Algorithm Theoretical Based Document (ATBD) - CESBIO

SO-TN-ESL-SM-GS-0001
Issue 1.a
Date: 31/08/2006
SMOS level 2 processor
Soil moisture ATBD
• Compute fractions FM 0 of various land types, which are used to drive the decision tree;
• Compute reference values (to be used as a priori) for each mean fraction or group of fractions FM;
• Apply incidence angle dependent fraction (FV) weighting to each L1c view for building the aggregated
forward model used in the iterative retrieval and to be used during retrieval.
Concerning tasks 1 and 2, use is made of an average MEAN_WEF function for which an analytical formulation is given
below and a tabulation will be described in the TGRD. The present section addresses point 3, where an incidence angle
dependant WEF function is needed.
The weighting functions are to be applied to the cells of DFFG working area and parameters analyzed over it.
3.2.2.4.1 Rigorous formulation of the WEF
For each snapshot, depending on whether the pixel is dominated by land surface or ocean (or possibly other
considerations), an apodization window is selected during the L1 processing. This quantity is close to the synthetic
antenna pattern, which drives the angular resolution of the SMOS interferometer.
The synthetic antenna pattern, also called Equivalent Array Factor EAF, is given by [99]:
Eq 69
where
• W is the apodisation function
• r is the fringe-washing factor (FWF) which accounts for the spatial decorrelation between antennas.
• u,v are the baseline coordinates in the frequency domain
• d is the antenna element spacing (= 0.875)
• f 0 is the central frequency (1413 MHz)
• ξ, η are the central director cosines (DC) coordinates; ξ', η' are running DC coordinates.
For the nominal processing, were it not for the FWF factor, the AF would be the same everywhere in the DC plane, i.e.
would not depend on ξ and η but only on ξ' and η' (For strip adaptive processing, a specific AF must be computed for
each node of the fixed grid: see [100]).
While its central part is centro-symmetric on antenna boresight (ξ = η = 0), the EAF has significant side-lobes which
are either positive or negative and are no longer symmetrical.
Furthermore, going from DC to polar coordinates, for nominal processing, the AF becomes elongated as the angular
distance to boresight increases.
The weighting function WEF is obtained through intersecting the AF with Earth surface. To this end, 2 further steps are
needed, which account for:
• a "smearing effect" due to integration along the track;
• the variation of the integrating element with incidence angle, as the fine grid area does not lie on a plane.
3.2.2.4.2 WEF approximations
The results of the SMS study [101] suggest that, for well-behaved APF functions (Blackmann or better), truncating the
exact APF to the main lobe does not generate significant errors for representative scenes. This is a worthwhile option to
be considered since it restricts the domain over which the APF and WEF (see below) should be computed.
Other approximations have been tested [102] through numerical retrieval simulation. As a result it is found possible to:
• ignore the FWF factor;
• approximate the APF by a centro-symmetric function, for which a simple analytical formula can be fitted;
• ignore both smearing effect and Earth sphericity on the scale of the fine grid area.
Then, the following simplified expression WEF A for the WEF is proposed:
WEFA
WEFA
( ρ )
DC
sinc
≈
1+
C
( C ⋅ρ )
( ρ ) = 0 otherwise
DC
WEF1
WEF3
⋅ρ
C
WEF2
DC
C
WEF4
DC
if C
WEF1
⋅ρ
DC
≤ π
Eq 70
where
.
71