Uncertainty analysis in groundwater modelling: An integrated
approach to account for conceptual model uncertainty
Rodrigo Rojas Mujica
Department of Earth and Environmental Sciences.
Katholieke Universiteit Leuven.
PhD public defence July 3, 2009
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Outline
1
Introduction
2
Objectives
3
Integrated uncertainty assessment approach
4
Value of conditioning data
5
Applications
Case I: Walenbos - Belgium
Case II: Pampa del Tamarugal - Chile
6
Conclusions
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
2 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Outline
1
Introduction
2
Objectives
3
Integrated uncertainty assessment approach
4
Value of conditioning data
5
Applications
Case I: Walenbos - Belgium
Case II: Pampa del Tamarugal - Chile
6
Conclusions
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
3 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
P
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
P
E description of EVT
process?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
P
E description of EVT
process?
SW-GW
interaction?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
Drainage
(Drains?, Rivers?)
P
E description of EVT
process?
SW-GW
interaction?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
Drainage
(Drains?, Rivers?)
P
E description of EVT
process?
R
(Point?, spatially distrib?,
Lateral flow?)
SW-GW
interaction?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
Drainage
(Drains?, Rivers?)
Q inflow? boundary
condition?
P
E description of EVT
process?
R
(Point?, spatially distrib?,
Lateral flow?)
SW-GW
interaction?
Q outflow? Boundary
condition?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
Drainage
(Drains?, Rivers?)
Q inflow? boundary
condition?
K (heterogeneous?,
Zonation?, homogeneous?)
Nr. Layers?
P
E description of EVT
process?
R
(Point?, spatially distrib?,
Lateral flow?)
SW-GW
interaction?
Q outflow? Boundary
condition?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model
Drainage
(Drains?, Rivers?)
P
E description of EVT
process?
Q inflow? boundary
condition?
K (heterogeneous?,
Zonation?, homogeneous?)
Nr. Layers?
R
(Point?, spatially distrib?,
Lateral flow?)
SW-GW
interaction?
Q outflow? Boundary
condition?
TOO MANY QUESTIONS!
⇒
LIMITED DATA AVAILABILITY
INCOMPLETE & FALSE? IDEA OF REALITY
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
4 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model uncertainty
Processes
Structure
• Geological description
• Kh/Kv zones
• Boundary conditions
• Initial conditions
• Data interpretation
• Implementation errors
• ...
• Steady – transient
• GW/SW interaction
• Boundary conditions
• Initial conditions
• Mathematical formulation
• ...
Parameters
Inputs
• Kh / Kv
• Ss / Sy
•…
• GW Recharge
• Rainfall
• EVT
•…
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conceptual model uncertainty
RAL
/ST
NTY
UAL ERTAI
T
P
NCE EL UNC
COStructure
M OD
U
UCT
RProcesses
• Geological description
• Kh/Kv zones
• Boundary conditions
• Initial conditions
• Data interpretation
• Implementation errors
• ...
Inputs
• Steady – transient
• GW/SW interaction
• Boundary conditions
• Initial conditions
• Mathematical formulation
• ...
Parameters
• Kh / Kv
• Ss / Sy
•…
• GW Recharge
• Rainfall
• EVT
•…
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
5 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Why is the assessment of conceptual model
uncertainty so important?
1
Uncertainty estimations based on a SINGLE conceptual model ⇒
likely to bias and “artificially” more conservative (under-dispersive).
2
For predictions beyond dataset used as calibration targets ⇒
conceptual model uncertainty >>> parametric uncertainty.
3
Uniqueness of sites vs. non-uniqueness of models ⇒ many models
will yield acceptable results.
4
Added-value to the making decision process.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
6 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Outline
1
Introduction
2
Objectives
3
Integrated uncertainty assessment approach
4
Value of conditioning data
5
Applications
Case I: Walenbos - Belgium
Case II: Pampa del Tamarugal - Chile
6
Conclusions
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
7 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Objectives
General objective
Develop an uncertainty assessment approach to explicitly account
for conceptual model uncertainty in groundwater modelling
applications
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
8 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Objectives
Specific objectives
1
Quantify conceptual model uncertainty for a hypothetical
groundwater system.
