the high-order stochastic sequential simulation framework · outline •limits of two-point...
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The High-order Stochastic Sequential
Simulation Framework:
A review with examples
Roussos Dimitrakopoulos
COSMO Stochastic Mine Planning Laboratory - http://cosmo.mcgill.ca/
Outline
• Limits of two-point statistical methods
• Spatial statistics: Definitions of cumulants, examples and interpretations
• Estimation of non-Gaussian conditional distributions with spatial cumulants
• High-order sequential simulation
• Conclusions
Limits of two-point statistics
• Second and higher order models
• Spatial cumulants
Limits of variograms
1
Variograms EW Variograms NS
2 3
0.4
0.8
1.2
10 20 30 400
3
12
lags
(h
)0.4
0.8
1.2
10 20 30 400
1
2
3
lags
(h)
Very different
patterns
same variogram
may share the
Widely different patterns, yet same statistics up to order 2Source:
SCRAF/Journel
Image 1
Image 2
Different third- and fourth-order cumulants
High-order spatial statistics
• Variogram
• two-point variances
• function of
lags & direction
• High-order
• considers joint
neighbourhoods
of n points
? h
?h1
h2
Second and high-order geostatistics
Multiple-point geostatistics (MPS)
A training image (TI)
is the model
Algorithms:
SNESIM
FILTERSIM
SIMPAT
……
Second and high-order geostatistics
Second and high-order geostatistics
• Multiple-point geostatistics (MPS)
• What if a lot of data and NOT relate to the TI?
• Applications with relatively ‘rich’ data sets?
An example:
Spatial statistics: Definitions of
moments and cumulants
• Concepts
• Definitions
• Spatial templates
• For a non-Gaussian process, cumulants provide a measure of non-Gaussianity
• Cumulants are invariant to additive constants
• For linear process, cumulants may be expressed as higher-order correlations
• Cumulants well define mathematical objects (e.g. covariance, variogram)
High-order spatial cumulants
First-order cumulant(the mean)
The first order cumulant for 3D ergodic stationary
random function (RF) Z(x) is:
where m is the mean of Z(x)
E Z(x) m Cum[Z(x)]
Second-order cumulant(the covariance)
The second order cumulant of a 3D ergodic stationary
RF Z(x) is given by:
For zero mean RF Z(x) the second order cumulant is
the centered covariance:
22( ) ( ) Cum[ ( ), ( )] ( ) zE Z x Z x h m Z x Z x h c h
2
1( ) ( ) ( ) ( ) ( ) Cum[ ( ), ( )] ( ) z
x x
E Z x Z x h C h Z x Z x h Z x Z x h c hN
h
Z(x): a 3D ergodic stationary random function (RF)
Third- to fifth-order spatial cumulants
The third-order cumulant
3 1 2 1 2
1 2
1 3
32 3
( , ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) 2 ( )
zc h h E Z x Z x h Z x h
E Z x E Z x h Z x h
E Z x E Z x h Z x h
E Z x E Z x h Z x h E Z x
For zero mean RF
c3
z (h1,h2) E Z(x)Z(x h1)Z(x h2) h1
h2
h3
The fourth-order cumulant (zero-mean RF)
4 1 2 3 1 2 3
2 1 2 2 3 2 2 2 3 1
2 3 2 1 2
( , , ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
z
z z z z
z z
c h h h E Z x Z x h Z x h Z x h
c h c h h c h c h h
c h c h h
Third- to fifth-order spatial cumulants
h
1h
2
h
3
The fifth-order cumulant (zero-mean RF)
5 1 2 3 4 1 2 3 4
2 1 3 3 2 4 2 2 2 3 3 1 4 1
2 3 3 2 1 4 1 2 4 3 2 1 3
( , , , ) { ( ) ( ) ( ) ( ) ( )}
- ( ) ( - , - ) - ( ) ( - , - )
- ( ) ( - , - ) - ( ) ( - , -
z
z z z z
z z z z
c h h h h E Z x Z x h Z x h Z x h Z x h
c h c h h h h c h c h h h h
c h c h h h h c h c h h h 1
2 2 1 3 3 4 2 3 1 3 2 4
2 4 1 3 2 3 2 3 2 3 1 4
2 4 2 3 1 3 2
)
- ( - ) ( , ) - ( - ) ( , )
- ( - ) ( , ) - ( - ) ( , )
- ( - ) ( , ) - (
z z z z
z z z z
z z z
h
c h h c h h c h h c h h
c h h c h h c h h c h h
c h h c h h c h4 3 3 1 2- ) ( , )zh c h h
Third- to fifth-order spatial cumulants
The third order cumulant
Experimental expression
1 2
1 2
1 2
,1 2
1 2
,
,,
1 2
;
,h h
h h
h hh h
h h
N
C h h3
3
3
T
a. and are two lags in two specific directions;
b. T is the set of replicates;
c. is the cardinality of T
d. is the cumulant at the node ( ).
