Jasper A. Vrugt

University of California Irvine, CEE & EES

University of Amsterdam, CGE Email: jasper@uci.edu

LECTURE 4

DATA ASSIMILATION

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

MATHEMATICAL FORMULATION

LETS USE A STATE SPACE FORMULATION

THE MEASUREMENT OPERATOR

THE ERROR RESIDUAL

SINGLE LAYER CANOPY INTERCEPTION MODEL

Rainfall, P Evaporation, E

Drainage, D

Storage, S

MAIN MODEL EQUATIONS

Vrugt et al., WRR, (2003)

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

INTERCEPTION MODELING (continued)

True rainfall

Rainfall data

Time [hours]

Sto

rage

[m

m]

Simulation

Unable to fit

DATA ASSIMILATION

True rainfall

Rainfall data

Time [hours]

Sto

rage

[m

m]

Simulation

REMEMBER

Blow up at time t

X

?

t +1

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

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

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)

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

Parameter and

State Estimation

Parameter Estimation

POSTERIOR MODEL PREDICTION RANGES

Vrugt et al., WRR, (2005)

-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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

-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)

RECENT DEVELOPMENTS

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

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.

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

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

SEQUENTIAL BAYES LAW

SEQUENTIAL BAYES LAW (PARAMETERS ASSUMED KNOWN!)

Doucet and Johansen, 2011; Vrugt et al., AWR, (2012)

SEQUENTIAL MONTE CARLO (SMC) METHODS

Doucet and Johansen, 2011; Vrugt et al., AWR, (2012)

CRUX OF SMC: RESAMPLING

Vrugt et al., AWR, (2012)

RESAMPLING WITH DREAM AT (t-1)

WITH METROPOLIS ACCEPTANCE PROBABILITY

SCHEMATIC ILLUSTRATION OF RESAMPLING

Vrugt et al., AWR, (2012)

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

BENCHMARK STUDY: LORENZ MODEL

Vrugt et al., AWR, (2012)

PARAMETERS ADDED TO STATE VECTOR – NOT RECOMMENDED!!

PARTICLE-DREAM WITH INVARIANT PARAMETERS

DREAM + PARTICLE-DREAM

PSEUDO-CODE OF P-DREAM(IP)

Vrugt et al., AWR, (2012)

CASE STUDY: SIMPLE LORENZ MODEL

Vrugt et al., AWR, (2012)

Vrugt et al., AWR, (2012)

CASE STUDY: HYDROLOGIC MODEL

FINAL REMARK ABOUT EFFICIENCY

DIRECTED UPDATED TOWARDS OBSERVATIONS

CAUTION: DETAILED BALANCE!!!

SOFTWARE: FACULTY.SITES.UCI.EDU/JASPER