2
Assess the value of prior knowledge about the alternative conceptual
models.
3
Assess the value of conditioning data to reduce conceptual model
uncertainty.
4
Apply the proposed methodology to real aquifer systems.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
9 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Outline
1
Introduction
2
Objectives
3
Integrated uncertainty assessment approach
4
Value of conditioning data
5
Applications
Case I: Walenbos - Belgium
Case II: Pampa del Tamarugal - Chile
6
Conclusions
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
10 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Generalized Likelihood Uncertainty Estimation
(GLUE) (Beven and Binley, 1992)
Steps to implement GLUE
1
Sample sets of parameters.
2
Assessment of simulators (conceptual model + parameter set) using
a likelihood function.
3
Select “behavioural” simulators based on a rejection criterion.
4
Rescale likelihood values for behavioural simulators such that they
sum up to 1.
5
Build cumulative predictive distributions.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Bayesian Model Averaging (BMA)
(Draper, 1995; Hoeting et al., 1999)
BMA provides a coherent framework to combine predictions from
multiple competing conceptual models based on their relative skill
to reproduce a given observed dataset D. BMA avoids having to
choose a conceptual model over the others, instead competing
models are assigned different weights for multi-model aggregation
(Wasserman, 2000).
Fundamental equation of BMA
p(∆|D) =
K
X
p(∆|D, Mk )p(Mk |D)
(1)
k=1
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Integrated uncertainty assessment approach
M1
M2
…
Mk
…
MK
Sample parameters
Assessment (GLUE-based likel.)
Define “behavioural” set
for each model
GLUE-based
model likelihoods
Rescale likelihoods &
build CPD for each model
GLUE-based
model weights
Multi-model aggregation using
BMA
Quantification
conceptual model uncertainty
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Hypothetical aquifer system
specified flux = 0
Y
3000
Bounded by constant
head and river
Evapotranspiration zone
5 pumping wells
(2500 m3 d −1 )
16 observation wells
Well-3
Obs-5
Obs-9
Obs-13
Well-1
Well-4
Obs-2
Obs-3
Obs-6
Obs-7
Well-5
Obs-10
Obs-14
Obs-11
Obs-15
river
Heterogeneous at grid
cell level (25 m × 25 m)
Obs-1
constant head = 46 m
3-dimensional aquifer
system
Well-2
Obs-4
0
0
0
700
1400
Obs-8
2100
Obs-12
2800
3500
Obs-16
4200
specified flux = 0
5000
5000 X
Z
50
Layer 1
Layer 2
Layer 3
15
10
-10
0
Evapotranspiration surface
5000 X
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Implementing the GLUE-BMA approach
1
Ensemble M of 7 alternative conceptual models (from one-layer to
three-layers models).
Conceptual
model
Spatial correlation
structure
Description
1Lhtg-L1
1Lhtg-L2
1Lhtg-L3
1Lhtg-AVG
2Lhtg
2LhtgQ3D
3Lhtg
Layer 1
Layer 2
Layer 3
Average of layers 1, 2 and 3
Layer 1 and 3
Layer 1 and 3
Layer 1, 2 and 3
One-layer model
One-layer model
One-layer model
One-layer model
Two-layer model not considering the aquitard
Two-layer model implicitly accounting for the aquitard
Three-layer model explicitly accounting for the aquitard
2
Prior ranges of parameters (RECH, EVT, RIVC, CH).
3
3 likelihood functions (Gaussian, Model Efficiency, and Triangular)
and rejection criterion of ± 5 m departure from observed heads.
4
Latin Hypercube Sampling (LHS) (McKay et al., 1979) of
parameters and inputs.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results Hypothetical aquifer
Summary of posterior model probabilities (model weights) for the seven
alternative conceptual models
Likelihood
p(Mk )
p(Mk |D)
Conceptual model
function
1Lhtg-L1
1Lhtg-L2
1Lhtg-L3
1Lhtg-AVG
2Lhtg
2LQ3Dhtg
3Lhtg
Total
GAUSS
TRIANG
MODEFF
(1/7)
0
0
0
(1/7)
0
0
0
(1/7)
0.1822
0.1771
0.1822
(1/7)
0.1888
0.1861
0.1896
(1/7)
0.1999
0.2018
0.2014
(1/7)
0.2113
0.2167
0.2126
(1/7)
0.2178
0.2184
0.2141
1.0
1.0
1.0
1.0
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results Hypothetical aquifer (cont.)
