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Stochastic Modelling and
Geostatisticsby
Dr. Amro Elfeki
2
Time Table• 02/10/2002 Introduction and probability theory of single and
multi-variate • 09/10/2002 Some termenology, real-domain and spectral
domain representation of stochastic processes.
• 16/10/2002 Stochastic models for site characterization: Discrete models
• 23/10/2002 Stochastic models for site characterization: Continuous models (1).
• 30/10/2002 Stochastic models for site characterization: Continuous models (2).
• 06/11/2002 Stochastic differential equations and methods of solution.
• 13/11/2002 MonteCarlo method.• 20/11/2002 Kriging and conditional simulations.
3
Time Series
0
20
40
60
80
100
120
140
160
1-1-
1978
1-3-
1978
1-5-
1978
1-7-
1978
1-9-
1978
1-11
-197
8
1-1-
1979
1-3-
1979
1-5-
1979
1-7-
1979
1-9-
1979
1-11
-197
9
Series1
4
Space Series
5
Outcrop (1)
6
Outcrop (2)
7
Boreholes
8
Site Characterization
• Deterministic Approach.
• Stochastic Approach.
9
Deterministic Approach
0 200 400 600 800 1000 1200 1400 1600-15
-10
-5
0
10
Stochastic Approach
• The word stochastic has its origin in the Greek adjective στoχαστικoς which means skilful at aiming or guessing .
• The stochastic approach is used to solve differential equations with stochastic parameters.
• This approach is a tool to evaluate the effects of
spatial variability of the hydrogeological parameters on flow and transport characteristics in porous formations.
11
Why we need the Stochastic Approach?
• The erratic nature of the hydrogeological parameters observed at field data.
• The uncertainty due to the lack of information about the subsurface structure which is known only at sparse sampled locations.
12
Concept of Stochastic Simulation
• Generation alternative equiprobable images of spatial distributions of objects or nodal values.
• Each alternative distribution is a stochastic simulation.
13
Types of Stochastic Models for Subsurface Characterization
D isc re te M od e ls C on tin u ou s M od e ls H yb rid M od e ls
S toch as tic M od e ls
14
Scales of Natural Variability
15
Definition of The Stochastic Process
A stochastic process can be defined as:
“a collection of random variables”.In a mathematical form:
the set {[x, Z(x,ζi)], x Rn }, i = 1,2,3...,m.
Z(x,ζ) is stochastic process, (random function), x is the coordinates of a point in n-dimensional space, ζ is a state variable (the model parameter), Z(x,ζi) represents one single realization of the stochastic process, i= 1,2,...,m (i: number of a realization of the stochastic process Z), Z(x0,ζ) = random variable, i.e., the ensemble of the realizations of the stochastic process Z at x0, and Z(x0,ζi)= single value of Z at x0. For simplification the variable ζ is generally omitted and the notation of this stochastic process is Z(x).
16
Realizations of A Stochastic Process
17
Uni-dimensional Stochastic Process
0 10 20 30 40Space or T im e Scale
-2.0
0.0
2.0
Par
amet
er
A stochastic process in which the variation of a property of a physical phenomenon is represented in one coordinate dimension is called uni-dimensional stochastic process. The coordinate dimension can be time as in time series, or space as in space series.
18
Spatial Random Fields
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
-3.3 -2.3 -1.3 -0.3 0.7 1.7 2.7
Y=Log (K )
Random fields are multi-dimensional stochastic processes.
19
Comparison in Terminology between
Statistical and Stochastic Theories Statistical Theory Stochastic Theory
Sample Realization
Population Ensemble
20
Probabilistic Description of Stochastic
Processes • Single Random Variable (Univariate).
• Multi Random Variables (Multivariate): Random Vector
21
Probability Distribution Function (Cumulative Distribution Function)
}{Pr)( zZ = zP Single random variable
where Pr{A} is a probability of occurrence of an event A, and P(A) is a cumulative distribution function of the event A, z is a value in deterministic sense.
The distribution function is monotonically nondecreasing.
1)(0)( = +P = -P
22
Probability Density Function (PDF)
dzzdP =
zz+zZ<z = zp
z
)(}Pr{lim)(0
The density function, p(z), of random variable Z is defined by,
Inversely, the distribution function can be expressed in terms of the density function as follows
z
-
dz zp = zP ')'()(
p(z) is not a probability, but must be multiplied by a certain region Δz to obtain a probability. P(z) is dimensionless, but p(z) is not. It has dimension of [z-1].
