where Ro is the range distance at t = x'/V. The integrand in Eq. (1), 
which may be interpreted as the contribution of a surface element at x' to 
the total received signal, can then be written as 
^R 
SG, E) = vint ot ja later joe IRF I-VE/R, (3) 
neglecting the constant-phase term 22 3IkRo which is common to all the 
surface elements considered in the integration. 
The complex image amplitude is obtained by convolving the recorded 
signal with the reference function h(x), i.e. 
ic) = f sto h(x — Vt) dt (4) 
where the reference function has the form 
(5) 
in which ap(x) is the amplitude weighting (which is normally chosen to 
match the antenna gain pattern aj) and b is the quadratic phase factor or 
focus parameter, which is normally chosen as 
b = k/R, (6) 
in order to cancel the quadratic time dependence in the phase of the 
recorded signal. If this is so chosen, the contribution of the surface 
element at x' to the complex image amplitude at x may be written as 
5 -jk(x'^-x*JR, 
i(x', x) s r(x', w)e (7) 
where 
r(x', w) foe. tJa(x', x, t) g Jet dt (8) 
with 
atx', x, t) = CHES - Vt)a, (x - Vt) (9) 
and 
w = 2kV(x' - x)/R, . (10) 
As discussed in Hasselmann et al (1985), r(x', w) may be interpreted as the 
finite-resolution Fourier transform of the surface reflectivity, with the 
resolution being determined by the width of the weighting function 
a(x', x, t). For stationary scenes, r(x, w) is centered at w = 0 and has 
a width inversely proportional to the width of a(x', x, t). For randomly 
moving surfaces, r(x', w) has non-zero values over a wider range of fre- 
quencies, which implies that the image is not a simple map of the radar 
reflectivity as in the case of a stationary surface. 
If the surface reflectivity is assumed to be spatially "white," i.e. 
428 
  
wh 
is 
Rar