SMOS L2 OS ATBD - ARGANS

43
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: 43
(the Kirchhoff regime) and small scale (the Bragg regime) roughness within a single
theoretical scheme.
The calculation yields the following expression for a dimensionless scattering cross section
o
o
for scattering of the wave of polarization into the wave of polarization
o :
2
2
2
1 2q
q
q q
qk
qi
r
k i (
n n
B n n e k
i ) (
0)
(
) (
)
( s , i ) ( , )
e
e
q q s i
1
k i o
( ns
ni
) r
dr
i
o
(2)
o
where ( q k , qi
) represent the vertical projections of the wave vectors and the kernel
functions B ( n , n ) are given in Appendix of [9]. These kernels are geometric
s i
o
functions of the dielectric constant: we used the Klein and Swift's model [10] to estimate the
dielectric constant of sea water at L-band.
Here, the function (r )
is defined by the relation:
2
iQ(
h(
r ) h(
r ) exp Q ( 0)
(
r r )
exp 1 2
1 2
where means averaging over the space homogeneous statistical ensemble of sea surface
roughness, described by the surface elevation signal h( r 1)
, and Q qk
qi
. For Gaussian
statistics represents the correlation function of roughness and can be expressed strictly in
terms of a roughness spectrum:
2
(
r)
W ( k ) exp i k r dk
0 0
where W (k )
is the directional wavenumber spectrum of the rough sea surface at surface
wavenumber vector k .
In the present work, sea surface statistics is assumed Gaussian and is obtained from the
sea surface spectrum model of Kudryavtsev al. [11]. In our approach, the calculation of
o
o
is performed using an azimuthal harmonic decomposition for the autocorrelation
function. Moreover, to calculate accurately the autocorrelation function, we introduced a
sufficiently dense net on the surface wavenumber vector plane within the range
3
3
10 k 10
rad/m, applying a uniform step with respect to log(k) rather than to k.
3.6.2. Mathematical description
3.6.2.1 Simplified scattered solar radiation contributions
An additional model simplification is used to estimate the amount of solar energy scattered
by the sea surface and impinging the MIRAS antenna. We assumed than within the solid
angle subtended by the sun as seen from any of the observed terrestrial targets, the local sun
direction ni is almost constant, so that, at any target T, the radiometric sunglint temperatures
T ( , )
of a sunglint Stokes vector component, can be approximated locally at polarization
ss ns
, by: