Lecture (6)Lecture (6)
Estimation Problem
(Kriging),
Conditional Simulations
and
Reduction of
Uncertainties

The Estimation ProblemThe Estimation Problem
The estimation problem is to determine the value of the quantity Zo
for point (xo,yo) which has not been measured.
Z2
*(x2,y2)
Z1 Z3
*(x1,y1) *(x3,y3)
Zo???
*(xo,yo)
By continuously modifying the position (xo, yo) we shall estimate
the whole field of Z.

Estimation MethodsEstimation Methods
Usual Estimation Methods:
1. Linear interpolation (by hand).
2. Trend surface analysis.
3. Least squares fitting.
The drawbacks of the methods:
1. They cannot give the confidence interval of the
estimation.
2. They do not take into account the spatial structure
of the phenomenon.

Estimation byEstimation by KrigingKriging
Assumptions:
• B.L.U.E.= Best Linear Unbiased Estimator.
Best = minimizing the estimation
(error) variance.
Linear = linear combination of weights.
Unbiased means E{Z*o}=E{Zo}
• Stationary = no trend assumed in the data.
• Normal distribution of the data.

SimpleSimple KrigingKriging ModelModel
The estimation of Z*o is a linear combination of all available measurements of Z,
[ ]*
1
n
o
o i i
i
Z Z
=
= λ∑
Z*o is the estimator,
Zi measurements at the n-points xi (i=1,2,3,…n).
λi is optimal weights to be computed.

Unbiased ConditionUnbiased Condition
{ } { }
{ }
{ }
{ } { }
*
1
1
( ) ,
o o
n
o
i i o
i
n
o
i i o
i
E Z E Z
assume m E Z
substitution
E Z E Z
Linearity
E Z E Z
=
=
=
= ∀
⎧ ⎫
λ =⎨ ⎬
⎩ ⎭
λ =
∑
∑
x x

Unbiased Condition (cont.)Unbiased Condition (cont.)
{ } { }
{ } { }
1
1
1
1
but from our assumption,
1
n
o
i i o
i
i o
n
o
i
i
n
o
i
i
n
o
i
i
E Z E Z
E Z E Z m
m m
m m
=
=
=
=
λ =
= =
λ =
λ =
λ =
∑
∑
∑
∑ Condition of unbiasedness

The Variance of the EstimatorThe Variance of the Estimator
( )
*
o oError of estimation,(Z -Z ) should be small
2
*2
o oSK
E Z Z⎧ ⎫σ = −⎨ ⎬
⎩ ⎭
•SK=Simple Kriging variance
•σ2=Variance about estimated point
Z is “true” value
Z* is the estimate produced by kriging
•Seek to minimize to determine weights (λ ’s)

Derivation of the Error VarianceDerivation of the Error Variance
( )
{ } { }2
1
2
*2
2
1
2
2
1 1
2
2
1
2
n
o
o i o i
i
o oSK
n
o
o i i
i
n n
o o
E o o i i i i
i i
n
o
E Z E Z Z E i i
i
E Z Z
E Z Z
Z Z Z Z
Z
=
=
=
= =
= − λ +
=
⎧ ⎫σ = −⎨ ⎬
⎩ ⎭
⎧ ⎫⎛ ⎞⎪ ⎪= − λ∑⎜ ⎟⎨ ⎬
⎝ ⎠⎪ ⎪⎩ ⎭
⎧ ⎫⎛ ⎞⎪ ⎪− λ + λ∑ ∑⎜ ⎟⎨ ⎬
⎝ ⎠⎪ ⎪⎩ ⎭
⎧ ⎫⎛ ⎞⎪ ⎪λ∑∑ ⎜ ⎟⎨ ⎬
⎝ ⎠⎪ ⎪⎩ ⎭