23
Graphical Representation of pdf and cdf
24
Log-normal Density
0.......,21)(
2
2)ln(
xex
xf x
xx
z
25
Derivation of a pdf from a Time Series Z(t)
t
zz+dz
dt1 dt2 dt3
T
3
1.1lim}Pr{lim)(
ii
T0z0z
tTz
= z
z+zZ<z = zp
3
1
1lim}Pr{i
iTt
T = z+zZ<z
p(z).z = prob. That Z(t) lies between z and z+z.
26
Joint Probability Density Function
z...zP
zzzz+zZ< z,...,z+zZ<zp
n
n
n
nnnn
zn
z
121
1111 )( = ...
}ΔΔ{ Prlim = )(
0..
01
zz
Here, z without index is a vector. Inversely, the joint distribution function can be expressed in terms of the joint density as follows,
nz
n
z
'dz'...dzpdpP-
1--
)'(... = )'( = )(1
zz'zzz
ijX
Y0
ZZ
1
p
2 3
27
Bivariate Normal pdf
28
Joint Probability Distribution Function
Consider Z as a random vector defined in a vector form as {Z1,Z2,…,Zn}T, where, Z1, Z2,… and Zn are single random variables. z is described by the joint distribution function of Z as,
)(Pr).( zZ ,...,zZ ,zZ = z .,,.z ,zP nn2211n21
1)(0)( = ...,+ ,+ ,+ ,+P
= ...,- ,- ,- ,-P
29
Graphical Representation of jpdf and jcdf
30
Marginal Probability Density Function The marginal probability density function is defined as follows,
dz...dzdz...dz dz p = zp ni+i-
n
i 1121
1
)(...)( z
-
-
dzzzp =zp
dzzzp =zp
1212
2211
),()(
),()(
In case of bivariate pdf:
31
Marginal Probability Distribution Function The marginal probability distribution function of a component Z1 of the random vector Z is obtained from the joint density function by the integration,
')(...
),,,Pr()(
12
1
dz dz...dz p =
<Z<-...<Z<-z<Z = zPz
n
n2111
z
The term between brackets is the marginal probability density function of the component Z1, and
P(z1) is called the marginal distribution function of the component Z1 of the random vector Z.
32
Conditional Probability Density Function The conditional density function of component Zn of the random vector Z given that the random components at n-1, n-2,…,1 have specified values is defined by
zz,..., z|z P = z,..., z | zp
n
n-nn-n
)()( 1111
the function p(zn zn-1,…,z1) can be expressed in a more convenient form as follows,
)()(),(
121121
z,...,z, zpp= z,...z, z | zp
n-n-n-n
z
where, p(z), is the joint density function of all the components of the vector Z. It can also be written,
)()( 321 z,..., z, z, z = pp nz
33
Conditional Probability Distribution Function (Cont.)
).()()(
z ,,..z ,zpz ..., ,z ,z ,zp = z,...z ,z | zp1-n21
n32112-n1-nn
If Z1, Z2,..., and Zn are independent random variables then the joint density function is the multiplication of the marginal density function of the individual random components. This can be expressed as follows,
)()()()( zp ...zp . zp = z ,...,z ,z ,zp n21n321
So, in conclusion, for independent random variables the following holds,
)()( zp= z,...z ,z | zp n12-n1-nn
34
Conditional Prob. Distribution Function
The conditional distribution function of one component Zn of a random vector Z given that the random components at n-1, n-2,…,1 have specified values is defined by,
}ΔΔPr{).(
11111111
121
z+zZ<z,..., z+zZ< z | zZ
= z.., ,z, z | zP
n-n-n-n-nn
n-n-n
where, the definition Prob{AB} is the conditional probability of event A given that, event B has occurred and is defined by,
}Pr{}{Pr}{Pr
B B A = B | A
where, AB is the conjunctive event of A and B.
35
Statistical Properties of Stochastic Processes
• Spatial or Temporal Properties.MeanVarianceCovariance
• Ensemble Properties.MeanVarianceCovariance
36
Spatial Average (Mean)
dZ| v |
= Zv ii )(
)(1x
xx
where, v(x) is the specified length, area or volume (for one, two or three dimensional space respectively) centred at x of measure v and index i is the i-th realization.
n
jjii Z
n Z
1
)(1x
where, n is the number of discrete points discretizing the volume v, index j is the j-th point on volume v.
Z
xv
Z
xZ 1
Z 2 Z i Z n
37
Spatial Mean Square Value
)(
22 )(1
x
xxv
ii dZ | v |
= Z
where, v(x) is the specified length, area or volume (for one, two or three dimensional space respectively) centred at x of measure v and index i is the i-th realization.
n
jjii Z
n Z
1
22 )(1x
where, n is the number of discrete points discretizing the volume v, index j is the j-th point on volume v.
