Figure 1. "Flat Earth" SAR geometry
which is not appropriate for
the orbital case.
and the resulting full bandwidth single look azimuth
resolution
Paza/C - 2 p ( 5 )
Although these are very familiar, the important point here
is that almost invariably, these expressions and quantities
that follow from them are applied to the orbital case
without further examination of the velocity parameter.
The subscript a/c has been attached to these quantities to
emphasize that they do not apply to the more general
case in spite of suggestions to the contrary frequently
found in the literature. It is appropriate to consider the
corresponding expressions for an orbital system.
ORBITAL SAR GEOMETRY
A sketch of the geometry encountered in the orbital case
is shown in Figure 2. It is obvious from this figure that
the rate V * at which the spacecraft moves along its
orbital path is greater than the rate V B at which the
footprint of the antenna beam moves along the surface of
the Earth. It is less obvious that this geometry has an
impact on the range variation between the radar and a
scattering object as the radar passes. The radar range
equation applicable to a narrow beam side looking radar
in circular orbit may be shown to be
* - *o +
K,c 2
(6)
which is the first order expression equivalent to Eq. 1.
(The effects of Earth rotation have been neglected, an
acceptable approximation for the purposes of this
discussion.) In comparing Eqs. 1 and 6, it is clear that the
change in range due to radar displacement x from
broadside is reduced by the beam to spacecraft velocity
ratio V B /V S/fc . This ratio is well known to those
concerned with the design of the azimuth matched filter
used in processors for orbital SAR data, but seems to
have been overlooked in other aspects of SAR work, such
as calibration.
Let the antenna beamwidth and bandwidth be defined in
terms of the two-way amplitude pattern w of the Doppler
modulated antenna. The available integration time T is
determined by the width R 0 ß of the antenna pattern on
the Earth surface and the rate V B at which the beam
footprint moves. Thus
T -
*oß
(7)
The Doppler bandwidth of the signals received through
the antenna depends on the spacecraft velocity according
to
2 ß V*
X
(8)
The corresponding time bandwidth product is
TB v ,c 2*0 ß 2
(9)
Note that the (inverse) relative velocity ratio enters the
time bandwidth product, a fundamental parameter arising
in SAR analysis, as in all pulse compression systems.
The full bandwidth single look resolution available is
given by the ratio of the width of the beam footprint to
the time-bandwidth product, so that
Paz -
Il _A_
K,c 2 ß
(10)
On comparison to Eq. 5, it is clear that the orbital
geometry leads to an improvement in available resolution
in proportion to the beam to spacecraft velocity ratio
(Raney 1986).
As an aside, note that the rate of change of Doppler
frequency in the orbital case is given by
B 2 V slc V B
T XR,
(11)
Figure 2. Spherical viewing geometry
needed for orbital SAR analysis.
710