Geoscience Reference
In-Depth Information
Substituting gives:
"
#
t
f
2N
f
y
2
f
sin B
ﾼ
cos B
f
t
f
1
þ ʷ
:
ð
6
:
79
Þ
In the same way, we get:
ﾼ
cos B
ﾼ
cos B
f
B
f
B
cos B
f
þ
sin B
f
B
f
B
2
3
t
f
2
t
f
ð
6
:
80
Þ
4
5
:
2M
f
N
f
y
2
2M
f
N
f
y
2
ﾼ
cos B
f
þ
sin B
f
ﾼ
cos B
f
1
þ
We replace l with Cartesian coordinates, and apply the second expression in
(
6.46
) to get:
9
=
l
00
cos B
f
ﾼ
ρ
00
ρ
00
6N
f
y
3
2t
f
þ ʷ
2
f
N
f
y
1
þ
þ
:
ð
6
:
81
Þ
l
00
cos
2
B
f
ﾼ
ρ
00
2
N
f
;
y
2
Inserting (
6.79
), (
6.80
), and (
6.81
) into (
6.78
), and neglecting the terms
containing l
3
4
or higher powers yields:
ʷ
2
4
3
5
2
4
3
5
2
4
3
5
f
ρ
00
N
f
y
ρ
00
6N
f
t
f
2N
f
1
3N
f
2
y
2
y
3
2t
f
þʷ
2
f
y
2
2
2
f
ʳ
00
ﾼ
1
þ
t
f
1
þʷ
1
þ
1
þ
3
ʷ
ﾼ
ρ
00
y
N
f
t
f
ρ
00
y
3
6N
f
ρ
00
y
3
2N
f
t
f
1
f
þ
ρ
00
y
3
t
f
1
f
:
2t
f
þʷ
f
t
f
1
þ
þʷ
þ
3
ʷ
3N
f
Finally one obtains:
,
ﾼ
ρ
00
y
N
f
t
f
ρ
00
y
3
3N
f
00
t
f
ʷ
2
f
ʳ
t
f
1 þ
ð6:82Þ
or, if calculated to terms containing y
5
, then:
ʳ
00
ﾼ
ρ
00
y
N
f
t
f
ρ
00
y
3
3N
f
þ
ρ
00
y
5
15N
f
t
f
ʷ
2
f
5t
f
þ
3t
f
t
f
1
þ
t
f
2
þ
:
ð
6
:
83
Þ
can achieve an accuracy of 1
00
if calculated from (
6.82
) and 0.001
00
from (
6.83
).
The subscript f indicates that the result is obtained from the footprint latitude, which
is similar to the meaning it carries in the formula for the inverse solution of the
Gauss projection.
ʳ
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