Geoscience Reference
In-Depth Information
This system of equations is a direct result of Eqs. (
81
) and (
82
) in a 3D medium,
and we need to solve all six differential equations simultaneously. The final output
is the positions of the points along the trajectory of the ray. For solving the above
ray-tracing system, the following initial conditions at the starting point (station) can
be used (Nafisi et al.
2012a
)
r
=
r
0
,
(92)
λ
=
λ
0
,
(93)
θ
=
θ
0
,
(94)
L
r
0
=
n
0
cos
z
0
,
(95)
L
=
n
0
r
0
sin
z
0
cos
a
0
,
(96)
θ
0
L
=
n
0
r
0
sin
z
0
sin
a
0
sin
θ
0
,
(97)
λ
0
where
a
0
and
z
0
are the initial geodetic azimuth and zenith angle, respectively. In the
3D case, gradients are important factors which can affect the bending of the ray path
and therefore of the total ray-traced delay. The gradient can be rewritten as
∂
T
(
, θ, λ)
∂
r
∂
(
, θ, λ)
∂θ
∂
(
, θ, λ)
∂λ
n
r
1
n
r
1
r
sin
n
r
∇
(
, λ, θ)
=
,
,
.
n
r
(98)
θ
r
From a practical point of view, we must find a sophisticated technique for computing
the gradients of the refractive index in Eqs. (
88
), (
89
) and (
90
), which are
n
r
=
∂
(
, θ, λ)
∂
n
r
∇
,
(99)
r
n
θ
=
∂
n
(
r
, θ, λ)
∂θ
∇
,
(100)
λ
=
∂
n
(
r
, θ, λ)
∂λ
∇
n
.
(101)
Taking the effects of the gradients on the ray-traced delay into account is important
for a ray-tracing algorithm. In particular in the case of symmetries, ray-tracing in
curvilinear coordinates system would be easier. However, in general the ray-tracing
systems in a curvilinear coordinates are more complex and sometimes fail. A typ-
ical example in atmospheric ray-tracing is the solution for ray-tracing in spherical
polar coordinates when
180
◦
. According to Cerveny (
2005
) a gen-
eral solution for removing such singularities is the use of transformations between
ray-tracing systems in various forms. Using standard transformation relations, a ray-
tracing system represented in curvilinear coordinates can be transformed into the
universal Cartesian coordinates, and after ray-tracing computations in this coordi-
nate system the results can be again transformed back to the curvilinear coordinates.
In another method suggested by Alkhalifah and Fomel (
2001
) a small constant para-
meter (
0
◦
or
θ
≈
θ
≈
δ
) is added in the denominator of fractions in the ray-tracing equations to
