Geoscience Reference
In-Depth Information
Ta b l e 3
Different cases for
Hamiltonian formalism
α
Parameter of interest
0
d
u
=
d
T
Travel time T along the ray
1
d
u
=
d
s
Arc-length s along the ray
d
T
n
2
2
d
u
=
d
σ
=
Natural variables along the ray
α
=
α
=
For
0, the parameter
u
represents travel time
t
along the ray. If
1, the
parameter
u
represents the arc-length
s
along the ray. In case of
α
=
2, the parameter
u
is equal to
dt
/
dn
, which represents the natural variables along the ray (Cerveny
2005
).
When applying ray-tracing for the determination of total delays along the ray path
the natural choice is
2 to construct a
tropospheric ray-tracing system (see Gegout et al. (
2011
)). Ray-tracing systems can
be expressed and solved in any curvilinear coordinate system, including spherical
coordinates. Selecting
α
=
1. However, it is also possible to use
α
=
α
=
1 and representing the function
H
in a spherical polar
coordinate system
(
r
, θ, λ)
,Eq.(
80
) can be rewritten as
L
r
+
1
2
1
r
2
L
2
1
r
2
sin
2
L
2
λ
H (
r
, θ, λ,
L
r
,
L
λ
,
L
θ
)
≡
θ
+
−
n
(
r
, θ, λ)
=
0
,
(84)
θ
where
r
is the radial distance,
θ
is the co-latitude, and
λ
is the longitude (0
≤
θ
≤
π,
∂
L
∂
L
∂θ
∂
L
∂λ
0
are the elements of ray directions.
Now, by substituting Eq. (
84
) into Eqs. (
81
) and (
82
), we obtain
≤
λ
≤
2
π
).
L
r
=
r
,
L
θ
=
and
L
λ
=
∂
d
r
d
s
=
1
β
L
r
,
(85)
d
d
s
=
1
β
L
θ
r
2
,
(86)
d
s
=
d
1
β
L
λ
r
2
sin
2
θ
,
(87)
L
2
θ
r
2
L
2
λ
r
2
sin
2
d
L
r
d
s
=
∂
n
(
r
, θ, λ)
∂
1
β
+
+
,
(88)
r
r
θ
L
2
λ
r
2
sin
3
d
L
θ
d
s
=
∂
n
(
r
, θ, λ)
∂θ
1
β
+
θ
,
(89)
d
L
λ
d
s
=
∂
n
(
r
, θ, λ)
∂λ
,
(90)
where
L
r
+
1
2
L
2
λ
r
2
sin
2
1
r
2
L
2
β
=
θ
+
=
n
(
r
, θ, λ).
(91)
θ
