Abstract

Full-wave solutions are given for the single- and double-scatter radar cross sections for two-dimensional random rough surfaces. High-frequency approximations are used for the double-scatter cross sections in order to express them as numerically tractable four-dimensional integrals. The major contributions to the double-scatter cross sections are associated with the quasi-parallel and quasi-antiparallel double-scatter paths. They come from the neighborhoods of specular points. The enhancement of the backscatter cross sections, which is associated with the quasi-antiparallel double-scatter paths, is observed for both the like- and cross-polarized cross sections.

© 2001 Optical Society of America

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References

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  1. E. Bahar, M. El-Shenawee, “Enhanced backscatter from one dimensional random rough surfaces—stationary phase approximations to full wave solutions,” J. Opt. Soc. Am. A 12, 151–161 (1995).
    [CrossRef]
  2. E. Bahar, Y. Zhang, “A new unified full wave approach to evaluate the scatter cross sections of composite random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 973–980 (1996).
    [CrossRef]
  3. M. I. Sancer, “Shadow-corrected electromagnetics scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
    [CrossRef]
  4. E. Bahar, M. El-Shenawee, “Vertically and horizontally polarized diffuse double scatter cross sections of one dimensional random rough surfaces that exhibit enhanced backscatter—full wave solutions,” J. Opt. Soc. Am. A 11, 2271–2285 (1994). An extensive list of references on backscatter enhancement is published here.
    [CrossRef]
  5. M. El-Shenawee, E. Bahar, “Double scatter cross sections for two dimensional random rough surfaces—high frequency approximation,” presented at the 1995 IEEE AP-S International Symposium and UNC/URSI Radio Meeting, Newport Beach, Calif., June 18–23, 1995.
  6. A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
    [CrossRef] [PubMed]
  7. A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
    [CrossRef]
  8. E. Bahar, E. M. Herzinger, M. A. Fitzwater, “Incoherent like and cross polarized cross sections of an anisotropic rough sea surface with swell,” J. Geophys. Res. [Oceans] 94, 2159–2169 (1989).
    [CrossRef]
  9. M. El-Shenawee, E. Bahar, “Double scatter radar cross sections for two dimensional random rough surfaces that exhibit backscatter enhancement,” presented at the meeting of the Applied Computational Electromagnetic Society, Monterey, Calif., March 18–22, 1996.

1996 (1)

E. Bahar, Y. Zhang, “A new unified full wave approach to evaluate the scatter cross sections of composite random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 973–980 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1990 (1)

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

1989 (1)

E. Bahar, E. M. Herzinger, M. A. Fitzwater, “Incoherent like and cross polarized cross sections of an anisotropic rough sea surface with swell,” J. Geophys. Res. [Oceans] 94, 2159–2169 (1989).
[CrossRef]

1969 (1)

M. I. Sancer, “Shadow-corrected electromagnetics scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

Bahar, E.

E. Bahar, Y. Zhang, “A new unified full wave approach to evaluate the scatter cross sections of composite random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 973–980 (1996).
[CrossRef]

E. Bahar, M. El-Shenawee, “Enhanced backscatter from one dimensional random rough surfaces—stationary phase approximations to full wave solutions,” J. Opt. Soc. Am. A 12, 151–161 (1995).
[CrossRef]

E. Bahar, M. El-Shenawee, “Vertically and horizontally polarized diffuse double scatter cross sections of one dimensional random rough surfaces that exhibit enhanced backscatter—full wave solutions,” J. Opt. Soc. Am. A 11, 2271–2285 (1994). An extensive list of references on backscatter enhancement is published here.
[CrossRef]

E. Bahar, E. M. Herzinger, M. A. Fitzwater, “Incoherent like and cross polarized cross sections of an anisotropic rough sea surface with swell,” J. Geophys. Res. [Oceans] 94, 2159–2169 (1989).
[CrossRef]

M. El-Shenawee, E. Bahar, “Double scatter radar cross sections for two dimensional random rough surfaces that exhibit backscatter enhancement,” presented at the meeting of the Applied Computational Electromagnetic Society, Monterey, Calif., March 18–22, 1996.