0.4
0.2
a
0
1000 2000 3000 4000
0.6
0.4
0.2
0
b
0
2500
5000
WBC inflow [m3 d-1]
Recharge inflow [m3 d-1]
Cumulative probability
0
1
0.8
1
0.8
0.6
0.4
0.2
0
d
0
500
1000
1Lhtg-AVG
1500
River gains [m d ]
3
1Lhtg-L3
-1
2Lhtg
Cumulative probability
0.6
Cumulative probability
1
0.8
Cumulative probability
Cumulative probability
Cumulative probability distributions and BMA
1
0.8
0.6
0.4
0.2
0
c
0
400
800
1200
WBC outflow [m3 d-1]
1
0.8
0.6
0.4
0.2
0
e
0
1000
2000
3000
EVT outflow [m d ]
3
2LQ3Dhtg
-1
3Lhtg
BMA
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results Hypothetical aquifer (cont.)
Predictive variance
100%
80%
60%
GAUSS
MODEFF
TRIANG
GAUSS
MODEFF
TRIANG
GAUSS
MODEFF
TRIANG
0%
GAUSS
MODEFF
TRIANG
20%
GAUSS
MODEFF
TRIANG
40%
WBC Recharge WBC River
EVT
inflow inflow outflow gains outflow
Within-model variance
Between-model variance
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Research questions . . .
1
Value of prior knowledge p(Mk ) to reduce predictive
uncertainty and the contribution of conceptual model
uncertainty?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Value of p(Mk ) to reduce uncertainty
Optimization problem
1
Postulate three sets of prior knowledge:
Prior Set 1 ⇒ total ignorance
Prior Set 2 ⇒ proper knowledge
Prior Set 3 ⇒ improper knowledge
2
Constrained maximum entropy ⇒ optimal values of p(Mk )
according to quantitative relationships among alternative conceptual
models.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results value of prior knowledge p(Mk )
Predictive variance and between-model variance as a function of the
optimized prior model probability sets
Prior Set 1
Prior Set 2
Prior Set 3
GW budget components
Predictive
variance
Between-model
variance
Predictive
variance
Between-model
variance
Predictive
variance
Between-model
variance
WBC inflow
463319.2
46584.7
Recharge inflow
516007.8
86870.7
WBC outflow
7624.7
342.8.
River gains
33893.9
8951.4
EVT outflows
158321.9
23414.6
280622.5
(39.4%)
312683.0
(39.4%)
3016.7
(60.4%)
19218.0
(43.3%)
82788.2
(47.7%)
17666.0
(62.3%)
43341.4
(50.1%)
173.4
(49.4%)
3871.3
(56.8%)
11495.5
(50.9%)
667218.6
(−44.0%)
681708.6
(−32.1%)
12172.6
(−59.6%)
48020.7
(−41.7%)
235107.8
(−48.5%)
65680.3
(−40.2)
79460.8
(8.5%)
319.1
(6.9%)
10321.4
(−15.3%)
22315.2
(4.7%)
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Research questions . . .
1
Value of conditioning data to reduce the predictive
variance and contribution of conceptual model
uncertainty?
2
Value of conditioning data to further constrain ensemble
M?
