equal to the inter-pulse spacing at the Earth’s 
surface, A - V B /f a . 
The correlation function for the reflecting neighbourhood 
in the scene is 
R d (m,n) 
o 0 A m - n 
0 m * n 
(20) 
Notice the subtle difference between this correlation 
function and that of the discrete isolated scatterer of Eq. 
14. The distributed scatterers are modelled as if they 
were replaced by a set of individual (point) reflectors 
having uniformly distributed random phase, spaced from 
each other by the surface inter-pulse distance. (The inter 
pulse spacing is generally smaller than the "resolution", 
sometimes by an appreciable margin.) Whereas the cells 
are statistically independent at the surface, the finite 
bandwidth of the radar and processor so that their 
energies spread over neighbouring cells in the image, thus 
reinforcing each other. This gives rise to a processing 
gain of relevance to the calibration question. Use of a 
well known and measurable quantity such as surface 
sample spacing avoids the insecurity encountered if the 
model were cast in terms of the azimuth resolution cell as 
is usually done. Resolution is not a fundamental 
parameter, but a consequence of optional and sometimes 
imprecise processor operations. 
The discrete scene correlation function of Eq. 20 may be 
substituted into Eq. 13 to find 
E[g d \ - o 0 A l/>(”)l 2 (21) 
The sum over the impulse response may be re-written as 
El 2 - Ip(°)I 2 
Ei^wi 2 
n 
U><0)| 2 
(22) 
in which the square of the number of integrated pulses 
appears as in the point scatterer case, and a sum over the 
normalized image impulse response which we let be p p . 
Thus, the processing gain G P may be defined as 
G,-1.2Ep ,00 (23) 
/1-1 
which describes the increase in mean signal level in the 
image over that predicted from the impulse response 
peak. The sum is over those cells neighbouring the 
reference cell whose image over-laps the image pixel 
being observed. In all cases the summation limit M needs 
to be sufficiently large to span the spread of the modified 
impulse response. 
The expected image power (per pixel) from a set of 
distributed scatterers may be written 
(g d ) - o 0 A K'Nfa (24) 
which should be compared to the response from an 
isolated point scatterer, Eq. 15. It may be shown that Eq. 
24 is robust in the presence of focus errors and multi 
looking. 
In order to illustrate behaviour typical of G P , let the 
antenna prefilter be Gaussian and of Doppler bandwidth 
B sampled by the pulse repetition frequency f a , and let 
the processor filter be matched. Then 
G p - 1 + 2 exp 
n-1 
2 „2) 
B z n 
f 
Ja 
(25) 
Eq. 24 is plotted in Figure 4. It shows for over-sampling 
ratios in excess of 1.2, that the approximation f a /B is 
satisfactory. For orbital situations in which the over- 
sampling ratio may be rather close to unity, the exact 
expression derived through the correlation function of the 
filters in use may be preferred. (Note that this 
formulation converges to the correct form for under 
sampling, a condition not satisfied by the usual 
approximation.) 
Processing gain 
Oversampling ratio 
Figure 4. Processing Gain as a function of 
the over-sampling ratio, and the 
conventional approximation. 
DISCUSSION 
The results above may used to find the usual signal to 
noise and clutter (distributed scene) to noise ratios. In 
both cases, it is worthwhile to note the way in which 
spacecraft and beam footprint velocities enter the 
expressions. The discussion focuses on the issue of 
calibration, for which comparisons of the clutter to the 
signal from a point reference signal are required. For 
compactness in the following, all expressions are derived 
assuming that only the chosen signals and/or noise are 
present. In practice, adequate signal to noise ratios are 
needed, and subtraction of noise and/or clutter is 
required. The points of this paper would not be served by 
going into details at that level, and good practical 
references are available (e.g., Gray et al. 1990). 
712