Geoscience Reference
In-Depth Information
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1
2
)
∝
e
−
δ
m
t
H
δ
m
p
(
m
|
d
(24)
2
E
(
)
=
∂
m
H
m
with
H
the Hessian matrix. Now, (24) has the form of a probability density function (PDF) for
normally distributed model errors
∂δ
m
∂δ
m
which we maximize through the minimization of
E
.We
can consequently identify the Hessian matrix with the inverse model covariance matrix
C
−
m
.
Taking the necessary derivatives gives the error bounds on the image:
δ
C
−
1
m
G
t
C
−
1
G
]
−
1
=
+[
Γ
Im
(25)
The
term in (12) represents the influence of the data on the final MaxEnt model. The greater
the influence, the smaller the uncertainties. The other term comes from the entropy prior
and ensures that the model variances will be very small where the model values themselves
are small.
Γ
This is obviously significant in view of the importance of suppressing spurious
artifacts.
4.2 Extensions
Hysell & Chau (2006) introduced two improvements to WD85 important for upper
atmospheric radar research. The first involves incorporating the overall two-way antenna
radiation pattern in the imaging analysis. Rather than attempting to remove the two-way
pattern from the effective brightness distribution through division, with the attendant
conditioning problems, we just acknowledge the influence of the pattern on the effective
brightness and modify the entropy metric accordingly to anticipate it. If Shannon's expression
favors a uniform brightness distribution, the expression that favors distributions that resemble
the beam shape is (Skilling, 1989)
S
=
−
∑
i
f
i
ln
(
f
i
/
p
i
F
)
where
p
i
represents the two-way radiation power pattern. Propagating this expression
through the preceding analysis alters only the brightness model:
Fp
i
e
−
[
h
λ
]
i
Z
f
i
=
(26)
p
i
e
−
[
h
λ
]
i
=
∑
i
Z
(27)
The remaining formalism is unchanged. The only restriction is that
p
i
should be positive. In
practice, the effect of the modification is to suppress spurious brightness outside in regions
where the radiation pattern is depressed.
The second improvement applies when heterogeneous antennas are used for reception. In
that case, the
p
i
in (26) can be made to match the radiation pattern of the transmitting antenna
array, which is common to all the received signals. The radiation patterns of the receiving
antennas are then explicitly incorporated into the expressions for the effective brightness,
B
eff
, associated with each baseline. In view of (1) or (6), this is done by modifying the point
