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MIN E 612: GEOSTATISTICAL METHODSMIN E 612: GEOSTATISTICAL METHODS

Lecture 16:Lecture 16:

Sequential Gaussian SimulationSequential Gaussian Simulation

• Estimation versus Simulation

• Smoothing of Kriging

• Covariance between Kriged Estimates

• Monte Carlo Simulation

• Sequential Gaussian Simulation

Estimation versus SimulationEstimation versus Simulation

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Petrophysical Property ModellingPetrophysical Property Modelling

Estimation:

• honors local data (with discontinuity)

• locally accurate• smooth appropriate for visualizing trends

• inappropriate for flow simulation

• no assessment of global uncertainty

Simulation:

• honors local data

• reproduces histogram

• honors spatial variability→ appropriate for flow simulation

• alternative realizations possible→ change random number seed

• assessment of global uncertainty is possible

Smoothing Effect of KrigingSmoothing Effect of Kriging

• Kriging is locally accurate and smooth, appropriate for visualizing trends,inappropriate for process evaluation where extreme values are important

• The “variance” of the kriged estimates is too small:

2{ ( )} ( )SK Var Y      u u

 – is complete variance

 – is zero at the data locations→ no smoothing

 – is variance far away from data locations→ completesmoothing

 – spatial variations of depend on the variogram and data spacing

2 

2 ( )SK 

    u

2 ( )SK 

    u   2 

2 ( )SK 

    u

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Smoothing Effect of KrigingSmoothing Effect of Kriging

• Missing variance is the kriging variance

• The idea of simulation is to correct the variance and get the rightvariogram

2 ( )SK 

    u

( ) ( ) ( )s

Y Y R u u u

  .

• Simulation reproduces histogram, honors spatial variability (variogram)→ appropriate for process evaluation

• Allows an assessment of uncertainty with alternative realizations

• We will see simulation later - recognize that the smoothing effect of

kriging can be quantified

Covariance Reproduction andCovariance Reproduction andKrigingKriging• Recall the simple kriging estimator:

and the corresponding simple kriging system:

• Let’s calculate the covariance between the kriged estimate and one of

the data values:

1

( ) ( )n

Y Y    

  

 

u u

1

( ) ( )n

C C      

 

  u u u u u

* *

1

1

1

( ), ( ) ( ), ( )

( ) ( )

( ) ( )

( , )

( , )

n

n

n

Cov Y u Y u E Y u Y u

 E Y u Y u

 E Y u Y u

C u u

C u u

 

     

     

     

 

 

 

 

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Covariance Reproduction andCovariance Reproduction and

KrigingKriging

• The covariance is correct! Note that the last substitution comes fromthe kriging equations shown at the top of the page.

• The kriging equations forces the covariance between the data valuesand the kriging estimate to be correct

 

 – between data values→ correct

 – between data values and predicted values→ correct

 – between predicted values→ incorrect

• Note also that variance of kriged estimates is too small

 Addition of Missing Variance Addition of Missing Variance

• The variance of our random function:

• Stationary variance should be constant everywhere:

•

2 (0)C    

2 2( )   A   u u

 

correct, the variance is too small:

the missing variance is the kriging variance !!

• We need to add back in the missing variance without changing thecovariance reproduction

2{ ( )} (0) ( )SK Var Y C       u u

2 ( )SK     u

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 Addition of Missing Variance Addition of Missing Variance

• Add an independent component with zero mean and the correctvariance:

• Covariance is unchanged:

( ) ( ) ( )s

Y Y R u u u

( ), ( ) ( ), ( )s sn

Cov Y u Y u E Y u Y u  

• note that , since R(u) is random

• Therefore

{ ( ) ( )} { ( )} { ( )} E R Y E R E Y   

u u u u

{ ( )} 0 0 { ( ) ( )} { ( )} { ( )} 0 E R E R Y E R E Y    u u u u u

{ ( ) ( )} { ( ) ( )} ( ) ( )}sCov Y Y Cov Y Y C Y     u u u u u u

 

1

1( ) ( ) ( ) ( )

n E Y u Y u E R u Y u

     

      

Sequential SimulationSequential Simulation• Transform data to standard normal distribution (all work will be done in

“normal” space)

• Go to a location and perform kriging to get mean and corresponding kriging

variance:

1

( ) ( )n

Y Y      

 

u un

• Draw a random residual R(u) that follows a normal distribution with mean of

0.0 and variance of

• Add the kriged estimate and residual to get simulated value:

• Note that Ys(u) could be equivalently obtained by drawing from a normal

distribution with mean Y*(u) and variance

1

( ) (0) ( )SK    C C    

 

u u u

2 ( )SK 

    u

( ) ( ) ( )s

Y Y R u u u

2 ( )SK 

    u

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Sequential SimulationSequential Simulation

• Add Ys(u) to the set of data to ensure that the covariance with thisvalue and all future predictions is correct

• A key idea of sequential simulation is to use previouslykriged/simulated values as data so that we reproduce the covariancebetween all of the simulated values!

