Geoscience Reference
In-Depth Information
A. For continuous variables,
n
∑
k
∑
(5)
(
)
−1
O
1
=
η
q
j
≤
x
j
≤
q
i
+1
i
=
1
j
=
1
(
)
where
is the number of x
i
that falls between quantiles q
i
j
and q
i+1
j
B. For categorical data, the objective function is to match the probability distribu-
tion for each class of:
η
q
j
≤
x
j
≤
q
i
+1
()
n
η
'
z
j
c
∑
(6)
O
2
=
−
k
j
j
=
1
where η'(x
i
) is the number of x that belongs to class j in sampled data, and k
i
is the
proportion of class j in X.
C. To ensure that the correlation of the sampled variables will replicate the origi-
nal data, another objective function is added:
k
∑
k
∑
(7)
O
3
=
c
ij
−
t
ij
i
=
1
j
=
1
where c is the element of C, the correlation matrix of X, and t is the equivalent element
of T, the correlation matrix of x.
4. Perform an annealing schedule (Press et al., 1992): M = exp[-∆O/T], where ∆O
is the change in the objective function, and T is a cooling temperature (between
0 and 1), which is decreased by a factor d during each iteration.
5. Generate a uniform random number between 0 and 1. If rand <M, accept the
new values; otherwise, discard changes.
6. Try to perform changes: Generate a uniform random number rand. If rand
< P, pick a sample randomly from x and swap it with a random site from un-
sampled sites r. Otherwise, remove the sample(s) from x that has the largest
η
(
)
and replace it with a random site(s) from unsampled sites r.
q
i
≤
x
j
≤
q
i
+1
End when the value of P is between 0 and 1, indicating that the probability of
the search is a random search or systematically replacing the samples that have
the worst fit with the strata.
7. Go to step 3 Repeat steps 3-7 until the objective function value falls beyond a
given stop criterion or a specified number of iterations.
Sequential Gaussian Simulation
In sequential simulation algorithm, modeling of the N-point cumulative density func-
tion (ccdf) is a sequence of N univariate ccdfs at each node (grid cell) along a random
path (Kyriakidis, 2001). The sequential simulation algorithm has the following steps
(Kyriakidis, 2001):
1. Establish a random path that is visited once and only once, all nodes {u
i
, i = 1,
Λ, N} discretizing the domain of interest Doman. A random visiting sequence
ensures that no spatial continuity artifact is introduced into the simulation by a
specific path visiting N nodes.
