Regularised
Inversion
and
Model Predictive Uncertainty
Analysis
PEST …
Input files
Model
Output files
writes model input files
Input files
PEST
Model
Output files
reads model output files
writes model input files
Input files
PEST
Batch or Script File
Output files
reads model output files
PEST
Input files
Input files
Model
Model
calibration conditions
predictive conditions
Output files Output files
PEST
Input files
Input files
Model
Model
calibration conditions
predictive conditions
Output files Output files
Maximise or minimise
key prediction while
keeping model
calibrated
value
Field or laboratory measurements and model output:-
Model output
calibration dataset
q2
q1
q3
etc
distance or time
prediction
value
Field or laboratory measurements and model output:-
Model output
calibration dataset
q2
q1
q3
etc
distance or time
Lower
predictive
limit
value
Field or laboratory measurements and model output:-
Model output
calibration dataset
q2
q1
q3
etc
distance or time
Upper
predictive
limit
value
Field or laboratory measurements and model output:-
Model output
calibration dataset
q2
q1
q3
etc
distance or time
Confidence
interval for
prediction
value
Field or laboratory measurements and model output:-
Model output
calibration dataset
q2
q1
q3
etc
distance or time
Predictive
uncertainty
interval
Traditional Parameter Estimation
• Principal of parsimony
• Employ no more parameters than can be estimated
• Calibration complexity dictated by calibration dataset.
Regularised inversion…
Advantages of Regularised Inversion
• The inversion process is able to put the heterogeneity
exactly where it is needed
• Maximum information content is extracted from the data
• Predictive error variance is thus minimised
• Parameterisation complexity determined by prediction
• Because complexity is retained in the system, we have
the ability to realistic assess predictive uncertainty
because we do not exclude the detail on which a
prediction can depend.
Two Principal Types of Regularisatoin
• “Tikhonov” – constrained minimisation
• Subspace methods – principal component analysis
SVD-Assist
Advantages
• Highly stable numerically.
• Highly efficient in model run requirements.
• Can adapt to noise content of data.
Hydraulic conductivity
Specific Yield
771
18
16
14
12
Feet
10
Measured
8
Modelled
6
4
2
0
-2
1
15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225
Points
Water levels
942
20
10
0
1
13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193
-10
Feet
Measured
Modelled
-20
-30
-40
-50
Points
Water levels
4609
20
15
10
Feet
Measured
Modelled
5
0
1
9
17 25 33 41 49 57 65 73 81 89 97 105 113 121 129
-5
Points
Water levels
3752
25
20
Feet
15
Measured
10
Modelled
5
0
1
12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177
-5
Points
Water levels
devilswb
Points
-1800000
-1600000
-1400000
Feet
-1200000
-1000000
Measured
Modelled
-800000
-600000
-400000
-200000
1
13 25 37 49 61 73 85 97 109 121 133 145 157 169 181
0
Snake River Inflow
crystal
Points
-5.00E+07
-4.50E+07
-4.00E+07
-3.50E+07
Feet
-3.00E+07
Measured
-2.50E+07
Modelled
-2.00E+07
-1.50E+07
-1.00E+07
-5.00E+06
1
14 27 40 53 66 79 92 105 118 131 144 157 170 183 196
0.00E+00
Snake River Inflow
Local Domain and Air Photo
Source
area
Recovery
Well
MTBE concentrations for an
elevation of:-
–35 ft-msl to –40 ft-msl
0 ft 200 ft 400 ft 600 ft 800 ft
Pilot Points and Observations
Source
area
Recovery
Well
• Pilot points – 58 per
layer, L1-L7, for HHK,
VHK, POR (crosses).
• Water level
observations (circles);
MTBE observations
(stars)
• Calibrated ‘mean’
particle.
Example Section Profile
-40
-60
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Profile across plume at IRM transect
190
200
Typical Concentration Profile
20.0
Figure 4
Elevation (feet MSL)
0.0
-20.0
-40.0
-60.0
-80.0
0
2,000
4,000
6,000
MTBE Concentration (ppb)
8,000
Observed MTBE
Modelled MTBE
-35 to -40 ft msl
0 ft 200 ft 400 ft 600 ft 800 ft
0 ft 200 ft 400 ft 600 ft 800 ft
20.0
Elevation (feet MSL)
0.0
-20.0
-40.0
-60.0
-80.0
1
10
100
1,000
MTBE Concentration (ppb)
10,000
20.0
Elevation (feet MSL)
0.0
-20.0
-40.0
-60.0
-80.0
1
100
10,000
MTBE Concentration (ppb)
1,000,000
20.0
Elevation (feet MSL)
0.0
-20.0
-40.0
-60.0
-80.0
1
10
100
1,000
MTBE Concentration (ppb)
10,000
20.0
Elevation (feet MSL)
0.0
-20.0
-40.0
-60.0
-80.0
1
10
100
1,000 10,000
MTBE Concentration (ppb)
Profile - data
Source Area
FLOW
20.00
10.00
0.00
-10.00
-20.00
-30.00
-40.00
-50.00
-60.00
-70.00
-80.00
-90.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
0.00
-500.00
Profile – data and modelled concentrations
Source Area
FLOW
20.00
10.00
0.00
-10.00
-20.00
-30.00
-40.00
-50.00
-60.00
-70.00
-80.00
-90.00
3000.00
2500.00
2000.00
1500.00
1000.00
500.00
0.00
-500.00
Simulated and Observed MTBE at the Recovery Well
5100
Observed
4600
Simulated
4100
MTBE (ppm)
3600
3100
2600
2100
1600
1100
600
100
0
20
40
60
80
100
Time Since System Start-up (Days)
120
140
Calibrated Horizontal and Vertical Hydraulic Conductivities
Ground
Water
Flow
The cost of uniqueness …..
