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·
=
,
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 ).
 
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