Ocean Remote Sensing Using GNSS-R - GNSS Remote Sensing: Theory, Methods and Applications - page 218

@F
@S ?

obs

geo mod

D S ? (9.3)

geo

and finally it should be inverted following a least-squares scheme.
Some comments about the procedure described above:

￿ Local vertical displacements could also force horizontal shifts of the specular
location, if condition 3 does not further apply (in fact this is what happened-
not made on purpose-on Fig. 9.2 ). This is a small effect, affecting only those
situations in which a very low altitude receivers is used and the vertical
component of the a-priori surface is wrong by meters. Simpler models can be
used in these cases as it will be explained below.
￿ The operator F can just be a straight propagator through an homogeneous
medium (straight rays), or a more sophisticated operator to account for atmo-
spheric gradients, such as ray-tracers ( Jones and Stephenson 1975 ). Horizontal
atmospheric gradients could bend the ray out of the plane P . The atmospheric
vertical gradients can also displace the horizontal location of the specular point.
This could still be solved by applying the same procedure iteratively, using each
i-solution as i + 1 a-priori where to find again the new i + 1 specular point.
￿ The operators could also take into account and try to solve for non-geometric
contributions tot the reflected delay (Eq. 9.1 ). In that case, the operator in Eq. 9.2
should output a more general delay (rather than geo ), and ancillary parameters
would enter as operator's input to help separating the different components of .
￿ Similarly, this inversion approach could be applied to differential delays, that
is, delays with respect to the direct radio-link (from T to R through the medium,
with no reflection off the Earth surface): D geo dir . In the equations above
geo should be replaced by and the operator implemented accordingly.
For low altitude receivers (aircrafts, or ground-based campaigns), simpler alti-
metric models can be applied. In these scenarios it is especially advantageous
to work with the differential delay , so that some of the instrumental and
atmospheric errors might cancel out. Moreover, the differential delay D geo
dir is a simple function of the elevation angle of observation e, and the (low) altitude
of the receiver above the Ocean surface H :

D 2H sin e (9.4)

Note that H here is not the Ocean height, but the vertical distance between the
surface and the receiver (see Fig. 9.3 ). This model applies when the incidence angle
over the region can be assumed as a constant value (parallel incidence), and the mean
surface locally horizontal (the altitude H is thus defined along the surface's local
normal direction). The Ocean instantaneous height can then be obtained subtracting
H from the receiver's altitude with respect to the reference ellipsoid: H ocean D

H ref

rcv H .
The following subsection details how to obtain and/or from the GNSS-R
waveforms when the range is measured through the delay of the code (delay of the