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Validity of Backscattering Models for Gaussian and Powerlaw Rough Surfaces

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Validity of Backscattering Models for Gaussian and Power-law Rough Surfaces ... warnick_at_ee.byu.edu. March 17, 2004. PIERS'04 Pisa, March 2004 ... – PowerPoint PPT presentation

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Title: Validity of Backscattering Models for Gaussian and Powerlaw Rough Surfaces


1
Validity of Backscattering Models for Gaussian
and Power-law Rough Surfaces
  • Karl F. Warnick, Floyd W. Millet and David V.
    Arnold
  • Department of Electrical and Computer Engineering
  • Brigham Young University
  • Provo, UT USA
  • warnick_at_ee.byu.edu
  • March 17, 2004

2
Physical Optics Validity criterion
  • Classical curvature criterion (Brehkovskikh,
    1952)
  • Surface undulations large relative to incident
    wavelength

EM/Acoustic Wavelength (?)
3
Power-law surfaces
  • Validity condition fails because radius of
    curvature rc ? infinity
  • Power spectrum ck-p, k gt 2?/L
  • Multiscale/fractal for p 3 (ocean surface)
  • Classical geometrical optics limit (RCS slope
    density) vanishes (slope variance is also
    infinite)
  • Empirically, PO can be accurate for some
    power-law surfaces
  • Fields not strongly affected by highly curved but
    small amplitude features
  • Effective spatial filter function has been
    conjectured (Brown, 1978, Rodriguez et al., 1992)
  • Validity condition for power-law surfaces is
    needed

4
Goals of this study
  • Backscattering, non-grazing angles
  • Power-law surface height power spectrum
  • New theoretical validity criterion for
    physical/geometrical optics
  • Numerical verification
  • Empirical validity criteria for
  • 2nd order Kirchhoff approximation or iterated PO
    (ITPO)
  • Small slope approximation (SSA)
  • Gaussian surface spectrum
  • Empirical validity criteria for GO, PO, ITPO, SSA

5
Physical optics integral
  • Scattering coefficient proportional to PO
    integral
  • where C(r) surface correlation function.
  • Asymptotically, it can be shown that
  • where the superscript denotes ?-fold convolution
    of normalized surface spectrum

6
Central limit theorem
  • The central limit theorem gives an estimate for
    multiple convolutions
  • Fast decay of ? ? Gaussian (classical geometrical
    optics RCS slope PDF)
  • Slow decay of ? ? ?-stable distribution
    (power-law surfaces)







Power-law tails
7
Central limit theorem
  • The scale parameter of the stable distribution is
    determined by
  • Upper limit in integral has physical significance
  • Near-nadir scattering is independent of spectrum
    above cutoff wavenumber ?
  • Filter function is implicit in physical optics
    approximation

8
Composite-model expansion
  • Long wave spectrum
  • Short wave spectrum

Long wave slope PDF (GO)
Long wave slope PDF
Short wave spectrum (like SPM/Bragg)
Scatt. Coeff.
Composite model
Near nadir
Bragg regime
Incidence angle
9
Validity condition (near nadir)
  • Define a filtered surface with spectrum set to
    zero for wavenumbers above cutoff
  • PO(full surface) PO(filtered surface)
  • Apply classical curvature criterion to filtered
    surface
  • PO(filtered surface) EM(filtered surface) if
    criterion met
  • Ensure that RMS height of spatial components
    above cutoff is much smaller than EM wavelength
  • EM(filtered surface) EM(full surface)
  • Validity condition (1D k-3, 2D k-4, k gt 2?/L)

10
Numerical Verification
  • What is actual error near boundary of region of
    validity?
  • Monte-Carlo-Method of Moments reference solution
  • Incoherent 2D scattering coefficient (1D
    surfaces)
  • 80 wavelength surfaces, tapered incident field
  • 50 realizations (.4 dB estimation error)
  • Error Model(dB) Reference(dB)
  • Display error over surface parameter space
  • roughness, dominant wavelength kh, kL

11
Typical scattering coefficients
12
Error - power law surface, 0º
Theoretical validity criterion
13
Power law surfaces, 45º
14
Sig. slope limit vs. incidence angle
(Limit corresponds to 2 dB error)
15
Gaussian surfaces previous studies
Bistatic
Near-nadir (Chen Fung, 1988) Invalid o
Valid
16
Gaussian surfaces, 0º
17
Gaussian surfaces, 45º
18
Comparison to classical curvature criterion
19
Summary conditions for error lt 2 dB
20
Conclusions
  • PO contains an inherent spatial filter, and can
    be valid for surfaces with infinite
    curvature/slope variance
  • Validity condition at backscattering is an upper
    bound on significant slope (power-law surfaces)
    or RMS slope (Gaussian surfaces)
  • Applies to PO and PO-like models (ITPO, SSA)
  • Remarkably, the conditions are frequency
    independent - if a surface meets the slope
    validity criterion, PO is valid at all
    frequencies
  • Power-law this might be expected due to
    multiscale nature of surface
  • Gaussian surfaces possible multiple scattering
    interpretation surface patches large enough to
    scatter rays must have low slope

21
Power law surfaces, 20º
22
Power law surfaces, 60º
23
Gaussian surfaces, 20º
24
Gaussian surfaces, 60º
25
RMS slope limit vs. incidence angle
26
Monte Carlo convergence
27
Surface curvature
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