Geoscience Reference
In-Depth Information
Fig. 10.4
1-D example of
conditioning by kriging
In earlier implementations of TB and because of hardware
limitations, the position of the original
N
lines was evident in
the resulting simulated image. The solution to avoid artifacts
is to use a very large number
N
of lines, which is currently
more practical than in years past. Artifacts, particularly if the
method is poorly implemented, can be a significant disad-
vantage.
conditioning data and
u
i
, i = 1, …, N, be the N nodes to be
simulated. The large covariance matrix (n + N)·(n + N) is par-
titioned into the data-to-data covariance matrix, the node-to-
node covariance matrix, and the two node-to-data covariance
matrices:
(
)
(
)
C
uu uu
−
C
−
¢
Y
α
β
Y
α
j
nn
⋅
nN
⋅
C
=
=⋅
LU
(
)
(
)
(
nN nN
+
)(
+
)
¢
¢
¢
C
uu uu
−
C
−
Y¡
β
Y¡
j
Nn
⋅
NN
⋅
10.2.3
LU Decomposition
The large matrix
C
is decomposed into the product of a
lower and an upper triangular matrix,
C
=
L
.
U
. A conditional
realization {y
(l)
(
u
i
), i = 1, …, N} is obtained by multiplica-
tion of
L
by a column matrix ω
(N+n)
ִ
1
(l)
of normal deviates:
When the total number of conditioning data
plus
the number
of nodes to be simulated is small (fewer than a few hundred)
and a large number of realizations is requested, simulation
through LU decomposition of the covariance matrix pro-
vides the fastest solution (Luster
1985
; Alabert
1987b
).
Let Y(
u
) be the stationary Gaussian RF model with co-
variance C
Y
(
u
). Let
u
α
, α = 1, …, n, be the locations of the
[
]
1
y
()
u
ω
L
0
α
n
⋅
11
1
()
l
()
l
y
=
=⋅ =
L
ω
⋅
()
l
u
¢
LL
()
l
y
()
ω
21
22
2
i
N
⋅
1
