IEEE Geoscience and Remote Sensing Magazine - March 2013 - 22

Interferometric Phase Error (stdv) (°)

110
100
90
80

50
40
30

8
16
32

20
10
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Coherence

FIGURE 13. Standard deviation of the interferometric phase as a
function of the coherence. The six curves illustrate the systematic
decrease of the phase noise with increasing look numbers.

is quite efficient in improving the phase estimates, but
SAR interferometry
it has to be traded against the
compares the phase
associated loss of spatial resolution. The interferogram
of two complex radar
in Fig. 12 was obtained using
images for highly
5 looks in range and 5 looks
accurate elevation
in azimuth, which reduced
measurements.
the phase noise standard
deviation by approximately a
factor of 5.
To further explore the peculiarities of SAR interferometry, we insert (27) into (28) and obtain after a division
by Th

T{
2mrB =
Th = mr0 sin ^i ih . (31)

This equation describes the sensitivity of the radar
interferometer to small height differences Th. It is obvious that the sensitivity can be improved by increasing
the length of the perpendicular baseline B = . However,
the maximum useful baseline length is constrained by
two factors.
A first limitation is baseline decorrelation. To understand this limitation, we have to remember that the
recorded SAR signal can be regarded as being composed
of the radar echoes from a large number of closely
spaced elementary point-like scatterers with random
amplitude and phase. Each of these scatterers contributes to the overall radar signal with an additional
phase shift, which is proportional to its distance from
the receiving antenna. If we consider now a fixed scatterer ensemble on the ground and vary the radar look
angle, it becomes clear that the relative phase between
22

the radar echoes from the individual scatterers changes.
The difference will be small for short baselines B =, but
with increasing baseline length B = the phase contributions from the elementary scatterers within each resolution cell will become more and more different between
the two SAR images. As a result, the correlation between
the two complex SAR images decreases systematically
with increasing baseline length until it completely vanishes. The baseline length for which the two SAR images
become completely decorrelated is known as the critical baseline B =,crit. For flat surfaces, this can be expressed
mathematically as [77], [78]

B =,crit =

mr0 tan ^i ih
. (32)
md r

For baselines that are smaller than the critical baseline,
the spatial surface decorrelation can be removed by a process known as range filtering at the cost of a degraded range
resolution [79]. Note that the length of the critical baseline
increases with decreasing range resolution. Modern SAR
systems have typically a rather large bandwidth and baseline decorrelation is therefore nowadays a less important
issue than it was with early SAR systems.
A second and from a practical point of view often more
restrictive limitation for the maximum useful baseline
length results from ambiguities in the phase-to-height conversion process. For this we consider again (31) and recall
that the interferometric measurement provides only phase
values which are ambiguous by integer multiples of 2r. As
a result, the height measurements are also ambiguous by
multiples of

h amb =

mmr0 sin ^i ih
. (33)
B=

Such ambiguities are usually resolved during phase
unwrapping, which exploits spatial correlations between
the height values arising from natural topography [13],
[52], [60], [71]. The right-hand side of Fig. 12 shows the
unwrapped phase of the Atacama interferogram, which
is scaled to terrain height in accordance with the factor given in (31). The accuracy of this phase (or height)
reconstruction process depends on several factors like
the signal-to-noise ratio, the surface and volume decorrelation (cf. Section V-A), the ground resolution, and,
most important, the actual terrain itself. The latter may
strongly limit the useful baseline length for rough terrain
like deep valleys, isolated peaks, tall forests, or mountains
with steep slopes. On the other hand, large baselines are
desired to achieve a sensitive radar interferometer with
a good phase-to-height scaling. This dilemma becomes
especially pronounced for state-of-the-art radar sensors,
which will provide a high range bandwidth and hence
enable coherent data acquisitions with long interferometric baselines. To illustrate this problem, we consider
a spaceborne radar interferometer like TanDEM-X [76],
which operates in the interferometric imaging mode
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