“myThesis” — 2007/2/2 — 20:08 — page xvi — #18
xvi LIST OF FIGURES
3.8 Relationship between the uncertainty measures and the number
of realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Flowchart of the multi-scale blocking McMC scheme . . . . . . 64
4.2 The conditioning data set and the reference ln K and head fields
(µ = 0, λ
x
= 50, σ
2
x
= 1.0) . . . . . . . . . . . . . . . . . . . . . 66
4.3 An MsBMcMC estimate by inverse-conditioning to head obser-
vations (λ
x
= 50, σ
2
x
= 1.0, σ
2
y
= 0.17) . . . . . . . . . . . . . . . 69
4.4 Effect of prior correlation length on the reproduction of refer-
ence fields (σ
2
x
= 1.0, σ
2
y
= 0.25, (A)λ
x
= 50, (B)λ
x
= 20) . . . . 73
4.5 Effect of prior correlation length on the performance of MsBM-
cMC (σ
2
x
= 1.0, σ
2
y
= 0.25, (A)λ
x
= 50, (B)λ
x
= 20) . . . . . . . 74
4.6 Effect of prior variances (σ
2
x
) on the reproduction of reference
fields (λ
x
= 50, (A)σ
2
x
= 0.5, σ
2
y
= 0.17, (B)σ
2
x
= 1.5, σ
2
y
= 0.25) 76
4.7 Effect of prior variances (σ
2
x
) on the performance of MsBMcMC 78
4.8 Effect of variances of head data (σ
2
y
) on the reproduction of
reference fields (λ
x
= 50, σ
2
x
= 1.0, (A)σ
2
y
= 1.00, (B)σ
2
y
=
0.25), (C)σ
2
y
= 0.17) . . . . . . . . . . . . . . . . . . . . . . . . 79
4.9 Effect of variances of head data (σ
2
y
) on the performance of
MsBMcMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.10 Effect of conditioning to hard data on the reproduction of ref-
erence fields (λ
x
= 50, σ
2
x
= 1.0, σ
2
y
= 0.17) . . . . . . . . . . . . 83
4.11 Effect of conditioning to hard data on the performance of Ms-
BMcMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.12 Identified mean ln K fields and residual distributions (λ
x
= 50,
σ
2
x
= 1.0, σ
2
y
= 0.17). Note that (A) and (B) are resulted from
3000 realizations, while (C) and (D) are based on 100 realizations. 86
5.1 The mean lnK fields due to conditioning to various source data 104
5.2 The variances of lnK fields due to conditioning to various source
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 The histogram of lnK distribution after conditioning to data
from various sources . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4 The mean head fields due to conditioning to data from various
sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5 The variances of head fields due to conditioning to data from
various sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6 A comparison of simulated macrodispersions due to condition-
ing to data from various sources . . . . . . . . . . . . . . . . . 112
5.7 A comparison on evolution history of contaminant plumes (A)
at the early time t
5%
and (B) at the late time t
95%
due to con-
ditioning to data from various sources . . . . . . . . . . . . . . 114
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