Stochastic modelling and uncertainty

1Lanzhou, June, 2007

Stochastic modelling and uncertainty

Jesus CarreraInstitute for Earth Sciences (IJA)

Higher council for Scientific Research (CSIC)Barcelona, Spain

[email protected]


Contents

• GW Modelling and uncertainty evaluation: – Is it needed?,– Why?,– What for?

• Basic Concepts– Modelling procedure– Sources of uncertainty– Evaluation of uncertainty

• Discussion– Examples

• Conclusion:– Conceptual model: surprise


Traditional use of models

1. Understanding the past

2. Evaluating the present

3. Assessing the future state of aquifers

Semi-quantitative answers are sufficient, but

Good qualitative answers require beingvery quantitative


Modeling = Accounting

Cell j

Recharge, ri

Cell i

Pumping, Qi

Storage var. ΔSi

Cell l

Cell n

Cell m

Lateral exchange, fij

filfim

fin


Modelling: future needs

• A model is the (water or solute mass) accounting system for water bodies

• A well managed company needs a reliableaccounting system. What about aquifers?

• If not, technical hidrogeology will continueto be a minor economic activity, despite ofthe importance of true hydrogeology.

But models need to be realistic, i.e., quantitatively accurate and reliable


Can models be accurate?

• Unknown parameters, extent and B.C.’s• Spatial Variability STOCHASTICS• Unknown actions. Pumping history is (one

of) the best guarded secrets of any country!

• But, (long?) records of heads, and concentrations, and environmental isotopes, and well logs, and geophysics, and geology, and,....

• Need to ensure consistency


690000 710000 730000 750000 770000

4080000

4100000

4120000

4140000

4160000

UTM

UTM

SEVILLA

GUADALQUIV

IR

0 20000 40000 60000 80000M3690000 710000 730000 750000 770000

4080000

4100000

4120000

4140000

4160000

UTM

UTM

SEVILLA

GUADALQUIV

IR

0 20000 40000 60000 80000

( ) 0h qBoundary and initial conditions

∇⋅ ∇ + =

+

T

2.- CONCEPTUALIZATION

M2690000 710000 730000 750000 770000

4080000

4100000

4120000

4140000

4160000

UTM

UTM

SEVILLA

GUADALQUIV

IR

0 20000 40000 60000 80000

M1

1.- REALITY

( ) ( )q x q==

pAh b

3.- DISCRETIZATION

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

150 175 200 225 250 275 300time (months)

head

leve

l (m

)

4.- CALIBRATION

How can this be achieved?

The trick:

We do not model reality but our view of it

… and then modify this view so as to fitactual observations


Site specific dataScience/knowledge

Conceptualization:

- Process Identification Equations

- Model Structure Identification GeometryCLASS

ICAL

HID

ROGE

OLO

GY

Discretization

Error Analysis

Calibration

Model selection New experiment

PredictionOPERATION !!!

Modelling Procedure


Uncertainty Analysis

Multiple computer “realizations”are simulated using a range of input values for uncertain parameters

0

2

4

6

8

10

12

-7 -6 -5 -4 -3 -2

Ksat_1.qpc

Cou

nt

log10

(Ksat

) (cm/s) for Layer 1

Stochastic Inputs Multiple Computer Simulations(Flow & Transport Model)

Ensemble of realizations yields probability distribution for “performance metric”

Distribution of Results(Multiple Simulations)

00.10.20.30.40.50.60.70.80.9

1

0.001 0.1 10 1000Peak Tritium Dose via the Air Pathway

(mrem/year)

Cum

ulat

ive

Prob

abilit

y

Met

ric =

10

mre

m/y

ear


Uncertainty evaluation procedure

Transmissivity

Other parameters (recharge, porosity, …)Deterministic, but uncertain,

ModelSimulation

Modelresults

Water deficitCapture from river

Travel time

Spatially variable

Uncertainty in output iscaused by uncertainty in input


Uncertainty evaluation procedureMonte Carlo Method

Transmissivity

Other parameters (recharge, porosity, …)Deterministic, but uncertain,

ModelSimulation

Modelresults

Water deficit,Capture from river,

Travel time

Uncertainty is evaluated by repeated simulations withvarying inputs according totheir uncertainty


Issues regarding uncertainty evaluation

• Sources of uncertainty• How to simulate inputs?• How to quantify inputs uncertainty?• How to condition on measurements?• How to account for correlation?• Which outputs should one look at?• How many simulations?• What about model and scenario uncertainty?• CPU time


Sources of uncertainty

• Parameter uncertainty and variability– Feasible to quantify both its value and

its effects on predictions• Conceptual model uncertainty

– Process: Feasible?– Structure: Feasible?

