Geoscience Reference
In-Depth Information
and let the angular lengths of the sides of the triangle be PX
=
a
,XN
=
b
and
NP
=
c
. Thus, the angular lengths
b
and
c
are known, but
a
is not:
b
=
90
−
λ
x
(2.4)
c
=
90
−
λ
p
(2.5)
Angle
A
is known, but
B
and
C
are not:
A
=
φ
p
−
φ
x
(2.6)
Equation (2.3)isused to obtain the magnitude of the velocity at point X:
v
=
ω
R
sin
a
(2.7)
The azimuth of the velocity,
β
is given by
β
=
90
+
C
(2.8)
To find the angles
a
and
C
needed for Eqs. (2.7) and (2.8), we use spherical
geometry. Just as there are cosine and sine rules relating the angles and sides of
plane triangles, there are cosine and sine rules for spherical triangles:
cos
a
=
cos
b
cos
c
+
sin
b
sin
c
cos
A
(2.9)
and
sin
a
sin
A
=
sin
c
sin
C
(2.10)
Substituting Eqs. (2.4)-(2.6) into Eq. (2.9)gives
cos
a
=
cos(90
−
λ
x
) cos(90
−
λ
p
)
+
sin(90
−
λ
x
) sin(90
−
λ
p
) cos(
φ
p
−
φ
x
)
(2.11)
This can then be simplified to yield the angle
a
,which is needed to calculate the
velocity from Eq. (2.7):
a
=
cos
−
1
[sin
λ
x
sin
λ
p
+
cos
λ
x
cos
λ
p
cos(
φ
p
−
φ
x
)]
(2.12)
Substituting Eqs. (2.5) and (2.6) into Eq. (2.10)gives
sin
a
sin(
φ
p
−
φ
x
)
=
sin(90
−
λ
p
)
sin
C
(2.13)
Upon rearrangement this becomes
C
=
sin
−
1
cos
λ
p
sin(
φ
p
−
φ
x
)
sin
a
(2.14)
Therefore, if the angle
a
is calculated from Eq. (2.12), angle
C
can then be
calculated from Eq. (2.14), and, finally, the relative velocity and its azimuth can
be calculated from Eqs. (2.7) and (2.8). Note that the inverse sine function of
Eq. (2.14)isdouble-valued.
1
Always check that you have the correct value for
C
.
1
An alternative way to calculate motion along a plate boundary and to avoid the sign ambiguities is
to use vector algebra (see Altman (1986)orCoxand Hart (1986), p. 154).