M. El-Shenawee, E. Bahar, “Double scatter cross sections for two dimensional random rough surfaces—high frequency approximation,” presented at the 1995 IEEE AP-S International Symposium and UNC/URSI Radio Meeting, Newport Beach, Calif., June 18–23, 1995.

Chen, J. S.

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

El-Shenawee, M.

E. Bahar, M. El-Shenawee, “Enhanced backscatter from one dimensional random rough surfaces—stationary phase approximations to full wave solutions,” J. Opt. Soc. Am. A 12, 151–161 (1995).
[CrossRef]

E. Bahar, M. El-Shenawee, “Vertically and horizontally polarized diffuse double scatter cross sections of one dimensional random rough surfaces that exhibit enhanced backscatter—full wave solutions,” J. Opt. Soc. Am. A 11, 2271–2285 (1994). An extensive list of references on backscatter enhancement is published here.
[CrossRef]

M. El-Shenawee, E. Bahar, “Double scatter cross sections for two dimensional random rough surfaces—high frequency approximation,” presented at the 1995 IEEE AP-S International Symposium and UNC/URSI Radio Meeting, Newport Beach, Calif., June 18–23, 1995.

M. El-Shenawee, E. Bahar, “Double scatter radar cross sections for two dimensional random rough surfaces that exhibit backscatter enhancement,” presented at the meeting of the Applied Computational Electromagnetic Society, Monterey, Calif., March 18–22, 1996.

Fitzwater, M. A.

E. Bahar, E. M. Herzinger, M. A. Fitzwater, “Incoherent like and cross polarized cross sections of an anisotropic rough sea surface with swell,” J. Geophys. Res. [Oceans] 94, 2159–2169 (1989).
[CrossRef]

Herzinger, E. M.

E. Bahar, E. M. Herzinger, M. A. Fitzwater, “Incoherent like and cross polarized cross sections of an anisotropic rough sea surface with swell,” J. Geophys. Res. [Oceans] 94, 2159–2169 (1989).
[CrossRef]

Ishimaru, A.

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

Maradudin, A. A.

Méndez, E. R.

Sancer, M. I.

M. I. Sancer, “Shadow-corrected electromagnetics scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

Zhang, Y.

E. Bahar, Y. Zhang, “A new unified full wave approach to evaluate the scatter cross sections of composite random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 973–980 (1996).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

M. I. Sancer, “Shadow-corrected electromagnetics scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

E. Bahar, Y. Zhang, “A new unified full wave approach to evaluate the scatter cross sections of composite random rough surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 973–980 (1996).
[CrossRef]

J. Acoust. Soc. Am. (1)

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

J. Geophys. Res. [Oceans] (1)

E. Bahar, E. M. Herzinger, M. A. Fitzwater, “Incoherent like and cross polarized cross sections of an anisotropic rough sea surface with swell,” J. Geophys. Res. [Oceans] 94, 2159–2169 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

Other (2)

M. El-Shenawee, E. Bahar, “Double scatter cross sections for two dimensional random rough surfaces—high frequency approximation,” presented at the 1995 IEEE AP-S International Symposium and UNC/URSI Radio Meeting, Newport Beach, Calif., June 18–23, 1995.

M. El-Shenawee, E. Bahar, “Double scatter radar cross sections for two dimensional random rough surfaces that exhibit backscatter enhancement,” presented at the meeting of the Applied Computational Electromagnetic Society, Monterey, Calif., March 18–22, 1996.

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Figures (10)

Fig. 1
Fig. 1

(a) Double-scatter electromagnetic wave, (b) double-scatter intensity for two-dimensional random rough surfaces, (c) double-scatter quasi-parallel case, (d) double-scatter quasi-antiparallel case.

