IEEE Geoscience and Remote Sensing Magazine - December 2014 - 33

factors of these gases (e.g., [153], [154]). Note that for reflected
signals, the integral along the ray in (35) includes the ray trajectory from the transmitter to the specular point, and from
the specular point to the receiver.
Monostatic radar altimeter missions use microwave radiometers aboard the same platform to determine the tropospheric delay over the measurement area. This approach is
complicated for spaceborne bistatic altimetric measurements
because of their wide range of areas that should be monitored
(very wide swath). Solutions are being investigated in the
frame of ESA's PARIS-IoD and GEROS-ISS missions.
In GNSS-R at lower receiver platforms, for which experimental data exist, the differential delay cancels most of the
atmospheric contribution, including the ionospheric one.
Only the effect coming from the troposphere below the receiver needs to be corrected. Sometimes the GNSS receiving
system is able to use direct signals to determine the zenith delay ( ZD). ZD is the atmospheric delay that would be induced
in a zenith observation from a receiver at a given altitude H
3
ZD ^H h = # N $ 10 -6 dh, and it can be estimated by combinH
ing information from different GNSS line-of-sight observations. The ZD at the receiver's altitude does not solve for the
tropospheric effect below the receiver, which is the one that
does not cancel out in differential GNSS-R and needs to be
corrected. A simple exponential model can then be used to
extrapolate from the relatively low receiver altitude down to
the reflecting surface. In [148] it is suggested that
ZD ^hh = ZD ^0 h e -h/hscale,

(42)

Generally, the observations are not at nadir, therefore
the equation above needs to be corrected with global mapping functions, m. This is an approximation used in Very
Long Baseline Interferometry (VLBI) and GNSS geodesy, for
which the tropospheric delay at a given elevation e is a function of the zenith delay: t tropo ^ e h = m ^e, lon, lat, time h $ ZD.
These global mapping functions depend on the elevation
angle, the geo-location and time [155]. A more rigorous
development of these terms separates the hydrostatic from
the wet component of the tropospheric delay (and their
corresponding mapping functions). The hydrostatic one is
a result of the induced dipole moment, and it has a typical value of approximately 2.3 meters at sea level. The wet
component is associated with the atmospheric water vapor,
it is due to the permanent dipole moment of the molecules
of water vapor, and has smaller values (0-0.3 m) but larger
variability in time and space.
december 2014

t iono -free =

f 12 t 1 - f 22 t 2
,
f 12 - f 22

(43)

where t i = 1, 2 are the observables as measured at fi = 1, 2 . The
disadvantage of using a linear combination of observables
is that the noise of the combination increases.
Other systematic effects are those induced by the surface roughness and the instrument itself, currently under
investigation. The electromagnetic bias (offset introduced
because of non-Gaussian surface elevation distribution
within the measurement zone) is now being modeled and
assessed [157], [158]. While on the instrumental side, the
bandwidth effect has been inspected in simulated data for
the PARIS IoD instrumental configuration [132].

(41)

where h scale is the vertical extent of the troposphere. If the
GNSS-R observation were at the nadir, and the GNSS receiver could estimate ZD (H r), then the remaining tropospheric delay in the differential observable Dt (line of sight
propagation subtracted) would be
t nadir
tropo = 2 $ ^ ZD ^ 0 h - ZD ^ H r hh
= 2 $ D ^H r h^e Hr /hscale - 1h .

Equation (40) includes the ionospheric effect. This effect
is usually separated from the tropospheric one because of
the dispersive nature of the ionospheric plasma. Then, the
combination of the observables at different GNSS transmitted frequencies are used to either mitigate or to isolate the
ionospheric effect. The ionospheric delay for a given carrier
frequency f is proportional to the integrated electron density along the propagation path, I , and the inverse square of
the frequency [156]: t iono ^ f h ? I/f 2 (the first-order approximation). The ionospheric effect, even in standard GNSS
applications, is one of the effects that is more difficult to
model accurately.
The ionospheric-free combination is one that eliminates
(or highly mitigates) its effects:

ieee Geoscience and remote sensing magazine

6) Altimetric PerformAnces
Table 4 compiles the precision achieved in GNSS-R experiments using both group-delay and phase-delay techniques.
The table compiles data from [75] and [159].
Various studies based on theoretical models have analyzed the expected altimetric performance of the In Orbit
Demonstrator for PARIS iGNSS-R spaceborne altimeter [74].
The resulting expected precision corresponding to observations at nadir and integrated for 100 km along the track are
13 cm, 17 cm, and 16 cm at GPS L1 band [74], [122], and
[132], respectively. In [132] other bands are also investigated
resulting in an expected precession in 30 cm at GPS L5, 13
cm, and 8 cm at GALILEO E1 and E5, respectively, for observations at nadir and integrated for 100 km along the track.
B. Winds and Waves
The use of GNSS reflected signals as wind scatterometers
was first suggested in [10]. In fact, both GNSS reflectometry and wind scatterometers measure surface roughness,
not wind speed directly, and it is generally assumed that
surface roughness is more closely correlated with the wind
stress on the sea surface rather than with wind speed measured at some altitude above the ocean surface (typically
at 10 m). Many ocean applications require the wind stress,
while meteorological applications often prefer the effective
wind at a certain altitude. The relationship between both
parameters, linked to the drag coefficient, is currently the
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2014