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
22 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Outline
1
Introduction
2
Objectives
3
Integrated uncertainty assessment approach
4
Value of conditioning data
5
Applications
Case I: Walenbos - Belgium
Case II: Pampa del Tamarugal - Chile
6
Conclusions
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
23 / 46
Hypothetical aquifer system
specified flux = 0
Y
Bounded by constant head and
river
3
W1b-(L3)W1a-(L1)
Obs-10
Obs-14
1
2
0
Obs-3
Obs-7
W5-(L3)
-1
-2
-3
lnK
Obs-6
Obs-13
Obs-9
W4-(L3)
-4
Obs-11
Obs-15
W2a-(L1)
W2b-(L3)
-5
-6
-7
Obs-4
Obs-8
Obs-12
Obs-16
-8
-9
0
0
700
1400
0
2100
2800
3500
4200
5000
5000 X
specified flux = 0
Z
b
Evapotranspiration zone
Layer 1
15
10
Layer 2
Layer 3
-10
5000 X
0
specified flux = 0
Y
3000
constant head = 46 m
c
5
Obs-1
W3a-(L3)
Obs-5
Obs-2
Obs-6
4
Obs-9
Obs-13
W1b-(L3)
Obs-10
Obs-14
Obs-11
Obs-15
specified flux = 0
32 observation wells ⇒
accounting for lowermost
aquifer
4
W3a-(L3)
Obs-5
W3b-(L1)
Obs-2
50
8 pumping wells
(2750 m3 d −1 ) (top and
bottom aquifers)
5
Obs-1
W4-(L3)
Obs-3
Obs-7
W5-(L3)
W2b-(L3)
Obs-4
Obs-8
Obs-12
3
2
1
0
-1
-2
-3
-4
Obs-16
-5
0
0
0
700
1400
2100
2800
specified flux = 0
Evapotranspiration surface
3500
4200
5000
5000 X
lnK
Heterogeneous at grid cell level
(25 m × 25 m)
constant head = 46 m
3-dimensional aquifer system
a
river
3000
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Implementing the GLUE-BMA approach Conditioning
1
Ensemble M of 7 alternative conceptual models (from one-layer to
three-layers models).
2
Prior ranges of parameters and inputs for sampling.
3
Gaussian likelihood function.
4
Markov Chain Monte Carlo (MCMC) sampling (Gilks et al., 1995)
of parameters and inputs.
5
Conditioning on 3 types of data: hydraulic conductivity data (k),
groundwater flow (GWF) and river discharge (RIV) observations.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conditioning of the K-field
Four cases analysed:
Location sampling points
Y
3000
P24
Unconditional
2
Case Cond-I 10 K meas.
3
4
Case Cond-II 20 K meas.
Case Cond-III 40 K meas.
P28
P05
P22
P40
P31
P18
P07
P25
P06
P32
P38
P35
P23
P10
P14
P15
P36
700
P09
1400
2100
Case Conditional-I
Case Conditional-II
Case Conditional-III
P01
P19
P37
P21
0
P33 P03
P02
P04
P17
0
P13
P30 P29
P12
P34
P20
0
P26
P08
P39
river
1
constant head = 46 m
P11
2800
3500
P16
P27
4200
5000
5000 X
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results conditioning
Table: Summary of model likelihoods and model weights for the conditioning
cases
Conditional-III
Unconditional
Conceptual models
1L-L1
1L-L2
1L-L3
1L-AVG
2L
2LQ3D
3L
p(Mk )
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
Total
Heads
p(D|Mk )
0
(0)
0
(0)
746.8
(0.188)
765.0
(0.193)
802.9
(0.202)
804.9
(0.203)
852.1
(0.215)
3971.3
(1.0)
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
Heads
p(D|Mk )
0
(0)
0
(0)
981.2
(0.216)
0
0
1148.8
(0.253)
1206.2
(0.265)
1208.8
(0.266)
4545.0
(1.0)
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
1.0
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results conditioning
Table: Summary of model likelihoods and model weights for the conditioning
cases
Conditional-III
Unconditional
