Abstract

On the basis of an analysis of the autocovariance of the complex heterodyne signal, some novel algorithms are derived and are investigated for use in determining, with high spatial resolution, Doppler-velocity coherent-lidar profiles in the case of rectangular and rectangularlike sensing laser pulses. These algorithms generalize other known Doppler-velocity estimators for the more complex case of nonuniform scattering and Doppler-velocity distribution within the pulse length. Algorithm performance and efficiency are studied and are illustrated by computer simulations. It is shown that the Doppler-velocity profiles can be determined with essentially better resolution in comparison with the use of other known estimation approaches, but at the expense of some increase in the number of statistical realizations (number of laser shots) required to reduce the speckle-noise effect. The minimum achievable resolution interval is shown to be much shorter than the pulse length.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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1997

1996

1995

1994

S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
[CrossRef]

1993

1992

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “High-resolution processing of long-pulse-lidar data,” NASA Conf. Publ. 3158, 637–640 (1992).

1991

1990

1989

R. Nordstrom, “A dual sensor role for pulsed CO2 laser,” Photonics Spectra 23, 89–98 (1989).

1987

1983

1982

1972

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).
[CrossRef]

1966

Ames, L. L.

Ancellet, G. M.

Angelova, M. D.

E. V. Stoykova, L. L. Gurdev, B. M. Bratanov, M. D. Angelova, D. V. Stoyanov, “Low Doppler velocity estimation in infrared coherent lidars,” in Proceedings of the 15th International Laser Radar Conference, Part II (Institute of Atmospheric Optics, Tomsk, Russia, 1990), pp. 413–417.

Bratanov, B. M.

E. V. Stoykova, L. L. Gurdev, B. M. Bratanov, M. D. Angelova, D. V. Stoyanov, “Low Doppler velocity estimation in infrared coherent lidars,” in Proceedings of the 15th International Laser Radar Conference, Part II (Institute of Atmospheric Optics, Tomsk, Russia, 1990), pp. 413–417.

Brockman, Ph.

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Gemeinschaftsausgabe Verlag Nauka, Moskau; BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany, 1989).

Calloway, R. S.

Churnside, J. H.

Cupp, R. E.

Dabas, A.

Ph. Salamitou, A. Dabas, P. Flamant, “Simulations in the time domain for heterodyne coherent laser radar,” Appl. Opt. 34, 499–506 (1995).
[CrossRef] [PubMed]

A. Dabas, Ph. Salamitou, D. Oh, M. Georges, J. L. Zarader, P. H. Flamant, “Lidar signal simulation and processing,” in Proceedings of the Seventh Conference on Coherent Laser Radar: Applications and Technology, P. H. Flamant, ed. (Ecole Polytechnique, Paris, 1993), pp. 221–228.

Dreischuh, T. N.

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “Deconvolution techniques for improving the resolution of long-pulse lidars,” J. Opt. Soc. Am. A 10, 2296–2306 (1993).
[CrossRef]

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “High-resolution processing of long-pulse-lidar data,” NASA Conf. Publ. 3158, 637–640 (1992).

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Intercomparison between two approaches for improving the resolution of Doppler-velocity coherent-lidar profiles,” in Tenth International School on Quantum Electronics: Laser Physics and Applications, P. A. Atanasow, D. V. Stroyanov, eds., Proc. SPIE3571, 272–276 (1999).
[CrossRef]

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Lidar profile deconvolution algorithms for some rectangular-like laser pulse shapes,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Raioux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996), pp. 135–138.

Flamant, P.

Flamant, P. H.

A. Dabas, Ph. Salamitou, D. Oh, M. Georges, J. L. Zarader, P. H. Flamant, “Lidar signal simulation and processing,” in Proceedings of the Seventh Conference on Coherent Laser Radar: Applications and Technology, P. H. Flamant, ed. (Ecole Polytechnique, Paris, 1993), pp. 221–228.

Forney, P.

Frehlich, R.

Frehlich, R. G.

Georges, M.

A. Dabas, Ph. Salamitou, D. Oh, M. Georges, J. L. Zarader, P. H. Flamant, “Lidar signal simulation and processing,” in Proceedings of the Seventh Conference on Coherent Laser Radar: Applications and Technology, P. H. Flamant, ed. (Ecole Polytechnique, Paris, 1993), pp. 221–228.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gurdev, L. L.