• Scenario uncertainty– ???


Probabilistic Performance Assessment Process

Formalization so as to ensure that all uncertainties are accounted for

Scenario 2Scenario 1

Select Select Reject

Scenario 3

Develop and Screen Scenarios

KsatClimate Change Defects

Estimate Parameter Ranges and Uncertainty

ClimateEvapotranspirationSource TermVadose ZoneSaturated ZoneHuman Exposure

Develop Models

PA_process.ai

Perform Calculations00E000E000E000E000E000E000E000E000E000E000E000E0000D63768118>I<FFF8FFF8FFF80038003800380038003800380038003800380038003800380038003800380038003003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380038003800380

Uncertainty AnalysisSensitivity AnalysisAlternative Designs

Risk/PerformanceCost/ScheduleRegulatory C ompliance

Interpret Results

Monitoring RequirementsEvaluate Design Options


Assigning probabilities to conceptual models

• Given that one wishes to predict L• That one has conjectured Nm models Mi• That one has evaluated the pdf of the

prediction for every model:

• The total pdf is given by

= ∫i i i i i if(L/M) f(L/ ,M)穎( /M)dp p p

=

= ∑Nm

i ii 1

f(L) f(L/M)稰(M) ?


How to evaluate P(Mi)?

1) Prescribe it: (e.g., equally probable)Even after data?

2) Estimate P(Mi/data) (Kashyap,1982; Carrera andNeuman, 1986; Medina and Carrera, 2004) fromexpected likelihood:

Strictly speaking:

Which is feasible by linearization (S=-2ln(Pi))

= = ∫i i i i i i iP P(M /data) f(data/ ,M)穎( /M)dp p p

=

= ∫∑

i i i i ii Nm

kk 1

f(data/ ,M)穎( /M)dP(M /data)

P(M /data)

p p p


Example: uncertainty in structure

S=1116

S=1321 = −i iP a積xp( S /2)

All models lead to goodfits, yet:

1000 orders ofmagnitude differencesin posterior probability

It is wrong!

S=4423


Example 2: Conceptual model=Surprise

Vigo


River Agrio

Aznalcollar dam


5.6

6.7

2.4

3.4

7.2

7.5

7.1

7.6

7.3

3.8

7.3

3.0

3.0

7.1

3.76.57.0

5.16.6

8.58.0

744000 746000

4149000

4151000

4153000

4155000

746250 746350 746450 746550 746650

4.23.8

6.0

6.5

3.8

3.8

3.83.84.67.0

4151500

4151600

4151700

4151800

4151900

4152000

4152100

4152200

Paleozóicopizarras

Miocenomargas azules

Cuaternario, T3limos arenosos

Cuaternario, T2arenas limosas

Cuaternario, T1arenas y gravas

sondeo mecánico

3.93.94.3

Carretera de Aznalcóllar

río Agrio

tailings

C8

2 pozo de brocal sin lodo

4.2

pozo de brocal con lodo

valor medido

A2 identificación punto

Cuaternario, T0(sin sedimentos)

A1A2

A3

A4

1CHG

J1

Characteristic pH after clean-up (5/1998 - 1/2000)

Geochemicalbarrier


Prior to barrier excavation

Based on:

Geophysics

41 EVS

Elect tomogr

Srf Mapping

Boreholes (27)

Trenches (10)

Hydraulic tests(3)


Prior to barrier excavation


After barrier


Revised Conceptual (Geological) model


The barrier (PRB) misses the aquifer!!!


Conclusions

• Uncertainty evaluation important• Parameter uncertainty can be evaluated (just

repeat simulations and examine outputs)• But, Conceptual model = Surprise (Bredehoeft

dixit). Always largest source of uncertaintyIf I knew the painting before hand, I would not paint it

(Picasso dixit)• Bayesian methods not yet mature for evaluating

posterior model probability• Do not believe in full evaluation of uncertainty


Stochastics and uncertainty

• Two different concepts• Uncertainty = ignorance

– Random variables, probabilities, pdf’s– Statistics

• In many natural phenomena, ignorance iscoupled to spatial variability– Random functions, variograms– Geostatistics, stochastics

Stochastic modelling and uncertainty

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