Fig. 5. Geometry for determining the effect of scatterer height on
received phase.
capability and has a higher cost per unit area, thus
being most useful for localized mapping. Timely
access to a given region can be limited by airspace
restrictions or simply the time required to transport
an instrument to the area. The much lower altitude of
airborne systems makes the area coverage rate much
smaller as well.
Airborne systems require high precision motion
compensation to overcome the defocusing and
mislocation effects resulting from path deviations
caused by vibration, atmospheric turbulence, and
winds. These effects are much reduced or absent
in spaceborne systems, although platform orbit and
attitude must still be carefully controlled. Spaceborne
systems are subject to dispersive ionospheric
propagation effects, principally variable path delays in
two-pass systems up to tens of meters, that are absent
in airborne systems [5]. Both air- and spaceborne
systems suffer potential errors due to differential delay
through the wet troposphere. For example, using 1995
Shuttle Imaging Radar-C (SIR-C) repeat-track data
(not the SRTM mission of 2000), Goldstein [30]
estimates rms path length variations of 0.24 cm at
both L and C band. For the baselines used in those
experiments, this translates into a 6.7 m rms elevation
estimate error.
V. BASIC INTERFEROMETRIC SAR RELATIONSHIPS
A. The Effect of Height on the Phase of a Radar Echo
Since IFSAR is based on phase measurements,
we begin our derivation of basic IFSAR equations
by considering the phase of a single sample of the
echo of a simple radar pulse from a single point
scatterer. Consider the geometry shown in Fig. 5,
which shows a radar with its antenna phase center
located at ground range coordinate y =0andan
altitude z = H meters above a reference ground
plane (not necessarily the actual ground surface).
The positive x coordinate (not shown) is normal to
the page, toward the reader. A scatterer is located at
position
P1 on the reference plane z = 0 at ground
range dimension y
1
. The reference ground plane,
in some standard coordinate system, is at a height
h
ref
, so that the actual elevation of the radar is h =
h
ref
+ H andofthescattererisjusth
ref
.However,
h
ref
is unknown, at least initially. The depression
angle of the LOS to
P1, relativ e to the local
horizontal, is à rad, while the range to
P1 is
R
0
=
q
y
2
1
+ H
2
=
y
1
cos Ã
=
H
sinÃ
: (6)
The radar receiver is coherent; that is, it has both
in-phase (I) and quadrature (Q) channels, so that it
measures both the amplitude and phase of the echoes.
Consequently, the transmitted signal can be modeled
as a complex sinusoid [31]:
¯
x(t)=Aexp[j(2¼F t + Á
0
)], 0 · t · ¿ (7)
where F is the radar frequency (RF) in hertz,
3
¿ is
the pulse length in seconds, A is the real-valued pulse
amplitude, and Á
0
is the initial phase of the pulse in
radians. The overbar on
¯
x indicates a signal on an RF
carrier. The received signal, ignoring noise, is
¯
y(t)=
ˆ
A½exp
½
j
·
2¼F
µ
t ¡
2R
0
c
¶
+ Á
0
¸¾
,
2R
0
c
· t ·
2R
0
c
+ ¿:
(8)
In (8), ½ is the complex reflectivity of
P1 (thus ¾,the
radar cross section (RCS) of
P1, is proportional to
j½j
2
)and
ˆ
A is a complex-valued constant incorporating
the original amplitude A, all radar range equation
factors other than ¾, and the complex gain of the radar
receiver.Weassumethat½ is a fixed, deterministic
value for now.
After d emodulation to remove the carrier and
initial phase, the baseband received signal is
y(t)=
ˆ
A½exp
µ
¡j
4¼F R
0
c
¶
=
ˆ
A½exp
µ
¡j
4¼
¸
R
0
¶
,
2R
0
c
· t ·
2R
0
c
+ ¿:
(9)
If this signal is sampled at a time delay t
0
anywhere in
the interval 2R=c · t
0
· 2R=c + ¿ (that is, in the range
gate or range bin corresponding to range R), the phase
3
We follow the practice common in digital signal processing
literature of denoting unnormalized frequency in hertz by the
symbol F, and reserving the symbol f for normalized frequency in
cycles, or cycles per sample. A similar convention is used for radian
frequencies − and !.
IEEE A&E SYSTEMS MAGAZINE VOL. 22, NO. 9 SEPTEMBER 2007 PART 2: TUTORIALS–RICH ARDS 9