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “Deconvolution techniques for improving the resolution of long-pulse lidars,” J. Opt. Soc. Am. A 10, 2296–2306 (1993).
[CrossRef]

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “High-resolution processing of long-pulse-lidar data,” NASA Conf. Publ. 3158, 637–640 (1992).

E. V. Stoykova, L. L. Gurdev, B. M. Bratanov, M. D. Angelova, D. V. Stoyanov, “Low Doppler velocity estimation in infrared coherent lidars,” in Proceedings of the 15th International Laser Radar Conference, Part II (Institute of Atmospheric Optics, Tomsk, Russia, 1990), pp. 413–417.

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Lidar profile deconvolution algorithms for some rectangular-like laser pulse shapes,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Raioux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996), pp. 135–138.

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Intercomparison between two approaches for improving the resolution of Doppler-velocity coherent-lidar profiles,” in Tenth International School on Quantum Electronics: Laser Physics and Applications, P. A. Atanasow, D. V. Stroyanov, eds., Proc. SPIE3571, 272–276 (1999).
[CrossRef]

Hale, C. P.

Hannon, S. M.

R. Frehlich, S. M. Hannon, S. W. Henderson, “Coherent Doppler lidar measurements of winds in the weak signal regime,” Appl. Opt. 36, 3491–3499 (1997).
[CrossRef] [PubMed]

S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
[CrossRef]

Harris, M. R.

M. R. Harris, D. V. Willets, “Performance characteristics of a TE CO2laser with a long excitation pulse,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 5–7.

Hawley, J. G.

Henderson, S. W.

Huffaker, A. V.

Huffaker, R. M.

A. V. Jelalian, R. M. Huffaker, “Laser Doppler techniques for remote wind velocity measurements,” in Specialist Conference on Molecular Radiation (Marshall Space Flight Center, Huntsville, Ala., 1967), pp. 345–358.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978).

Jelalian, A. V.

A. V. Jelalian, R. M. Huffaker, “Laser Doppler techniques for remote wind velocity measurements,” in Specialist Conference on Molecular Radiation (Marshall Space Flight Center, Huntsville, Ala., 1967), pp. 345–358.

Kavaya, M. J.

Klein, S. H.

Lottman, B. T.

Magee, J. R.

Measures, R. M.

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

Menzies, R. T.

Miller, K. S.

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).
[CrossRef]

Nordstrom, R.

R. Nordstrom, “A dual sensor role for pulsed CO2 laser,” Photonics Spectra 23, 89–98 (1989).

Oh, D.

A. Dabas, Ph. Salamitou, D. Oh, M. Georges, J. L. Zarader, P. H. Flamant, “Lidar signal simulation and processing,” in Proceedings of the Seventh Conference on Coherent Laser Radar: Applications and Technology, P. H. Flamant, ed. (Ecole Polytechnique, Paris, 1993), pp. 221–228.

Otto, R. G.

Post, M. J.

Robinson, P. A.

Rochwarger, M. M.

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).
[CrossRef]

Salamitou, Ph.

Ph. Salamitou, A. Dabas, P. Flamant, “Simulations in the time domain for heterodyne coherent laser radar,” Appl. Opt. 34, 499–506 (1995).
[CrossRef] [PubMed]

A. Dabas, Ph. Salamitou, D. Oh, M. Georges, J. L. Zarader, P. H. Flamant, “Lidar signal simulation and processing,” in Proceedings of the Seventh Conference on Coherent Laser Radar: Applications and Technology, P. H. Flamant, ed. (Ecole Polytechnique, Paris, 1993), pp. 221–228.

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Gemeinschaftsausgabe Verlag Nauka, Moskau; BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany, 1989).

Siegman, A. E.

Steakley, B. C.

Stone, R.

Stoyanov, D. V.

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “Deconvolution techniques for improving the resolution of long-pulse lidars,” J. Opt. Soc. Am. A 10, 2296–2306 (1993).
[CrossRef]

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “High-resolution processing of long-pulse-lidar data,” NASA Conf. Publ. 3158, 637–640 (1992).

E. V. Stoykova, L. L. Gurdev, B. M. Bratanov, M. D. Angelova, D. V. Stoyanov, “Low Doppler velocity estimation in infrared coherent lidars,” in Proceedings of the 15th International Laser Radar Conference, Part II (Institute of Atmospheric Optics, Tomsk, Russia, 1990), pp. 413–417.