Derivation of the Error Variance (cont.)Derivation of the Error Variance (cont.)
2
2
1 1 2 2
1 1 1 1
2
1 1 1
( ) ( )( )
( ) ( )
...
n n n n
o o o o o o o
i i i i i i n n i i
i i i i
n n n
o o o
i i j j i i
i j i
a b a b a b
a a b b a b
aa ab ba bb
Z Z Z Z Z Z Z
Z Z Z
= = = =
= = =
+ = + +
= + + +
= + + +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
λ =λ λ +λ λ λ λ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞
λ = λ λ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∑ ∑ ∑ ∑
∑ ∑ ∑
1 1
n n
o o
j i j i
j i
Z Z
= =
= λ λ∑ ∑

Derivation of the Error Variance (cont.)Derivation of the Error Variance (cont.)
{ } { }
{ } { }
{ } { } { }
2 2
1
2 2
1
2 2
1 1 1
2 2
1 1
2
2
1
2
1 1
2
2 ( , ) ( ,
n
o
SK o i o i
i
n
o
SK o i o i
i
n n n
o o o
SK o i o i j i j i
i j i
n n
o o o
SK o i o i j i i j
i j
n
o
E Z E Z Z E i i
i
n n
o o
E Z E Z Z E j i j i
j i
E Z E Z Z E Z Z
Cov Z Z Cov Z Z
Z
Z Z
=
=
= = =
= =
σ = − λ +
=
σ = − λ +
= =
σ = − λ + λ λ
σ =σ − λ + λ λ
⎧ ⎫⎛ ⎞⎪ ⎪λ∑∑ ⎜ ⎟⎨ ⎬
⎝ ⎠⎪ ⎪⎩ ⎭
⎧ ⎫
λ λ∑ ∑∑ ⎨ ⎬
⎩ ⎭
∑ ∑ ∑
∑ ∑ 1
)
n
i=
∑

Minimizing The Error VarianceMinimizing The Error Variance
( )
2
2
1 1 1
1 11
1 1
0
2 ( , ) ( , ) 0
( , )2 ( , )
0 0
( , ) ... ( , ) ... ( , )
2
SK
o
k
n n n
o o o
o i o i j i i jo
i i jk
n nn
o oo
j i i ji o i
i ji
o o
k k
o o o
o k o k n o n
Cov Z Z Cov Z Z
Cov Z ZCov Z Z
Cov Z Z Cov Z Z Cov Z Z
= = =
= ==
∂σ
=
∂λ
⎛ ⎞∂
σ − λ + λ λ =⎜ ⎟
∂λ ⎝ ⎠
⎛ ⎞⎛ ⎞
∂ λ λ∂ λ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠− + =
∂λ ∂λ
∂ λ + + λ + + λ
−
∂λ
∑ ∑ ∑
∑ ∑∑
1 1
1 1
1
1
1
( , ) ... ( , ) ... 0
2 ( , ) ( , )
o
k
n n
o o o o
i i k i i ko
i ik
o n
o
o k i io
ik
Cov Z Z Cov Z Z
Cov Z Z Cov Z Z
= =
=
∂ ⎛ ⎞
+ λ λ + + λ λ + =⎜ ⎟∂λ ⎝ ⎠
∂λ ⎛ ⎞
− + λ⎜ ⎟∂λ ⎝ ⎠
∑ ∑
∑ 1 1
1
1 1
( , )
... ( , ) ( , ) ... 0
n
o o
i io
ik
o n n
o o ok
i i k k i i ko o
i ik k
Cov Z Z
Cov Z Z Cov Z Z
=
= =
∂ ⎛ ⎞
+ λ λ +⎜ ⎟∂λ ⎝ ⎠
∂λ ∂⎛ ⎞ ⎛ ⎞
+ λ + λ λ + =⎜ ⎟ ⎜ ⎟∂λ ∂λ⎝ ⎠ ⎝ ⎠
∑
∑ ∑
We seek the weights λi that minimize the error variance.

