Geography Reference
In-Depth Information
As an example, if we take a distance range of 1000 to 2000 m and there are 346 pairs
of observations separated by a distance within that band then
p
(
h
), the number of
paired observations, is 346. Note that for each pair (e.g. observations 23 and 37), the
semivariance is calculated twice: once with respect to the i rst location and once with
respect to the second. We then calculate the squared dif erence between each of these
paired values. h e i rst value in each pair is given by
z
(
x
i
) and the value separated from
it by the specii ed lag
h
(in this example the distance is 1000 to 2000 m and we are
concerned with all directions) is given by
z
(
x
i
+
h
). h eir squared dif erence is there-
fore given by {
z
(
x
i
) -
z
(
x
i
+
h
)}
2
. h e summed values are then divided by two, hence the
term 'semivariance. Putting this together, the experimental variogram for lag
h
is
computed from:
p
()
h
1
{
}
Â
2
ˆ
()
g
h
=
z
( )
x
-
z
(
x
+
h
)
(9.9)
i
i
2()
p
h
i
=
1
As an example, if our lag is 5 ± 2.5 km (i.e. 2.5 to 7.5 km) and we have two values sepa-
rated by 6.2 km, then these paired observations qualify and we compute the squared
dif erence. If the two values are 26.2 and 43.3 then their squared dif erence is:
{
}
{
}
{
}
2
2
2
z
(
x
)
z
-+ = -
(
x
h
)
26.3
43.4
=-
17.1
=
292.41
i
i
In the same way we compute the squared dif erence for all other pairs separated by
2.5 to 7.5 km and at each stage add the computed value to the previous values com-
puted for that lag. Once this is done, we multiply the summed values by
1
/
(2
p
(
h
))
.
Figure 9.10 gives an example of an experimental variogram estimated from the pre-
cipitation data introduced in Section 9.5. h e Gstat sot ware (Pebesma and Wesseling,
1998; Pebesma, 2004) was used to estimate the variogram. h e lags are 0-5000 m,
5000-10,000 m and so on in groups of 5000 up to 60,000 m. In this case, data values are
600
500
400
300
200
100
Semivariance
0
0
10000
20000
30000
40000
50000
60000
Lag (m)
Figure 9.10
Omnidirectional variogram of July 2006 precipitation amount in Northern Ireland.
Search WWH ::
Custom Search