SMOS L2 OS ATBD - ARGANS

178
ICM-CSIC
LOCEAN/SA/CETP
IFREMER
SMOS L2 OS
Algorithm Theoretical
Baseline Document
4.14.1.2. Mathematical description of algorithm
Doc: SO-TN-ARG-GS-0007
Issue: 3 Rev: 9
Date: 25 January 2013
Page: 178
Mathematically equation 4.14.1 (including model error) can be written as follows:
2 meas mod
T 1
meas mod
prior T 1
prior
( T ( ,
P )) C ( T T ( ,
P )) ( P P ) C ( P P ) [4.14.2]
Tb b j TB
b b j j j Pj
Where the Tb meas are the Nm observations performed at different incidence angles, T
represents the transposition operation, and CTb is the variance/covariance matrix for Tb. The
diagonals of this matrix are the quadratic sum of the radiometric sensitivity of Tb
measurements and of the model error. In this first approach, off diagonal elements are
neglected in the antenna frame.
Pj are different parameters that should be retrieved, Pj prior are the a priori knowledge of the
parameters (obtained from models or satellites, the auxiliary information), and CPj is the
variance/covariance matrix of these parameters. The diagonal of the matrix are the
uncertainties on the a priori parameters.
Finally the above equation can be expressed as follows:
2
T 1
( X mod) C ( X X mod)
[4.14.3]
X z
Where CZ matrix is built by aligning along the main diagonal the matrixes CTb and CPj; the
vector X has a Nm+Np length and consists on:
1 2 Tb
Tb
.....
Tb
X
P
p
.....
p
Nm 1 2 prior
prior
prior
Np
j
j
[4.14.4]
Where Pprior is the a priori information of the parameter; X_mod has the same length as X and
is defined as: