Geoscience Reference
In-Depth Information
u
i
(up, zenith)
P
j
X
ij
'
z
ij
s
ij
u
ij
P
i
n
i
A
ij
(north)
e
ij
n
ij
e
i
(east)
Fig. 5.12. Measurement quantities in the local level system
direction, are thus represented in the global system by
⎡
⎤
⎡
⎤
⎡
⎤
−
sin Φ
i
cos Λ
i
−
−
sin Λ
i
cos Λ
i
0
cos Φ
i
cos Λ
i
cos Φ
i
sin Λ
i
sin Φ
i
⎣
⎦
,
⎣
⎦
,
⎣
⎦
,
n
i
=
sin Φ
i
sin Λ
i
cos Φ
i
e
i
=
u
i
=
(5-64)
where the vectors
n
i
and
e
i
span the tangent plane at
P
i
(Fig. 5.11). The
third coordinate axis of the local level system, i.e., the vector
u
i
, is orthog-
onal to the tangent plane and has the direction of the plumb line.
Now the components
n
ij
,e
ij
,u
ij
of the vector
x
ij
in the local level system
are introduced. These coordinates are sometimes denoted as ENU (east,
north, up) coordinates. Considering Fig. 5.12, these components are obtained
by a projection of vector
X
ij
onto the local level axes
n
i
,
e
i
,
u
i
. Analytically,
this is achieved by scalar products. Therefore,
⎡
⎤
⎡
⎤
n
ij
e
ij
u
ij
n
i
·
X
ij
e
i
·
X
ij
u
i
·
⎣
⎦
=
⎣
⎦
x
ij
=
(5-65)
X
ij
is obtained. Assembling the vectors
n
i
,
e
i
,
u
i
of the local level system as
columns in an orthogonal matrix
D
i
, i.e.,
⎡
⎤
−
−
sin Φ
i
cos Λ
i
sin Λ
i
cos Φ
i
cos Λ
i
⎣
⎦
,
D
i
=
−
sin Φ
i
sin Λ
i
cos Λ
i
cos Φ
i
sin Λ
i
(5-66)
cos Φ
i
0
sin Φ
i
relation (5-65) may be written concisely as
x
ij
=
D
i
X
ij
.
(5-67)