Geoscience Reference
In-Depth Information
Nx
−
Nx
−
Nx x
−−
N
j
j
k
k
−
α=−
1
+
/
,
(3.3)
jk
n
n
n
n
where
j
≠
k
and α=α
jk
j
, when
j
=
k
. The estimator of the variance is
α−αα
αα
∑
K
∑
K
·
jk
j k
Var(
ˆ
)
**
τ=
yy
zz
(3.4)
jK
jk
j
=
1
k
=
1
j k
In Figure 3.1, which uses blue-winged teal as an example (Brown, 2011;
Smith et al., 1995), the initial sample is 10 quadrats,
n
= 10; the threshold con-
dition is
y
i
≥ 1; and the definition of the neighborhood is the four surrounding
quadrats. Only one quadrat in the initial sample triggered adaptive selection
of the surrounding quadrats.
The final sample size is 16. Note that many more than 16 quadrats are
assessed because the 4 neighboring quadrats around any occupied quadrat
are checked. Only quadrats in the initial sample or in a selected network are
used in calculating the sample estimators, and the other quadrats are the
edge units. With a simple condition like
y
i
≥ 1, the quadrats only need to be
checked to see if ducks are present or absent, but with a condition like
y
i
≥ 10,
ducks within the units would need to be counted to know whether there
were less than 10, something that may be more time consuming than simply
checking for presence.
The Horvitz-Thompson estimate using Equation (3.1) of the population
total from the sample in Figure 3.1 is
∑
187
yz
*
ˆ
kk
τ=
α
k
1
=
k
13753 1
0.3056
⋅
=
++ +
0
0
…
=
45003.
The only nonzero term in this equation is for the network of size 7. The values
of all the other terms are zero. The other nine networks that were selected
in the initial sample were only one quadrat in size and
y
* = 0. All the other
networks that were not selected have
z
k
= 0.
The initial intersection probability, Equation (3.2), for the size 7 network is
calculated as
200 7
−
200
=
α= −
1
/
0.3056
.
1
10
10