Minimizing The Error Variance (cont.)Minimizing The Error Variance (cont.)
( )1
1
1
2 ( , ) ( , )
o
n
o
o k i io
i
k
Cov Z Z Cov Z Z
=
∂λ
− + λ
∂λ
∑ ( )
( ) ( )
1 1
1
1 1
1 1 2 2
1
( , )
... ( , ) ( , ) ... 0
2 ( , ) ( , )
2 ( , ) ( , ) ( , ) ...
( , ) ( , ) ... 0
n
o o
i io
i
k
o
n n
o o ok
i i k k i i ko o
i i
k k
o
o k i i k
o o
o k k k
n
o o
i i k k k k
i
Cov Z Z
Cov Z Z Cov Z Z
Cov Z Z Cov Z Z
Cov Z Z Cov Z Z Cov Z Z
Cov Z Z Cov Z Z
=
= =
=
∂
+ λ λ +
∂λ
∂λ ∂
+ λ + λ λ + =
∂λ ∂λ
− + λ
− + λ + λ +
+ λ + λ + =
∑
∑ ∑
∑
1 1
1
1
( , ) 0
because ( , ) ( , ),so,
2 ( , ) 2 ( , ) 0
( , ) ( , )
n n
o
i k i
i i
i k k i
n
o
o i ik k
i
n
o
i i ok k
i
Cov Z Z
Cov Z Z Cov Z Z
Cov Z Z Cov Z Z
Cov Z Z Cov Z Z
= =
=
=
+ λ =
=
− + λ =
λ =
∑ ∑
∑
∑

KrigingKriging Matrix FormMatrix Form
1 1 1 2 1 1 1
2 1
1
**
( , ) ( , ) ... ( , ) ( , )
( , ) . ... . . .
. . ... . . .
( , ) . ... ( , ) ( , )
Once this system is solved we get the estimate,
n o
n n n n o n
o o
Cov Z Z Cov Z Z Cov Z Z Cov Z Z
Cov Z Z
Cov Z Z Cov Z Z Cov Z Z
Y m Z
λ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥=
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
λ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
= + =
1
2 *
0
1
var( ) (0) ( , )
n
o
i i
i
n
o
SK o o i i
i
m Z
Z Z Cov Cov Z Z
=
=
+ λ
σ = − = − λ
∑
∑

Some Useful Remarks RegardingSome Useful Remarks Regarding KrigingKriging
*
*
= (value measured) I.e. =1, =0,
( ) 0 (no Uncertainty in a measured Point)
If the error is assumed to be normally distributed,
then
1. Exact Interpolator:
2. Confidence Intervals:
k k
k k k i
k
Z Z i k
Var Z Z
λ λ ≠
− =
( )*
*
1
we can estimate 95% confidence intervals of the estimation as,
=
= 2
3. The Kriging system does not depend explicitly on the measurement values
. Only the locations of t
o o
n
o
o i i
i
i i
Var Z Z
Z Z
Z x
=
σ −
λ ± σ∑
he measurement points are needed
to compute the weights.
4. Drawing contour maps: by solving the Kriging system,
one can estimate at any point .
5. Kriging estimator is generally much smoother tha
o oZ x
n
the actual spatially variable field.

SimpleSimple KrigingKriging ExampleExample
Computer code SIMKRIG.exe
Data files: SIMKRIG.dat,
Lx, Ly 100., 50.
Dx,Dy, 1., 1.
λx , λy 20., 10.
SKFIELD.dat
No. of
Measurements mean_k sd_k
10 10 50
X_COR Y_COR Y
0 0 10
100 0 15
0 50 12
100 50 20
30 25 15
60 25 10
80 10 20
10 5 13
50 50 16
50 0 16
10 15
12 20
15 10
20
13
16
16
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
Data
Kriged Map
Error variance
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
5
30
55
80
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5

LinearLinear GeostatisticsGeostatistics
• Simple Kriging.(mean is known and const.)
• Ordinary Kriging.(mean is unknown and const.)
• Non-stationary Kriging. Use moving neighborhood or Universal
Kriging.
• Co-Kriging (Coregionalization): If more than one variable is
sampled.
• Ref.
Deutsch, C. V. and Journel, A. G. 1992, GSLIB; Geostatistics
Software Library and user’s guide: Oxford Univ. Press, New
York, 340p.
'
*
1 1
n n
o i i j j
i j
Z Z
= =
= λ + ν ϕ∑ ∑

Conditional SimulationsConditional 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.

