Abstract

We develop a bistatic model for airborne lidar returns collected by an imaging array from underwater objects, incorporating additional returns from the surrounding water medium and ocean bottom. Our results provide a generalization of the monostatic model by Walker and McLean. In the bistatic scheme the transmitter and receiver are spatially separated or are not coaligned. This generality is necessary for a precise description of an imaging array such as a CCD, which may be viewed as a collection of receiver elements, with each transmitter-element pair forming a bistatic configuration. More generally, the receiver may consist of photomultiplier tubes, photodiodes, or any of a variety of optical receivers, and the imaging array can range in size from a CCD array to a multiple-platform airborne lidar system involving multiple aircraft. The majority of this research is devoted to a derivation of the bistatic lidar equations, which account for multiple scattering and absorption in the water column. We then describe the application of these equations to the modeling and simulation of an imaging array. We show an example of a simulated lidar return and compare it with a real ocean lidar return, obtained by a CCD array.

© 2002 Optical Society of America

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References

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  1. P. J. Shargo, N. Çadallı, A. C. Singer, D. C. Munson, “A tomographic framework for LIDAR imaging,” in Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, Salt Lake City, Utah, 7–11 May 2001 (Institute of Electrical and Electronics Engineers, New York, 2001).
  2. N. Çadallı, P. J. Shargo, A. C. Singer, D. C. Munson, “Three-dimensional tomographic imaging of ocean mines from real and simulated lidar returns,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frovin, G. D. Gilbert, eds., Proc. SPIE4488, 155–166 (2002).
    [CrossRef]
  3. R. E. Walker, J. W. McLean, “Lidar equations for turbid media with pulse stretching,” Appl. Opt. 38, 2384–2397 (1999).
    [CrossRef]
  4. J. W. McLean, J. D. Freeman, R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37, 4701–4711 (1998).
    [CrossRef]
  5. I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
    [CrossRef]
  6. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, The Netherlands, 1976).
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  8. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley, New York, 1984).
  9. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  10. J. R. Apel, Principle of Ocean Physics (Academic, Orlando, Fla., 1987).
  11. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).
  12. R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).
  13. N. Çadallı, “Signal processing issues in reflection tomography,” Ph.D. dissertation (University of Illinois, Urbana-Champaign, Ill., 2001).
  14. A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  15. R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
    [CrossRef] [PubMed]
  16. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  17. J. W. McLean, J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt. 35, 3261–3269 (1996).
    [CrossRef] [PubMed]

1999

1998

1997

1996

1995

Apel, J. R.

J. R. Apel, Principle of Ocean Physics (Academic, Orlando, Fla., 1987).

Çadalli, N.

N. Çadallı, “Signal processing issues in reflection tomography,” Ph.D. dissertation (University of Illinois, Urbana-Champaign, Ill., 2001).

P. J. Shargo, N. Çadallı, A. C. Singer, D. C. Munson, “A tomographic framework for LIDAR imaging,” in Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, Salt Lake City, Utah, 7–11 May 2001 (Institute of Electrical and Electronics Engineers, New York, 2001).

N. Çadallı, P. J. Shargo, A. C. Singer, D. C. Munson, “Three-dimensional tomographic imaging of ocean mines from real and simulated lidar returns,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frovin, G. D. Gilbert, eds., Proc. SPIE4488, 155–166 (2002).
[CrossRef]

Ciervo, A. P.

Freeman, J. D.

Hall, G. J.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, The Netherlands, 1976).

Katsev, I. L.

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Lutomirski, R. F.

McLean, J. W.

Measures, R. M.

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley, New York, 1984).

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

Munson, D. C.

N. Çadallı, P. J. Shargo, A. C. Singer, D. C. Munson, “Three-dimensional tomographic imaging of ocean mines from real and simulated lidar returns,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frovin, G. D. Gilbert, eds., Proc. SPIE4488, 155–166 (2002).
[CrossRef]

P. J. Shargo, N. Çadallı, A. C. Singer, D. C. Munson, “A tomographic framework for LIDAR imaging,” in Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, Salt Lake City, Utah, 7–11 May 2001 (Institute of Electrical and Electronics Engineers, New York, 2001).

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Polonsky, I. N.

Prikhach, A. S.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Shargo, P. J.

P. J. Shargo, N. Çadallı, A. C. Singer, D. C. Munson, “A tomographic framework for LIDAR imaging,” in Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, Salt Lake City, Utah, 7–11 May 2001 (Institute of Electrical and Electronics Engineers, New York, 2001).