Conceptual models
1L-L1
1L-L2
1L-L3
1L-AVG
2L
2LQ3D
3L
p(Mk )
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
Total
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
0
(0)
0
(0)
0
(0)
0
(0)
746.8
(0.188)
620.0
(0.177)
765.0
(0.193)
590.3
(0.169)
802.9
(0.202)
739.5
(0.211)
804.9
(0.203)
751.2
(0.215)
852.1
(0.215)
799.0
(0.228)
3971.3
(1.0)
3500.0
(1.0)
Heads + GWF + RIV
p(D|Mk )
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
0
(0)
0
(0)
0
(0)
0
(0)
981.2
(0.216)
781.5
(0.198)
0
(0)
0
(0)
1148.8
(0.253)
1028.9
(0.261)
1206.2
(0.265)
1063.7
(0.270)
1208.8
(0.266)
1065.2
(0.270)
4545.0
(1.0)
3939.2
(1.0)
Heads + GWF + RIV
p(D|Mk )
1.0
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results conditioning
Table: Summary of model likelihoods and model weights for the conditioning
cases
Conditional-III
Unconditional
Conceptual models
1L-L1
1L-L2
1L-L3
1L-AVG
2L
2LQ3D
3L
p(Mk )
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
Total
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
746.8
(0.188)
620.0
(0.177)
592.5
(0.182)
765.0
(0.193)
590.3
(0.169)
507.5
(0.156)
802.9
(0.202)
739.5
(0.211)
680.9
(0.209)
804.9
(0.203)
751.2
(0.215)
696.0
(0.213)
852.1
(0.215)
799.0
(0.228)
783.7
(0.240)
3971.3
(1.0)
3500.0
(1.0)
3260.5
(1.0)
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
981.2
(0.216)
781.5
(0.198)
595.4
(0.177)
0
(0)
0
(0)
0
(0)
1148.8
(0.253)
1028.9
(0.261)
823.8
(0.245)
1206.2
(0.265)
1063.7
(0.270)
959.2
(0.286)
1208.8
(0.266)
1065.2
(0.270)
979.7
(0.292)
4545.0
(1.0)
3939.2
(1.0)
3358.2
(1.0)
1.0
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
27 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results conditioning
Table: Summary of model likelihoods and model weights for the conditioning
cases
Conditional-III
Unconditional
Conceptual models
1L-L1
1L-L2
1L-L3
1L-AVG
2L
2LQ3D
3L
p(Mk )
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
Total
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
746.8
(0.188)
620.0
(0.177)
592.5
(0.182)
765.0
(0.193)
590.3
(0.169)
507.5
(0.156)
802.9
(0.202)
739.5
(0.211)
680.9
(0.209)
804.9
(0.203)
751.2
(0.215)
696.0
(0.213)
852.1
(0.215)
799.0
(0.228)
783.7
(0.240)
3971.3
(1.0)
3500.0
(1.0)
3260.5
(1.0)
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
981.2
(0.216)
781.5
(0.198)
595.4
(0.177)
0
(0)
0
(0)
0
(0)
1148.8
(0.253)
1028.9
(0.261)
823.8
(0.245)
1206.2
(0.265)
1063.7
(0.270)
959.2
(0.286)
1208.8
(0.266)
1065.2
(0.270)
979.7
(0.292)
4545.0
(1.0)
3939.2
(1.0)
3358.2
(1.0)
1.0
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
27 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results conditioning
Table: Summary of model likelihoods and model weights for the conditioning
cases
Conditional-III
Unconditional
Conceptual models
1L-L1
1L-L2
1L-L3
1L-AVG
2L
2LQ3D
3L
p(Mk )
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
(1/7)
Total
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
0
0
0
0
0
0
0
0
0
0
0
0
746.8
(0.188)
620.0
(0.177)
592.5
(0.182)
765.0
(0.193)
590.3
(0.169)
507.5
(0.156)
802.9
(0.202)
739.5
(0.211)
680.9
(0.209)
804.9
(0.203)
751.2
(0.215)
696.0
(0.213)
852.1
(0.215)
799.0
(0.228)
783.7
(0.240)
3971.3
(1.0)
3500.0
(1.0)
3260.5
(1.0)