Representation of a Conditional SimulationRepresentation of a Conditional Simulation

Methods of ConditioningMethods of Conditioning
Direct Methods
"Metrical Methods"
Indirect Methods
"Kriging Method"
Methods of Conditioning

Indirect Conditioning byIndirect Conditioning by KrigingKriging
(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,
( - )cs kd us kusZ Z Z Z= +
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.

Graphical Illustration of Conditioning byGraphical Illustration of Conditioning by
KrigingKriging
(1) A kriged map is generated from
the data:
(2) Unconditional simulation is
generated from the data:
(3) Allocation pseudo
measurements.
(4) Kriged map is generated from the
pseudo measurements:
(5) A pseudo error =kriged map(step
4) - the unconditional simulation
in step 2.
(6) Conditional simulation = the
pseudo error in step 5 + the
kriged map in step 1:
kdZ
usZ
kusZ
csZ
( - )cs kd us kusZ Z Z Z= +

Program to perform Conditional SimulationProgram to perform Conditional Simulation
byby KrigingKriging
0 20 40 60 80 100 120 140 160 180 200
0
20
40
60
80
100
3.5
10 15
12 20
15 10
20
13
16
16
1614
13
2010
15
20
3.5
10 15
12 20
15 10
20
13
16
16
1614
13
2010
15
20
0 20 40 60 80 100 120 140 160 180 200
0
20
40
60
80
100
0 20 40 60 80 100 120 140 160 180 200
0
20
40
60
80
100
3.5
10 15
12 20
15 10
20
13
16
16
1614
13
2010
15
20
0 20 40 60 80 100 120 140 160 180 200
0
20
40
60
80
100
Data
Kriged Map
Single Realization
of Conditional Simulation
CSTBM.DAT
20, 20, 1 corr.length x, corr.length y, MC
100, 1, 0 No. of TB lines, Dz, RLD (0 OR 1)
0.25, 100 Dk, Max. Harmonics
19999, 20 Seed, ib
Csfield.dat
200 100 Lx, ly
1 1 dx,dy
17 no. of measurements
X_coor Y_coor Y
0 0 10
100 0 15
0 50 12
100 50 20
30 25 15
60 25 10
80 10 20
10 5 13
50 50 16
50 0 16
200 100 16
0 100 14
200 0 13
200 25 20
0 25 10
100 100 15
80 40 20

Conditioning in OneConditioning in One--dimensional Markovdimensional Markov
ChainChain
i0 1 i+1i-1 N2
SkSl Sq
)1(
)(
)1(
)(
1
11
11
1
1
1
1
1
11
1
1
1
1
1
)(Pr
)Pr().|(Pr
)(Pr).|(Pr).|(Pr
)(Pr
),(Pr
),(Pr).|(Pr
)(Pr
),(Pr
),(Pr),|(Pr
)(Pr
),(Pr
),,Pr(
)(Pr
)(Pr
+−
−
+−
−
−
−−
−−
−
−
−
−
−
−−
−
−
−
−
−
=
====
===
=====
====
==
====
====
==
=====
====
==
===
====
===
iN
lq
iN
kqlk
qlk
iN
lq
lk
iN
kq
qNliki
liliqN
lilikikiqN
qNliki
qNli
kilikiqN
qNliki
qNli
kilikiliqN
qNliki
qNli
qNkili
qNliki
qNliki
p
pp
p
p
pp
SZ,SZ|SZ
SZSZSZ
SZSZSZSZSZ
SZ,SZ|SZ
SZSZ
SZSZSZSZ
SZ,SZ|SZ
SZSZ
SZSZSZSZSZ
SZ,SZ|SZ
SZSZ
SZSZSZ
SZ,SZ|SZ
SZ,SZ|SZ

