Geoscience Reference
In-Depth Information
where
e
2
=(
a
2
b
2
)
/a
2
. Dividing this equation by the above expression for
−
p
, we find
=
1
− e
2
tan
ϕ,
Z
p
N
N
+
h
(5-31)
so that
1
− e
2
−
1
tan
ϕ
=
Z
p
N
N
+
h
.
(5-32)
Given
X, Y, Z
, and hence
p
, Eqs. (5-29) and (5-32) may be solved iteratively
for
h
and
ϕ
. As a first approximation, we set
h
= 0 in (5-32), obtaining
tan
ϕ
(1)
=
Z
e
2
)
−
1
.
p
(1
−
(5-33)
Using
ϕ
(1)
, we compute an approximate value
N
(1)
by means of (5-21). Then
(5-29) gives
h
(1)
.Now,asasecondapproximation,weset
h
=
h
(1)
in (5-32),
obtaining
1
− e
2
−
1
N
(1)
N
(1)
+
h
(1)
tan
ϕ
(2)
=
Z
p
.
(5-34)
Using
ϕ
(2)
, improved values for
N
and
h
are found, etc. This procedure is
repeated until
ϕ
and
h
remain practically constant.
The result for
λ
is immediately obtained from the first two equations of
(5-27):
λ
=arctan
Y
X
.
(5-35)
Many other computation methods have been devised. One example for
the transformation of
X, Y, Z
into
ϕ, λ, h
without iteration but with an
inherent approximation is
ϕ
=arctan
Z
+
e
2
b
sin
3
θ
p
,
−
e
2
a
cos
3
θ
λ
=arctan
Y
(5-36)
X
,
p
cos
ϕ
−
h
=
N,
where
θ
=arctan
Za
pb
(5-37)
is an auxiliary quantity and
e
2
=(
a
2
b
2
)
/a
2
,
e
2
=(
a
2
b
2
)
/b
2
−
−
(5-38)