Z
xv
Z
xZ 1
Z 2 Z i Z n
38
Spatial Variance
where, v(x) is the specified length, area or volume (for one, two or three dimensional space respectively) centred at x of measure v and index i is the i-th realization.
where, n is the number of discrete points discretizing the volume v, index j is the j-th point on volume v.
dZ - Z | v |
= Z - Z = S = ZVarv
iiiiZi i )(
222 )(1
)(][x
xxx
n
ji
2Z Z
-n S i
1i
2Z - )(
11
x j
)( 2ii
22Z Z - Z S i
39
Spatial Covariance
The covariance is a measure of the mutual variability of a pair of realizations; or in other words, it is the joint variation of two variables about their common mean.
)xv(
iiiiii dZ-Z +Z-+Z| v |
=Z,ZCov xxxsxsxxsx )()()()(1))()((
Z - Z Z - Zn
Z,ZCovn
j=iiiiii
)(
1
)()()()()(
1))()((s
jjjj xxs+xs+xs
xs+x
where, n(s) is the number of points with lag s.
40
Ensemble Statistical Properties
(Mathematical Expectation) The average of statistical properties over all possible realizations of the process at a given point on the process axis.
41
Ensemble Average (Mean)
dzz pz = ZE
)()()}({ xx oo
xo is the coordinate of a given point on the space axis, p(z) is probability density function of the process Z(x) at location xo, and E{.} is the expected value operator.
Zm
ZE m
ii
1
)(1)}({ xx oo
m is the number of realizations.
42
Ensemble Mean Square Value
dzz pz = ZE
)()()}({ 22 xx oo
m
iiZ
m ZE
1
22 )(1)}({ xx oo
43
Ensemble Variance
dzz pZ-E Z= Z-EZ=E σ Z
)()}({)()}({)( 222)( xxxx ooooxo
m
iiZ Z - E Z
m-σ1
22)( })({)(
11
xx ooxo
})({})({ 222)( xx ooxo
ZE - Z=E σ Z
44
Ensemble Covariance
dzdz, zzp=, ZZCov
)()())()(())()(( xs+xxs+xxs+x
p(z(x+s),z(x)) is the joint probability density function of the process Z(x) at locations x+s and x.
m
ijjjj Z - E Z.Z - E Z
m, Z+ZCov
1
})({)(})({)(1))()(( xxs+xs+xxsx
45
Some Terminology
• Stationarity (Statistical Homogeneity).
• Non-stationarity.• Intrinsic Hypothesis.• Ergodicity.
46
Stationarity
The stochastic process is said to be second-order stationary (weak sense) if:
1) The mean value is constant at all points in the field, i.e., the mean does not depend on the position. μ = ZE Z)}({ x
2) The covariance depends only on the difference between the position vectors of two points (xi-xj)= sij the separation vector, and does not depend on the position vectors xi and xj themselves. )()}({)()}({)())()(( sxxxxxx jjiiji =CovZ-EZ Z-EZ=E, ZZCov
σ = = Cov)Z(Var Z2)0(x
ijX
Y0
ZZ
1
p
2 3
47
Non-StationarityA stochastic process is called non-stationary, if the moments of the process are variant in space, i.e., from one position to another.
48
Example of Non-stationarity
0 20 0 40 0 600 80 0 1 00 0 1 20 0 14 0 0 16 0 0 1 800 2 00 0-4 00
-2 00
0
-10 -8 -6 -4 -2 0 2 4 -5 .0 -3 .0 -1 .0 1.0 3 .0
0 20 0 40 0 600 80 0 1 00 0 1 20 0 14 0 0 16 0 0 1 800 2 00 0-4 00
-2 00
0
0 .0 0.8 1 .5 2 .3 3.0
0 20 0 40 0 60 0 80 0 10 0 0 12 0 0 14 0 0 16 0 0 180 0 20 0 0-400
-200
0
- 8 - 6 - 4 - 2 0 2 4
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0H o r i z o n t a l D i s t a n c e ( m )
-4 00
-2 00
0
Dep
th (m
)
1 2 3 4
L og (H yd raulic C on d uc tiv ity m /day) L o g (H ydra u lic C o nd u c tivity m /d a y)
L og (H yd raulic C on d uc tiv ity m /day)L og (H yd ra ulic C on d uctiv ity m /da y)
(a ) N o n-S ta tio n a rity in T h e M ea n .
(b ) N o n -S ta tio na r ity in T h e V ar ia n ce .
(c ) N o n -S ta tio n arity in C o rrela tio n L en g ths.
(d ) G lo ba l N o n - S ta tio n a rity .
G eo lo g ica l S truc tu re .