Heads
p(D|Mk )
Heads + GWF
p(D|Mk )
Heads + GWF + RIV
p(D|Mk )
0
0
0
0
0
0
0
0
0
0
0
0
981.2
(0.216)
781.5
(0.198)
595.4
(0.177)
0
0
0
0
0
0
1148.8
(0.253)
1028.9
(0.261)
823.8
(0.245)
1206.2
(0.265)
1063.7
(0.270)
959.2
(0.286)
1208.8
(0.266)
1065.2
(0.270)
979.7
(0.292)
4545.0
(1.0)
3939.2
(1.0)
3358.2
(1.0)
1.0
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
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Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results conditioning
Predictive variance and conditioning
Conditional-II
EVT outflows
Variance [m3 d-1]2
40
20
0
Variance [m3 d-1]2
4
80 x10
60
a
Conditional-III
4
Heads
Heads+GWF
Heads+GWF+RIV
60
40
20
0
b
Heads
Heads+GWF
Heads+GWF+RIV
4
25 x10
4
25 x10
20
20
Variance [m3 d-1]2
Variance [m3 d-1]2
Recharge inflows
80 x10
15
10
5
0
c
Heads
Heads+GWF
Heads+GWF+RIV
Within-model variance
15
10
5
0
d
Heads
Heads+GWF
Heads+GWF+RIV
Between-model variance
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
28 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Outline
1
Introduction
2
Objectives
3
Integrated uncertainty assessment approach
4
Value of conditioning data
5
Applications
Case I: Walenbos - Belgium
Case II: Pampa del Tamarugal - Chile
6
Conclusions
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
29 / 46
North [m]
168000 170000 172000 174000 176000 178000 180000 182000 184000 186000 188000
Case I - Study area
Demer
River
Motte
River
LEGEND
Molenbeek
River
Rivers and Streams
Walenbos Nature Reserve
Observation Wells
0
2000
4000
6000 m
Velp
River
181000
184000
187000
East [m]
190000
Figure: Local aquifer underlying Walenbos Nature Reserve - Belgium
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Implementing the GLUE-BMA approach in
Walenbos
Ensemble M of 3 conceptual models.
Extension of BMA to account for scenario uncertainties.
3 scenarios ⇒ average recharge ± 2σR̄ .
Dataset D 51 obs. heads.
MCMC sampling of K, river and drain conductances.
Calibration using UCODE-2005 (Poeter et al., 2005) to obtain
calibrated parameters and model selection criteria (AIC, AICc, BIC
and KIC) used to approximate model weights (Ye et al., 2008).
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
31 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results case I - Walenbos
Cumulative predictive distributions for
groundwater flow components
alternative conceptual models
1
a
0.6
0.4
0.2
x103
0
0
1
2
3
4
2210.5
(1/3)
0.355
1966.5
(1/3)
0.315
M3
2058.1
(1/3)
0.330
1
d
0.4
0.2
x10
0
2
4
6
3
x103
2
4
6
8
0.8
0.8
0.4
0.2
0
1
e
0.2
x10
0
3
Velp gains [m d ]
-1
Drain outflows [m d ]
1
0.8
2
3
0.8
f
0.6
0.4
0.2
3
92 94 96 98 100 102
-1
1
Velp losses [m3 d-1]
0.4
8
x103
0
10
0.6
3
c
0.6
Demer gains [m3 d-1]
0.6
0
0.2
0
probability
p(D|Mk )
p(Mk )
p(Mk |D)
M2
0.4
0
probability
M1
probability
1
0.8
1
b
0.6
5
Demer losses [m3 d-1]
Conceptual models
0.8
probability
0.8
probability
posterior model probabilities for
1
probability
prior model probabilities, and
probability
Integrated model likelihoods,
x103
0
0
0.4
0.8
1.2
Walenbos outflows [m3 d-1]
M1
g
M2
0.6
M3
0.4
BMA scenario S2
0.2
x103
0
3
4
5
6
Walenbos inflows [m3 d-1]
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
32 / 46