Fig. 2
Fig. 2

Single-scatter vertical-to-vertical radar cross section. Mean square height (Rayleigh parameter β) is the variable parameter 110β394. The mean square slope (mss) value is 0.5. The following parameters apply to Figs. 210: incident angle of 10°, relations permittivity of -9.888312/j1.051766, and wavelength of 0.633 μm.

Fig. 3
Fig. 3

Single-scatter vertical-to-vertical radar cross section. Mean square slope (mss) is the variable parameter, and the curves from top to bottom correspond to mss values of 0.25, 0.4, 0.5, 0.65, 0.75, 0.85, and 1.0. The value of β is 394.

Fig. 4
Fig. 4

Single-scatter like-polarized radar cross section for vertical to vertical (lower curve) and horizontal-to-horizontal (upper curve) polarization. The mss and β values are 0.5 and 394, respectively.

Fig. 5
Fig. 5

Double-scatter radar cross section. Mean square height (Rayleigh parameter β) is the variable parameter (quasi-parallel and quasi-antiparallel contributions added). The polarization is vertical to vertical. The mss value is 0.5. Lowest curve, β=10. Data saturate as β increases.

Fig. 6
Fig. 6

Double-scatter radar cross section. The variable parameter is the mss (quasi-parallel and quasi-antiparallel contributions added). The polarization is vertical to vertical. The value of β is 394. Lowest curve, mss=0.25. Data saturate as mss increases.

Fig. 7
Fig. 7

Double-scatter radar cross sections for both like-polarized and both cross-polarized cases (quasi-parallel and quasi-antiparallel contributions added). The mss and β values are 0.5 and 394.105, respectively. Cross-polarized data are lower near grazing angles.

Fig. 8
Fig. 8

Single- and double-scatter radar cross sections added. The variable parameter is β. The polarization is vertical to vertical. The mss value is 0.5. Lowest curve, β=10. Data saturate as β increases.

Fig. 9
Fig. 9

Single- and double-scatter radar cross sections added. The variable parameter is the mss, and the curves from top to bottom correspond to mss values of 0.25, 0.4, 0.5, 0.65, 0.75, 0.85, and 1.0. The value of β is 394. The polarization is vertical to vertical.

Fig. 10
Fig. 10

Single- and double-scatter radar cross sections added for both like-polarized and both cross-polarized cases. The mss and β values are 0.5 and 394, respectively. Upper curves, HH followed by VV. Lowest curves, VH and HV merge.

Equations (72)

