Lecture7 Uncertainty

8/12/2019 Lecture7 Uncertainty

1/28

1

Uncertainty in geological models

Irina Overeem

Community Surface Dynamics Modeling System

University of Colorado at Boulder

September 2008


2/28

2

Course outline 1

Lectures by Irina Overeem:

Introduction and overview

Deterministic and geometric models

Sedimentary process models I

Sedimentary process models II

Uncertainty in modeling


3/28

3

Outline

Example of probability model

Natural variability & non-uniqueness

Sensitivity tests

Visualizing Uncertainty

Inverse experiments


4/28

4

Case Study

Lotschberg base tunnel, Switzerland

Scheduled to be completed

in 2012

35 kmlong

Crossing the entire the

Lotschberg massif

Problem: Triassic evaporite

rocks


5/28

Variability model

Empirical Variogram model

Z u m u u u


6/28

09 June 2014 6

Geological model

2 2

20.000 1 exp1500 500

h hh

,h h h


7/28

Probability combined into singlemodel

Probability model

Probability profile along the tunnel


8/28

09 June 2014 8

Which data to use for constraining which part of the model?

Do the modeling results accurately mimic the data (reality)?

Should the model be improved?

Are there any other (equally plausible) geological scenariosthat could account for the observations?

We cannot answer any of the above questions without knowinghow to measure the discrepancy between data and modelingresults, and interpret this discrepancy in probabilistic terms

We cannot define a meaningful measure without knowledge ofthe natural variability (probability distribution of realisationsunder a given scenario)

Natural variability and non-uniqueness


9/28

Scaled-down physical models

Assumption: scale invariance of major geomorphological features(channels, lobes) and their responses to external forcing (baselevelchanges, sediment supply)

This permits the investigation of natural variability through experiments

(multiple realisations)

Delta /Shelf

Fluvialvalley


10/28

Model Specifications

Initial topography Sea-level curve

Discharge and sediment input rate constant(experiments by Van Heijst, 2001)

P 0 300 600 900 1200 1500 1800

Time [min]

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

410

420

Sealevel[mm

]

Three snapshots:

t = 900, 1200, 1500 min


11/28

09 June 2014 11

Natural variability: replicate experiments

T1 T2 T3

T1 T2 T3


12/28

09 June 2014 12

Squared difference topo grids

T1 T2 T3

T1 T2 T3

Time

Realisations

0 .0 0 5 00 .0 0 1 00 0. 00 1 50 0. 00 2 00 0. 00 2 5 00 .0 0 3 00 0. 00

T900(1) - T1200(1)

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

0 .0 0 5 00 .0 0 1 00 0. 00 1 50 0. 00 2 00 0. 00 2 50 0. 00 3 00 0. 00

T1200(1) - T1500(1)

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

0 .0 0 5 00 .0 0 1 00 0. 00 1 50 0. 00 2 00 0. 00 2 5 00 .0 0 3 00 0. 00

T900(2) - T1200(2)

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

0 .0 0 5 00 .0 0 1 00 0. 00 1 50 0. 00 2 00 0. 00 2 50 0. 00 3 00 0. 00

T1200(1) - T1200(2)

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

0 .0 0 5 00 .0 0 1 00 0. 00 1 5 00 .0 0 2 00 0. 00 2 50 0. 00 3 00 0. 00

T900(1) - T900(2)

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

0 .0 0 5 00 .0 0 1 00 0. 00 1 50 0. 00 2 00 0. 00 2 50 0. 00 3 00 0. 00

T1200(2) - T1500(2)

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

0 .0 0 5 00 .0 0 1 00 0. 00 1 50 0. 00 2 00 0. 00 2 50 0. 00 3 00 0. 00

T1500(1) - T1500(2)

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00


13/28

13

Forecasting / Hindcasting /Predictability?

Sensitivity to initial conditions (topography)

Presence of positive feedbacks (incision)

Other possibilities: complex response, negative feedbacks,insensitivity to external influences

Dependent on when and where you look within a complex system


14/28

14

Sensitivity tests

Run multiple scenarios with ranges of plausible inputparameters.

In the case of stochastic static models the plausibleinput parameters, e.g. W, D, L of sediment bodies, aresampled from variograms.

In case of proces models, plausible input parameterscan be ranges in the boundary conditions or in themodel parameters (e.g. in the equations).


15/28

15

Visualizing uncertainty by usingsensitivity tests

Variability and as such uncertainty in SEDFLUX output isrepresented via multiple realizations. We propose to associatesensitivity experiments to a predicted base-case value. In thatway the stratigraphic variability caused by ranges in the

boundary conditions is evident for later users.

Two main attributes are being used to quantify variability:

TH= deposited thickness and GSD = predicted grain size.

We use the mean and standard deviation of both attributes tovisualize the ranges in the predictions.