Results case I - Walenbos
Variance contributions for groundwater flow
Summary of posterior model
components
probabilities for different model
GLUE- Variance [%]
BMA 0 20 40 60 80 100
selection criteria
Conceptual models
M1
M2
M3
p(Mk )
(1/3)
(1/3)
(1/3)
p(D|Mk )
Rank
p(Mk |D)
2210.5
1
0.355
1966.5
3
0.315
2058.1
2
0.330
AIC
Rank
p(Mk |D)
74.59
1
0.596
78.93
3
0.068
75.73
2
0.337
AICc
Rank
p(Mk |D)
75.92
1
0.845
81.54
3
0.051
80.12
2
0.104
BIC
Rank
p(Mk |D)
84.25
1
0.972
92.46
2
0.016
93.11
3
0.012
KIC
Rank
p(Mk |D)
−5.99
3
0.085
−6.68
2
0.119
−10.48
1
0.796
Demer (L)
Demer (G)
Velp (L)
Velp (G)
Drain (O)
Walenbos (O)
Walenbos (I)
26
KIC
Demer (L)
Demer (G)
Velp (L)
Velp (G)
Drain (O)
Walenbos (O)
Walenbos (I)
25
67
44
5
1
22
75
78
41
20
BIC 0
55
76
76
8 16
76
12
3
8 1
4
20
20
62
28
43
1
60
20 40 60 80 100
12
47
3
85
32
11
20
21
60
5
14
8
61
6
32
35
3
6
66
28
47
20 40 60 80 100
28
76
18
9
28
55
5 17
37
0 20 40 60 80 100
5
66
AICc 0
Demer (L)
Demer (G)
Velp (L)
Velp (G)
Drain (O)
Walenbos (O)
Walenbos (I)
69
Variance [%]
AIC
91
52
7 3
25
26
69
36
12
90
0 20 40 60 80 100
36
52
15
22
4
57
74
36
10
54
39
18 8
43
5
62
63
2
Within models &
within scenarios
Between models &
between scenarios
Between scenarios
Towns/well-fields
Vegetation
112
0
Obs. wells
North [m]
7760000
I
II
III
V
PTA
a
0
50000
7700000
VI
VII
3500 masl
100000
Canchones
Pica
Salar de
Pintados
7720000
IV
IQUIQUE
El Carmelo
Pozo Almonte
7740000
Pacific Ocean
Pampa del Tamarugal Basin
La Noria aquifer
Huara
7780000
ARICA
Pampa del Tamarugal
Aquifer (PTA)
Dolores
7800000
Pacific Ocean
Chile
7820000
Case II - Study area
150000 m
Pintados
Oficina Victoria
b
Salar de
Bellavista
400000
Cerro Gordo
420000
440000
East [m]
geological
section
460000
Figure: Location regional aquifer Pampa del Tamarugal - Chile
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Implementing the GLUE-BMA approach in
Pampa del Tamarugal
Ensemble M of 8 conceptualizations covering major features of
groundwater models independently developed in past studies.
Two recharge mechanisms ⇒
Recharge originating from eastern sub-basins (likely)
Recharge from eastern sub-basins + recharge from deep fissures in
basement rocks (unlikely?)
Dataset D 42 obs. heads.
MCMC sampling of parameters evaporation, transpiration,
constant-head elevations, recharge rates (two mechanisms),
connection local aquifer.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
35 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results case II - Pampa del Tamarugal
Integrated model likelihoods and posterior model probabilities for the alternative
conceptual models
Conceptual models
p(D|Mk )
p(Mk )
p(Mk |D)
M1a
M1b
M2a
M2b
M3a
M3b
M4a
M4b
597.3
(1/8)
0.1088
590.9
(1/8)
0.1076
620.4
(1/8)
0.1130
656.5
(1/8)
0.1196
741.6
(1/8)
0.1351
726.5
(1/8)
0.1323
759.3
(1/8)
0.1382
797.4
(1/8)
0.1452
Total
5498.84
1.0
1.0
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
36 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Results case II - Pampa del Tamarugal
Predictive variance for groundwater flow components
% variance
0
20
40
60
80
100
Recharge inflows
21
79
Evaporation outflows
81
19
Transpiration outflows
71
29
Rech. Chacarilla sub-basin
24
76
Outflows to La Noria aq.