N. Çadallı, P. J. Shargo, A. C. Singer, D. C. Munson, “Three-dimensional tomographic imaging of ocean mines from real and simulated lidar returns,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frovin, G. D. Gilbert, eds., Proc. SPIE4488, 155–166 (2002).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Singer, A. C.

N. Çadallı, P. J. Shargo, A. C. Singer, D. C. Munson, “Three-dimensional tomographic imaging of ocean mines from real and simulated lidar returns,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frovin, G. D. Gilbert, eds., Proc. SPIE4488, 155–166 (2002).
[CrossRef]

P. J. Shargo, N. Çadallı, A. C. Singer, D. C. Munson, “A tomographic framework for LIDAR imaging,” in Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, Salt Lake City, Utah, 7–11 May 2001 (Institute of Electrical and Electronics Engineers, New York, 2001).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Walker, R. E.

Zege, E. P.

Appl. Opt.

J. Opt. Soc. Am. A

Other

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, The Netherlands, 1976).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley, New York, 1984).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

J. R. Apel, Principle of Ocean Physics (Academic, Orlando, Fla., 1987).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).

N. Çadallı, “Signal processing issues in reflection tomography,” Ph.D. dissertation (University of Illinois, Urbana-Champaign, Ill., 2001).

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

P. J. Shargo, N. Çadallı, A. C. Singer, D. C. Munson, “A tomographic framework for LIDAR imaging,” in Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, Salt Lake City, Utah, 7–11 May 2001 (Institute of Electrical and Electronics Engineers, New York, 2001).

N. Çadallı, P. J. Shargo, A. C. Singer, D. C. Munson, “Three-dimensional tomographic imaging of ocean mines from real and simulated lidar returns,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frovin, G. D. Gilbert, eds., Proc. SPIE4488, 155–166 (2002).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Bistatic lidar geometry.

Fig. 2
Fig. 2

Time-resolved lidar return at monostatic and bistatic receivers.

Fig. 3
Fig. 3

Simulated CCD return.

Fig. 4
Fig. 4

Real CCD return from CEFT data.

Fig. 5
Fig. 5

Simulated CCD return with noise.

Equations (56)

