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Comparing Probabilis/c Inversion Techniques on a Highly Parameterized Ground Water Model
Sean A. McKenna, Hongkyu Yoon, David B. Hart
Geoscience Research and Applica/ons
Sandia Na/onal Laboratories
SIAM UQ 2012 Mee/ng April, 2012
This material is based upon work supported as part of the Center for Fron/ers of Subsurface Energy Security, an Energy Fron/er Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy
Sciences under Award Number DE-‐SC0001114. Sandia Na/onal Laboratories is a mul/-‐program laboratory managed and operated by Sandia Corpora/on, a wholly owned subsidiary of Lockheed Mar/n
Corpora/on, for the U.S. Department of Energy's Na/onal Nuclear Security Administra/on under contract DE-‐AC04-‐94AL85000.
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Mo/va/on
• Quan/fy uncertainty in inverse parameter es/mates – Basis for probabilis/c predic/ve modeling – Computa/onally efficient and accurate – Robust for strongly heterogeneous media
• Culebra dolomite as a mo/va/ng example
Faster, Cheaper and At Least As Good!
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Culebra Dolomite • The Culebra Dolomite near the WIPP site
(NM,USA) • Predic/ve performance measure is advec/ve
transport /me to a prescribed boundary • Observa/on data include two decades of
steady-‐state head measurements and transient pumping test results.
n Exis/ng Mul/ple Star/ng Point (MSP) results (Hart et al., 2009) n Start with a large number of equally likely
indicator simula/on “seed” fields n 200 seed fields serve as the start of 200
independent calibra/ons
~ 30 km
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4
Heterogeneous Field Parameteriza/on
• “Pilot Points” (similar to a kernel-‐based approach) • Choose loca/ons in the model domain and update their proper/es to produce beder fit to measured heads (“calibra/on points”)
• Spread influence of each point to neighboring model cells by using the spa/al covariance func/on as a weigh/ng scheme
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5
Pilot Point Example
Simple example of five pilot points used to update a log10 transmissivity field
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6
Pilot Point Example (Cont.)
-11
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-3
0
20
40
60
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100
0
20
40
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log1
0 T
(m2 /s
)
E-W distance (m
eters)
N-S Distance (meters)
-11 -10 -9 -8 -7 -6 -5 -4 -3
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T (m
2 /s)
E-W DIstance (m
eters)
N-S DIstance (meters)
-11 -10 -9 -8 -7 -6 -5 -4 -3
No Pilot Points 5 Pilot Points
Pilot points and variogram provide a means of calibra;ng the heterogeneous field to match pressure data
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Parameter Es/ma/on
y = Xp+!
! = (y" Xp)TQ(y" Xp)
p = (XTQX)!1XTQy
!p = (XTQX)"1XTQr
Xij =!yi!pj
Linear Model
Non-‐Linear Model
Assumes that the inverse of: XTQX exists and can be uniquely es/mated If # parameters (n) > # observa/ons (m), this is not the case
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Regulariza/on
XTQX ! (XTQX +! 2TTST ) Tikhonov
XTQX ! (V1E1V1T ) Truncated SVD
XTQX ! V1V2[ ]E1 00 E2
"
#$$
%
&''V1V2[ ]T =V1E1V1T +V2E2V2T
Hybrid Tikhonov-‐TSVD Regulariza/on:
XTQX !V1(ZTQZ +! 2DTSD) Where: D = TV1
Tonkin and Doherty, 2009, Water Resources Research
y =Gp Goal: G
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Super Parameters
XTQX ! V1V2[ ]E1 00 E2
"
#$$
%
&''V1V2[ ]T =V1E1V1T +V2E2V2T
Resolu;on matrix: R
y =Gp
G = V1E1V1T( )XTQ
p =GXp+G!Super Parameters
Goal
Goal achieved
Mapping from base parameters to super parameters
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Sampling Null Space p! p = !(I ! R)p+G! Unknown error between true and
es/mated parameters
C(p! p) = !(I ! R)C(p)(I ! R)T +GC(!)GT
Use informa/on in C(p) and C(ε) to define C(p-‐ )
Parameter error from calibra/on null space (V2)
Errors of es/mates in solu/on space (V1) driven by measurement noise
Moore, C. and Doherty, J., 2005. The role of the calibra;on process in reducing model predic;ve error, Water Resources Research, Volume 41, No 5. Tonkin M., and Doherty, J., 2009. Calibra;on-‐constrained Monte Carlo analysis of highly parameterized models using subspace techniques, Water Resources Research, 45,
(p ! pST ") =V2V2T (p ! pST ) Project stochas/c parameter differences that
have zero impact on calibra/on into null space
10
p
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Ques/ons • 1) Can NSMC approach approximate ensemble predic/ons obtained with MSP runs?
