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 Rayleighcriterion from the information provided. • Show all of the stepsleading 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 wavelengthk < 0.3 or  < 0.048  • average surface slope  the standard deviation times the wavenumberrms 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) andpq 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. 87-98 1982 • Davies H; "The reflection of electromagnetic waves from a rough surface," Proc. IEE, 101(part IV), pp. 209-214, 1954 • Ruck GT; Barrick DE; Stuart WD; Kirchbaum CK; Radar Cross Section, Vol. 2, 1970