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Lidar profile deconvolution algorithms for some rectangular-like laser pulse shapes,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Raioux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996), pp. 135–138.

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Intercomparison between two approaches for improving the resolution of Doppler-velocity coherent-lidar profiles,” in Tenth International School on Quantum Electronics: Laser Physics and Applications, P. A. Atanasow, D. V. Stroyanov, eds., Proc. SPIE3571, 272–276 (1999).
[CrossRef]

Stoykova, E. V.

E. V. Stoykova, L. L. Gurdev, B. M. Bratanov, M. D. Angelova, D. V. Stoyanov, “Low Doppler velocity estimation in infrared coherent lidars,” in Proceedings of the 15th International Laser Radar Conference, Part II (Institute of Atmospheric Optics, Tomsk, Russia, 1990), pp. 413–417.

Swanson, D.

Targ, R.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in Turbulent Atmosphere (Nauka, Moscow, 1967).

Thomson, J. A.

S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
[CrossRef]

Wang, J. Y.

Willets, D. V.

M. R. Harris, D. V. Willets, “Performance characteristics of a TE CO2laser with a long excitation pulse,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 5–7.

Yura, H. T.

Zarader, J. L.

A. Dabas, Ph. Salamitou, D. Oh, M. Georges, J. L. Zarader, P. H. Flamant, “Lidar signal simulation and processing,” in Proceedings of the Seventh Conference on Coherent Laser Radar: Applications and Technology, P. H. Flamant, ed. (Ecole Polytechnique, Paris, 1993), pp. 221–228.

Zarifis, V.

Appl. Opt.

IEEE Trans. Inf. Theory

K. S. Miller, M. M. Rochwarger, “A covariance approach to spectral moment estimation,” IEEE Trans. Inf. Theory IT-18, 588–596 (1972).
[CrossRef]

J. Mod. Opt.

S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
[CrossRef]

J. Opt. Soc. Am. A

NASA Conf. Publ.

L. L. Gurdev, T. N. Dreischuh, D. V. Stoyanov, “High-resolution processing of long-pulse-lidar data,” NASA Conf. Publ. 3158, 637–640 (1992).

Opt. Lett.

Photonics Spectra

R. Nordstrom, “A dual sensor role for pulsed CO2 laser,” Photonics Spectra 23, 89–98 (1989).

Other

A. V. Jelalian, R. M. Huffaker, “Laser Doppler techniques for remote wind velocity measurements,” in Specialist Conference on Molecular Radiation (Marshall Space Flight Center, Huntsville, Ala., 1967), pp. 345–358.

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Lidar profile deconvolution algorithms for some rectangular-like laser pulse shapes,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Raioux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996), pp. 135–138.

E. V. Stoykova, L. L. Gurdev, B. M. Bratanov, M. D. Angelova, D. V. Stoyanov, “Low Doppler velocity estimation in infrared coherent lidars,” in Proceedings of the 15th International Laser Radar Conference, Part II (Institute of Atmospheric Optics, Tomsk, Russia, 1990), pp. 413–417.

M. R. Harris, D. V. Willets, “Performance characteristics of a TE CO2laser with a long excitation pulse,” in Coherent Laser Radar: Technology and Applications, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 5–7.

A. Dabas, Ph. Salamitou, D. Oh, M. Georges, J. L. Zarader, P. H. Flamant, “Lidar signal simulation and processing,” in Proceedings of the Seventh Conference on Coherent Laser Radar: Applications and Technology, P. H. Flamant, ed. (Ecole Polytechnique, Paris, 1993), pp. 221–228.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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

V. I. Tatarski, Wave Propagation in Turbulent Atmosphere (Nauka, Moscow, 1967).

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Gemeinschaftsausgabe Verlag Nauka, Moskau; BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany, 1989).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978).

T. N. Dreischuh, L. L. Gurdev, D. V. Stoyanov, “Intercomparison between two approaches for improving the resolution of Doppler-velocity coherent-lidar profiles,” in Tenth International School on Quantum Electronics: Laser Physics and Applications, P. A. Atanasow, D. V. Stroyanov, eds., Proc. SPIE3571, 272–276 (1999).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Model of the alternating radial wind-velocity profile [Eq. (16)] as a function of range.

Fig. 2
Fig. 2

Model of the maximum-resolved signal power profile Φ(z) used in the simulations of the case of alternating wind velocity profile.