• 2) Given a set of previously run models, what is an effec/ve means of expanding the predic/ve ensemble?
• 3) In the absence of exis/ng calibra/ons, can a mean-‐field representa/on serve as an ini/al star/ng point?
NSMC = Null Space Monte Carlo
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Three Property Fields • Simultaneous es/ma/on of three spa/ally correlated property fields Transmissivity Anisotropy Stora/vity
Loca;ons in blue are fixed values and red are loca;ons where parameters are adjusted. For transmissivity, 2 fields (Zones 0 and 1) are es;mated, then combined 1380 observa;ons, ≈ 1200 parameters 12
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MSP Calibra/on Process
0
5000
10000
15000
20000
25000
30000
35000
0 10 20 30 40 50
PEST Iteration Number
Phi (
sum
of s
quar
ed
resi
dual
s)
200 star/ng fields are calibrated to 1380 steady-‐state head and transient drawdown observa/ons Computa/on /me is several months
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Approach 1: MSP
n Stochastic Inverse Modeling (~ Ramarao et al (1995, WRR))
• • •
Random seed fields
Calibrate model(s) to observed data
• • •
Prediction with calibrated fields
Pro
babi
lity
Prediction
Observed
Predicted
EXPENSIVE!!!!
n Conceptual model is stochastic – poorly known location and extent of highly fractured zones. Leads to multiple, equally probable starting points (seed fields for inversion). Calibrate each seed field.
Up to 50 itera/ons
Advec0ve Travel Times
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Approach 2: NSMC
n Null-Space Monte Carlo Method (~ Tonkin & Doherty (2009,WRR))
Single field
Calibrate
• • •
Prediction (may involve 1-2 iterations for inverse modeling) • • •
Calibration-constrained random fields
Pro
babi
lity
Prediction
Observed
Predicted
Not so expensive: 1-‐2 itera/ons
n Shift the approach: Calibrate one seed field and then modify that field to create probabilistic distribution of results
Advec0ve Travel Times
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Ground Water Parameter Es/ma/on n Calibrated parameters (>1200 parameters in total) transmissivity (T), horizontal hydraulic anisotropy,
storativity (S), and a section of recharge n 200 multiple random seed fields are calibrated to
observed data, and then 100 best fields are selected for travel-time analysis
• • Pilot points
Calibration-based five fields
Travel time-based five fields
Mean field of 200 random seed fields
Best/fastest Worst/slowest 25th Percentile in 200 calibrated multiple fields
50th 75th
NSMC Analysis (3 groups -11 fields)
NSMC generation of 200 random fields. From each calibrated field
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Travel Time and Obj. Func/on Objec/ve Func/on Selec/ons Travel Time Selec/ons
5 calibrated fields selected with each criterion 100 NSMC realiza/ons shown for each calibrated field
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Travel Time Distribu/ons
00.10.20.30.40.50.60.70.80.91
1000 10000 100000
Prob
ability
Advective-‐only Travel Times (yr)
100 Selected Fields from multiple seed fields100 Selected Fields from NSMC fields with r102100 Selected Fields from NSMC fields with r880100 Selected Fields from NSMC fields with r151
00.10.20.30.40.50.60.70.80.91
1000 10000 100000
Prob
ability
Advective-‐only Travel Times (yr)
100 Selected Fields from multiple seed fields100 Selected Fields from NSMC fields with r120100 Selected Fields from NSMC fields with r515100 Selected Fields from NSMC fields with r910
Objec/ve Func/on Selec/ons Travel Time Selec/ons
3 calibrated fields selected with each criterion (low, median, high) Original MSP calibra/on results 100 NSMC realiza/ons shown for each calibrated field
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Mean field of 200 mul/ple seed fields can be used to get a calibrated model for NSMC method
Stochastic Inverse Method NSMC method with mean field
Measured steady head (m)
Mod
eled
stea
dy h
ead
(m)
!"""#
!""""#
!"""""#
$""# !"""# !$""# %"""# %$""# &"""# &$""#
'()*+,#'-.