Model grid
Dimensions of model domain
500m by 800m
Boundary
Q = 50 m3/day
H = 0.0
Particle release point
Reality
Reality
True time = 3256.24 days
True exit point = easting of 206.78
12 head observations
Reality
Exit time = 3256
Exit point = 206
Calibration to 12 observations (no noise)
Exit time = 7122 [true=3256]
Exit point = 241 [true=206]
Reality
12 obs
This model (with its three parameters)…
Calibration to 12 observations
Zone-based calibration
Exit time = 6364 [true=3256]
Exit point = 244 [true=206]
… does not even acknowledge the detail upon
which a critical prediction will depend,
whereas this model ….
Calibration to 12 observations (no noise)
Exit time = 7122 [true=3256]
Exit point = 241 [true=206]
Reality
12 obs
Another important point…
… does .
The former model will grossly under-estimate
predictive variance.
Calculation of Model Predictive Error Variance…..
Parameter space
Increasing number of parameter combinations
Estimable
parameter
combinations
Increasing number of parameter combinations
Unestimable
parameter
combinations
Error variance
calculable from
measurement
error C(h)
Increasing number of parameter combinations
Error variance
supplied by
hydrogeologists
C(p)
Error variance
calculable from
measurement
error C(h)
Error variance
supplied by
hydrogeologists
C(p)
model
prediction
Therefore total “possible model error” depends on
both C(h) and C(p)
σ2 = yt(I-R)tC(p)(I-R)y + ytGC(h)Gy
Error variance
calculable from
measurement
error C(h)
Error variance
supplied by
hydrogeologists
C(p)
model
prediction
Where do we draw the line on what
we try to estimate?
Error variance
calculable from
measurement
error C(h)
Error variance
supplied by
hydrogeologists
C(p)
model
prediction
Predictive error variance vs dimensions of calibrated
parameter space
Predictive error variance
Total
“Measurement” term
“Null space” term
Number of singular values
Optimising Data Acquistion…..
Schematic block diagram illustrating model layers and boundary conditions
The prediction
Pumping from layer 3 - 2050
Measurements
Observation wells
Layer 1
Layer 2
Layer 3
Water levels
Parameters
Parameters included in analysis
Hydraulic conductivity – layer 1
Hydraulic conductivity – layer 2
Hydraulic conductivity – layer 3
VCONT – layer 2
VCONT – layer 3
Specific yield – layer 1
Specific yield – layer 2
Primary storage capacity – layer 2
Primary storage capacity – layer 3
Riverbed conductance
Recharge
Pre-calibration contribution to predictive error variance
400
350
300
250
200
150
100
CR
Recharge
VCONT2
VCONT1
Sc3
Sc2
Sy3
Sy2
Sy1
K3
K2
0
K1
50
Predictive error variance vs dimensions of
calibrated parameter space
Varance of predictive error (ft^2)
1200
1000
800
Minimum = 418 ft2
at 160 singular values
600
400
200
0
1
10
Number of singular values
100
Contribution to pre- and post-calibration predictive
variance by selected parameter types
400
300
200
Pre-cal
100
Post-cal
0
K3
VCONT1
VCONT2
Optimization of data acquisition:How can I deepen the minimum in the predictive
variance curve?
σ2 = yt(I-R)tC(p)(I-R)y + ytGC(h)Gy
Reduction in predictive variance if VCONT2 characterization at each
point is reduced from 0.74 to 0.37 (maximum reduction = 112.7ft2)
Locations of proposed layer 2-3 differential head measurements
(reduction in predictive error variance = 230 ft2)
400
350
300
250
200
150
100
50
0
NT
2
VC
O
NT
1
VC
O
K3
Pre-cal
Post-cal
Geophysics
Extra_wells
Predictive error variance vs dimensions of
calibrated parameter space
Varance of predictive error (ft^2)
1200
Previous minimum = 418 ft2
at 160 singular values
1000
800
600
400
New minimum = 188 ft2
at 190 singular values
200
0
10
100
Number of singular values
Error variance of an existing model…..
IBOUND array
Riverbed K parameters
Log of K
(K ranges from 1e-4 to 500)
All lateral Inflow Zones
(red cells are fixed head – except for zone 1)
Management zones
19
5
6
9
8
10
3
11
7
16
12
2
13
15
4
14
Number of cells
Head error variance