +1
X
SIZE OF STATE UPDATES DEPENDS DIRECTLY ON SIZE OF MODEL AND MEASUREMENT ERROR
t
f
t
a
t yyy ~22
2
22
2
REMEMBER
![Page 10: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/10.jpg)
3
2
X = measurement
= model
1
x
t
x
t+1
= updated
Cmax
0
bexp Alpha
(1-Alpha)
Rq Rq Rq
Rs
ANOTHER CONCEPTUAL EXPLANATION USING ANOTHER MODEL
Vrugt et al., WRR, (2005)
![Page 11: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/11.jpg)
Y(t)
Forcing (Input Variables)
System invariants (Parameters)
Output (Diagnostic Variables)
f p(Yt)
U(t)
X(t)
Observations
p(Ot)
Update rule
DREAM p(M)
p(Ut)
State (Prognostic Variables)
p(Xt)
Ensemble Kalman Filter
Vrugt et al., WRR, (2005); Vrugt et al., GRL, (2005); Vrugt et al., JHM, (2006)
SIMULTANEOUS OPTIMIZATION AND DATA ASSIMILATION
![Page 12: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/12.jpg)
Parameter and
State Estimation
Parameter Estimation
POSTERIOR MODEL PREDICTION RANGES
Vrugt et al., WRR, (2005)
![Page 13: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/13.jpg)
-0.2
0
0.2
0.4
0.6
Au
toc
orr
ela
tio
n
(A) SCEM-UA
0 5 10 15 20 25
-0.2
0
0.2
0.4
0.6
Au
toc
orr
ela
tio
n
Lag [d]
(B) SODA
Significantly less auto-correlation between residuals with recursive state updating
AUTOCORRELATION BETWEEN RESIDUALS
Vrugt et al., WRR, (2005)
![Page 14: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/14.jpg)
0
0.05
0.1
0.15
0.2
0.25
0.3(A)
SC
EM
-UA
Marg
inal
po
ste
rio
r d
en
sit
y (B) (C) (D) (E)
250 300 350 400 4500
0.05
0.1
0.15
0.2
0.25
0.3(F)
Cmax
SO
DA
Marg
inal
po
ste
rio
r d
en
sit
y
0.4 0.6 0.8 1 1.2 1.4
(G)
bexp
0.8 0.85 0.9 0.95
(H)
Alpha
0.02 0.04 0.06 0.08
(I)
Rs
0.38 0.4 0.42 0.44 0.46
(J)
Rq
[mm] [-] [-] [d] [d]
MARGINAL POSTERIOR PARAMETER DISTRIBUTIONS
Vrugt et al., WRR, (2005)
![Page 15: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/15.jpg)
0 100 200 300 400-150
-100
-50
0
50
100
150(A) DRIVEN
Me
an
en
se
mb
le o
utp
ut
inn
ov
ati
on
[
m3/s
]
0 50 100 150 200 250-150
-100
-50
0
50
100
150(B) NONDRIVEN QUICK
1 1.5 2 2.5 3-1.5
-1
-0.5
0
0.5
1
1.5(C) NONDRIVEN SLOW
Mean ensemble streamflow prediction [m3/s]
Byproduct of Data Assimilation is the time series of output/state innovations: info about model structural errors?
INSIGHTS INTO MODEL STRUCTURAL ERRORS?
