Surface and Volume Clutter - MATLAB Simulations for Radar Systems Design - page 266

The low grazing angle region extends from zero to about the critical angle.
The critical angle is defined by Rayleigh as the angle below which a surface is
considered to be smooth, and above which a surface is considered to be rough;
Denote the root mean square (rms) of a surface height irregularity as ,
then according to the Rayleigh criteria the surface is considered to be smooth if

h rms

4π h rms
λ

π
---

----------------- sin ψ g <

(6.2)

Consider a wave incident on a rough surface, as shown in Fig. 6.3. Due to
surface height irregularity (surface roughness), the Ðrough pathÑ is longer than
the Ðsmooth pathÑ by a distance 2 h rms sin ψ g . This path difference translates

into a phase differential ∆ψ :

2π
λ

------

∆ψ = 2 h rms sin ψ g

(6.3)

smooth
path

rough
path

ψ g

s m ooth surface level

h rms

ψ g

Figure 6.3. Rough surface definition.

The critical angle ψ gc is then computed when ∆ψ = π (first null), thus

4π h rms
λ

----------------- sin ψ gc = π

(6.4)

or equivalently,

λ
4 h rms

ψ gc = asin -------------

(6.5)

In the case of sea clutter, for example, the rms surface height irregularity is