Coupled Markov Chain “CMC” in 2DCoupled Markov Chain “CMC” in 2D
Dark Grey (Boundary Cells)
Light Grey (Previously Generated Cells)
White (Unknown Cells)
i-1,j i,j
i,j-1
1,1
Nx,Ny
Nx,1
1,Ny
Nx,j
, , 1, , 1
, 1, , 1 ,,
Unconditioinal Coupled Markov Chains
: Pr( | , ) . 1,...
Conditioinal Coupled Markov Chains
: Pr( | , , )x
h v
lk mk
lm k i j k i j l i j m h v
lf mf
f
i j k i j l i j m N j qlm k q
h
lk
.p p
p Z S Z S Z S k n
.p p
p Z S Z S Z S Z S
.p
− −
− −
= = = = = =
= = = = = =
∑
( )
( )
, 1,... .
x
x
h N i v
kq mk
h h N i v
lf fq mf
f
.p p
k n
. .p p p
−
−
=
∑

Coupled Markov Chain (application)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 0.140
3 0.032 0.000 0.623 0.000 0.000 0.238 0.000 0.107
3 0.000 0.025 0.000 0.662 0.013 0.000 0.000 0.299
5 0.111 0.000 0.000 0.074 0.519 0.000 0.000 0.296
6 0.000 0.000 0.026 0.032 0.006 0.084 0.000 0.851
7 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

Coupled Markov Chain (application cont.)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

Application of C_CMC Single RealizationsApplication of C_CMC Single Realizations
1
2
3
4
5
6
7
8
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 50 100 150 200 250 300
-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 50 100 150 200 250 300
-80
-60
-40
-20
0

Application of C_CMC Single RealizationsApplication of C_CMC Single Realizations
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
0 50 100 150 200
-10
-5
0
1 2 3 4
Lithology Coding
0 80 160 240
-10
-5
0
1
2
3
4
2 boreholes
9 boreholes
25 boreholes
31 boreholes

Many Realizations: Hypothetical ExampleMany Realizations: Hypothetical Example
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
1
2
3
4
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0

Probability MapsProbability Maps
⎪
⎩
⎪
⎨
⎧ =
=
.0
1
)(
otherwise
SZif
ZI
kij
ijk
Let the realizations be numbered 1,…, MC, and let Zij
(R) be the
lithology of cell (i,j) in the Rth realization. The empirical relative
frequency of lithology Sk at cell (i,j) is:
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
1
2
3
4
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
( )∑=
=
=
=
MC
R
R
k
k
R
k
ij ji
ji
ZI
MCMC
SZ
1
)(
)(
,
, 1}{#
π
k
ijπPlots of

Procedure for Extracting a Final GeologicalProcedure for Extracting a Final Geological
ImageImage
⎪
⎩
⎪
⎨
⎧ =
=
.0
1
)(
otherwise
SZif
ZI
kij
ijk
( )∑=
=
=
=
MC
R
R
k
k
R
k
ij ji
ji
ZI
MCMC
SZ
1
)(
)(
,
, 1}{#
π
Let the realizations be numbered 1,…, MC, and let Zij
(R) be the
lithology of cell (i,j) in the Rth realization. The empirical relative
frequency of lithology Sk at cell (i,j) is:
}...,,max{ 21 n
ijijij
l
ij ππππ =
In the final image Z* the lithology at cell (i, j) will be the lithology
which occurs most frequently in the MC realizations. So, if Sl is such
that
Zij
*= Sl.

Program WELLLOGProgram WELLLOG
Input Data for preInput Data for pre--processingprocessing
Boreho01.dat
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
.
.
Wells
16 no. of boreholes
1, 0.0 well no. and X_Coor.
2, 21 .
3, 34 .
4, 39 .
5, 46
6, 56
7, 76
8, 96
9, 126
10, 152
11, 172
12, 194
13, 214
14, 234
15, 256
16, 276
borehol 1 borehol 2 borehol 3 borehol 4 borehol 4 borehol 6 borehol 7 borehol 8 borehol 9 borehol 10 borehol 11 borehol 12 borehol 13 borehol 14 borehol 15
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
Boreho02.dat
Boreho02.dat
Boreho03.dat
Boreho04.dat
.
0 50 100 150 200 250
-10
-5
0
0
1
2
3
4
5
T
T=p
v
lq
n
q
v
lkv
lk
∑=1GEO.DAT
7 No. of states
276, 14.2 Lx, Ly
3, 0.1 Dx, Dy