• Visit all locations in random order (to avoid artifacts of limited search)

• Back-transform all data values and simulated values when model ispopulated

• Create another equiprobable realization by repeating with differentrandom number seed

Why Gaussian Simulation?Why Gaussian Simulation?

• Local estimate is given by kriging

• Mean of residual is zero and variance is given by kriging; however,what “shape” of distribution should we consider?

• Advantage of normal / Gaussian distribution is that the global N(0,1)distribution will be preserved if we always use Gaussian distributions

• Could derive theory in terms of the multivariate Gaussian distribution(see early lectures)

• Transform data to normal scores in the beginning (before variography)

• Simulate 3-D realization in “normal space”

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Why Gaussian Simulation?Why Gaussian Simulation?

• Back-transform all of the values when finished

• Price of mathematical simplicity is the characteristic of maximum spatial

entropy, i.e., low and high values are disconnected. Not appropriate forpermeability.

• Of all unbounded distributions of finite variance, the Gaussian distributionmax m zes en ropy:

• Consequences:

- maximum spatial disorder beyond the variogram

- maximum disconnectedness of extreme values

- median values have greatest connectedness

- symmetric disconnectedness of extreme low / high values

( ) ( ) H lnf z f z dz

Steps in Sequential GaussianSteps in Sequential GaussianSimulationSimulation

1. Transform data to “normal space”

2. Establish grid network and coordinate system (-space)

3. Assign data to the nearest grid node (take the closest of multiple dataassigned to the same node)

4. Determine a random path through all of the grid nodes

• construct the conditional distribution by kriging

• find nearby data and previously simulated grid nodes

• draw simulated value from conditional distribution

5. Check results

• honor data?

• honor histogram: N(0,1) standard normal with a mean of zero anda variance of one?

• honor variogram?

• honor concept of geology?

6. Back transform

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Some SGSIM RealizationsSome SGSIM Realizations

• Variogram is very important

• Assumes that the property is

“stationary”

Ergodic FluctuationsErgodic Fluctuations

• Expect some statistical fluctuation in the input statistics

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Review of Main PointsReview of Main Points

• Kriged estimates are too smooth and inappropriate for most engineeringapplications

• Simulation corrects for this smoothness and ensures that the variogram /covariance is honored

• There are many different simulation algorithms sequential Gaussian issimple and most widely used

• Kriging forces the covariance between the data values and kriged estimatesto be correct

• Total variance at each location should be stationary variance

• Sequential simulation:

 – Add random component to kriged estimates without changing covariance

 – Simulated values are kriged estimates plus random component (that corrects forsmoothing / missing variance)

 – Add simulated data values to data so that covariance between the simulatedvalues is correct, that is, proceed sequentially

 – Use Gaussian / normal distribution for residuals so that final histogram is correct

 – Consequence of Gaussian distribution is maximum entropy / disorder beyondvariogram

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• Principle of Simulation

• Prerequisites

• MCS from High Dimension Distributions

• Sequential Gaussian Simulation

• Some Implementation Details

Fundamentals of Geostatistics

Sequential Gaussian Simulation

1

Centre for Computational Geostatistics (CCG)

• Estimation is locallyaccurate and smooth,appropriate for visualizingtrends, inappropriate forengineering calculationswhere extreme values areimportant, and does notassess global uncertainty

• Simulation reproduceshistogram, honors spatialvariability (variogram), appropriate for flowsimulation, allows anassessment of uncertaintywith alternative realizationspossible

2

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Preliminary Remarks

• Kriging is smooth – but is entirely based on the data

• Kriging does not reproduce the histogram and variogram

• Simulation is designed to draw realizations that reproduce the data,

histogram and variogram• The average of multiple simulated realizations is very close to kriging

•  Al though mult ip le simulated real izat ions average to kriging, theaverage behavior of multip le realizations may be very different

from that of the kriged model

• The advantages of simulation are:

 – Heterogeneity

 – Uncertainty

• The price is non-uniqueness

3

Prerequisites

• Work within “homogeneous” lithofacies/rock-type classification – Model lithofacies first

 – Sequence stratigraphic framework: Zrel vertical coordinate space

• Clean data: positioned correctly, manageable outliers,

• Need to understand “special” data: – trends

 – production data

 – seismic data• Considerations for areal grid size:

 – practical limit to the number of cells

 – need to have sufficient resolution so that the upscaling is meaningful

 – this resolution is required even when the wells are widely spaced(simulation algorithms fill in the heterogeneity)

• Work with grid nodes and not grid blocks: – assign a property for the entire cell knowing that there are “sub-cell”

features

4

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Central Idea

• MCS or simulation from a univariate distribution is easy

• MCS from a very high dimensional distribution is more complex

 – Must ensure the relationship between all values

 – Require a multivariate distribution model to sample from

• Adopt the Gaussian model because it is the only practical one

 – Work within facies

 – Normal score transform all variables• Sequential simulation can be thought of as a recursive decomposition

of a multivariate Gaussian distribution by Bayes Law5

Bayes Law

• The arithmetic ofprobability:

