Geoscience Reference
In-Depth Information
S
Ê
Á
ˆ
˜
◊◊
j
=- -
◊
11
S
, else
Ê
Ë
ˆ
¯
dj
1
/
J
Equation 17.12 performs the scaling at the grid cell level, and hence creates an “other” map,
denoted
. Equation 17.13 performs the scaling at the stratum level, and hence creates an “other”
map, denoted
A
dnj
.
The logic of the scaling is as follows, where the word “paint” can be substituted for the word
“category” to continue the painting analogy. If the quantity of category
E
d
◊
j
j
in the comparison map
is less than 1/
J
, then more of category
j
must be added to the comparison map. In this case, category
j
is increased in cells that are not already 100% members of category
j
. If the quantity of category
j
in the comparison map is more than 1/
J
, then some of category
j
must be removed from the
comparison map. In that case, category
.
For expressions in the “medium information” column of Figure 17.3, the other maps have the
same quantities as the comparison map. For the expressions in the “perfect information” column,
the other maps are derived such that the proportion of membership for each of the
j
is decreased in cells that have some of category
j
categories
matches perfectly with the proportions in the reference map. This adjustment is necessary to answer
the question, What would be the agreement between the reference map and the comparison map,
if the scientist would have had perfect information of quantity during the production of the
comparison map? The adjustment holds the level of information of location constant while adjusting
each grid cell such that the quantity of each of the
J
categories in the landscape matches the
quantities in the reference map. The logic of the adjustment is similar to the scaling procedure
described for the other maps in the “no information of quantity” column of Figure 17.3.
Equation 17.14 and Equation 17.15 give the necessary mathematical adjustments to scale the
comparison map to express perfect information of quantity:
J
Ê
ˆ
R
BS
S
=
◊◊
◊◊
j
,
if
RS
£
(17.14)
Á
˜
dnj
dnj
◊◊
j
◊◊
j
Ë
j
¯
S
R
Ê
Á
ˆ
˜
◊◊
j
=- -
11
S
,
else
Ê
Ë
ˆ
¯
dnj
◊◊
j
FS
R
Ê
ˆ
=
,
if
R
£
◊◊
◊◊
j
j
S
(17.15)
Á
˜
dj
◊
dj
◊
◊◊
j
◊◊
j
S
Ë
¯
S
R
Ê
Á
ˆ
˜
◊◊
j
=- -
◊
11
S
, else
Ê
Ë
ˆ
¯
dj
◊◊
j
Equation 17.14 performs this scaling at the grid cell level, and hence creates an “other” map,
denoted
. Equation 17.15 performs this scaling at the stratum level, and hence creates an “other”
map, denoted
B
dnj
.
There are five levels of information of location: no, stratum, medium, perfect within stratum,
and perfect, denoted, respectively, as N(
F
d·j
). Figure 17.3 shows the
differences in the 15 mathematical expressions among these various levels of information of
location. In N(
x
), H(
x
), M(
x
), K(
x
) and P(
x
) rows, the mathematical expressions of Figure 17.3 consider the
reference map at the grid cell level, as indicated by the use of all three subscripts:
x
), H(
x
), and M(
x
d
,
n
, and
j
. In
the K(
) row, the mathematical expressions consider the reference map at the stratum level, as
indicated by the use of two subscripts:
x
) row, the expressions consider the reference
map at the study area level, as indicated by the use of one subscript:
d
and
j
. In the P(
x
j
. In the M(
x
) row, the