Program WELLLOG Output Data for preProgram WELLLOG Output Data for pre--
processingprocessing
Vertical Sampling Interval=0.1 m
0 50 100 150 200 250
-10
-5
0
0
1
2
3
4
5
S. 1 2 3 4 5 6 7
1 .879 .103 .009 .000 .009 .000 .000
2 .026 .911 .046 .009 .003 .000 .005
3 .003 .030 .897 .044 .010 .000 .016
4 .000 .006 .094 .869 .031 .000 .000
5 .000 .000 .003 .010 .961 .000 .026
6 .009 .014 .009 .005 .000 .963 .000
7 .000 .000 .000 .000 .000 .000 1.00
Well.res
Well.grd

Program F_CMC2 Input DataProgram F_CMC2 Input Data
GEOSIM
7, 1, 30 No. of sataes,Seed , MC
276.00, 14.20 Lx, Ly
3.0, 0.10 Dx, Dy
Hor. Transitions
0.922 0.015 0.015 0.015 0.015 0.015 0.001
0.015 0.922 0.015 0.015 0.015 0.015 0.001
0.015 0.015 0.922 0.015 0.015 0.015 0.001
0.015 0.015 0.015 0.922 0.015 0.015 0.001
0.015 0.015 0.015 0.015 0.922 0.015 0.001
0.015 0.015 0.015 0.015 0.015 0.922 0.001
0.001 0.001 0.001 0.001 0.001 0.001 0.992
Vert. Transitions
0.879 0.103 0.009 0.000 0.009 0.000 0.000
0.026 0.911 0.046 0.009 0.003 0.000 0.006
0.003 0.031 0.897 0.044 0.010 0.000 0.016
0.000 0.005 0.094 0.869 0.031 0.000 0.000
0.000 0.000 0.003 0.010 0.961 0.000 0.026
0.009 0.014 0.009 0.005 0.000 0.962 0.000
0.001 0.001 0.001 0.001 0.001 0.001 0.994
WELLSIM
16 no. of boreholes
1, 0.0 well no. and X_Coor.
2, 21 .
3, 34 .
4, 39 .
5, 46
6, 56
7, 76
8, 96
9, 126
10, 152
11, 172
12, 194
13, 214
14, 234
15, 256
16, 276
WELLS
16 no. of boreholes
1, 0.0 well no. and X_Coor.
2, 21 .
3, 34 .
4, 39 .
5, 46
6, 56
7, 76
8, 96
9, 126
10, 152
11, 172
12, 194
13, 214
14, 234
15, 256
16, 276
Well.grd

Horizontal Transition Probability MatricesHorizontal Transition Probability Matrices
S. 1 2 3 4 5 6 7
1 .500 .100 .100 .100 .100 .100 .000
2 .100 .500 .100 .100 .100 .000 .100
3 .100 .100 .500 .100 .100 .000 .100
4 .100 .100 .100 .500 .100 .000 .100
5 .100 .100 .100 .100 .500 .100 .000
6 .001 .001 .001 .001 .001 .994 .001
7 .001 .001 .001 .001 .001 .001 .994
S. 1 2 3 4 5 6 7
1 .879 .103 .009 .000 .009 .000 .000
2 .026 .911 .046 .009 .003 .000 .005
3 .003 .030 .897 .044 .010 .000 .016
4 .000 .006 .094 .869 .031 .000 .000
5 .000 .000 .003 .010 .961 .000 .026
6 .009 .014 .009 .005 .000 .963 .000
7 .001 .001 .001 .001 .001 .001 .994
S. 1 2 3 4 5 6 7
1 .922 .015 .015 .015 .015 .015 .003
2 .015 .922 .015 .015 .015 .015 .003
3 .015 .015 .922 .015 .015 .015 .003
4 .015 .015 .015 .922 .015 .015 .003
5 .015 .015 .015 .015 .922 .015 .003
6 .015 .015 .015 .015 .015 .922 .003
7 .001 .001 .001 .001 .001 .001 .994

Program FCMC2 Output DataProgram FCMC2 Output Data
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0 1
2
3
4
5
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
a
b
c
d
e