0 20 0 40 0 60 0 80 0 10 0 0 12 0 0 140 0 16 0 0 180 0 20 0 0-40 0
-20 0
0
49
Intrinsic Hypothesis
The intrinsic hypothesis assumes that even if the variance of Z(x) is not finite, the variance of the first-order increments of Z(x) is finite and these increments are themselves second-order stationary. This hypothesis postulates that:
(1) the mean is the same everywhere in the field; and (2) for all distances, s, the variance of the increments,
{Z(x+s)-Z(x)} is a unique function of s so independent of x. A stochastic process that satisfies the stationarity of order two also satisfies the intrinsic hypothesis, but the converse is not true.
)(2)()(
0})()({2 sxsx
xsx
γ = -Z+ZE
= -Z+ZE
(s) is called the semi-variogram,
50
Intrinsic Hypothesis (cont.)
From practical point of view:
1. The intrinsic hypothesis is appealing, because it allows the determination of the statistical structure, without demanding the prior estimation of the mean.
2. For a stationary random process, where both a covariance and a sime-variogram are exist, it is easy to show the relationship between them as, )()0()( ss - Cov = Covγ
51
Comparison between Intrinsic Hypothesis and Second-order Stationarity
Intrinsic Hypothesis Second-order Stationarity
Less strict than 2nd order stationarity
More strict than Intrinsic hypothesis
Variogram Correlogram
If the phenomenon does not have a finite variance, the variogram will never have a horizontal asymptotic value.
0})()({ = -Z+ZE xsx
-Z+ZEγ 2)()(21)( xsxs
Z = ZE })({ x
Z+ZECov ZZ )(.)()( xsxs
finitebemustσ Z2
52
Ergodicity
Ergodicity is a statistical property which implies that:
(spatial statistics) are equivalent to (ensemble statistics).
This equivalence is achieved when the size of the space domain is sufficiently large or tends to infinity.
It is theoretical defined, but practically impossible.
Z E Z i )}({ xo
σ S Z2
Z i
2)( xo
53
Real (Lag) Domain Representation
of Stochastic Processes
Properties of stationary stochastic processes may be represented in a lag domain:
- auto-correlation function of the lag s, or - cross-correlation function of s.
Correlogram: represents the correlation coefficients between the values of the process versus the lag s.
54
Spatial Auto-Correlation It is a measure of the spatial correlation structure of a process.
σ
, Z+ZCov = ρZ
ZZ 2
)()()( xsxs
The auto-correlation function has the following properties:
)(- = )(0)(
1)0(
ZZ ss
ZZ
ZZ
ZZ
= =
Z - Z Z - Zn
Z,ZCovn
j=iiiiii
)(
1
)()()(
1))()((s
jj xs+xs
xs+x
55
Calculation of Auto-Correlation Function
56
High and Poor Correlations
57
Some Auto-Correlation of Series
58
2D Isotropic Exponential Auto Correlation
59
Statistical Isotropy and Anisotropy
A multi-dimensional stochastic process is said to be
Isotropic, if the process does not have a preferred direction, i.e., the variability in the process is the same in all directions.
Anisotropic, if the variability changes from one direction to another.
Isotropic Anisotropic
60
Integral ScaleThe integral scale Iz of autocorrelation function is defined as,
0
)( ss dρ = I ZZz
which implies that the average distance over which the process is autocorrelated in space. For practical applications, the integration is calculated over a certain limits [0, So] where, So is the smallest value of s at which the autocorrelation function becomes practically zero.
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
61
Correlation Scale (Range)
The correlation scale is defined as the distance over which the process is autocorrelated in space. It is calculated as the distance at which the autocorrelation function tends to zero. There are various ways, some authors suggest the threshold value taken as e-1 to others.
In case of 1D of linear auto-correlation,
sif =sρ
sifλ
| s | = sρ
ZZ
ZZ
0)(
1)(
one finds that the integral scale is related to the correlation length by the formula,
2 = I z
62
Correlation Range
63
Spatial Cross-Correlation
)())()(()(
22 21σ σ
, Y+ZCov = ρYZ
/ZYxsxs
The spatial cross-correlation represents a relation between two stochastic processes. It defines the degree of which two stochastic process are correlated as a function of separation lag.
positive or negative or zero
64
A Typical Variogram
65
Sime-Variogram Models
Sime-variogram models
]5.05.1[)( 3ssCsγ
ssCsγ )sin(1)(
s Csγ .)(
]1[)( se Csγ
]1[)(2ss e Cγ
66
Variogram Example Calculation
-Z+ZEγ 2)()(21)( xsxs
67
Uncorrelated, Orthogonal, Independent Random Vectors
)().(),(
0
jiji
jTi
jTij
Ti
ZpZ pZZpiftIndependen
ZZE ifOrthogonal
ZE Z E ZZE ifedUncorrelat
-Independence is a stronger condition than uncorrelatedness.
