Geoscience Reference
In-Depth Information
In Fig.
7.3
, O
new
—X
new
Y
new
Z
new
and O
old
—X
old
Y
old
Z
old
are two geodetic
Cartesian coordinate systems (geodetic spatial rectangular coordinate systems).
Their coordinate origins do not coincide, i.e., there are three translation parameters,
Δ
Z
0
, denoting the components of the old coordinate origin with respect
to the new coordinate origin along the three coordinate axes. The coordinate axes of
the two systems are mutually non-parallel, generally less than 1
00
for classical
geodetic coordinate systems, meaning that Euler angles
X
0
,
Δ
Y
0
,
Δ
ʵ
z
exist (known
also as the three rotation parameters). Apparently, these two systems can be made
coincident under translation and rotation. According to (
7.7
), we get:
ʵ
x
,
ʵ
y
,
2
3
2
3
2
3
2
3
X
Y
Z
Δ
X
0
1
ʵ
Z
ʵ
Y
X
Y
Z
4
5
new
ﾼ
4
5
þ
4
5
4
5
old
:
Δ
Y
0
ʵ
Z
1
ʵ
X
ð
7
:
12
Þ
Δ
Z
0
ʵ
Y
ʵ
X
1
For various reasons, there will also be a difference in scale while establishing the
two systems. Assume that S
new
and S
old
are the measurements of the same distance
in space in the new and old coordinate systems; then we can define
S
new
S
old
Δ
m
ﾼ
ﾼ
m
1
S
old
as the scale factor of the two coordinate systems. Here,
m is homogeneous and is
independent of the point position and direction. Hence, the old coordinates can be
improved in accordance with the scale of
Δ
the new coordinate system, as
X
mZ
old
.
In (
7.12
), considering the effect of the scale factor means improving the (X, Y,
Z)
T
old
in accordance with the above relations, namely:
ﾼ
X
old
+
Δ
mX
old
, Y
ﾼ
Y
old
+
Δ
mY
old
and Z
ﾼ
Z
old
+
Δ
2
4
3
5
2
4
3
5
þ
2
4
3
5
2
4
3
5
:
X
Y
Z
Δ
X
0
1
ʵ
Z
ʵ
Y
X
old
þ Δ
mX
old
new
ﾼ
Δ
Y
0
ʵ
Z
1
ʵ
X
Y
old
þ Δ
mY
old
Δ
Z
0
ʵ
Y
ʵ
X
1
Z
old
þ Δ
mZ
old
Disregarding the second-order small quantities and rearranging gives:
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
,
X
Y
Z
X
Y
Z
0
ʵ
Z
ʵ
Y
X
Y
Z
Δ
X
0
ﾼ
ð
1
þ Δ
m
Þ
þ
ʵ
Z
0
ʵ
X
þ
Δ
Y
0
ð
7
:
13
Þ
ʵ
Y
ʵ
X
0
Δ
Z
0
new
old
old
which is called the Bursa-Wolf transformation model (or, simply, Bursa model; see
Thomson 1976) with seven transformation parameters
Δ
X
0
,
Δ
Y
0
,
Δ
Z
0
,
ʵ
X
,
ʵ
Y
,
ʵ
Z
,
and
Δ
m. The linear equation in terms of these parameters is:
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