SMOS L2 OS ATBD - ARGANS

130
ICM-CSIC
LOCEAN/SA/CETP
IFREMER
SMOS L2 OS
Algorithm Theoretical
Baseline Document
Doc: SO-TN-ARG-GS-0007
Issue: 3 Rev: 9
Date: 25 January 2013
Page: 130
The upper hemisphere pole corresponds to the unit normal to the Earth surface at the target
latitude and longitude. Denoting the right ascension and declination of the projection of this
point in the celestial frame by ( , nn)
, we can remove the explicit dependence on time in (45)
by introducing ( , nn)
as independent variables and expressing the scattered galactic noise as
s
T p( n,
n,
s,
s,
u10,
w)
.
However, this parametrization is not optimal for representing the functional dependence of
the scattered signal, since we know that the dominant source of scattered signal is associated
with noise in the specular direction. Therefore, we seek to represent the scattered signal in
terms of the location in the sky of the specular direction, which we denote ( spec , spec)
. In
order to represent the scattering solution in terms of these variables, we must find a mapping
between ( spec , spec)
and ( , nn)
. This mapping will necessarily involve s and s , so that we
can write the mapping function as
T: ( n, n,
s,
s)
( spec,
spec,
spec,
uh),
(46)
where spec is the incidence angle of the specular direction in the upper hemisphere altitudeazimuth
frame, and where we have introduced the angle uh,
which represents the orientation
of the upper hemisphere at the specular point ( spec , spec)
. The mapping operator T can be
seen to be that function which rotates the unit normal vector in the upper hemisphere frame
into the unit vector in the specular direction.
uhmust be defined so as to allow construction of an inverse mapping operator T -1 that maps
a specular direction ( spec , spec)
uniquely into an upper hemisphere unit normal ( , nn)
. To
facilitate a definition of uh,
we first establish alt-azimuth coordinate systems and associated
basis vectors in both the upper hemisphere and celestial frames along the line of sight in the
specular direction. Detailed definitions of reference frames and associated transformation
used in the derivation of uh are given in Appendix B. These transformation can be
performed with the use of CFI. The basis vectors are analogous to horizontal and vertical
polarization basis vectors used to describe electromagnetic plane waves. In the upper
hemisphere frame, which is the topocentric frame whose origin is the surface target, also
called the earth alt-azimuth frame, we define the 'horizontal' basis vector hˆ u= nˆ
u
rˆ
/ nˆ
urˆ
,
where nˆ u is the unit normal to the surface at the target and rˆ is directed outward towards the
specular direction from the target. Next, we define a 'vertical' basis vector by vˆ u= hˆ
u
rˆ
/ hˆ
urˆ
.
If we let spec and spec be the specular azimuth and altitude, respectively, of rˆ in the upper
hemisphere frame, then we have
ˆu
u
u
h = sin
specxˆ
cos
specyˆ
,
(47)
u
u
u
u
vˆ
= cos
specsin
specxˆ
sin
specsin
specyˆ
sin
speczˆ
,
where xˆ u , yˆ u , and zˆ u are basis vectors for the topocentric Earth frame that determines the
upper hemisphere. Analogous basis vectors can be defined in the celestial frame as
hˆ
c=
sin
specxˆ
ccos
specyˆ
c,
(48)
vˆ
c=
cos
specsin
specxˆ
csin
specsin
specyˆ
csin
speczˆ
c,