Geoscience Reference
In-Depth Information
provide numerical stability. The value of this constant must be chosen carefully;
otherwise we may get unreasonable results. Other stability solutions have been sug-
gested by Schneider (
1993
) and Fowler (
1994
). In addition, the variation of the size
of the grid cells in polar coordinates can causes a stability problems, especially in
presence of small-scale inhomogeneities (Fowler
1994
).
By substituting Eq. (
84
) into Eq. (
83
) and solving the equation, we can obtain this
well-known equation to calculate the optical path length
L
L
=
n
(
r
, θ, λ)
d
s
.
(102)
S
Once the position of a point along the ray path has been determined by Eqs. (
85
)-
(
90
), the refractivity index
n
and the optical path length
L
can be calculated using
Eq. (
102
). As mentioned before, the total delay is defined as the difference between
the optical path length
L
and the straight line distance
G
Δ
=
−
.
L
L
G
(103)
3D ray-tracing can be easily reduced to 2D ray-tracing by substituting
∂
n
∂θ
=
0 and
∂
n
0 into Eqs. (
85
)-(
90
). In this case, we assume that the ray does not leave a
plane of constant azimuth angle. For the 2D system, the coupled partial derivatives
in Eqs. (
85
)-(
90
) reduce to four equations
∂λ
=
d
r
d
s
=
1
β
L
r
,
(104)
d
d
s
=
1
β
L
θ
r
2
,
(105)
L
2
θ
r
2
L
2
λ
r
2
sin
2
d
L
r
d
s
=
∂
n
(
r
, θ, λ)
∂
1
β
+
+
,
(106)
r
r
θ
L
2
λ
r
2
sin
3
d
L
1
β
θ
d
s
=
θ
.
(107)
Equation (
91
) remains valid also for 2D ray-tracing systems. For a horizontally strat-
ified atmosphere, further simplifications can be applied to improve the calculation
speed (Thayer
1967
). According to Böhm (
2004
), we can develop a 1D ray-tracing.
Figure
7
shows the geometry of this method for a troposphere with
k
different refrac-
tivity layers. The geocentric distances can be estimated by adding the radius of the
Earth (
R
e
) to the heights of each layer
r
i
=
R
e
+
h
i
.
(108)
In this method, the elevation angles (
e
i
) are with respect to a horizontal plane of
the station, whereas
θ
i
show the angles between the ray path and the tangents to the
