IEEE Geoscience and Remote Sensing Magazine - December 2014 - 31
between the transmitter, the surface and the receiver, and
its incident angle is equal to the reflected one. Then, with
a good apriori knowledge of the Earth surface S ref (lon,
lat), such as the geoid or the mean sea surface, and both
vR and Tv positions, it is possible to have a"good a-priori
estimation of the specular point position S 0, and we can
assume that only a small vertical correction needs to be
"
"
v, vR) = S 0 (S ref , Tv, vR) + H $ ut v (ut v beadjusted: v
S = F (H; S 0, T
ing a unitary vector pointing towards the local vertical
direction) [75], [144].
Then, solving the bistatic altimetric solution is reduced
to finding the vertical correction H such that it better fits
the corrected data Dtl :
(32)
r
The correction terms t ra, t da , t rough
, and t rins are briefly
commented on in Section VII.A.4), while f rd is unmodelled
noise and residual errors.
3) Flat Earth approximation
altimEtric invErsion
When the receiver platform is at a relatively low altitude
above the surface, it is possible to assume that the Earth
is locally flat. The altitude to which this condition holds
depends on the incidence angle of the observation and
the tolerance of the user to the errors this might induce.
Fig. 23 shows the errors made in the vertical component of the specular point location when assuming a
flat Earth, with respect to assuming a spherical Earth.
Only incidence angles in up to 50° (down to 40° elevation) have been included; beyond these angles the error
increases quickly. The horizontal errors are slightly larger (16% at 50° incidence, and increasing to more than
500% at an 80° incidence angle).
Under this assumption, the differential geometric term
Dt g can be simplified to a function of the elevation angle
of observation:
Dt g = 2A sin ^ e h - t ant,
Dtl + t ant
+ e.
2sin ^ e h
(34)
4) phasE-DElay altimEtry mEasurEmEnts
In precise GNSS positioning techniques, the accurate but
not very precise group-delay pseudorange observables are
complemented with the phase-delay ones, of much better
precision but inaccurate. The phase-delay observables are
very precise measures of range variations. If both transmitter and receiver were in static positions, the electromagnetic
field would oscillate at the transmitted carrier frequency,
as ~ $ t. This is usually fully or partially compensated in
the receiving system (down-conversion to baseband or
intermediate band). In transmitter and receiver static conditions, the phase term determined by v
k $ vr (being v
k and
vr propagation wavenumber and range vectors) does not
change. However, as either/both the transmitter or/and receiver move, this term determines the phase at which
the
""
"
down-converted field is received: z ^ t h = z ^t 0h + e i k Dr ^ t h (Dr
being the variation in vr after t 0). This phase can also be
expressed as a pseudorange t z ^ t h = (m/2r) z ^ t h . Note that
this observable is very precise: for each full cycle variation
of the phase, the pseudorange observable only changes one
m (+ 20 cm in GNSS L-band frequencies). This information would be very valuable in GNSS-R altimetry, but unfortunately most of the GNSS-R observations correspond
to diffuse scattering, in which the phase shifts and jumps
randomly, impeding its use as range-change tracker. The
exceptions are:
◗ very calm waters (it eventually might happen, not as a
general case, but e.g., over lakes, ponds, harbors [23],
[25], [145]);
12
50
10
(33)
where A is the altitude of the receiver above the reflecting
surface. Because the precise positioning of the receiver is
usually made with an up-looking antenna, here we assume
that A is the altitude of this antenna above the reflecting
surface. The height of the surface with respect to a reference surface (ellipsoid, geoid, or other topographic models) can then be obtained with precise knowledge of the receiver's position: H = R v - A (R v being the vertical distance
between the reference surface and the receiver). Note that
t ant is not the antenna's offset vector itself, but its projection into the direction from where the signal reaches the
" " "
receiver, -kt s =-k s/ |k s|(k s being the scattering wavenumber vector). Then, t ant = t ant $ ^-kt sh . These concepts are illustrated in Fig. 24.
december 2014
A=
ieee Geoscience and remote sensing magazine
Vertical Error (m)
Dt' = Dt - ^t ra - t da h - t rrough - t rins
"
v, vR, t ant j + e rd
Dt' = Dt g + e rd = F ` H; S 0, T
Equation (31) then can be directly inverted into
8
6
40
4
30
2
20
10
0
1
2
3
4
5
6
7
8
Receiver Altitude (km)
9
10
Figure 23. Vertical component of the error in the specular point
location due to the assumption of local flat Earth, for different
receiver altitudes (x-axis) and incidence angles (different curves).
31
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