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Basics in Geostatistics 3Geostatistical Monte-Carlo methods:
Conditional simulation
Hans Wackernagel
MINES ParisTech
NERSC • April 2013
http://hans.wackernagel.free.fr
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Basic concepts
Geostatistics
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 2 / 34
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Concepts
Geostatistical model
The experimental variogram serves to analyze thespatial structure of a regionalized variable z(x).It is fitted with a variogram model which is thestructure function of a random function.The regionalized variable (reality) is viewed as onerealization of the random function Z(x).
Kriging: Best Linear Unbiased Estimation of point values(or spatial averages) at any location of a region.
Conditional simulation: generate an ensemble of realizationsof the random function, conditional upon data.Statistics not linearly related to data can becomputed from this ensemble.
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Concepts
Geostatistical model
The experimental variogram serves to analyze thespatial structure of a regionalized variable z(x).It is fitted with a variogram model which is thestructure function of a random function.The regionalized variable (reality) is viewed as onerealization of the random function Z(x).
Kriging: Best Linear Unbiased Estimation of point values(or spatial averages) at any location of a region.
Conditional simulation: generate an ensemble of realizationsof the random function, conditional upon data.Statistics not linearly related to data can becomputed from this ensemble.
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Concepts
Geostatistical model
The experimental variogram serves to analyze thespatial structure of a regionalized variable z(x).It is fitted with a variogram model which is thestructure function of a random function.The regionalized variable (reality) is viewed as onerealization of the random function Z(x).
Kriging: Best Linear Unbiased Estimation of point values(or spatial averages) at any location of a region.
Conditional simulation: generate an ensemble of realizationsof the random function, conditional upon data.Statistics not linearly related to data can becomputed from this ensemble.
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Limitations of linear geostatistics
Adequate for Gaussian random functions: in practice thedistribution function is often skew.
Probing with two-point statistics (covariance function,variogram): other tools are also available.
Need for non-linear estimates,e.g. for estimating probability of exceeding:
environmental threshold,cut-off grade in mining.
Conditional simulation techniques address all these aspects.
Gaussian conditional simulation generates an ensemble ofrealizations on which non-linear statistics can be readilycomputed.
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Random functions: spatial correlation structure
Stationary random function:
mean, variance and spatial distribution function exist,
spatial correlation is described by the covariance function:
C(h) = E[(Z(x+h)−m) · (Z(x)−m)]
the variogram of a stationary random function is given bythe formula:
γ(h) = C(0)− C(h)
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Gaussian conditional simulationClassical approach
1 Simulate realizations of a stationary Gaussian randomfunction with known covariance function C(h).
2 Condition the realizations using simple kriging.
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Gaussian simulation
1) Unconditional simulation of a Gaussian RF
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 7 / 34
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Simulation of a Gaussian random functionTurning bands method (TBM)
The simulation of realizations of a GRF can be done simply:determine the 1D covariance function of a corresponding 2D or 3Dcovariance model,generate directions θ1, . . . , θKsimulate realizations of 1D processes Y1, . . . ,YK along lines in thosedirections,project a given point on the lines and combine the correspondingsimulated values to obtain the simulated value of the 3D process atthat point:
Y(x) =1√K
K∑k=1
Yk(< x, θk >) for x ∈ D.
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1D covariance function corresponding to2D or 3D isotropic covariance
The following formulas rely on Bochner’s theorem,in 3D:
C3D(h) =∫ 1
0C1D(t h)dt and C1D(h) =
ddh
(hC3D(h))
in 2D:
C2D(h) =1π
∫ π
0C1D(h sin θ)dθ and C1D(h) = 1+ h
∫ π/2
0
dC2D
dh(h sin θ)dθ
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Example: exponential covariance function
The 1D model associated to a 3D exponential covariance is:
C1D(h) =(
1− ha
)exp
(−ha
)with h,a ≥ 0
Migration method: compute Poisson points, split intervalsinto halves set to ±1 (mean interval length is 2a):
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TBM: exponential covariance
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Gaussian simulation
2) Conditional simulation of a Gaussian RF
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 12 / 34
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Best linear unbiased estimation (BLUE): kriging
Estimation of a value Z? at a location x0 in geographical spaceis performed using a linear combination of weights wα withdata at neighboring locations xα, α = 1, . . . ,n.