Vrugt et al., WRR, (2005)
![Page 16: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/16.jpg)
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
3.5
Time [days]
Nor
malized
Tra
cer
Con
c. (x
10
3)
Unplanned 14 hr
flow interruption
Planned 7-day
flow interruption Planned 14-day
flow interruption
Bromide
Pentafluorobenzoate
Lithium
YUCCA MOUNTAIN SUBSURFACE FLOW AND TRANSPORT MODEL
Vrugt et al., GRL, (2005)
![Page 17: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/17.jpg)
x = 30 meters
Injection well Production well
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
3.5
Time [days]
No
rm
alized
Tracer C
on
c. (x
10
3)
Unplanned 14 hr
flow interruption
Planned 7-day
flow interruption Planned 14-day
flow interruption
Bromide
Pentafluorobenzoate
Lithium
DATA COLLECTION AND TRACERS
Vrugt et al., GRL, (2005)
![Page 18: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/18.jpg)
Injection well
Modeling forced-gradient cross-hole tracer experiments
Define volume size nodes
Define exit fluxes (qi)
Solve advection – dispersion equation
time
Con
c. Production well
- ×
N
i
i
N
i
i i t
out t
q
q C
C
1
1
,
, ) (
CONCEPTUAL MODEL: RESIDENCE TIME DISTRIBUTION
Vrugt et al., GRL, (2005)
![Page 19: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/19.jpg)
Nodal concentration update according to:
i
N
i
iouttouttit
itititq
qCCZ
KCC-
-
1
,,,
,,,
)~
)((
Parameter estimation using adaptive MCMC
The Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm
HOW TO DO PARAMETER AND STATE ESTIMATION?
Vrugt et al., GRL, (2005)
![Page 20: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/20.jpg)
0 25 50 75 100 1250
0.5
1
0.25
0.75
Nor
maliz
ed L
ithium
C
onc.
(x 1
03)
Time [days]
(B) SCEM-UA -- No state updating
Time [days]
Time [days]
0 25 50 75 100 1250
0.5
1
0.25
0.75
Nor
maliz
ed L
ithium
C
onc.
(x 1
03) (A) SODA -- State updating
Kf = 0.20 – 0.24
n = 0.63 – 0.64
Kf = 0.01 – 0.08
n = 0.64 – 0.68
Which parameter values to use for transport predictions?
MODEL PREDICTION UNCERTAINTY RANGES
Vrugt et al., GRL, (2005)
![Page 21: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/21.jpg)
0
0.2
0.4
0.6
0.8
1
Norm
aliz
ed P
ara
mete
r R
ange
sx sy
DB
r
DP
FB
A
DLi
Kf n 0.2
0.30.4
0.5
0
0.2
0.40.05
0.1
0.15
0.2
0.25
f Li
fBr
fPFBA
Nonsorbing tracers show similar parameter values. Most trade-off appears between the fitting of the sorbing and nonsorbing tracers
► Sorbing and nonsorbing tracers provide conflicting information
PARETO SOLUTION SET (WITH AMALGAM)
Vrugt et al., VZJ, (2008)
![Page 22: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/22.jpg)
0
1
2
3
No
rmali
zed
Bro
mid
e
C
on
c.
(x 1
03)
(A) Tracer - Bromide
0
1
2
3
4
No
rmali
zed
PF
BA
Co
nc.
(x 1
03)
(B) Tracer - Pentafluorobenzoate
0 25 50 75 100 1250
0.2
0.4
0.6
0.8
1
Time [days]
No
rmali
zed
Lit
hiu
m
C
on
c.
(x 1
03)
(C) Tracer - Lithium
MODEL PREDICTION UNCERTAINTY RANGES
Vrugt et al., VZJ, (2008)
![Page 23: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/23.jpg)
-1
0
1(A) No State Updating
Tracer - Bromide
-1
0
1
Au
toco
rr.
Resid
uals
(B) SODA
-1
0
1(A) No State Updating
Tracer - Lithium
0 5 10 15 20 25-1
0
1
Au
toco
rr.
Resid
uals
(B) SODA
Lag
PARAMETER ESTIMATION .VS. DATA ASSIMILATION
Vrugt et al., VZJ, (2008)
![Page 24: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/24.jpg)
RECENT DEVELOPMENTS
![Page 25: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/25.jpg)
PARTICLE-MARKOV CHAIN MONTE CARLO
Vrugt et al., AWR, (2012)
X = measurement = model = updated
x t
x
t+1
Particle-DREAM Joint Parameter and State Estimation
f
![Page 26: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/26.jpg)
BAYESIAN ANALYIS
Bayes, Thomas (1763). "An Essay towards solving a Problem in the Doctrine of Chances.“, Philosophical Transactions of the Royal Society of London, 53, 370–418.