84
16
Recharge deep fissures
30
70
GW outflows Cerro Gordo
45
55
within-models
between-models
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
37 / 46
Results case II - Pampa del Tamarugal
37 15 10 10 15 18 12 12 44 38 24 45 12
Observation wells
100
58 60
42 42 20
6
33
9
64 18
7
40
7780000
P4
P5
P8
162
P6
D-60
P7
133
P9
P10
237
P15
P14
P13
P11
P10
P7
P4
P12
between-models
P12
276
P14
P13
281
286
290
Synthetic piezometers
within-models
P11 315
263
58 58 80 94 67 91 36 82 93
P3
b
0
42 40
P2
20
P3
7720000
60
P1
% variance
80
Synthetic
piezometer
P2
North [m]
7760000
133
A-13
315
D-60
294
C-30
290
286
281
276
263
a
237
0
63 85 90 90 85 82 88 88 56 62 76 55 88
Observation
wells
P1
C-30
7740000
40
20
A13
7800000
60
162
% variance
80
7700000
100
7820000
Predictive variance for groundwater head estimations
400000
294
420000
440000
East [m]
P15
460000
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Outline
1
Introduction
2
Objectives
3
Integrated uncertainty assessment approach
4
Value of conditioning data
5
Applications
Case I: Walenbos - Belgium
Case II: Pampa del Tamarugal - Chile
6
Conclusions
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
39 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conclusions
1
GLUE-BMA ⇒
General
Results conditional on ensemble M.
No restrictions in number and nature of conceptual models.
No limitations in the shape of prior distributions.
Flexible
Model performance can be expressed through alternative functions.
Bayesian nature allows easy updating when new information
available.
Allows for conditioning based on different types of data.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
40 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conclusions
2
Hypothetical aquifer application ⇒
Limited influence of selection of alternative likelihood functions.
Despite similar model weights ⇒ predictive distributions varied
substantially in shape, central moment and spread.
Low information content of head measurements to discriminate
between conceptual models.
Despite known and controlled conditions ⇒ significant contribution
of conceptual model uncertainty (up to 30%).
3
Prior knowledge about conceptual models ⇒
Including proper prior knowledge ⇒ reduction predictive variance.
Increase in predictive variance ⇒ likely caused by improper prior
knowledge.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
41 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conclusions
4
Value of conditioning data ⇒
Low information content of 16 extra head measurements to
discriminate between models.
Drastically improved the performance of the GLUE-BMA approach
⇒ better definition of likelihood surfaces, increase in integrated
model likelihoods, better discrimination between conceptual models,
important reductions of predictive variances.
Complement information provided by head measurements ⇒
information content K, RIV, GWF.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
42 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conclusions
5
Application to Walenbos ⇒
Despite small differences in model weights ⇒ predictive distributions
varied drastically in shape, central moment and spread.
Critical differences between GLUE-BMA and criteria-based
multi-model methodologies ⇒ ranking of models, calculation of
model weights and uncertainty estimations.
Important contribution of conceptual model uncertainty to
predictive uncertainty (up to 75%), even for a case when a single
model was prefered (BIC) (up to 36%).
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
43 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Conclusions
6
Application to Pampa del Tamarugal ⇒
Low information content of head measurements to discriminate
between recharge mechanisms.
Apparent spatial relationship between uncertainty in head
estimations and areas affected by recharge fronts.
Significant contribution of conceptual model uncertainty to
predictive variances (up to 79%).
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
44 / 46
Intro
Objectives
Integrated uncertainty assessment
Value of conditioning data
Applications
Conclusions
Suggestions for future research
1
Methodology to efficiently define ensemble M.
2
Protocol to efficiently assign prior model probabilities (weights) ⇒
Combine expert-based knowledge and maximum entropy principle.
3
How to efficiently complement information content of head
observations with other sources of (soft) information (tracer test
data, environmental isotopes, geophysics, etc.)?
4
Improve on the sampling of the GLUE-BMA approach (SCEM-UA,
Hybrid Monte Carlo (HMC), etc.)
5
Extend the application of GLUE-BMA to the transient case.
Uncertainty analysis in groundwater modelling: An integrated approach to account for conceptual model uncertainty
45 / 46
Uncertainty analysis in groundwater modelling: An integrated
approach to account for conceptual model uncertainty
Rodrigo Rojas Mujica
Department of Earth and Environmental Sciences
Katholieke Universiteit Leuven
PhD public defence July 3, 2009