Fig. 3
Fig. 3

Model of the wind-vortex-like distribution [Eq. (18)] of the radial (Doppler) velocity along the line of sight.  

Fig. 4
Fig. 4

Probability density distribution of the heterodyne signal power normalized to its mean value [|I(t)|2/P(t)], compared with the expected negative exponential distribution (dashed curve).

Fig. 5
Fig. 5

One simulated signal realization corresponding to the models in Figs. 1 and 2, at λ=10.6 μm and τ=4 μs (lp=1200 m). The in-phase J(t) (solid curve) and quadrature Q(t) (dashed curve) components of the signal are presented. The sampling interval employed is Δto(Δzo)=0.1 μs (15 m).

Fig. 6
Fig. 6

Alternating radial wind velocity profiles vr(z) restored in the case of rectangular pulse (with λ=10.6 μm and τ=4 μs) by use of (a) algorithm (14a), (b) algorithm (14b), and (c) algorithm (14a) and the lidar dead-zone scanning technique. The sampling interval employed is Δto(Δzo)=0.1 μs(15 m). The number of signal realizations employed is (a) and (b) N=2000 and (c) N=500. Cov^(t, θ) and vr(z) are smoothed by a moving average with (a) and (b) 8Δto-(8Δzo)-wide window and (c) 4Δto-(4Δzo)-wide window. The original profile of v(z) [described by Eq. (16)] is given for comparison by the dashed curve. The dotted curves in (a) and (b) represent the result obtained with the PP algorithm.

Fig. 7
Fig. 7

Wind-vortex Doppler-velocity profiles vr(z) restored in the case of rectangular pulse (with τ=0.2 μs and λ=2 μm) by use of (a) algorithm (14a) and (b) algorithm (14b). The sampling interval employed is Δto(Δzo)=0.01 μs(1.5 m), and the number of signal realizations is N=1000. Cov^(t, θ) and vr(z) are smoothed by a moving average with (a) 6Δto- (6Δzo)-wide window and (b) 4Δto- (4Δzo)-wide window. The original profile of v(z) (dashed curve) and the profile obtained with the PP algorithm (dotted curve) are given for comparison.

Fig. 8
Fig. 8

Wind-vortex Doppler-velocity profiles vr(z) restored in the case of (inset) a rectangularlike pulse, with λ=2 μm, τ=0.2 μs, and τr=0.01 μs (dashed curve), 0.02 μs (dashed–dotted curve), 0.04 μs (dotted curve), 0.06 μs (dashed–double-dotted curve), by use of (a) algorithm (14a) and (b) algorithm (14b). The sampling interval employed is Δto(Δzo)=0.01 μs(1.5 m). The number of signal realizations employed is N=1000. Cov^(t, θ) and vr(z) are smoothed by a moving average with (a) 8Δto- (8Δzo)-wide window and (b) 4Δto- (4Δzo)-wide window. The original profile of v(z) is given for comparison by the solid curve.