68
Spectral (Frequency) Domain Representation of Stochastic Processes
Properties of stochastic processes can be represented in the frequency domain, relating:
“ the square of amplitude of each sine or cosine component fitting the process versus ordinary frequency or its angular frequency or wave number”.
In this respect, the stochastic process is considered as made up of oscillations of all possible frequencies.
The diagram used for this presentation is called priodgram.
69
Decomposition of a Random Signal
0 1 2 3 4 5 6Frequency
0
20
40
60
80
10 0
(Am
plitu
de)^
2
70
Auto-Power (Variance) Spectral Density Function (Auto-PSD)
The term power is commonly seen in the literature. Its origin comes from the field of electrical and communication engineering:
power dissipated in an electrical circuit is proportional to the mean square voltage applied.
The adjective spectral denotes a function of frequency.
The concept of density comes from the division of the power (variance) of an infinitesimal frequency interval by the width of that interval.
The power spectrum describes the distribution of power (variance) with frequency of the random processes, and as such is real and non-negative.
71
(Variance) Spectral Density Function
The auto-power spectrum (spectral density function) for a process Z(x) is given by,
||z L
= z.zL
= SL
*
LZZ )(
1lim)()(1lim)( 2ωωωω
where, z(ω) is Fourier transform of the process Z(x), which is expressed as,
deZπ
= z -i
xxω xω)(21)(
and z*(ω) is the conjugate of z(ω) and ω is the angular frequency vector.
72
Calculation of Power Spectrum from a Signal
73
Properties of the Spectral Density Function
)()(
)(
0)(2
ωω
ωω
ω
S S
σ = dS
S
ZZZZ
Z- ZZ
ZZ
74
Cross-Power (Variance) Spectral Density
Function (Cross-PSD) The cross-PSD is defined between a pair of stochastic process. Cross-PSD is in general complex. The magnitude of the cross-PSD describes whether frequency components in a process are associated with large or small amplitudes at the same frequency in another process, and the phase of the cross-PSD indicates the phase lag or lead of one process with respect to the other one for a given frequency component. This expressed mathematically as,
)()(1lim)( ωωω y.z L
=S *
LZY
where, y*(ω) is the conjugate of y(ω), and y(ω) is Fourier transform of the process Y(x)
75
Relation between AutoCovariance Functions and AutoSpectral Density Functions
The covariance functions and spectral density functions are Fourier transform pairs. This can be expressed in mathematical forms using
Wiener-Khinchin relationships,
σ = dS = C
deS = C
deCπ
= S
Z- ZZZZ
-
iZZZZ
-
-iZZZZ
2)()0(
)()(
)(21)(
ωω
ωωs
ssω
sω
sω
76
Relation between Auto Correlation and Power Spectrum (examples)
77
Relation between Cross-Covariance Functions and Cross-Spectral Density
Functions
For cross-PSD and cross-correlation these relations are,
-
iZYZY
-
-iZYZY
deS =C
deC π
= S
ωωs
ssω
sω
sω
)()(
)(21)(
78
Summary of A Random Variable
79
Monte-Carlo Sampling
80
Generation of a Random Variable
81
Models of The Stochastic Approach
D isc re te M od e ls C on tin u ou s M od e ls H yb rid M od e ls
S toch as tic A p p roach
82
Mosaic Facies (Discrete) Models
Types of discrete models:• Object-based Models.• Sequential based Models:
-Markov Chains in 1-D, 2-D etc.-Markov Random Fields.-Sequential Indicator Simulation Models. -Random Lines Models.
In this approach one is aiming to construct - formation geological units, its geometric characteristics. - lithologies.- units dimensions (length, thickness, and width), - orientations and frequency of occurrence, etc..
83
Object-based Models
84
Sequential-based Models
-Markov Chains in 1-D.-Markov Chain in 2-D.
85
Theory of One-dimensional Markov ChainSS S
i0 1 i+ 1i-1 N2
l k q
,:)Pr()Pr(
p S Z | S ZS Z ,..., S Z ,S Z ,S Z | S Z
lkl1-iki
p0r3-in2-il1-iki
................
..
1
21
11211
nnn
lk
n
pp
pp
ppp
p 1,...,01
ppn
klklk
w p kN
lkN
)(lim1,0
...,
1
1
n
kkk
klkl
n
l
ww
n ,1 k ,w p wMarginal prob.
Transition prob.