,1 2,1 23
1 2
1 2
1 2, 1
,1 2 3
1( ) ( ) ( ),
{ ; ; } ,
h h
h hN
k k kh h k
h hk k k
C Z x Z x h Z x hN
x x h x h
Z(x0 )
Z(x0 h2)
0 1( )Z x h
Z(x1 h2)
Z(x1) 1 1( )Z x h
c3
z (h1,h2) E Z(x)Z(x h1)Z(x h2)
Calculation of spatial cumulants
The fourth-order cumulant
4 1 2 3 1 2 3
2 1 2 2 3 2 2 2 3 1
2 3 2 1 2
( , , ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
z
z z z z
z z
c h h h E Z x Z x h Z x h Z x h
c h c h h c h c h h
c h c h h
1 3( )Z x h
Z(x1)
1 1( )Z x h
Z(x1 h2)
Calculation of spatial cumulants
Spatial templates
z(x) z(x1)
z(x2)
h1
h2
z(x)
z(x1)
z(x2)
h1
h2
z(x)
z(x1)
z(x2)
h2
h1
z(x) z(x1)
z(x2)
h1
h2
z(x)
z(x1)
z(x2)
h1
h2
+45o
Third-order templates (2D)
X
Y
Examples and interpretations
• Binary images
• A diamond pipe
75 m75 m
Example 1: 2D binary image
Original image Third-order cumulant
h1
h2
75 m75 m75 m
3 1 2 1 2
1 2
1 3
32 3
( , ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) 2 ( )
zc h h E Z x Z x h Z x h
E Z x E Z x h Z x h
E Z x E Z x h Z x h
E Z x E Z x h Z x h E Z x
Third-order
Original image Third-order cumulant
h1
h2
125 m125 m
125 m125 m
Example 1: 2D binary image
125 m
Original image fifth-order cumulant
5 1 2 3 4 1 2 3 4
2 1 3 3 2 4 2 2 2 3 3 1 4 1
2 3 3 2 1 4 1 2 4 3 2 1 3
( , , , ) { ( ) ( ) ( ) ( ) ( )}
- ( ) ( - , - ) - ( ) ( - , - )
- ( ) ( - , - ) - ( ) ( - , -
z
z z z z
z z z z
c h h h h E Z x Z x h Z x h Z x h Z x h
c h c h h h h c h c h h h h
c h c h h h h c h c h h h
1
2 2 1 3 3 4 2 3 1 3 2 4
2 4 1 3 2 3 2 3 2 3 1 4
2 4 2 3 1 3 2
)
- ( - ) ( , ) - ( - ) ( , )
- ( - ) ( , ) - ( - ) ( , )
- ( - ) ( , ) - (
z z z z
z z z z
z z z
h
c h h c h h c h h c h h
c h h c h h c h h c h h
c h h c h h c h4 3 3 1 2- ) ( , )zh c h h
Fifth-order
Original image
125 m125 m h1
h2
h4
h3
5 { , } vector space 5 1 2
1 2
2 3 2 1 1 3 2 1 1 3 1 2
2
( , ,0,0)
{ ( ) ( ) ( ) ( ) ( )}
- (0)( ( - , - ) ( - , - ) ( , ))
-
z zx y
z z z z
c c h h
E Z x Z x Z x Z x h Z x h
c c h h h c h h h c h h
c
1 3 2 2 3 2
2 2 3 1 1 3 1
2 2 1 3
( )( ( , ) 2 ( ,0))
- ( )( ( , ) 2 ( ,0))
- ( - ) (0,0)
z z z
z z z
z z
h c h h c h
c h c h h c h
c h h c
Example 1: 2D binary image
Example 2: 2D continuous image
Original image Third-order cumulant
78 000 nodes
10 000 nodes
h1
h2
(Srivastava and Isaaks, 1989)
460 nodes
Example 3: 3D image - diamond pipe
(Fox diamond pipe, Ekati mine, NWT, Canada)
(Nowicki et al., 2004)
Example 3: 3D image - diamond pipe
(third-order)
x
y
z
x
y
y
z
x
z
Example 3: 3D image - diamond pipe
2D Cross-sections
Fourth-order Fifth-order
3rd-Order 4th-Order 5th-Order
z
x
y
z
x
y
Example3: Drill hole data, diamond pipe
A high-order sequential
simulation algorithm
• Estimating conditional distributions
• Examples
0 }n = {d
1 0( ; | )Zf z u( )
1 1( ) ( )lZ zu u
( )1 1, ( )}l
n z = {d u
2 1( ; | )Zf z u( )
2 2( ) ( )lZ zu u
...