![Page 27: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/27.jpg)
P(A|B)
PRIOR, LIKELIHOOD, EVIDENCE, POSTERIOR
= P(A) P(B|A)
P(B)
PRIOR CONDITIONAL PROBABILITY (= LIKELIHOOD)
EVIDENCE POSTERIOR
IN OUR CASE WE USE THE FOLLOWING NOTATION
IMAGINE WE HAVE SOME DATA “B” AND WE LIKE TO ESTIMATE “B”
BAYES LAW TELLS US TO DO THE FOLLOWING
![Page 28: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/28.jpg)
SEQUENTIAL BAYES LAW
DERIVATION OF SEQUENTIAL BAYES LAW
Doucet and Johansen, 2011; Vrugt et al., AWR, (2012)
X = measurement = model = updated
x t
x
t+1
Particle-DREAM
f
![Page 29: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/29.jpg)
SEQUENTIAL BAYES LAW
SEQUENTIAL BAYES LAW (PARAMETERS ASSUMED KNOWN!)
Doucet and Johansen, 2011; Vrugt et al., AWR, (2012)
![Page 30: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/30.jpg)
SEQUENTIAL MONTE CARLO (SMC) METHODS
Doucet and Johansen, 2011; Vrugt et al., AWR, (2012)
![Page 31: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/31.jpg)
CRUX OF SMC: RESAMPLING
Vrugt et al., AWR, (2012)
RESAMPLING WITH DREAM AT (t-1)
WITH METROPOLIS ACCEPTANCE PROBABILITY
![Page 32: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/32.jpg)
SCHEMATIC ILLUSTRATION OF RESAMPLING
Vrugt et al., AWR, (2012)
![Page 33: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/33.jpg)
PSEUDO-CODE OF PARTICLE-DREAM
BUT WHAT TO DO WITH PARAMETERS?
TWO DIFFERENT POSSIBILITIES
P-DREAM(VP) STATE AUGMENTATION
P-DREAM(IP) OUTSIDE DREAM LOOP
VARIABLE PARAMETERS (STATE AUGMENTATION) NOT RECOMMENDED!!!!
CAUTIONARY NOTE
Vrugt et al., AWR, (2012)
PARTICLE - DREAM
![Page 34: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/34.jpg)
BENCHMARK STUDY: LORENZ MODEL
Vrugt et al., AWR, (2012)
PARAMETERS ADDED TO STATE VECTOR – NOT RECOMMENDED!!
![Page 35: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/35.jpg)
PARTICLE-DREAM WITH INVARIANT PARAMETERS
DREAM + PARTICLE-DREAM
PSEUDO-CODE OF P-DREAM(IP)
Vrugt et al., AWR, (2012)
![Page 36: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/36.jpg)
CASE STUDY: SIMPLE LORENZ MODEL
Vrugt et al., AWR, (2012)
![Page 37: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/37.jpg)
Vrugt et al., AWR, (2012)
CASE STUDY: HYDROLOGIC MODEL
![Page 38: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/38.jpg)
FINAL REMARK ABOUT EFFICIENCY
DIRECTED UPDATED TOWARDS OBSERVATIONS
CAUTION: DETAILED BALANCE!!!
![Page 39: LECTURE 4 DATA ASSIMILATION...University of California Irvine, CEE & EES University of Amsterdam, CGE Email: jasper@uci.edu LECTURE 4 DATA ASSIMILATION true true input response observed](https://reader035.vdocument.in/reader035/viewer/2022062510/6128174bffd97312124db5d4/html5/thumbnails/39.jpg)
SOFTWARE: FACULTY.SITES.UCI.EDU/JASPER