86
Example on One-dimensional Markov Chain
87
Coupled Markov Chain
i0 1 i+ 1i-1 N2
S lS f
i0 1 i+ 1i-1 N2
S kS q
X-Chain
Y-Chain
kflmmilif1+ik1+i p S Y ,S X | S Y ,S X ,)Pr(
},...,,,...,,,....{. 2121, nnmflkkflm ssssssppp Transition probabilities
mllm www .Marginal probabilities
88
Coupled Chain on a Lattice Dark Grey (Boundary Cells)Light Grey (Previously Generated Cells)W hite (Unknown Cells)
i-1 ,j i,ji,j-1
1 ,1
N x ,N y
N x ,1
1 ,N y
N x ,j
nkp . p
p . p SZSZSZ p
p . p C
SYSYSXSXCS Z ,S Z | S Z
SYSYSXSXS Z ,S Z | S Z
SYSYS Z | S Z
SXSXS Z | S Z
f
vmf
hlf
vmk
hlk
mjiljikjiklm
n
f
vmf
hlf
mjkjlikimjiljikji
mjkjlikimjiljikji
mjkjmjikji
likiljikji
,...1.),|Pr(:
)Pr()Pr()(Pr
)Pr()Pr()(Pr
)Pr()(Pr
)Pr()(Pr
1,,1,,
1
1
111,,1,
111,,1,
11,,
1,1,
89
Coupled Chain Prob. In Two-State ModelTwo-State Model
1
1 22
1
2 11
1
2 22
1
1 11
2
1 11
2
1 22
2
1 21
2
1 12
2
2 11
2
2 22
2
2 21
2
2 12
1
1 21
1
1 12
1
2 12
1
2 21
P(ij,lk) = P(i,l) P (j,k) , i,j,l,k = 1,2 h v
The Transition Probabilities o f The Coupled C hain
k= l k<> lP(11,11) P(11,22) P (11,21) P(11,12) P(12,11) P(12,22) P (12,21) P(12,12)P(21,11) P(21,22) P (21,21) P(21,12) P(22,11) P(22,22) P (22,21) P(22,12)
90
Example of Two-State ModelNumerical Exam ple of Two-State Model
The Transition Probabilities of The Coupled Chain
k=l k<>lP(11,11) =0.250 P(11,22) =0.250 P(11,21) =0.250 P(11,12) =0.250 P(12,11) =0.250 P(12,22) =0.250 P(12,21) =0.250 P(12,12) =0.250P(21,11) =0.125 P(21,22) =0.375 P(21,21) =0.375 P(21,12) =0.125 P(22,11) =0.125 P(22,22) =0.375 P(22,21) =0.375 P(22,12) =0.125
P(ij,lk) = P(i,l) P(j,k) , i,j,l,k = 1,2 h v
The Horizontal Chain The Vertical Chain 1 2 1 2 1 0.50 0.50 1 0.50 0.50 2 0.25 0.75 2 0.50 0.50
Normalized Transition Probabilities of The Coupled Chain
k=l k<>lP(11,11) =0.500 P(11,22) =0.500 P(11,21) =0.000 P(11,12) =0.000 P(12,11) =0.500 P(12,22) =0.500 P(12,21) =0.000 P(12,12) =0.000P(21,11) =0.250 P(21,22) =0.750 P(21,21) =0.000 P(21,12) =0.000 P(22,11) =0.250 P(22,22) =0.750 P(22,21) =0.000 P(22,12) =0.000
P(ij,kk) = P(i,f) P(j,f)
P(i,k) P(j,k)i,j,k =1,2
f
91
One-dimensional Markov Chain Conditioned on Future States
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92
Coupled Markov Chain (application) Two-dimensional Cross-sectional Panel of the Fluvial Succession of the Medial
Area of the Tόrtola Fluvial System, Spain
Length of The Section (m) = 648. Depth of The Section (m) = 115.Sampling interval in X-axis (m) = 9. Sampling interval in Y-axis (m) = 2.5
Horizontal Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.893 0.009 0.005 0.000 0.000 0.000 0.000 0.093 2 0.000 0.796 0.011 0.000 0.000 0.000 0.000 0.194 3 0.000 0.000 0.989 0.000 0.000 0.000 0.000 0.011 4 0.006 0.000 0.013 0.885 0.000 0.000 0.000 0.096 5 0.074 0.000 0.000 0.074 0.593 0.037 0.000 0.222 6 0.000 0.013 0.000 0.000 0.000 0.946 0.000 0.040 7 0.040 0.000 0.000 0.000 0.000 0.000 0.940 0.020 8 0.007 0.006 0.002 0.007 0.005 0.005 0.001 0.968
Vertical Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.591 0.000 0.000 0.000 0.014 0.000 0.042 0.353
2 0.011 0.753 0.097 0.000 0.000 0.000 0.000 0.1403 0.032 0.000 0.623 0.000 0.000 0.238 0.000 0.1073 0.000 0.025 0.000 0.662 0.013 0.000 0.000 0.2995 0.111 0.000 0.000 0.074 0.519 0.000 0.000 0.2966 0.000 0.000 0.026 0.032 0.006 0.084 0.000 0.8517 0.120 0.000 0.000 0.100 0.000 0.000 0.360 0.420
8 0.029 0.008 0.039 0.017 0.003 0.031 0.010 0.863
1 2 3 4 5 6 7 8
93
Coupled Markov Chain (application cont.)
