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
ᄐ
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
,
ᄐ
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
ᄐ
ᄐ
:
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
ᄐ
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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