• Must restandardize multivariate probabilities once we know somethinghas occurred

• Using definition of conditional probability:

• Apply Bayes Law recursively

( and ) ( and )( | ) ( | )

( ) ( )

( | ) ( )( | )

( )

P A B P A BP A B P B A

P B P A

P B A P AP A B

P B

 

( , ) ( | ) ( )

( , , ) ( | , ) ( | ) ( )

P A B P B A P A

P A B C P C A B P B A P A

6

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Sequential Conditional Simulation

• Consider N dependent events Ai, i=1,…,N that we want to simulate

• From recursive application of Bayes law:

• Proceed sequentially to jointly simulate the N events A j:

 – Draw A1 from the marginal P(A1)

 – Draw A2 from the conditional P(A2 |A1=a1)

 – Draw A3 from the conditional P(A3 | A1=a1 ,A2=a2)

 –  …

 – Draw AN

from the conditional P(AN

| A1=a

1,A

2=a

2,…,A

N-1=a

N-1)

• This is theoretically valid with no approximations/assumptions

)()|(),...,|(),...,|(

),...,(),...,|(),...,|(),...,(),...,|(),...,(

11221111

2121111

11111

 AP A AP A A AP A A AP

 A AP A A AP A A AP

 A AP A A AP A AP

 N  N  N  N 

 N  N  N  N  N 

 N  N  N  N 

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More Sequential Simulation

• Sequential approach requires:

 – Knowledge of the (N-1) conditional distributions

 –  N simulations with increasing conditioning (larger and larger systems)

 – Solution to size problem is to retain only the nearest conditioning values• Condition to prior data:

 – Partition N into n prior and N-n events to be simulated

 – Start at n+1 and require N-n conditional probabilities

• Sequential indicator simulation for categorical variables is common

• Sequential Gaussian simulation is widely used:

 – Determine each conditional distribution by multivariate Gaussian model

 – Virtually all continuous variable simulation is Gaussian based

 – ANSI standard algorithm

 – Equivalent to turning bands, matrix methods, moving averagemethods,…

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Multivariate Gaussian Distribution

• Where:

 – d is the dimensionality of vector x

 –      is a (d x 1) vector of mean values,

 –    is a (d x d) matrix of covariance values, and

 – || is the determinant of .

• The diagonal elements of are the variances of the correspondingcomponent of x.

• The basic approach (whether stated clearly or not):

 – Transform variables one-at-a-time to a Gaussian distribution (see next)

 – Establish mean and variance values (0 and 1 for standard Gaussian)

 – Establish the covariances between the variables

 – Perform mathematical calculations to infer at unsampled locations, simulate, …

 – Back transform to original units

2[ / 2]1( )2

 x f x e

 

9

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Sequential Simulation

• Transform data to standard normal distribution (declustering/debiasingmust be used to get a representative distribution)

1. Go to a location u and perform kriging to get mean and correspondingkriging variance (require a representative local mean and variogram)

2. Draw a random residual R(u) that follows a normal distribution with mean of0.0 and variance of 2SK(u)

3. Add the simulated value to the set of data

4. Loop over all locations

• Back transform results when finished

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Some Implementation Details

• The sequence of grid nodes to populate:

 – Random to avoid potential for artifacts

 – Multiple grid to constrain large scale structure

• Assign data to grid nodes for speed

• Keep data at their original locations if working with a small area

• Search to variogram range and keep closest 20-40 data to calculatelocal mean and variance

• Could make the mean locally varying

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Check Results

• Check in normal units and in original units

• Reproduce data at their locations?

• Reproduce histogram, N(0,1), over many realizations?

• Reproduce variogram?• Average of many realizations equal to the kriged value?

• Expect statistical fluctuations in the output statistics particularlywhen the domain size is small relative to the variogram

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Different Simulation Algorithms

• There are alternative Gaussian simulation techniques:

 – Matrix Approach LU Decomposition of an N x N matrix

 – Turning Bands simulate the variable on 1-D lines and combine in 3-D

 – Spectral and other Fourier techniques

 – Fractal techniques

 – Moving Average techniques

• All Gaussian techniques are fundamentally the same since theyassume the multivariate distribution is Gaussian

• Sequential Gaussian Simulation widely used and recommendedbecause it is theoretically sound and flexible in its implementation

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Review of Main Points

• Continuous variable modeling:

 – Work within homogeneous stationary subsets

 – Grid system that facilitates data integration

• Estimation is not appropriate for most engineering applications• Simulation is useful for quantifying uncertainty on a response

variable (reproduce spatial variability)

• Sequential simulation decomposes the multivariate distributioninto a sequence of univariate conditional distributions

• Gaussian simulation is used because of its “nice” properties

 – All conditional distributions are Gaussian in shape

 – Conditional mean and variance are given by normal equations(called simple kriging in geostatistics)

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