0 50 100 150 200 250 300-80
-60
-40
-20
0
0 50 100 150 200 250 300
-80
-60
-40
-20
0
1 2 3 4 5 6 7 8
0 50 100 150 200 250 300
-80
-60
-40
-20
0
94
Application of C_CMCM
1
2
3
4
5
6
7
8
0 50 100 150 200 250 30 0-80
-60
-40
-20
0
0 50 10 0 150 200 250 300-80
-60
-40
-20
0
0 50 10 0 150 200 250 300-80
-60
-40
-20
0
0 50 100 150 200 250 300-80
-60
-40
-20
0
0 50 100 150 200 250 300-80
-60
-40
-20
0
0 50 100 15 0 200 250 300-80
-60
-40
-20
0
95
Random Lines Model
96
Random Sets Mosaic Model
97
Continuous Models
- Multi-variate Method.
- Nearest Neighbour Method.
- Turing Bands Method.
98
Multivariate Normal Method
ijX
Y0
ZZ
1
p
2 3
99
Nearest Neighbour Method
100
Turning Bands Method
101
Comparison Between Various Methods
102
Hybrid Models
103
Stochastic Differential Equations (SDEs)
Stochastic differential equation (SDE) = Differential equations for random functions (stochastic processes)
= Classical differential equation (DE) +
Random functions, coefficients, parameters and boundary or initial values
Ω = yΦyxK
y +
xΦyxK
x yyxx
0),(),(
104
Solving SDEs
Analytical Approaches G reen 's F u n c tion A p p roach
P ertu rb a tion M eth odS p ec tra l M eth od
Num erical ApproachesM on teC arlo M eth od
S o lvin g S D E s
105
Spectral Method
The dependent variable and parameter in a stochastic differential equation are represented in terms of its mean or expected value denoted with an angle brackets, and some fluctuations around the mean denoted by a prime, as follows:
Φ + ΦΦ = where, Y is written as the perturbed parameter, Y is the mean or expected value of the parameter, E{Y}, Y´ is a perturbation around the mean value of the parameter, so E{Y´}= 0. Similarly, Φ is the perturbed variable, Φ is the mean or expected value of the variable, E{Φ}, and Φ` is a perturbation around the mean value of the variable, E{Φ`}= 0.
Y + YY =
106
Spectral Method (Cont.1)
Assumptions:1. The perturbations are relatively small compared to the mean
value, so that second order terms involving products of small perturbations can be neglected.
2. The stochastic inputs parameters and the outputs variables are second order stationary so that they can be expressed in terms of the representation theorem.
Procedure:1. Introducing the expressions into the differential equation.2. Taking the expected value of the equation results in two new
equations, one for the first moment (mean) and the other for the perturbations.
3. The first is a deterministic differential equation, which can be solved analytically to get the solution for the mean of the dependent variable as a function of the mean of the parameter.
4. The second equation is transformed in the spectral domain by using Fourier-Stieltjes representation theorem.
107
Spectral Method (Cont.2)
)(kkx dZ e = Y Yi
)(kkx dZ e = Φ Φi
5. The following integral transformation is used,
Where k is wave number vector, x is space dimension vector, Z(k) is a random function with orthogonal increments, i.e., non-overlapping differences are uncorrelated and dZ(k) is complex amplitudes of the Fourier modes of wave number k. The spectral density function SYY(k) of Y’ is related to the generalized Fourier amplitude, dZY by
0)}()({ 2121 k k if , = kdZ.kdZE *YY
k = k if dk, kS = kdZ.kdZE YY*YY 21121 )()}()({
The asterisk, *, denotes the complex conjugate.
