  Remote sensing of soil moisture and vegetation water content from space often
requires underdetermined inversion of a zeroth-order approximation of the
forward radiative transfer equation in L-band---known as the $\tau$-$\omega$
model. This paper shows that the least-squares (LS) inversion of the model is
not strictly convex due to its saddle point structure. It is demonstrated that
the widely used unconstrained damped least-squares (DLS) inversion of the model
could lead to biased and physically unrealistic retrievals---chiefly because of
the existing preferential solution spaces that are characterized by the
eigenspace of the model Hessian. In particular, the numerical experiments show
that for sparse (dense) vegetation with a shallow (deep) optical depth, the DLS
tends to overestimate (underestimate) the soil moisture and vegetation water
content for a dry (wet) soil. This paper proposes a new Constrained
Multi-Channel Algorithm (CMCA) that confines the retrievals by an a priori
information of the soil type and vegetation density. Unlike the existing
algorithms, the presented approach can account for slowly varying dynamics of
the vegetation water content over croplands through a temporal smoothing-norm
regularization. We also demonstrate that depending on the resolution of the
constraints, the algorithm leads to super-resolved soil moisture retrievals
with much higher resolution than the spatial resolution of the radiometric
observations.