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L0, ρ, s, t=Qδρ-ρδs-sδt.
Lz, ρ, s, t=Qkz, ρ-ρ+zs, s-s, τ,
Lz, ρ, s, t=Q    ψρθs×kz, ρ-ρ+zs, s-s, τdρds,
Lz, ρ, s, t=Q2π4   ΨκΘq+κz×Kz, κ, q, τ×exp-jκ · ρ+q · sdκdq.
Lz, ρ, s, t=Q2π4   ΨTXκΘTXq+κz×Kz, κ, q, τexpjκ · ρTX×expjq+κz · sTX×exp-jκ · ρ+q · sdκdq
z tan θ2+H tan θ1zθ2+Hθ1=z+Hmθ2.
LTXH+z, ρ, s, t=QTaTaw2π4   ΨTXκ ×ΘTXq/m+κH+z/m×Kz, κ, q, τexpjκ · ρTX×expjq/m+κH+z/m · sTX×exp-jκ · ρ+q · sdκdq,
LbackH+z, ρ, s, t=Rπ LTXH+z, ρ, s, tds = QTaTawRπ2π2 ΨTXκΘTXκH+z/m×Kz, κ, 0, τexpjκ · ρTX+H+z/msTX×exp-jκ · ρdκdq,
Ldifz, ρ, s, t=12π2 Kz, κ, -zκ, τ×exp-jκ · ρ-zsdκ.
LdifH+z, ρ, s, t=12π2 Kz, κ, -zκ, τ×exp-jκ · ρ+H+z/msdκ.
LRXρ, s, t=TaTawm2 LbackH+z, ρ, s, t*ρ *τ LdifH+z, ρ, s, t,
LRXρ, s, t=QTa2Taw2Rm2π2π2ΘTXκH+z/m×ΨTXκexp-jκ·ρ×Kz, κ, 0, τ *τ Kz, κ, -zκ, τ×expjκ ·ρTX+H+z/msTX×exp-jH+z/mκ · sdκ,
Pbt=ARXΩRX  LRXρ, s, t×ψRXρθRXsdρds,
ψRXρ=ψRXρ-ρRX, θRXs=θRXs-sRX
Pbt=Cb2π2 ΨTXκΘTXκH+z/m×Ψ*RXκΘ*RXκH+z/m×Kz, κ, 0, τ *τ Kz, κ, -zκ, τ×expjκ · ρTX-ρRX+H+z/msTX-sRXdκ,
Pbζ=Cb2π2 ΨTXκΘTXκH+D/m×Ψ*RXκΘ*RXκH+D/m×KD, κ, 0, τ *τ KD, κ, -Dκ, τ×expjκ · ρTX-ρRX+H+D/msTX-sRXdκ,
Pwt=Cw2π20c2mt-2H/c ΨTXκΘTXκH+z/m×Ψ*RXκΘ*RXκH+z/m×Kz, κ, 0, τ *τ Kz, κ, -zκ, τ×expjκ·ρTX-ρRX+H+z/msTX-sRXdκdz,
Pwζ=Cw2π2 ΨTXκΘTXκH+ζ/m ×Ψ*RXκΘ*RXκH+ζ/m×expjκ · ρTX-ρRX+H+ζ/msTX-sRX× c2mL1L2Kζ, κ, 0, τ *τ Kζ, κ,-ζκ, τdτdκ.
Lback=AtRtπ δρ-ρt LTXH+z, ρ, s, tds,
LbackH+z, ρ, s, t=AtRtπ δρ-ρtQTaTaw2π2 ×   ΨTXκΘTXκH+z/m×Kz, κ, 0, τexp-jκ · ρ×expjκ · ρTX+H+z/msTXdκ.
LRXρ, s, t=QAtTa2Taw2Rtm2π  ΨTXκ1 ×ΘTXκ1H+z/mexp-jκ1 · ρt×Kz, κ1, 0, τ *τ Kz, κ2, -zκ2, τ×exp-jκ1 · ρTX+H+z/msTX×exp-jH+z/mκ2 · s×exp-jκ2 · ρ-ρtdκ12π2dκ22π2,
Ptt, ρt=ARXΩRX   LRXρ, s, t×ψRXρ-ρRXθRXs-sRXdρds.
Ptt, ρt=Ct   ΨTXκ1ΘTXκ1H+z/m×Ψ*RXκ2Θ*RXκ2H+z/m×Kz, κ1, 0, τ *τ Kz, κ2, -zκ2, τ×expjκ1 · ρTX+H+z/msTX×exp-jκ2 · ρRX+H+z/msRX×exp-jρt · κ1-κ2dκ12π2dκ22π2,
Ptζ, ρt=Ct   ΨTXκ1ΘTXκ1H+D/m×Ψ*RXκ2Θ*RXκ2H+D/m×KD, κ1, 0, τ *τ KD, κ2, -Dκ2, τ×expjκ1 · ρTX+H+D/msTX×exp-jκ2 · ρRX+H+D/msRX×exp-jρt · κ1-κ2dκ12π2dκ22π2,
Ptrgtζ, ρ= Ptζ, ρtψtρ-ρtdρt.
Psζ, ρt=Cs   ΨTXκ1ΘTXκ1H+ζ/m×Ψ*RXκ2Θ*RXκ2H+ζ/m×expjκ1 · ρTX+H+ζ/msTX×exp-jκ2 · ρRX+H+ζ/msRX×exp-jρt · κ1-κ2×c2m02ζ-Dm/cKζ, κ1, 0, τ*τ Kζ, κ2, -ζκ2, τdτ dκ12π2dκ22π2,
Pshdwζ, ρ= Psζ, ρtψtρ-ρtdρt.
kz, ρ, s, τ=δρδsδτexp-a+bz+1-exp-bzexp-az+cτ/m×gz, τhz, ρ, s, τ,
gz, τ=μσ2Γμ2/σ2μτσ2μ2/σ2-1 exp-μτσ2,
