Lecture7 Uncertainty
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Transcript of Lecture7 Uncertainty
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Uncertainty in geological models
Irina Overeem
Community Surface Dynamics Modeling System
University of Colorado at Boulder
September 2008
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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
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Outline
Example of probability model
Natural variability & non-uniqueness
Sensitivity tests
Visualizing Uncertainty
Inverse experiments
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Case Study
Lotschberg base tunnel, Switzerland
Scheduled to be completed
in 2012
35 kmlong
Crossing the entire the
Lotschberg massif
Problem: Triassic evaporite
rocks
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Variability model
Empirical Variogram model
Z u m u u u
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Geological model
2 2
20.000 1 exp1500 500
h hh
,h h h
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Probability combined into singlemodel
Probability model
Probability profile along the tunnel
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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
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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
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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
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Natural variability: replicate experiments
T1 T2 T3
T1 T2 T3
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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
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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
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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).
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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.
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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
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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).
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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
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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)
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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 -
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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
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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
Data courtesy G.J.Weltje, Delft University of Technology
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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)
Data courtesy G.J.Weltje, Delft University of Technology
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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)
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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
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01/1402/1403/1404/1405/1406/1407/1408/1409/1410/1411/1412/1413/1414/14
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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
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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!