( ) ( )1, ( ),..., ( )}l l
N n Nz z {d u u
1( ; | )Z N N Nf z u( )( ) ( )l
N NZ zu u
High-order sequential simulation
Sequential simulation
• Multivariate Legendre series
• The conditional density of Z0 given Z1=a1,…,Zn=an is
given by
= g(ci0i1…in) and ci1i2…in = cum(Xi00,X
i11,…,X
inn)NN iiiL ,,..., 10
),(...)(
1
),...,,()(
1)/(
0,,...,,
0000
10
0
00
0110
10
0
zPLdzf
zzzfdzf
zf
iiiii
iiiD
n
D
Z
NN
NN
x
x
Z
Z
Z
Legendre
cumulantsLegendre
polynomials
Order of the
approximation
High-order sequential simulation
• Multivariate Legendre-like orthogonal splines
• The conditional density of Z0 given Z1=a1,…,Zn=an is
given by
= g(ci0i1…in) and ci1i2…in = cum(Xi00,X
i11,…,X
inn)NN iiiL ,,..., 10
Legendre
cumulants
Legendre-like
orthogonal
splines
Order of the
approximation
High-order sequential simulation
)..(...)(
1
),...,,()(
1)/(
0,,...,,
0000
10
0
00
0110
10
0
zSLdzf
zzzfdzf
zf
iiiii
iiiD
n
D
Z
NN
NN
x
x
Z
Z
Z
?i0
i1
i2i3
i4
i5
i0=i1=…=in=1
Multiple point statistics
(Journel, 1990’s, and on)
Estimating conditional distributions
),(...)(
1
),...,,()(
1)/(
0,,...,,
0000
10
0
00
0110
10
0
zPLdzf
zzzfdzf
zf
iiiii
iiiD
n
D
Z
NN
NN
x
x
Z
Z
Z
Calculating cumulants when simulating
h3
u0+h3
h2
u0+h2
u0+h1h1
u0
Node to Simulate 1, order = 6,
calculate cumulants from data
Node to Simulate 2, order = 6,
calculate up to order 4 from data,
and the rest from a Training
Image
h4
h5
u0+h4
u0+h5
h2
u0+h2
h3
u0+h3
u0+h1h1
u0
Using the first the cumulants
c1, c2 and c3 of the true
distribution.
Estimating conditional distributions
Using cumulants up to
orders 12 and 25
Mixture of Gaussians
),( yxf
).()(),( ,
12
0
12
0
12 yPxPLyxf nmnm
nm
).()(),( ,
0
12
0
12 yPxPLyxf nmnm
m
nm
L1,1 . . L1,12
. .
. .
L12,1 . . L12,12
L1,1
.
.
L12,1 . . L12,12
Bivariate lognormal
Estimating conditional distributions
Simulation of complex
geological patterns
• Testing the method
Realiz
atio
n 2
T
rue im
age
Realiz
atio
n 1
Third cum
Third cum
Variograms NSHistograms Training image
Variograms NS
Test - Sparse data and a ‘disoriented’ TI
Sample Data location(85 data)
Training image 2Training image 1True image
Sample Data location
(25 data)
Test - Very sparse data and a ‘disoriented’ TI
39
Simulation of a gold deposit
40
Geological Domains Drillhole data
cdf in each domain Proportion of samples per domain
A gold deposit
3rd-order and 4th- order cumulant maps (data)
A gold deposit
42
Realization Cdf reproduction
55m
75m
Template dimensions:
Order of approximation:
7
20 Realizations
cdfs
Declustered
Data cdf
A gold deposit
Cumulants of Realizations
Declustered
3rd-order
4th-order
A gold deposit
E-Type
P[Au>0.25
ppm]
Au ppm
5
2.5
0
Prob.1
0.5
0
A gold deposit
Some implications for
mine production forecasts
• Open pit mine schedule
Comments:
• Both schedules above are
stochastic
• Schedules are physically different
(and pit limits)
• High-order simulations lead to 40%
higher NPV
• More ore for less waste
Do the methods of modelling uncertainty matter
to mine production scheduling ?
Read is based on high-order simulations (HOSIM)
Blue is based on second-order simulations (SGSIM)
Metal Production
Cumulative NPV
47
Current Research:
High-order stochastic simulation
via statistical learning
in
reproducing kernel Hilbert space
Statistical learning paradigm
• Learning functional dependency from data
• No reliance on parametric models
• Model complexity and generalization
• Minimizing training error complexity
• Minimizing test error generalization
• High-order simulation
• Learning from the input (capture the regularity)
• Adapt to the new data (generalize to the unseen)
• A framework to manipulate the model complexity48
High-order simulation in kernel space
49
Training imageSample data
Replicates
RKHS
Match the high-order
spatial statistics in RKHS
Distribution
Regularize to a “nicer” space
Some References
Minniakhmetov I, Dimitrakopoulos R, Godoy M (2018) High-order spatial
simulation using Legendre-like orthogonal splines. Mathematical
Geosciences, 50(7): 753-780 (Includes source code of related program)
Minniakhmetov I, Dimitrakopoulos R (2016) Joint high-order simulation of
spatially correlated variables using high-order spatial statistics.
Mathematical Geosciences,DOI:10.1007/s11004-016-9662-x
de Carvalho J P, Dimitrakopoulos R, Minniakhmetov I (2019) High-order
block support spatial simulation and application at a gold deposit.
Mathematical Geosciences, DOI 10.1007/s11004-019-09784-x
Mustapha H, Dimitrakopoulos R, Chatterjee S (2011) Geologic
heterogeneity representation using high-order spatial cumulants for
subsurface flow and transport simulations. Water Resources Research, 47,
doi:10.1029/2010WR009515