1. Surface scattering Chris Allen (callen@eecs.ku.edu)
Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm

2. Outline Factors affecting scattering Simple  models
More complex  models
Where to find more information

3. Factors affecting surface scattering
The scattering characteristics of a surface are represented by the scattering coefficient, 
For surface scattering, several factors affect 
Dielectric contrast
Large contrast at boundary produces large reflection coefficient
Air (r = 1), Ice (r ~ 3.2), (Rock (4  r  9), Soil (3  r  10),
Vegetation (2  r  15), Water (~ 80), Metal (  )
Surface roughness (measured relative to )
RMS height and correlation length used to characterize roughness
Incidence angle, ()
Surface slope
Skews the () relationship
Polarization
VV  HH » HV  VH

4. Factors affecting surface scattering
Surface roughness (measured relative to )
RMS height and correlation length used to characterize roughness
 is the surface height standard deviation
ℓ is the surface correlation length

5. Surface roughness and scattering
Rayleigh criteria for “smoothness”
Phase difference between two reflected rays < /2
Which leads to the following constraint on RMS height
Frauenhofer criteria for “smoothness”
Phase difference between two reflected rays < /8
Which leads to the following constraint on RMS height

6. Surface roughness and scattering
The Rayleigh criterion states that if the phase difference  (due to propagation) between two reflected rays shown (see Fig. 2.1) is less than /2 radians, then the surface may be considered smooth.
From the geometry we know (eqn 1)
which, if set < /2, leads to the Rayleigh criterion for a surface to be
considered smooth, that is (eqn 2)
Derive the Rayleigh criterion from the information provided.
Show all of the steps leading to eqns (1) and (2)

7. Surface roughness and backscatter
Backscatter is the special case where o = s, o = s

8. Backscatter from bare soil
Note: At 1.1 GHz,  = 27.3 cm

9. Simple  models
For purposes of radar system design, simple models for the backscattering characteristics from terrain can be used.
A variety of models have been developed.
Below are some of the more simple models that may be useful.
() = (0) cosn()
where  is the incidence angle and n is a roughness-dependent variable.
n = 0 for a very rough (Lambertian) surface [() = (0)]
n = 1 for a moderately rough surface [() = (0) cos ()]
n = 2 for a moderately smooth surface [() = (0) cos2 ()] or
() = (0) e – / o
where  is the incidence angle and o is a roughness-dependent angle.
In both model types (0) depends on the target characteristics

10. More complex  models Less simple backscattering models
A is the illuminated area
k is the wavenumber, k = 2/
ℓ is the surface correlation length
r is the permittivity of medium 2 relative to medium 1
r is the permeability of medium 2 relative to medium 1
(0) is the 2nd derivative of correlation coefficient at the origin
 is the incidence angle
 is the surface height standard deviation
2|(0)| is the mean-squared surface slope
Backscattering assumed throughout, unless specified otherwise o = s, o = s
r = 1 also assumed

11. More complex  models
Small-perturbation model – or – Incoherent scattering from a slightly rough surface
constraints:
rough surface-height standard deviation << incident wavelength k < 0.3 or  < 
average surface slope  the standard deviation times the wavenumber rms slope < 0.3 or  < 0.21ℓ

12. More complex  models
Small-perturbation model – or – Incoherent scattering from a slightly rough surface

13. More complex  models
Coherent reflection coefficients for rough planar surface
Incoherent scattering from a very rough planar surface
constraints:
radius of curvature >>  , isotropic roughness, ℓ << A
shadowing and multiple scattering ignored
where s = 4 2 / ℓ2

14. More complex  models
Incoherent scattering from a very rough planar surface

15. More complex  models
Incoherent Kirchhoff surface scattering – or – Geometric optics model
constraints:
ℓ > 1.6 
ℓ2 > 2.76  
 > 0.25 
shadowing and multiple scattering ignored
where p and q represent the transmit and receive polarizations, hence pp represents co-polarized backscattering (hh or vv) and pq represents cross-polarized backscattering (vh or hv)

16. More complex  models
Incoherent Kirchhoff surface scattering – or – Geometric optics model
2|(0)| is the mean-squared surface slope
– or –
2|(0)| = m2

17. Where to find more information
Ulaby FT; Moore RK; Fung AK; Microwave Remote Sensing, Vol. 2, Artech House, 1982
Fung AK; "Review of random surface scatter models," Proc. SPIE, vol. 358, Applications of Mathematics in Modern Optics, pp
Davies H; "The reflection of electromagnetic waves from a rough surface," Proc. IEE, 101(part IV), pp , 1954
Ruck GT; Barrick DE; Stuart WD; Kirchbaum CK; Radar Cross Section, Vol. 2, 1970