16/28

Visualizing Grainsize Variability

For any pseudo-well thegrain size with depth is

determined, and attached

to this prediction the

range of the grain size

prediction over the

sensitivity tests.

Depth zones of high

uncertainty in the

predicted core are

typically related to strong

jumps in the grain size

prediction.

Facies shift

causes a

strong jump inGSDHigh uncertainty zone

associated with variation

of predicted jump in GSD


17/28

Visualizing Thickness Variability

For any sensitivity test the

deposited thickness versuswater depth is determined

(shown in the upper plot).

Attached to the base case

prediction (red line) the range

of the thickness prediction

over the sensitivity tests is

then evident. This can be

quantified by attaching the

associated standard

deviations over the different

sensitivity tests to the modelprediction (lower plot).


18/28

Visualizing X-sectional grainsize variability

Depthin10cmbi

ns

ofGrainsizeinmic

ron

This example collapses

a series of 6 SedFlux

sensitivity experiments

of one of the tunable

parameters of the 2D-

model (BW= basinwidth

over which the sediment

is spread out).

The standard deviation

of the predicted

grainsize with depth

over the experiments

has been determined

and plotted with

distance. Red colorreflects low uncertainty

(high coherence

between the different

experiments) and yellow

and blue color reflects

locally high uncertainty.

i li i f i b bili


19/28

Visualizing facies probabilitymaps

Probabilistic output

of 250 simulationsshowing change ofoccurrence of threegrain-size classes:

sandy depositssilty depositsclayey deposits

highchance

lowchance

After Hoogendoorn,Overeem and Storms

(in prep. 2006)


20/28

09 June 2014 20

Inverse Modeling

Simplest case: linear inverse problems

A linear inverse problem can be described by:

d= G(m)

where Gis a linear operatordescribing the explicitrelationship between data and model parameters, and is arepresentation of the physical system.

In an inverse problemmodel values need to be

obtained the values from the observed data.
http://www.answers.com/topic/linear-map-1http://www.answers.com/topic/datum-1http://www.answers.com/topic/datum-1http://www.answers.com/topic/linear-map-1


21/28

21

Inverse techniques to reduceuncertainty by constraining to data

Stochastic, static models areconstrained to well dataExample: Petrel realization

Process-response models canalso be constrained to well dataExample: BARSIM

Data courtesy G.J.Weltje, Delft University of Technology


22/28

22

Inversion: automated reconstruction of geologicalscenarios from shallow-marine stratigraphy

Inversion scheme (Weltje & Geel, 2004): result of manyexperiments

Forward model: BARSIM Unknowns: sea-level and sediment-supply scenarios Parameterisation: sine functions

SL: amplitude, wavelength, phase angle SS: amplitude, wavelength, phase angle, mean

The truth : an arbitrary piece of stratigraphy, generated byrandom sampling from seven probability distributions



23/28

23

Our goal: minimization of anobjective function

An Objective Function (OF) measures the distancebetween a realisation and the conditioning data

Best fit corresponds to lowest value of OF

Zero value of OF indicates perfect fit:

Series of fullyconditionedrealisations

(OF = 0)



24/28

09 June 2014 24

1. Quantify stratigraphy and data-modeldivergence: String matching of permeability logs

The discrepancy between a candidate solution (realization) and the data isexpressed as the sum of Levenshtein distances of three permeability logs.

Stratigraphic data: permeability logs

(info on GSD + porosity)

Objective function: Levenshtein distance

(string matching)


25/28

09 June 2014 25

2. Use a Genetic Algorithm as goal seeker:a global Darwinian optimizer

Each candidate solution (individual) isrepresented by a string of seven numbers inbinary format (a chromosome)

Its fate is determined by its fitness value(proportional inversely to Levenshteindistance between candidate solution anddata)

Fitness values gradually increase insuccessive generations, because preference

is given to the fittest individuals


26/28

09 June 2014 26

01/1402/1403/1404/1405/1406/1407/1408/1409/1410/1411/1412/1413/1414/14


27/28

27

Conclusions from inversion experiments

Typical seven-parameter inversion requires about 50.000 modelruns !

Consumes a lot of computer power

Works well within confines of toy-model world: few localminima, sucessful search for truth

Automated reconstruction of geological scenarios seems feasible,given sufficient computer power (fast computers & models) and

statistically meaningful measures of data-model divergence


28/28

28

Conclusions and discussion

Validation of models in earth sciences is virtuallyimpossible, inherent natural variability is a problem.

Uncertainty in models can be quantified by makingmultiple realizations or by defining a base-case andassociating a measure for the uncertainty.

Probability maps of facies occurence generated bymultiple realizations are a powerfull way of conveying theuncertainty.

In the end, inverse experiments are the way to go!

Lecture7 Uncertainty

Documents

Transcript of Lecture7 Uncertainty