Effect of Number of Boreholes on SiteEffect of Number of Boreholes on Site
CharacterizationCharacterization
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
-1 0
-5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
-1 0
-5
0
1
2
3
4
5
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 1 0
- 5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
-1 0
-5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
-1 0
-5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
-1 0
-5
0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
-1 0
-5
0

Application of CS_CMC at MADE siteApplication of CS_CMC at MADE site
0 50 100 150 200 250
-10
-5
0
0
1
2
3
4
5

Simulation of the MADE1 ExperimentSimulation of the MADE1 Experiment
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0
0.1
1
10
100
0 50 100 150 200 250
-10
-5
0
1
2
3
4
5
0 50 100 150 200 250
-10
-5
0

Effect of the Number of Boreholes on theEffect of the Number of Boreholes on the
Simulated PlumeSimulated Plume
0 50 100 150 200 250
-10
-5
0
0
0.1
1
10
100
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
49 days
0 50 100 150 200 250
-10
-5
0
594 days
0 50 100 150 200 250
-10
-5
0
49 days
0 50 100 150 200 250
-10
-5
0
279 days
0 50 100 150 200 250
-10
-5
0
594 days
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0
0 50 100 150 200 250
-10
-5
0

Quantification of Uncertainties using MCQuantification of Uncertainties using MC
Classifications of Uncertainties:
Geological Uncertainty:
Geological configuration.
Parameter Uncertainty:
Conductivity value of each unit.
0 50 100 150 200 250 300
-50
0
Single realization of the geological structure used in the experimentsFigure 1.

Geological and Parameter UncertaintiesGeological and Parameter Uncertainties
1 2 3 4
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
time = 1600 days
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-40
-20
0
0 50 100 150 200 250 300
-40
-20
0
Geology is Certain and Parameters are Uncertain
Geology is Uncertain and Parameters are Certain
0 0.01 0.1 1

MonteMonte--Carlo Results for GeologicalCarlo Results for Geological
UncertaintyUncertainty
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0.00 0.01 0.10 1.00
time = 200 days
time = 1000 days
time = 2000 days
time = 3000 days
Concentration in mg/l
time = 1600 days
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0.00 0.01 0.10 1.00
time = 200 days
time = 1000 days
time = 2000 days
time = 3000 days
Concentration in mg/l
time = 1600 days
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0 50 100 150 200 250 300
-50
0
0.00 0.01 0.10 1.00
time = 200 days
time = 1000 days
time = 2000 days
time = 3000 days
Concentration in mg/l
time = 1600 days

MonteMonte--Carlo Method ResultsCarlo Method Results
-50
0
-50
0
0 50 100 150 200 250 300
-50
0
-50
0
-50
0
0 50 100 150 200 250 300
-50
0
-50
0
-50
0
0 50 100 150 200 250 300
-50
0
-50
0
-50
0
0 50 100 150 200 250 300
-50
0
-50
0
-50
0
0 50 100 150 200 250 300
-50
0
Concentration in (mg/l)
-50
0
-50
0
0 50 100 150 200 250 300
-50
0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
(a)
(b)
(c)
(a)
(b)
(c)
(c)
(c)
(c)
(c)
(b)
(b)
(b)
(b)
(a)
(a)
(a) (a)
Concentration field in set 1L: (a) Single realization,
(b) Ensemble concentration,
(c) Standard deviation on concentration.
Figure 2 Concentration field in set 1H: (a) Single realization,
(b) Ensemble concentration,
(c) Standard deviation on concentration.
Figure 5
Concentration field in set 2L: (a) Single realization,
(b) Ensemble concentration,
(c) Standard deviation on concentration.
Concentration field in set 2H: (a) Single realization,
(b) Ensemble concentration,
(c) Standard deviation on concentration.
Concentration field in set 3L: (a) Single realization,
(b) Ensemble concentration,
(c) Standard deviation on concentration.
Concentration field in set 3H: (a) Single realization,
(b) Ensemble concentration,
(c) Standard deviation on concentration.
Figure 3 Figure 6
Figure 4 Figure 7