Geoscience Reference
In-Depth Information
we can see that X, Y, and Z are the functions of L, B, H, a, and f (or e 2 ). When these
variables are differentiated respectively as dL,dB,dH,da, and df, by taking the total
differential of dX,dY, and dZ, one gets:
2
4
3
5
2
4
3
5 þ
,
dX
dY
dZ
dL
dB
dH
da
df
J
A
ð
7
:
15
Þ
with
2
4
3
5
X
X
X
L
B
H
Y
Y
Y
J ᄐ
L
B
H
Z
Z
Z
L
B
H
2
3
ð
N
þ
H
Þ
cos B sin L
ð
M
þ
H
Þ
sin B cos L
cos B cos L
4
5 ,
ð
N
þ
H
Þ
cos B cos L
ð
M
þ
H
Þ
sin B sin L
cos B sin L
0
ð
M
þ
H
Þ
cos B
sin B
This is called a Jacobian matrix. For the solution to its inverse matrix, readers
can refer to the literature (e.g., Zhu 1986).
In addition:
2
3
2
3
X
X
N
a cos B cos L
M
cos B cos L sin 2 B
4
5
4
5
a
f
1
f
N
a cos B sin L
M
Y
Y
cos B sin L sin 2 B
A
:
1
f
a
f
sin B
N
a
M
Z
Z
e 2
cos 2 B
e 2 sin 2 B
1
sin B 1
þ
1
f
a
f
It follows from ( 7.15 ) that:
2
4
3
5
2
4
3
5
,
dL
dB
dH
dX
dY
dZ
da
df
J 1
J 1 A
ð
7
:
16
Þ
where
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