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. 
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