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(no sum implied). Maximizing with respect to
Λ
yields one more equation relating that term
to the others.
2
t
C
− λ
λ
=
4
ΣΛ
0
(13)
The resulting system of 2
M
1 coupled, nonlinear equations for the Lagrange multipliers can
be solved numerically. (The algorithm implemented here uses the hybrid method of Powell
(1970).) Finally, (8) yields the desired image. The algorithm is robust and converges in practice
when provided with data uncontaminated by interference. An analytic form of the required
Jacobian matrix can readily be derived from (11).
+
4.1 Error analysis
Defining
ρ
12
as the normalized cross-correlation of the signals from receivers 1 and 2, an
obvious estimator of
ρ
12
is:
1
m
m
i
1
v
1
i
v
2
i
∑
=
ρ
12
ˆ
=
m
∑
(14)
i
2
m
i
|
v
1
i
|
|
v
2
i
|
2
∑
=
=
1
1
where the numerator and denominator are computed from the same
m
statistically
independent, concurrent samples. The error covariance matrix for interferometric
cross-correlation or cross-spectral visibility estimates derived from this estimator was given
by Hysell & Chau (2006):
2
+
δ
2
e
r
12
e
r
34
=
(
δ
)
/2
(15)
2
−
δ
2
e
i
12
e
i
34
=
(
δ
)
/2
(16)
2
+
δ
2
e
r
34
e
i
12
=
(
δ
)
/2
(17)
e
r
12
e
i
34
=
(
δ
2
2
−
δ
)
/2
(18)
where
e
r
12
stands for the error in the estimate of the real part of the correlation of the signals
from spaced receivers 1 and 2, for example, and where the indices may be repeated depending
on the interferometry baselines in question. Also,
1
m
1
2
ρ
34
(
ρ
13
ρ
23
+
ρ
14
ρ
24
)
ρ
13
ρ
24
−
2
δ
=
(19)
1
2
ρ
12
(
ρ
13
ρ
14
+
ρ
23
ρ
24
)
−
2
4
ρ
12
ρ
34
1
2
2
2
+
|
ρ
13
|
+
|
ρ
14
|
+
|
ρ
23
|
+
|
ρ
24
|
and
1
m
1
2
ρ
34
δ
2
ρ
14
ρ
23
−
(
ρ
13
ρ
23
+
ρ
14
ρ
24
)
=
(20)
1
2
ρ
12
(
ρ
13
ρ
14
+
ρ
23
ρ
24
)
−
2
,
4
ρ
12
ρ
34
1
2
2
2
+
|
ρ
13
|
+
|
ρ
14
|
+
|
ρ
23
|
+
|
ρ
24
|
