Geoscience Reference
In-Depth Information
For the GRS algorithm, first-order inclusion probabilities are easy to obtain.
These are the scaled interval lengths used in the final systematic sampling
step. That is, the first-order GRS probability of unit
i
being selected is its
associated scaled value of
x
i
. The second-order inclusion probabilities for a
GRS are more difficult to obtain. In theory, second-order GRS inclusion prob-
abilities are computable, but the computations required are time consuming.
Consequently, second-order inclusion probabilities are usually approxi-
mated, either by a formula or by simulation. Stevens (1997) gave formulas
for special cases of GRTS samples that can also be used to approximate sec-
ond-order inclusion probabilities for GRS samples. Second-order inclusion
probabilities in other cases will usually be approximated by simulation. The
simulation needed to approximate second-order inclusion probabilities rep-
licates the GRS algorithm and tallies the number of times each pair of units
occurs in the sample. R code to implement this type of simulation appears
on the topic's web site.
10.4.2.4 Balanced Acceptance Samples
As the name implies, balanced acceptance samples (BAS) are a form of accep-
tance sampling that ensures spatial balance. In this section, the algorithm
for drawing an equiprobable BAS sample in two dimensions is described
and illustrated. BAS algorithms that sample
n
-dimensional resources and
can incorporate variable inclusion probabilities are possible but are beyond
the scope of this topic. The description here is a condensed version of the
description in Robertson et al
.
(2013).
10.4.2.4.1 Halton Sequence
The Halton sequence (Halton, 1960) is well known in mathematics as a
pseudorandom number generator and as a method that evenly distributes
points throughout a space. The Halton sequence serves sampling purposes
by effectively mapping
d
-dimensional space to a one-dimensional sequence
of numbers. The
d
-dimensional Halton sequence is, in fact, a combination of
n
one-dimensional sequences (each being van der Corput sequences), and
these one-dimensional sequences are defined first.
The sequences that spread sample locations throughout a one-dimensional
resource are constructed by choosing a number base, say,
p
(
p
≥ 2) and then
reversing the base
p
representation of the integers 1, 2, and so on. For exam-
ple, the base 3 representation of 10 is 101 (i.e.,
2
). The reversed
base
p
representation, often called the radical inverse, of interest here is
1(3) 0(3) 110
++=
1
3
0
3
1
3
10
27
ϕ=
(10) 0.101
=++= =
0.3704
.
3
1
2
3