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

Eo(r, t)=PA(r)f(t-z/c)expjωo(t-z/c)+0t-z/cδωo(t)dt+ϕo(t-z/c),
Ebi[ρt, ri(ti), t]
=A[ri(ti)]fo[t-2zi(ti)/c]expjωo[t-2zi(ti)/c]+0t-2zi(ti)/cδωo(t)dt+ϕo[t-2zi(ti)/c]
×ai[ri(ti)]G[ρt; ri(ti)],
Eb(ρt, t)=iEbi[ρi, ri(ti), t].
Eh(ρt, t)=Ah(ρt)exp{j[ωht+ϕh(t)]},
I(t)=J(t)+jQ(t)=KEb(ρt, t)Eh*(ρt, t)dρt,
I(t)=K exp[j(ωo-ωh)t-ϕh(t)]iaiA[ri(ti)]×fo[t-2zi(ti)/c]expj-2ωozi(ti)/c+0t-2zi(ti)/cδωo(t)dt+ϕo[t-2zi(ti)/c]Ah*(ρt)G[ρt; ri(ti)]dρt.
I(t)=K exp[-jϕh(t)]iaiA(ri)fo(t-2zi/c)×exp(-2jkDizio)×exp( j{ωmit+ϕch[(t-2zio/c)χi]+ϕo[(t-2zio/c)χi]})Ah*(ρt)G[ρt; ri(ti)]dρt,
ϕch[(t-2zio/c)χi]=0(t-2zio/c)χiδωo(t)dt
I(t)=K exp[-jϕh(t)]lfo(t-2zl/c)×exp( j{ωmlt+ϕch[(t-2zl/c)χl]+ϕo[(t-2zl/c)χl]})dA(zl),
dA(zl)=nexp(-2jkDlzoln)A(ρln, zl)anlAh*(ρt)G(ρt; ρln, zl)dρt,
dA(zl)=Φo1/2(zl)(Δzo)1/2wl,
Φo(zl)(Δzo)=|dA(zl)|2=D[dAr(zl)]+D[dAj(zl)]
P(t)=l=l1+1l2f(t-2zl/c)Φ(zl)Δzo
P(t)=ϕ(t)ct/2f(t-2z/c)Φ(z)dz,
I(t)=exp[-jϕh(t)]l=l1+1l2[f(t-2zl/c)Φ(zl)Δzo]1/2×w(zl)exp{jωmlt+jϕch[(t-2zl/c)χl]+jϕo[(t-2zl/c)χl]},
I(t)=exp[-jϕh(t)]ϕ(t)ct/2[f(t-2z/c)]1/2×exp{jωm(z)t+jϕch[(t-2z/c)χ(z)]+jϕo[(t-2z/c)χ(z)]}dA(z),
dA(z)dA(z)=Φ(z)δ(z-z)dzdz,
ωm(z)=ωoχ(z)-ωh, χ(z)=1-2v(z)/c,
Cov(t, θ)=I*(t)I(t+θ)=ϕ(t+θ)ct/2dzfo(t-2z/c)fo(t+θ-2z/c)Φ(z)×exp{j[ωm(z)θ+Δϕch(t, θ, z)]}×ξ(t, θ, z)γz(2ωoθ/c),
Δϕch(t, θ, z)=ϕch[(t-2z/c)χ(z)]-ϕch[(t+θ-2z/c)χ(z)],
ξ(t, θ, z)=exp{jϕo[(t-2z/c)χ(z)]-jϕo[(t+θ-2z/c)χ(z)]},
γz(y)=exp[-jyv˜(z)]p[v˜(z)]dv˜
Cov(t, θ)=c(t+θ-τ)/2ct/2dzΦ(z)exp[jωm(z)θ]γz(2ωoθ/c).
Re Cov(t, θ)=J(t)J(t+θ)+Q(t)Q(t+θ)
Im Cov(t, θ)=J(t)Q(t+θ)-J(t+θ)Q(t).
Cov(t, θ)=P(t)=c(t-τ)/2ct/2 Φ(z)dz.
ωm(z=ct/2)={Φ[c(t-τ)/2]ωm[c(t-τ)/2]+Im[(2/c)Covtθ(t, θ=0)]}×[Φ(ct/2)]-1,
ωm(z=ct/2)=θ-1×arctan(2/c)Im Covt(t, θ)+Φ[c(t+θ-τ)/2]γ sin{ωm[c(t+θ-τ)/2]θ}(2/c)Re Covt(t, θ)+Φ[c(t+θ-τ)/2]γ cos{ωm[c(t+θ-τ)/2]θ},
ωm(z=ct/2)=θ-1arctan[Im Covt(t, θ)/Re Covt(t, θ)],
ωm(z=ct/2)=[Im Covθ(t, θ=0)]/Cov(t, θ=0),
ωm(z=ct/2)=θ-1arctan[ImCov(t, θ)/ReCov(t, θ)],
v(z)=v(z-zo)=v1(z)sin{4π(z-zo)/[q1λ+q2(z-zo)]}+vo,
v1(z)=q3(z-zo)/(zs-zo)+q4,
q1=1.5×108,q2=0.3,q3=10 m/s,
q4=3 m/s,vo=3 m/s,zs=8700 m.
Φ(z=ct/2)=W0forttoB1(t-to)-3exp[-B2/(t-to)]+B3sin2[2π(t-to)/T]forto<tB2+toB1(t-to)-3exp[-B2/(t-to)]fort>B2+to,
v(z)=C[(z-zo)-a]exp{-[(z-zo)-a]2/b2}/b2,
Cov^(t=tl2=2zl2/c, θ=mΔto)
=N-1k=1NIk*(tl2)Ik(tl2+mΔto),
f(ϑ)=0forϑ0D[1-exp(-ϑ/τr)]for0ϑτexp[-(ϑ-τ)/τr]forτ<ϑ,

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