Geoscience Reference
In-Depth Information
ρ
12
representing the complex correlation of the signals from spaced receivers 1 and 2, for
example. In practice, these terms must be based on experimental estimates. The overall
stability of error estimators based on data with statistical errors themselves has not been
considered.
4.1.1 Added noise
The formulas above were derived in the absence of system noise but can easily be generalized
to include noise. The normalized correlation function error covariances for signals in the
presence of noise are still given by (19) and (20), only substituting the factor
+
S
N
ρ
Sii
→
(21)
S
wherever correlation terms with repeated indices appear. Here,
S
and
N
refer to the signal
and noise power, respectively.
On the whole, this analysis shows that the error covariance matrix is diagonally dominant
only in cases where either the signal-to-noise ratio or the coherence is small. These limits are
seldom applicable to coherent scatter, however. Even the longest interferometry baseline at
Jicamarca, nearly 100 wavelengths long, very often exhibits high coherence, and even small,
portable coherent scatter radars typically run in the high SNR limit. Since the error covariance
is not diagonally dominant in general, neglecting off-diagonal terms misrepresents statistical
confidence and could lead to image distortion.
In practice, it is expedient to diagonalize the error covariance matrix computed using the
formulas above and to apply the corresponding similarity transformation to forward problem
stated in (6) (Hysell & Chau, 2006). We find that the error variances that result fall into
two groups with relatively smaller and larger values, respectively. The former correspond
roughly to errors associated with measuring interferometric coherence, and the latter to errors
associated with interferometric phase.
4.1.2 Error propagation
Error propagation through MaxEnt can be treated as follows (see for example Hysell (2007);
Silver et al. (1990)). Using Bayes' theorem, we can cast the MaxEnt optimization problem
posed in (10) as one of maximizing the posterior probability of a model image,
m
, based on
visibility data
d
, which are related linearly through
d
=
Gm
, in the form
1
2
e
t
C
−
1
e
e
S
/
Γ
e
−
(
|
) ∝
p
m
d
(22)
e
−
E
≡
(23)
where the entropy
S
is the prior probability and the chi-squared model prediction error is
transitional probability. The constant
weights the two probabilities and must be adjusted
according to some criteria. In the variational approach to the optimization problem outlined
above, the Lagrange multiplier
Γ
Λ
plays the role of
Γ
. That variable is controlled by
Σ
, and so
there is always an adjustable free parameter.
Consider small departures
m
about the maximum probability (minimum
E
) solution. In the
neighborhood of a maximum, the gradient of the argument
E
vanishes, and we can always
δ