Kriging with known mean m (simple kriging):
Z?(x0) = m+n∑
α=1
wα (Z(xα)−m)
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Conditional simulation of Gaussian RF
ZCS(x) = Z?(x)︸ ︷︷ ︸kriged from data
+ (ZS(x)− Z?S(x))︸ ︷︷ ︸simulated kriging error
( − )
+
=
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Conditional simulation and krigingComparison with kriging
Simulation (left) Samples (right)
Simple kriging (left) Conditional simulation (right)
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Conditional simulation and kriging
Conditionally on the data,
the mean of conditional simulations is equal to the kriging:
E[ZCS(x) |Z(xα), α= 1, . . . ,n
]= Z?(x),
the variance of the conditional simulations is the krigingvariance:
var(ZCS(x) |Z(xα), α= 1, . . . ,n) = var(ZS(x)− Z?S(x)) = σ2K(x).
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Gaussian simulationwith non-Gaussian data
1 Fit of a Gaussian anamorphosis function Z(x) = ϕ(Y(x)).2 Transform the data to
Gaussian values: Y(xα) = ϕ−1(Z(xα)).3 Fit the variogram of
the Gaussian random function Y(x).4 Simulate realizations YS(x).5 Condition YS(x) with Y(xα), thus obtaining YCS(x).6 Transform the result to
the initial scale: ZCS(x) = ϕ(YCS(x)).7 Compute various statistics on
the ensemble of realizations.
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Case study
Simulating Yeu islandThe island is located off the south-west coast of Bretagne
Measurements of elevation in the sea (depths).
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 18 / 34
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Kriging the elevation data
Negative kriging estimates are set to zero (below sea level).
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Conditional simulation of elevation9 realizations of Yeu island
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Simulation profiles along island
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Probability that elevation is above sea level
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Estimation of surface (km²) of Yeu island
2
Real From kriging Sim. min Sim. mean Sim. maxSurface 23.3 22.9 15.4 23.2 31.9
From conditional simulation the volume is estimated to be: 0.188 km3
(as compared to the value deduced from kriging results: 0.169 km3)
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Conclusion
Summary
Gaussian random function simulations
Adequate for simulating Gaussian random functions
Anamorphosis to apply them to non-Gaussian data
Satisfy the need for non-linear estimates,e.g. for estimating probability of exceeding:
environmental thresholdcut-off grade in mining
Generate an ensemble of realisations on which non-linearstatistics are readily computed.
However, in a number of applications there is a need forstochastic models beyond the random functions framework...
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 24 / 34
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Conclusion
Summary
Gaussian random function simulations
Adequate for simulating Gaussian random functions
Anamorphosis to apply them to non-Gaussian data
Satisfy the need for non-linear estimates,e.g. for estimating probability of exceeding:
environmental thresholdcut-off grade in mining
Generate an ensemble of realisations on which non-linearstatistics are readily computed.
However, in a number of applications there is a need forstochastic models beyond the random functions framework...
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 24 / 34
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References
M Armstrong, A G Galli, H Beucher, G Le Loc’h, D Renard,B Doligez, R Eschard, and F Geffroy.Plurigaussian Simulations in Geosciences.Springer-Verlag, Berlin, 2nd edition, 2011.
JP Chilès and P Delfiner.Geostatistics: Modeling Spatial Uncertainty.Wiley, New York, 2nd edition, 2012.
C Lantuéjoul.Geostatistical Simulation: Models and Algorithms.Springer-Verlag, Berlin, 2002.
