Geoscience Reference
In-Depth Information
Since the flattening
f
is small, we may expand (2-149) as
N
=
a
2
b
(1 +
e
2
cos
2
ϕ
)
−
1
/
2
=
a
2
b
1
···
1
2
e
2
cos
2
ϕ
−
(5-55)
f
cos
2
ϕ
f
cos
2
ϕ
=
a
(1 +
f
···
−
···
)=
a
(1 +
f
−
···
)(1
)
yielding
=
a
(1 +
f
sin
2
ϕ
)
N
(5-56)
and
b
2
a
2
N
=(1
=
a
(1
)
a
(1 +
f
sin
2
ϕ
2
f
+
f
sin
2
ϕ
)
−
2
f
···
···
)
−
(5-57)
and
e
2
=2
f
b
=
a
(1
−
f
)
,
···
.
(5-58)
Thus, Eqs. (5-53) are approximated by
X
=
x
0
+(
a
+
af
sin
2
ϕ
+
h
)cos
ϕ
cos
λ,
Y
=
y
0
+(
a
+
af
sin
2
ϕ
+
h
)cos
ϕ
sin
λ,
(5-59)
2
af
+
af
sin
2
ϕ
+
h
)sin
ϕ.
Z
=
z
0
+(
a
−
Now we can form the partial derivatives in (5-54), for instance,
∂X
∂a
.
=cos
ϕ
cos
λ,
=(1+
f
sin
2
ϕ
)cos
ϕ
cos
λ
(5-60)
since we may neglect the flattening in these coecients. This amounts to
using for the coecients, and only for them, a spherical approximation anal-
ogous to that of Sect. 2.13. Similarly, all coe
cients are easily obtained as
partial derivatives, and Eqs. (5-54) become
δX
=
δx
0
−
a
cos
ϕ
sin
λδλ
+cos
ϕ
cos
λ
(
δh
+
δa
+
a
sin
2
ϕδf
)
,
a
sin
ϕ
cos
λδϕ
−
δY
=
δy
0
−
a
sin
ϕ
sin
λδϕ
+
a
cos
ϕ
cos
λδλ
+cos
ϕ
sin
λ
(
δh
+
δa
+
a
sin
2
ϕδf
)
,
(5-61)
δZ
=
δz
0
+
a
cos
ϕδϕ
+sin
ϕ
(
δh
+
δa
+
a
sin
2
ϕδf
)
−
2
a
sin
ϕδf.
These formulas give the changes in the rectangular coordinates
X, Y, Z
in
terms of the variation in the position (
x
0
,y
0
,z
0
)andthedimensions(
a, f
)
of the ellipsoid and in the ellipsoidal coordinates
ϕ, λ, h
referred to it.
