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amounts to reducing the candidate space of model solutions by imposing a priori information.
The information may involve expectations about the variance of the model solution (its
“roughness” or regularity) or something more specific, such as the range of admissible
numerical values it can assume. A priori information may be incorporated implicitly through
the inclusion of damping terms in the inversion algorithm (e.g. damped least squares,
Tikhonov regularization, etc.) or explicitly using a Bayesian formalism. Other desirable
properties, including model-data consistency, model resolution, and data resolution, can also
be optimized.
A common approach to radar imaging is based on the linear constrained minimum variance
(LCMV) (or sometimes minimum variance distortionless response (MVDR)) principle and
was introduced by Capon (1969). Consider the column vector x with n entries corresponding
to complex voltage samples from n sensors, each at a coordinate r n measured from some
reference point. Suppose the objective is to discriminate echoes arriving along a wavevector
k from other echoes, noise, interference, etc., by forming an appropriated weighted sum of
the voltage samples, y
w x , prior to detection. If the weights are all unity, the signals
detected by the sensors from a point source designated by k would be proportional to the
column vector with elements e i k · r 1 , e i k · r 2 ,
=
, e i k · r n . After incoherent integration, the output
···
of the detector with arbitrary weights will be
y 2
w
xx
w Rw
|
| =
=
w
(3)
where w is the column vector composed of the weights, R is the signal covariance matrix,
constructed from the measured visibilities, and † denotes the complex conjugate transpose.
Capon's LCMV strategy is to optimize the weights by minimizing the output of the detector
while maintaining unity gain in the direction of the point source, viz.
: w Rw
w e
w =
+ γ (
)
arg min
w ,
1
(4)
γ
where
is a Lagrange multiplier. The unity-gain constraint is imposed to prevent the trivial
solution. The output of the optimal detector using the weights thus found for a given bearing
can readily be shown to be
γ
) 1 . Imaging then is performed by computing the
optimized detector output for all possible bearings. The algorithm is essentially a linear beam
former, where nulls are adaptively aligned with sources that are not aligned with the bearing
of interest.
2
e R 1 e
|
|
=(
y
Capon's LCMV method is simple to implement and execute computationally. While there is
no provision for error handling in the algorithm posed above, the remedy is to precompute
the visibility error covariance matrix (see below) and then transform (3) through similarity
transformation into a space where that matrix is the identity. The method is equally well suited
for imaging continua and point targets, making it a superior choice for geophysical remote
sensing compared to point-targeting algorithms like MUSIC (Schmidt, 1986) or CLEAN.
However, there is no guarantee that the brightness distribution found will be consistent with
the visibility data within the tolerance of the specified error bounds. The a priori information
contained within the method is moreover far from explicit, making it hard to assess its validity.
Any number of alternative imaging methods exist that can minimize or constrain the model
prediction error while managing issues arising from the mixed determined or ill conditioned
nature of the inverse problem. In the next section of the paper, we turn our attention to the
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