G. Matheron.Random Sets and Integral Geometry.John Wiley & Sons, New York, 1975.
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RGeoS case-study
Conditional simulation example with RGeoS9.1.2
Code from the document Doc2D.pdf on the site:http://rgeos.free.fr
Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 26 / 34
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Loading the soil pollution data setlibrary(RGeoS)data(Exdemo_2D_pollution.table)DAT=Exdemo_2D_pollution.table
data.db = db.create(DAT,flag.grid=FALSE,ndim=2,autoname=F)
data.db = db.locate(data.db,"Zn","z",1)plot(data.db,pch=21,bg.in="black",title="Zn Sample locations")# suppress two outliershist(DAT$Zn,n=20)data.db = db.sel(data.db,Zn<20)hist(DAT$Zn[DAT$Zn<20])
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Zn Sample locations Histogram of DAT$Zn
DAT$Zn
Fre
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010
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Structural analysisdata.vario = vario.calc(data.db,lag=1,nlag=10)plot(data.vario,npairdw=TRUE,npairpt=TRUE)
data.4dir.vario =vario.calc(data.db,lag=1,nlag=10,dir=c(0,45,90,135))
plot(data.4dir.vario,title="Directional variograms")
data.model = model.auto(data.vario,
struct=c("Spherical","Exponential"),title="Modelling omni-directional variogram")
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Kriging
# for KRIGING use all 102 data (suppress selection)data.db = db.sel(data.db)#kriging gridgrid.db = db.grid.init(data.db,nodes=c(100,90))# defining unique neighborhooddata.unique = neigh.input(ndim=2)data.db = db.locate(data.db,seq(8,9))data.db = db.locate(data.db,Zn,z)grid.db = kriging(data.db,grid.db,data.model,data.unique,radix="KU")
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Plot kriging results
# plot estimates with contour lines for std deviations
plot(grid.db,name.image="KU.Zn.estim", col=topo.colors(20),title = "Estimation (Unique Neighborhood)")plot(grid.db,name.contour="KU.Zn.stdev",nlevels=10,add=TRUE)plot(data.db,pch=21,bg.in=1,add=TRUE)# separate plot of std deviations
plot(grid.db,name.image="KU.Zn.stdev", col=topo.colors(100),title = "Std deviation (Unique Neighborhood)",zlim=c(0,2.5))plot(grid.db,name.contour="KU.Zn.stdev",nlevels=10,add=TRUE)
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Anamorphosis
## anamorphosis (normal score transform & Hermite poly)data.anam=anam.fit(data.db,"Zn")# transform z to Gaussian ydata.db = anam.z2y(data.db,"Zn",anam=data.anam)data.g.vario = vario.calc(data.db,nlag=10,lag=1)plot(data.g.vario,npairdw=TRUE,npairpt=TRUE)data.g.model = model.auto(data.g.vario,struct=c("Exponential"))
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Conditional simulation10 realizations (100 turning bands)
grid.db = simtub(data.db,grid.db,data.g.model,data.unique,nbsimu=10,nbtuba=100)# transform back from Gaussian Y to Zgrid.db = anam.y2z(grid.db,ngrep="Simu.Gaussian.Zn",anam=data.anam)
plot(grid.db,name.image="Raw.Simu.Gaussian.Zn.S1",col=topo.colors(20))plot(data.db,pch=21,bg.in=1,add=TRUE)
plot(grid.db,name.image="Raw.Simu.Gaussian.Zn.S10",col=topo.colors(20))plot(data.db,pch=21,bg.in=1,add=TRUE)
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Mean and standard deviationof 10 simulations
## plot mean of simulations
grid.db <- db.compare(grid.db,ngrep="Raw.Simu.Gaussian.Zn",fun="mean")
plot(grid.db,col=topo.colors(20),zlim=c(3,13))plot(data.db,bg.in=1,add=TRUE,pch=21)## standard deviation of simulations
grid.db <- db.compare(grid.db,ngrep="Raw.Simu.Gaussian.Zn",fun="stdv")
plot(grid.db,col=topo.colors(100),zlim=c(0,2.5))plot(data.db,bg.in=1,add=TRUE,pch=21)
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Exporting the resultsPlotting them with the lattice package
grid.db # to find out the names of columnsSmean=db.extract(grid.db,"mean"); Sstdv=db.extract(grid.db,"stdv")x1=db.extract(grid.db,"x1"); x2=db.extract(grid.db,"x2")library(lattice) # a standard graphical package in R
levelplot(Smean~x1*x2,main="Mean of 10 simulations",col.regions=topo.colors)
levelplot(Sstdv~x1*x2,
main="Std deviation of 10 simulations",col.regions=rainbow(20, start=.5, end=0.01))
Mean of 10 simulations
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