Geoscience Reference
In-Depth Information
where
X
ij
=
X
j
− X
i
,
Y
ij
=
Y
j
−
Y
i
,
(5-72)
Z
ij
=
Z
j
−
Z
i
have been introduced accordingly. The relation (5-71) may also be expressed
as
δs
ij
=
X
ij
s
ij
δX
i
)+
Y
ij
s
ij
δY
i
)+
Z
ij
s
ij
(
δX
j
−
(
δY
j
−
(
δZ
j
−
δZ
i
)
(5-73)
if the differentials are replaced by differences.
Azimuths
Again the same principle applies: the measured azimuth
A
ij
as a function of
the local level coordinates is given in (5-69). If
n
ij
,e
ij
,u
ij
, the components
of
x
ij
, are substituted by (5-65), the relation
tan
A
ij
=
e
ij
/n
ij
(5-74)
−
X
ij
sin Λ
i
+
Y
ij
cos Λ
i
=
−
X
ij
sin Φ
i
cos Λ
i
−
Y
ij
sin Φ
i
sin Λ
i
+
Z
ij
cos Φ
i
is obtained. After a lengthy derivation, the relation
δA
ij
=
sin
ϕ
i
cos
λ
i
sin
α
ij
−
sin
λ
i
cos
α
ij
s
ij
sin
z
ij
(
δX
j
−
δX
i
)
+
sin
ϕ
i
sin
λ
i
sin
α
ij
+cos
λ
i
cos
α
ij
s
ij
sin
z
ij
(
δY
j
−
δY
i
)
(5-75)
cos
ϕ
i
sin
α
ij
s
ij
sin
z
ij
−
(
δZ
j
−
δZ
i
)
+cot
z
ij
sin
α
ij
δ
Φ
i
+(sin
ϕ
i
−
cos
α
ij
cos
ϕ
i
cot
z
ij
)
δ
Λ
i
is obtained. Approximate values are sucient
in the coe
cients
, denoted by
ϕ, λ, α, z
instead of Φ
,
Λ
,A,z
.
Directions
Measured directions
R
ij
are related to azimuths
A
ij
by the orientation un-
known
o
i
. The relation reads
R
ij
=
A
ij
−
o
i
,
(5-76)