Regularised Inversion Model Predictive Uncertainty Analysis

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Regularised Inversion Model Predictive Uncertainty Analysis

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


Input files

Input files

PEST

Model

calibration conditions

Model

predictive conditions

Output files

Output files


Input files

Input files

PEST

Model

calibration conditions

Model

predictive conditions

Output files

Maximise or minimise

key prediction while

keeping model

calibrated

Output files


value

Field or laboratory measurements and model output:-

Model output

calibration dataset

prediction

q 1

q 2

q 3

etc

distance or time


value

Field or laboratory measurements and model output:-

Model output

calibration dataset

Lower

predictive

limit

q 1

q 2

q 3

etc

distance or time


value

Field or laboratory measurements and model output:-

Model output

calibration dataset

Upper

predictive

limit

q 1

q 2

q 3

etc

distance or time


value

Field or laboratory measurements and model output:-

Model output

calibration dataset

Confidence

interval for

prediction

q 1

q 2

q 3

etc

distance or time


value

Field or laboratory measurements and model output:-

Model output

calibration dataset

Predictive

uncertainty

interval

q 1

q 2

q 3

etc

distance or time


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


Feet

771

18

16

14

12

10

8

6

Measured

Modelled

4

2

0

-2

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225

Points

Water levels


Feet

942

20

10

0

1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193

-10

-20

Measured

Modelled

-30

-40

-50

Points

Water levels


Feet

4609

20

15

10

5

Measured

Modelled

0

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129

-5

Points

Water levels


Feet

3752

25

20

15

10

Measured

Modelled

5

0

-5

1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177

Points

Water levels


Feet

devilswb

-1800000

Points

-1600000

-1400000

-1200000

-1000000

-800000

Measured

Modelled

-600000

-400000

-200000

0

1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181

Snake River Inflow


Feet

crystal

-5.00E+07

Points

-4.50E+07

-4.00E+07

-3.50E+07

-3.00E+07

-2.50E+07

-2.00E+07

Measured

Modelled

-1.50E+07

-1.00E+07

-5.00E+06

0.00E+00

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196

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

Recovery

Well

Source

area

• 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 90 100 110 120 130 140 150 160 170 180 190 200

Profile across plume at IRM transect


Elevation (feet MSL)

Typical Concentration Profile

20.0

Figure 4

0.0

-20.0

-40.0

-60.0

-80.0

0 2,000 4,000 6,000 8,000

MTBE Concentration (ppb)


Observed MTBE

-35 to -40 ft msl

Modelled MTBE

0 ft 200 ft 400 ft 600 ft 800 ft

0 ft 200 ft 400 ft 600 ft 800 ft


Elevation (feet MSL)

20.0

0.0

-20.0

-40.0

-60.0

-80.0

1 10 100 1,000 10,000

MTBE Concentration (ppb)


Elevation (feet MSL)

20.0

0.0

-20.0

-40.0

-60.0

-80.0

1 100 10,000 1,000,000

MTBE Concentration (ppb)


Elevation (feet MSL)

20.0

0.0

-20.0

-40.0

-60.0

-80.0

1 10 100 1,000 10,000

MTBE Concentration (ppb)


Elevation (feet MSL)

20.0

0.0

-20.0

-40.0

-60.0

-80.0

1 10 100 1,000 10,000

MTBE Concentration (ppb)


Profile - data

FLOW

Source Area

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

FLOW

Source Area

20.00

10.00

0.00

-10.00

-20.00

-30.00

-40.00

-50.00

-60.00

-70.00

-80.00

3000.00

2500.00

2000.00

1500.00

1000.00

500.00

0.00

-90.00

-500.00


MTBE (ppm)

Simulated and Observed MTBE at the Recovery Well

5100

4600

4100

Observed

Simulated

3600

3100

2600

2100

1600

1100

600

100

0 20 40 60 80 100 120 140

Time Since System Start-up (Days)


Calibrated Horizontal and Vertical Hydraulic Conductivities

Ground

Water

Flow


The cost of uniqueness …..


Dimensions of model domain

500m by 800m

Model grid


Boundary

Q = 50 m 3 /day

H = 0.0


Particle release point


Reality


Reality


True time = 3256.24 days

True exit point = easting of 206.78


12 head observations


Exit time = 3256

Exit point = 206

Reality


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

Unestimable

parameter

combinations

Increasing number of parameter combinations


Error variance

calculable from

measurement

error C(h)

Error variance

supplied by

hydrogeologists

C(p)

Increasing number of parameter combinations


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)

σ 2 = y t (I-R) t C(p)(I-R)y + y t GC(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

Predictive error variance vs dimensions of calibrated

parameter space

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


K1

K2

K3

Sy1

Sy2

Sy3

Sc2

Sc3

VCONT1

VCONT2

Recharge

CR

Pre-calibration contribution to predictive error variance

400

350

300

250

200

150

100

50

0


Varance of predictive error (ft^2)

Predictive error variance vs dimensions of

calibrated parameter space

1200

1000

800

600

Minimum = 418 ft 2

at 160 singular values

400

200

0

1 10 100

Number of singular values


Contribution to pre- and post-calibration predictive

variance by selected parameter types

400

300

200

100

0

K3 VCONT1 VCONT2

Post-cal

Pre-cal


Optimization of data acquisition:-

How can I deepen the minimum in the predictive

variance curve?


σ 2 = y t (I-R) t C(p)(I-R)y + y t GC(h)Gy


Reduction in predictive variance if VCONT 2 characterization at each

point is reduced from 0.74 to 0.37 (maximum reduction = 112.7ft 2 )


Locations of proposed layer 2-3 differential head measurements

(reduction in predictive error variance = 230 ft 2 )


400

350

300

250

200

150

100

50

0

K3

VCONT1

VCONT2

Pre-cal

Post-cal

Geophysics

Extra_wells


Varance of predictive error (ft^2)

Predictive error variance vs dimensions of

calibrated parameter space

1200

1000

800

Previous minimum = 418 ft 2

at 160 singular values

600

400

200

0

New minimum = 188 ft 2

at 190 singular values

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

3

5 6

8

9

10

7

11

2

16

12

4

15

14

13


Head error variance

Number of cells

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