Geoscience Reference
In-Depth Information
·
∇
=
,
e
L
0
(77)
·
∇
=
.
h
L
0
(78)
By eliminating
h
between Eqs. (
75
) and (
76
), and considering Eq. (
77
), the following
differential equation is obtainedwhich is independent of the amplitude vectors
e
and
h
2
2
∇
L
=
n
(
r
)
,
(79)
where
∇
L
com
pr
ises the components of the ray directions,
L
is the optical path
c
√
εμ
length,
n
is the refractivity index of a medium, and
r
is the position vector.
This equation is the well-known Eikonal equation. The surfaces
L
=
(
r
)
=
constant
are
called geometrical wave surfaces or the geometrical wave-fronts.
4.1.2 Hamiltonian Formalism of Eikonal Equation
The above mentioned Eikonal equation is a partial differential equation of the first
order for
n
and it is possible to express it in many alternative forms. In general,
the Eikonal equation can be written in the Hamiltonian canonical formalism as fol-
lows (Born and Wolf
1999
; Cerveny
2005
;Nafisietal.
2012a
)
(
r
)
)
α
1
α
2
H (
r
,
∇
L
)
=
(
∇
L
·
∇
L
)
−
n
(
r
=
0
,
(80)
d
r
d
u
=
∂H
∂
∇
L
,
(81)
d
L
d
u
∇
=−
∂H
∂
r
,
(82)
d
L
d
u
=
∇
∂H
∂
∇
·
L
.
L
(83)
Here
is a scalar value related to the parameter of interest
u
(see Table
3
). In
general it is a real number but in our applications we can consider it to be an integer.
H (
α
is called Hamiltonian function or just the Hamiltonian. In a 3D space
this system consists of seven equations. Six equations are obtained from Eqs. (
81
)
and (
82
) must be solved together. The result of these six equations is
r
r
,
∇
L
)
which
is the trajectory of the signal in space. The seventh equation, i.e. Eq. (
83
), can be
solved independently and yields the optical path length.
=
r
(
u
)
4.1.3 Ray-Tracing System for Tropospheric Modeling
Equations (
80
)-(
82
) can be used for constructing a ray-tracing system for any specific
application by simply selecting a correct value for the scalar
α
(Nafisi et al.
2012a
).