Geoscience Reference
In-Depth Information
Z
b
geocenter
Y
x
0
x
0
z
0
a
y
0
ellipsoid center
X
Fig. 5.8. Translation problem
First we ask how the rectangular coordinates
X, Y, Z
change if we vary
the ellipsoidal coordinates
ϕ, λ, h
by small amounts
δϕ, δλ, δh
and if we also
alter the geodetic datum, namely, the reference ellipsoid
a, f
and its position
x
0
,y
0
,z
0
,by
δa, δf
and
δx
0
,δy
0
,δz
0
.Notethat
δx
0
,δy
0
,δz
0
correspond to a
small translation (parallel displacement) of the ellipsoid,
its axis remaining
parallel to the axis of the earth
.
The solution of this problem is found by differentiating (5-53):
δX
=
δx
0
+
∂X
∂a
δa
+
∂X
∂f
δf
+
∂X
∂ϕ
δϕ
+
∂X
∂λ
δλ
+
∂X
∂h
δh ,
δY
=
δy
0
+
∂Y
∂a
δa
+
∂Y
δf
+
∂Y
∂ϕ
δϕ
+
∂Y
∂λ
δλ
+
∂Y
∂h
δh ,
(5-54)
∂f
δZ
=
δz
0
+
∂Z
∂a
δa
+
∂Z
δf
+
∂Z
∂ϕ
δϕ
+
∂Z
∂λ
δλ
+
∂Z
∂h
δh ,
∂f
since, according to Taylor's theorem, small changes can be treated as differ-
entials.
In these differential formulas we shall be satisfied with an approximation.