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Gdf(r¯)=k02πj3exp(-jk0r)rD2(n^f, n^)nyf-ny×exp(jk0n^f  rs2)×exp[-jk0n^  (rs2-rs1)] D1(n^,n^i)-nyi+ny×exp(-jk0n^i  rs1) dnydnz(1-ny2-nz2)1/2×U(rs1)U(rs2)dxs1dzs1dxs2dzs2Gi(0),
n^i=nxie^x+nyie^y+nzie^z,
n^f=nxfe^x+nyfe^y+nzfe^z,
r=xe^x+ye^y+ze^z,
n^=nxe^x+nye^y+nze^z.
rs1=xs1e^x+h(xs1, zs1)e^y+zs1e^z,
rs2=xs2e^x+h(xs2, zs2)e^y+zs2e^z.
σT=4πr2AGfGi2,
σT=k0616Aπ5×D2(n^f, n^)D1(n^, n^i)D2*(n^f, n^)D1*(n^, n^i)(nyf-ny)(nyf-ny)(-nyi+ny)(-nyi+ny)×U(rs1)U(rs1)U(rs2)U(rs2)×exp{jk0[nxf(xs2-xs2)+nyf(h2-h2)+nzf(zs2-zs2)]}exp{-jk0[nx(xs2-xs1)+ny(h2-h1)+nz(zs2-zs1)]}×exp{jk0[nx(xs2-xs1)+ny(h2-h1)+nz(zs2-zs1)]}exp{-jk0[nxi(xs1-xs1)+nyi(h1-h1)+nzi(zs1-zs1)]}×dnydnz(1-ny2-nz2)1/2dnydnz(1-ny2-nz2)1/2×dxs1dzs1dxs2dzs2dxs1dzs1dxs2dzs2.
xd1=xs1-xs1, xd2=xs2-xs2,
zd1=zs1-zs1,zd2=zs2-zs2,
xa1=(xs1+xs1)/2,xa2=(xs2+xs2)/2g,
za1=(zs1+zs1)/2,za2=(zs2+zs2)/2.
σT=k0616Aπ5×D2(n^f, n^)D2(n^, n^i)D2*(n^f, n^)D1*(n^, n^i)(nyf-ny)(nyf-ny)(-nyi+ny)(-nyi+ny)×U(ra1)U(ra2)exp{jk0[nxfxd2-nx(0.5xd2-0.5xd1+xa2-xa1)+nx(-0.5xd2+0.5xd1+xa2-xa1)-nxixd1]}×exp{ jk0[nzfzd2-nz(0.5zd2-0.5zd1+za2-za1)+nz(-0.5zd2+0.5zd1+za2-za1)-nzizd1]}exp{ jk0[nyf(h2-h2)-ny(h2-h1)+ny(h2-h1)-nyi(h1-h1)]}×dnydnz(1-ny2-nz2)1/2dnydnz(1-ny2-nz2)1/2×dxd1dzd1dxd2dzd2dxa1dza1dxa2dza2.
h1=ha1+(xd1hxa1+zd1hza1)/2,
ha1=hy(xa1, zd1),hxa1=h/x,
hza1=h/z at x=xa1 andz=za1,
h1=ha1-(xd1hxa1+zd1hxa1)/2;
h1-h1=xd1hxa1+zd1hxa1.
δ(hx-hxs)=12π-exp[ixd(hx-hxs)]dxd.
δ(hxa1-hxa1s)δ(hxa2-hxa2s)δ(hza1-hza1s)
×δ(hza2-hza2s)
δ(ha1-ha1s)δ(ha2-ha2s),
hxa1s=--nxi+nx+nx2/-nyi+ny+ny2,
hxa2s=--nxf+nx+nx2/-nyf+ny+ny2,
hza1s=--nzi+nz+nz2/-nyi+ny+ny2,
hza2s=--nzf+nz+nz2/-nyf+ny+ny2.
σTPQ=k02πAR,S=V,HD2PS(n^f, n^)D1SQ(n^, n^i)D2*PR(n^f, n^)D1*RQ(n^, n^)(nyf-ny)(nyf-ny)(nyi-ny)(nyi-ny) U(ra1)U(ra2)δ(ha1-ha1s)×δ(ha2-ha2s)exp[jk0(ra1-ra2)(n^-n^)] dnydnzdnydnzdxa1dxa2dza1dza2nxnx[nyf-(ny+ny)/2]2[nyi-(ny+ny)/2]2,
rai=xaie^x+hie^y+zaie^z(i=1, 2).
hai(x, z)=hxaie^x+hzaie^z,
hais(x, z)=hxaise^x+hzaise^z.
