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

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

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

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

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

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)

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

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

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

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

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  < 0.048  average surface slope  the standard deviation times the wavenumber rms slope < 0.3 or  < 0.21ℓ

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

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

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

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)

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

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