Page 141 - Fundamentals of Radar Signal Processing

(2.75)

where K (·) is the modified Bessel function of the second kind and order a–1
a–1
a nd . Thus, the product formulation suggests that modulation of a
standard Rayleigh variable by a central chi-distributed geometric term can
account for observed sea clutter distributions. Additional information on the K
distribution is given in App. A.
More recent research has begun to bridge the gap between the physics of

scattering and the apparent success of compound clutter models of the type
promoted by Ward and Jakeman and Pusey. Sangston summarizes the work on
extensions of the “many scatterer” physical model that leads to the Rayleigh
distribution (Sangston, 1994). Specifically, consider the model of Eq. (2.50),
but let the number of scatterers N be a random variable instead of a fixed
constant. This representation is referred to as a number fluctuations model.
Depending on the choice of the statistics of the number N of scatterers

contributing to the return at any given time, this modified version of Eq. (2.50)
can result in K, Weibull, gamma, Nakagami-m, or any of a number of other
distributions in the class of so-called Rayleigh mixtures.
Much of the work in compound RCS models has been performed in the
context of sea clutter analysis, and empirical sea clutter data have often been
observed to exhibit non-Rayleigh statistics such as Weibull, K, and log-normal

distributions. The number fluctuation model is intuitively appealing in this case
because it can be related to the physical behavior of waves. Specifically,
scattering theory suggests that the principal scatterers on the ocean surface are
the small capillary waves, as opposed to the large swells. These small
scattering centers tend to cluster near the crest of the swells, with fewer of them
in between. In other words, they are nonuniformly distributed over the sea
surface. Consequently, a radar illuminating the sea will receive echoes from a

variable number N of scatterers as the crests of the swells move into and out of
a given resolution cell. By summing echoes from a variable number of
scatterers, the number fluctuation model predicts the Weibull and K
distributions and provides a link between a phenomenological model of sea
scatter and these empirically observed statistics.
All of the statistical models described in Sec. 2.2.5 apply to the scattering

observed from a single resolution cell. That is, they represent the variations in
RCS observed by measuring the same region of physical space multiple times,
for example by transmitting multiple pulses in the same direction and measuring
the received power at the same delay after each transmission. Another use of the
product model of Eq. (2.74) is to describe the spatial variation of clutter