Geoscience Reference
In-Depth Information
These parameters must be calculated for all layers, and then it is possible to find the
total slant delay (
Δ
L
d
) as follow
n
−
1
Δ
L
d
=
S
i
N
i
.
(116)
i
=
1
This equation can be divided into two terms for finding hydrostatic and non-
hydrostatic components of the delay separately. By inserting
h
i
(along the zenith
direction) instead of
S
i
(along the slant path) an equation for total zenith delay will
be derived. As mentioned before, the bending effect is a part of the total delay and
must be estimated separately. This parameter is zero for the zenith direction and
n
−
1
Δ
L
b
=
1
(
S
i
−
cos
(
e
i
−
e
k
)
S
i
).
(117)
i
=
for a slant direction. Implementation of the above ray-tracing system can vary sub-
stantially, with different degrees of complexity and accuracy. Examples of ray-tracing
algorithms in atmospheric studies are given by Bean and Thayer (
1959
), Thayer
(
1967
), Budden (
1985
), Davis (
1986
), Mendes (
1999
), Pany (
2002
), Böhmand Schuh
(
2003
), Thessin (
2005
), Hulley (
2007
), Hobiger et al. (
2008
), Nievinski (
2009
),
Wijaya (
2010
), Gegout et al. (
2011
) and Nafisi et al. (
2012a
). Several ray-tracing
algorithms were compared by Nafisi et al. (
2012b
). Also it is possible to express the
Eikonal equation and the ray-tracing system in curvilinear non-orthogonal coordi-
nates systems. For details see Cerveny et al. (
1988
).
4.2 Mapping Functions and Gradients
In the analysis of space geodetic data the troposphere path delay
Δ
L
(
e
)
at the ele-
L
z
and an
vation angle
e
is usually represented as the product of the zenith delay
Δ
elevation-dependent mapping function
mf
(
e
)
with
L
z
Δ
L
(
e
)
=
Δ
·
mf
(
e
).
(118)
This concept is not only used to determine a priori slant delays for the observa-
tions, but the mapping function is also the partial derivative to estimate residual
zenith delays. Typically, the zenith delay is estimated with a temporal resolution of
20-60 min in VLBI and GPS analysis. In VLBI analysis—when there is only one
observation at a time at a station—this allows the zenith delays to be estimated in
a least-squares adjustment. In the analysis of space geodetic observations not only
the zenith delays are estimated, but also other parameters like the station clocks and
the stations heights (Fig.
8
). The partial derivatives of the observed delays w.r.t. the
