Geoscience Reference
In-Depth Information
The base 3 representation of 11 is 102, and the reversed base 3 representation
of 11 is
2
3
0
9
1
27
19
27
(
ϕ=
11
0.201
=++==
0.7037
.
3
A property of these reversed base
p
sequences is that any contiguous sub-
sequence is evenly distributed over the interval [0,1] (Wang and Hickernell,
2000).
A
d
-dimensional Halton sequence is simply
d
one-dimensional reversed
base sequences, one for each dimension, but with one extra condition. The
extra condition, which in fact ensures the spatial balance, is that bases of
all one-dimensional sequences must be pairwise coprime. Pairwise coprime
means all bases must be prime and unique (no repeats). Although any set of
coprime bases will satisfy the definition of a Halton sequence, for the pur-
poses of sampling two-dimensional space, the BAS algorithm chooses
p
1
= 2
and
p
2
= 3. For the purposes of sampling, stochasticity of the selected points
is added to the Halton sequence by starting each one-dimensional sequence
in a random place. A Halton sequence started in random placed in each
dimension is called a randomized Halton sequence.
10.4.2.4.2 Equal Probability BAS Design
To draw an equiprobable BAS sample, first define a rectangular bounding
box surrounding the study area. Generate a randomized Halton sequence
inside the bounding box and take points, in order, until
n
locations in the
study area are obtained. Points landing outside the study area are discarded.
A realization of an equiprobable BAS design in the Canadian province of
Alberta is shown in Figure 10.8. The points displayed in Figure 10.8 were
drawn using the R package SDraw.
10.4.2.4.3 Unequal Probability BAS Design
The BAS design described in the previous section can be modified to draw
unequal probability samples in continuous space. Drawing an unequal
probability sample is accommodated by adding a dimension proportional to
inclusion probabilities and implementing an acceptance sampling scheme.
See the work of Robertson et al
.
(2013) for a description.
10.4.2.4.4 Inclusion Probabilities for BAS
Robertson et al
.
(2013) noted that the first- and second-order inclusion prob-
abilities for regions in the study area can be computed by enumerating all
possible randomized Halton sequences. Although enumerating all possible
Halton sequences is not practical for large problems, it is computationally
feasible in many cases, especially if the computations are processed in paral-
lel. If it is not feasible to enumerate all randomized Halton sequences, first-
