Geoscience Reference
In-Depth Information
spread function for the given baseline
j
such that
h
ij
1,2
i
are the radiation
amplitude patterns for the antennas at either end of the baseline. Given the principle of
pattern multiplication, characteristics of the radiation pattern common to all the receiving
antennas can equally well be incorporated in
p
i
instead of
→
h
ij
2
i
, where
℘
℘
℘
1
i
℘
1,2
i
.
4.3 Super-resolution
That radar imaging resolution does not need to be diffraction limited can be appreciated
by considering coherence (normalized visibility) measurements made with a single
interferometry baseline in the high signal-to-noise ratio, high coherence limit. As shown by
Farley & Hysell (1996), the mean-squared error for the coherence estimate in this limit is
N
S
+
2
,
N
2
S
2
1
m
1
2
+
O
N
S
,
2
δ
=
,
···
(28)
2
,
m
is the number of statistically independent samples used, and where
S
and
N
are the signal and noise powers, respectively. Even given a finite number of samples,
the coherence estimate for a highly coherent target can be arbitrarily accurate given a high
enough signal-to-noise ratio. This means that the angular width of narrow targets can be
measured arbitrarily well, regardless of the baseline spacing, if
S
/
N
is sufficiently high.
Insofar as imaging, an inverse method that accounts for the effects of diffraction in the forward
model (i.e. the point spread function) need not be diffraction limited.
≡
−|
|
where
1
V
On the basis of information theory pertaining to the rate of information transmission through
a noisy channel, Kosarev (1990) investigated the resolution limit for spectral analysis, deriving
Shannon's resolution limit:
1
3
SR
=
log
2
(
1
+
S
/
N
)
(29)
This metric represents the maximum achievable resolution improvement over the
diffraction-limited, noise-free case for non-parametric signal processing methods. Kosarev
(1990) argued that there is no contradiction between this limit and the Heisenberg uncertainty
principle. Kosarev (1990) furthermore performed numerical tests, comparing a spectral
recovery algorithm based on maximum likelihood with entropy prior probability. Over the
S
/
N
range from 10-50 dB, the algorithm was able to achieve the Shannon limit. At Jicamarca,
the longest interferometry baseline is nearly 100 wavelengths long, and the diffraction limited
resolution is consequently about 0.5
◦
. In practice, useful resolution at about the 0.1
◦
-level can
be obtained with strong backscatter.
4.4 Optimal sensor placement
The placement of sensors (receiving antennas or antenna groups) on the ground is typically
constrained by practical consideration. If the sensors are subarrays of a fixed phased array, as
in the case of the Jicamarca Radio Observatory in Peru or the MU Radar in Japan, a number
of modules set by the number of receivers available will be selected from the total available
in such a way as to avoid redundant baselines. To avoid ambiguity, baseline lengths can
be selected such that the interferometry sidelobes are not illuminated by the transmitting
antenna. Baseline orientations may be selected to accommodate anisotropies in the scatterers.
As a rule, uniform sampling of visibility space seems to be conducive to artifact reduction,
although there may be good reasons to deviate from it.