σTPQk02πA P2(n^i)P2(n^f)P,Q=V,HD2PS(n^f, n^)D1SQ(n^, n^)D2*PR(n^f, n^)D1*RQ(n^, n^i)(nyf-ny)(nyf-ny)(nyi-ny)(nyi-ny)s×exp[jk0(ρa1-ρa2)  (n^-n^)]exp[-k02h2(ny-ny)2]p(ha1s, ha2s)[1-P2(|ny|)]×[1-P2(|ny|)] dnydnzdnydnzdxa1dza1dxa2dza2nxnx[nyf-(ny+ny)/2]2[nyi-(ny+ny)/2]2.
exp[ik0(ny-ny)(ha1-ha2)]
exp[ik0(ny-ny)ha1]exp[-ik0(ny-ny)ha2]=|χ2[k0(ny-ny)]|.
ραi=rαi-hαieγ
ρa1-ρa2ρad,(ρa1+ρa2)/2ρaa.
-LmLm-LmLmexp[(jk0ρad)  (n^-n^)]dxaddzad
=2Lm sinc[k0Lm(nx-nx)]2Lm sinc[k0Lm(nz-nz)],
σdpPQ=(2k0Lm)2π P2(n^i)P2(n^f) R,S=V,H[D2PS(n^f, n^)D1SQ(n^, n^i)D2*PR(n^f, n^)D1*RQ(n^, n^i)](nyf-ny)(nyf-ny)(-nyi+ny)(-nyi+ny)s×p(hxa1shxa2shza1shza2s)[nyf-(ny+ny)/2]2[-nyi+(ny+ny)/2]2 [1-P2(|ny|)][1-P2(|ny|)]×sin[k0Lm(nx-nz)] sinc[k0Lm(nz-nz)]exp[-h2k02(ny-ny)2] sin θsin θdθdθdϕdϕ.
nx=sin θ cos ϕ,ny=cos θ,nz=sin θ sin ϕ,
0<θ<π,ϕ0<ϕ<2π+ϕ0,
JJ=sin2 θcos ϕsin2 θcos ϕ.
dnydnydnzdnz/nxnx=sin θsin θdθdθdϕdϕ.
0θθ,θθπ,
0ϕϕ,ϕϕ2π.
σdpPQ=σ1PQ+σ2PQ=2 Reσ1PQ.
xd1=xs1-xs2,xd2=xs2-xs1,
zd1=zs1-zs2,zd2=zs2-zs1,
xa1=(xs1+xs2)/2,xa2=(xs2+xs1)/2,
za1=(zs1+zs2)/2,za2=(zs2+zs1)/2.
h2=ha2-(xd2hxa2+zdzhza2)/2;
h1-h2=xd1hxa1+zd1hza1.
σdaPQ=(2k0Lm)2π P2(n^i)P2(n^f) R,S=V,HD2PS(n^f, n^)D1SQ(n^, n^i)D2*PR(n^f, n^)D1*RQ(n^, n^i)(nyf-ny)(nyf-ny)(-nyi+ny)(-nyi+ny)s×p(hxa1s, hxa2s, hza1s, hza2s)16/(nyf-nyi+ny-ny) (nyf-nyi-ny+ny)2[1-P2(|ny|)][1-P2(|ny|)]×sinc[k0Lm(nxf+nxi-nx-nx)]sinc[k0Lm(nzf+nzi-nz-nz)]exp[-h2k02(nyf-ny-ny+nyi)2]×sin θsin θdθdθdϕdϕ.
0θπ-θ,π-θθπ,
0ϕϕ+π,ϕ+πϕ2π.
hxa1s=-(nxf-nxi+nx-nx)/(nyf-nyi+ny-ny),
hxa2s=-(nxf-nxi+nx-nx)/(nyf-nyi+ny-ny),
hza1s=-(nzf-nzi+nz-nz)/(nyf-nyi+ny-ny),
hxa2s=-(nzf-nzi+nz-nz)/(nyf-nyi+ny-nyi),
n^s1=n^-n^i,n^s1=n^-n^i,
n^s2=n^f-n^,n^s2=n^f-n^.
n^sap1=(n^s1+n^s1)/2,n^sap2=(n^s2+n^s2)/2,
n^saa1=(n^s1+n^s2)/2,n^saa2=(n^s1+n^s2)/2,
n^sdp=n^s1-n^s1=n^s2-n^s2=n^-n^=0.
n^sda=n^s1-n^s2=n^s1-n^s2=n^+n^-n^f-n^i0.
χ[k0(ny-ny)]=exp[-h2k02(ny-ny)2/2],
χ[k0(nyf+nyi-ny-ny)]
=exp[-h2k02(nyf+nyi-ny-ny)2/2]
σsPQ=4π|DPQ(n^f, n^i)|2p(hxs, hzs)/(nyf-nyi)4,
P=Q,σsPQ=0,PQ,
VVH,VHH,
HHV,HVV.

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