108
Spectral Method (Cont.3)
6. By using the above representation and substituting them into the stochastic differential equation of perturbation, one can get the spectrum of the variable as a function of the spectrum of the parameter.7. The spectral density function is the Fourier transform of its auto-covariance function, which can be expressed mathematically as follows:
ssk ks dC eπ
= S ΦΦ-i
ΦΦ )(21)(
where s = lag vector of the auto-covariance function. 8. By using Wiener-Khinchin theorem, one can write,
kks ks dS e = C ΦΦ-i
ΦΦ )()(
)0(2ΦΦΦ = Cσ
109
Perturbation Method
The parameter,Y, (e.g. conductivity) and the variable, Φ, (e.g. head) can be expressed in a power series expansion as,
......Yβ + Y + βYY = o 22
1
......β + + β = o 22
1where, β is a small parameter (smaller than unity). These expressions are introduced in the differential equations of the system to get a set of equations in terms of zero- and higher-order expressions of the factor β. The equation that is in terms of zero β corresponds to the mean head. The equation that is in terms of first-order of β corresponds to the head perturbation. In practice, only two or three terms of the series are usually evaluated.
110
MonteCarlo Method1. Assumption of the pdf of the model parameters or joint pdf. The
pdfs are based on some field tests and/or laboratory tests. 2. Generation of random fields of the hydrogeological parameters to
represent the heterogeneity of the formation.3. By using a random number generator, one generates a realization
for each one of these parameters. The parameter generation can be correlated or uncorrelated depending on the type of the problem.
4. With this parameter realization a classical numerical flow or/and transport model is run and a set of results is obtained.
5. Another random selection of the parameters is made and the model is run again, and so on.
6. It's necessary to have a very large number of runs, and the output model results corresponding to each input is obtained which can be represented mathematically by the stochastic process Φ(x,ζi).
7. Statistical analysis of the ensemble of the output (i.e. Φ(x,ζi) for i = 1,2,...m, can be made to get the mean, the variance, the covariance or the probability density function for each node with a location x in the grid.
111
Comparison between Analytical and MonteCarlo Methods
Item Analytical MonteCarloSolution defined over a
continuumdefined over a grid.
Stationarity of the variables
input and output variables should be stationary
no need for stationarity assumption.
Probability distribution of input variables
no need to define PDF of the input variable in some applications.
the PDF of the input variables must be known.
Handling variability limited to small variability.
not limited to small variability.
112
Comparison between Analytical and MonteCarlo Methods (cont.1)
Item Analytical MonteCarloLinearity versus non-linearity
based on linearized theories or weakly-nonlinearity.
it can address both cases.
Outcome of the method
closed form solution of moments.(limited only for the first two moments)
numerical values used to calculate moments of the independent variables. (One can calculate the complete PDF).
113
Comparison between Analytical and MonteCarlo Methods (cont.2)
Item Analytical MonteCarloSpatial structure of the variability
simple forms of auto-covariance models
simple and compound (nested) forms of auto-covariances.
Sources of errors number of simplifying assumptions such as, the form of mean and covariance function, the geometry of the domain and the boundary conditions.
sampling (finite number of realizations) and discretization errors are introduced because of approximation of the governing equations.
Time and computer effort
limited (to calculate the values).
time consuming.
114
Comparison between Analytical and MonteCarlo Methods (cont.3)
Item Analytical MonteCarloperforming conditioning to field measurements
difficult easy
handling more than one stochastic variable
if it is possible, it is too difficult.
it is easy to handle more than one variable.
115
Kriging
116
Kriging Example
117
Conditional Simulations
From a practical point of view, it is desirable that the random fields not only reproduce the spatial structure of the field but also honour the measured data and their locations.
This requires an implementation of some kind of conditioning, so that the generated realizations are constrained to the available field measurements.
118
Conditional Simulations (cont.)
119
Methods of Conditioning
D irec t M eth od s"M etrica l M eth od s "
In d irec t M e th od s"K rig in g M eth od "
M eth od s o f C on d it ion in g
120
Indirect Conditioning by Kriging
(1) A kriged map is generated from the field data with the sampled locations which will be smoother than reality. (2) An unconditional simulated field is generated by TBM from the data which reproduces the spatial structure of the underlying random function.(3) Allocation of the unconditional values (pseudo measurements) at the sites of measurements is done on the simulated map in step 2.(4) Another kriged map is generated from the pseudo measurements.(5) A pseudo error is calculated by subtracting the kriged map in step 4 from the unconditional simulation in step 2.(6) The conditional simulation map is generated by adding the pseudo error in step 5 to the kriged map in step 1. So,
)Z - Z( + Z = Z kususkdcs
Zcs is the required conditional simulation, Zkd is the kriged map from the real data, Zus is the unconditional simulation, Zkus is the kriged map with the pseudo measurements.
121
Example of Conditioning by Kriging
122
Correlation Matrix
123
Fractals
124
Pdf of Sine wave
125