+#/0(12#
345+67*+#89:67;:#/<=-2#
!"#$%&''%()*+,-*./01%
2''%")+)34)5%6%7,)+51%
$*+,8(*4)5%9/5)+%:,4;%9)*0%<)+5%
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 10000 100000
Prob
ability
Advective-‐only (no dispersion/diffusion) Travel Times (yr)n = 0.16; b = 4m
Multiple initial seed fieldsNSMC with 5 calibrated models (Phi)NSMC with 5 calibrated models (Travel Time)NSMC with mean field
Comparison of travel /mes for different methods shows the effec/veness of the NSMC method
Effects of high T channel
~ 20x speed up
~ 80x speed up
40 RFs from each calibrated model
100 Selected fields
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Te has a similar distribu/on for both fields, but S.D. of Te distribu/on is quite different
100 selected Fields MSP (Multiple Starting Point)
100 selected Fields (NSMC method with the mean field)
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Te distribu/on along three transects capture calibrated Te trends
Calibrated model with mean field NSMC 100 selected fields
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Change Trunca/on Threshold?
!"#!!"$!!!"$#!!"%!!!"%#!!"&!!!"&#!!"'!!!"'#!!"#!!!"
!(!!" !(%#" !(#!" !()#" $(!!"
!"#$%&&'(%
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!(!!" !(#)" !()!" !(*)" '(!!"
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&)*$%+,
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'(!+,!$"
'(!+,!)"
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("!#$!)%
("&#$!)%
*"!#$!)%
!"!!% !"*&% !"&!% !"+&% ("!!%
!"#$%&'!()
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.$%"#&&'/#&(0"#123'*45(-'/67'42,(123'
("!#,!)%
("!#,!&%
!"!#$!!%&"!#$!'%("!#$!'%)"!#$!'%'"!#$!'%*"!#$!'%+"!#$!'%,"!#$!'%-"!#$!'%
!"!!% !"(*% !"*!% !",*% &"!!%
!"#$%&'!()
%'*+%#",-'
.$%"#&&'!"#$%&'!()%'/01'23,(435'
&"!#.!'%
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Decrease threshold from 1.0E-‐04 to 1.0E-‐05, ~70% increase in number of super parameters
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Conclusions n Can NSMC approach approximate ensemble predic/ons obtained with MSP runs? n Yes, but the calibra/on constraint will bias es/mates and predic/ons to values proximal to the ini/al calibra/on
n Given a set of previously run models, what is an effec/ve means of expanding the predic/ve ensemble? n Select final ensemble from larger set of NSMC realiza/ons using calibra/on quality and non-‐uniform sampling from NSMC realiza/ons
n Without exis/ng calibra/ons, can a mean-‐field representa/on serve as an ini/al star/ng point? n Yes, NSMC realiza/ons provide good match to observa/ons and reasonable approxima/on of MSP predic/ve distribu/on