Kz, κ, q, τ=δτexp-a+bz+1-exp-bzexp-az+cτ/m×gz, τHz, κ, q, τ.
Hz, κ, q, τ=exp-τcmzq2+zq · κ+ κ2z2,
Hz, κ, 0, τ=Hz, κ, -zκ, τ.
Kz, κ1, 0, τ *τ Kz, κ2, -zκ2, τ=exp-2az×δτexp-2bz+exp-bz×1-exp-bzexp-acτ/mgz, τ×Hz, κ1, 0, τ+Hz, κ2, -zκ2, τ+1-exp-bz2 exp-acτ/m×gz, τHz, κ1, 0, τ *τ gz, τ×Hz, κ2, -zκ2, τ.
θis=1πθi,xθi,yexp-sx2/θi,x2-sy2/θi,y2,
Θiq=exp-14qx2θi,x2+qy2θi,y2.
Pwζ=CwL1L2c2mexp-2aζδτexp-2bζexp-X1+jκ · X2dκ2π2I1ζ +2 exp-bζ1-exp-bζexp-acτ/mgζ, τ ×exp-X1+jκ · X2Hζ, κ, 0, τdκ2π2I2ζ +1-exp-bζ2 exp-acτ/m ×exp-X1+jκ · X2gζ, τHζ, κ, 0, τ *τ gζ, τHζ, κ, -ζκ, τdκ2π2I3ζdτ,
X1=¼H+ζ/m2κx2θRX,x2+θTX,x2+κy2θRX,y2+θTX,y2,
X2=ρTX-ρRX+H+ζ/msTX-sRX.
X2,i=ρTX,i-ρRX,i+H+ζ/msTX,i-sRX,i,
12πexp-ax2expjbxdx=12πaexp-b2/4a.
Hζ, κ, 0, τ=exp-13m ζτcκ2,
I2ζ=πH+ζ/m2θRX,x2+θTX,x2+4ζτc3mH+ζ/m21/2×θRX,y2+θTX,y2+4ζτc3mH+ζ/m21/2-1×exp-ρTX,x-ρRX,x+H+ζ/msTX,x-sRX,x2H+ζ/m2θRX,x2+θTX,x2+4ζτc3m×exp-ρTX,y-ρRX,y+H+ζ/msTX,y-sRX,y2H+ζ/m2θRX,y2+θTX,y2+4ζτc3m.
gz, τ Hz, κ, 0, τ *τ gz, τ Hz, κ, -zκ, τ= gζ, τHζ, κ, 0, τ×Hζ, κ, -zκ,τ-τgζ, τ-τdτ,
Hζ, κ, 0, τHζ, κ, -zκ, τ-τ=exp-13m ζτcκ2exp-13mζτ-τcκ2=exp-13m ζτcκ2=Hζ, κ, 0, τ.
I3ζ=gζ, τ *τ gζ, τI2ζ,
gζ, τ *τ gζ, τ=μσ2Γ2μ2/σ2μτσ22μ2/σ2-1×exp-μτσ2
Pbζ=Cb exp-2aDδτexp-2bDI1D+2 exp-bD1-exp-bD×exp-acτ/mgD, τI2D+[1-exp-bD]2 exp-acτ/mI3D,
Y1=¼H+ζ/m2κ1,x2θTX,x2+κ1,y2θTX,y2, Y2=¼H+ζ/m2κ2,x2θRX,x2+κ2,y2θRX,y2, Y3=κ1,xρTX,x+H+ζ/msTX,x-ρt,x+κ1,yρTX,y+H+ζ/msTX,y-ρt,y, Y4=-κ2,xρRX,x+H+ζ/msRX,x-ρt,x-κ2,yρRX,y+H+ζ/msRX,y-ρt,y,
Psζ, ρt=Cs02ζ-Dm/cc2mexp-2aζδτexp-2bζexp-Y1+jY3dκ12π2J1ζ ×exp-Y2+jY4dκ22π2J2ζ+exp-bζ1-exp-bζexp-acτ/mgζ, τJ1ζ×exp-Y2+jY4Hζ, κ2, -ζκ2, τdκ22π2J3ζ+exp-bζ1-exp-bζexp-acτ/m×gζ, τJ2ζexp-Y1+jY3Hζ, κ1, 0, τdκ12π2J4ζ+1-exp-bζ2×exp-acτ/m  exp-Y1-Y2+jY3+jY4pζ, κ1, κ2, τdκ12π2dκ22π2J5ζdτ,
pζ, κ1, κ2, τ=gζ, τHζ, κ1, 0, τ*τ gζ, τHζ, κ2, -ζκ2, τ.
J1ζ=πH+ζ/m2θTX,xθTX,y-1×exp-ρTX,x+H+ζ/msTX,x-ρt,x2H+ζ/m2θTX,x2×exp-ρTX,y+H+ζ/msTX,y-ρt,y2H+ζ/m2θTX,y2.
J3ζ=πH+ζ/m2θRX,x2+4ζτc3mH+ζ/m21/2×θRX,y2+4ζτc3mH+ζ/m21/2-1×exp-ρRX,x+H+ζ/msRX,x-ρt,x2H+ζ/m2θRX,x2+4ζτc3m×exp-ρRX,y+H+ζ/msRX,y-ρt,y2H+ζ/m2θRX,y2+4ζτc3m.
pζ, κ1, κ2, τ=c02xc1xc2xc3yΓy τy-1 exp-c3τ
pζ, κ1, κ2, τgζ, τ *τ gζ, τ×exp-τZ1exp-τZ2.
J5ζ=gζ, τ *τ gζ, τJ3ζJ4ζ.
Ptζ, ρt=Ct exp-2aDδτexp-2bDJ1DJ2D+exp-bD1-exp-bD×exp-acτ/mgD, τJ1DJ3D+exp-bD1-exp-bD×exp-acτ/mgD, τJ2DJ4D+1-exp-bD2 exp-acτ/mJ5D,

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