Abstract

Remote lidar sensing in the photon-counting mode is now the commonly accepted method for studying atmospheric processes in the lower and free atmosphere. However, when processing signals obtained from lidar measurements, investigators necessarily face the problem of achieving accuracy in reconstructing the atmospheric parameters despite the presence of inhomogeneous noise in the measured signals. We propose an optimal method of linear regression (OMLR) of signals. The accuracy of the the method for the reconstructed signal is estimated. An example of application of the OMLR to the reconstruction of the temperature profile from the data obtained with a Raman lidar at the Siberian Lidar Station of the Institute of Atmospheric Optics (Tomsk, Russia) is given. The proposed method is distinguished by simplicity of interpretation of the criteria used, based on careful adherence to statistical principles. This method is shown to be an efficient auxiliary tool for the processing of measured data.

© 2002 Optical Society of America

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References

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  1. V. E. Derr, Remote Sensing of the Troposphere (U.S. Government Printing Office, Washington, D.C., 1972).
  2. W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).
  3. E. D. Hinkly, ed., Laser Monitoring of the Atmosphere (Springer-Verlag, Berlin, 1976).
    [CrossRef]
  4. G. Darmois, “Sur les lois de probabilité à estimation exhaustive,” C. R. Acad. Sci. 200, 1265–1266 (1935).
  5. S. S. Wilks, Mathematical Statistics (Wiley, New York, 1962).
  6. W. G. Cochran, Sampling Techniques (Wiley, New York, 1977).
  7. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).
  8. M. Abramovitz, I. Stegun, Handbook of Mathematical Functions [Dover, New York, 1974; (Vol. 55 of National Bureau of Standards Applied Mathematics Series, Washington, D.C., 1964)].
  9. H. B. Mann, A. Wald, “On the choice of number of class intervals in the application of the chi-squared test,” Ann. Math. Stat. 13, 306–317 (1942).
    [CrossRef]
  10. J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (Elsevier, New York, 1968).
  11. S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Measurements of the vertical temperature profile by means of the spontaneous Raman scattering channel of the atmospheric laser sounding station,” Atmos. Oceanic Opt. 5, 384–385 (1992).
  12. A. Cohen, Y. A. Cooney, K. N. Geller, “Atmospheric temperature profiles from lidar measurements of rotational Raman and elastic scattering,” Appl. Opt. 15, 2896–2900 (1976).
    [CrossRef] [PubMed]
  13. S. N. Volkov, B. V. Kaul, “Method for determination of light backscattering and extinction coefficients in tropospheric aerosol layers using elastic- and Raman-backscatter-based lidar,” Atmos. Oceanic Opt. 7, 864–869 (1994).
  14. S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Some problems in smoothing lidar signals,” Atmos. Oceanic Opt. 13, 702–706 (2000).

2000

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Some problems in smoothing lidar signals,” Atmos. Oceanic Opt. 13, 702–706 (2000).

1994

S. N. Volkov, B. V. Kaul, “Method for determination of light backscattering and extinction coefficients in tropospheric aerosol layers using elastic- and Raman-backscatter-based lidar,” Atmos. Oceanic Opt. 7, 864–869 (1994).

1992

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Measurements of the vertical temperature profile by means of the spontaneous Raman scattering channel of the atmospheric laser sounding station,” Atmos. Oceanic Opt. 5, 384–385 (1992).

1976

1942

H. B. Mann, A. Wald, “On the choice of number of class intervals in the application of the chi-squared test,” Ann. Math. Stat. 13, 306–317 (1942).
[CrossRef]

1935

G. Darmois, “Sur les lois de probabilité à estimation exhaustive,” C. R. Acad. Sci. 200, 1265–1266 (1935).

Abramovitz, M.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions [Dover, New York, 1974; (Vol. 55 of National Bureau of Standards Applied Mathematics Series, Washington, D.C., 1964)].

Cochran, W. G.

W. G. Cochran, Sampling Techniques (Wiley, New York, 1977).

Cohen, A.

Cooney, Y. A.

Darmois, G.

G. Darmois, “Sur les lois de probabilité à estimation exhaustive,” C. R. Acad. Sci. 200, 1265–1266 (1935).

Derr, V. E.

V. E. Derr, Remote Sensing of the Troposphere (U.S. Government Printing Office, Washington, D.C., 1972).

Dryard, D.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Eadie, W. T.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Geller, K. N.

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

James, F. E.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Kaul, B. V.

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Some problems in smoothing lidar signals,” Atmos. Oceanic Opt. 13, 702–706 (2000).

S. N. Volkov, B. V. Kaul, “Method for determination of light backscattering and extinction coefficients in tropospheric aerosol layers using elastic- and Raman-backscatter-based lidar,” Atmos. Oceanic Opt. 7, 864–869 (1994).

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Measurements of the vertical temperature profile by means of the spontaneous Raman scattering channel of the atmospheric laser sounding station,” Atmos. Oceanic Opt. 5, 384–385 (1992).

Kowalik, J.

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (Elsevier, New York, 1968).

Mann, H. B.

H. B. Mann, A. Wald, “On the choice of number of class intervals in the application of the chi-squared test,” Ann. Math. Stat. 13, 306–317 (1942).
[CrossRef]

Osborne, M. R.

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (Elsevier, New York, 1968).

Roos, M.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Sadoulet, B.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Shapranov, V. A.

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Some problems in smoothing lidar signals,” Atmos. Oceanic Opt. 13, 702–706 (2000).

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Measurements of the vertical temperature profile by means of the spontaneous Raman scattering channel of the atmospheric laser sounding station,” Atmos. Oceanic Opt. 5, 384–385 (1992).

Shelefontyk, D. I.

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Some problems in smoothing lidar signals,” Atmos. Oceanic Opt. 13, 702–706 (2000).

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Measurements of the vertical temperature profile by means of the spontaneous Raman scattering channel of the atmospheric laser sounding station,” Atmos. Oceanic Opt. 5, 384–385 (1992).

Stegun, I.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions [Dover, New York, 1974; (Vol. 55 of National Bureau of Standards Applied Mathematics Series, Washington, D.C., 1964)].

Volkov, S. N.

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Some problems in smoothing lidar signals,” Atmos. Oceanic Opt. 13, 702–706 (2000).

S. N. Volkov, B. V. Kaul, “Method for determination of light backscattering and extinction coefficients in tropospheric aerosol layers using elastic- and Raman-backscatter-based lidar,” Atmos. Oceanic Opt. 7, 864–869 (1994).

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Measurements of the vertical temperature profile by means of the spontaneous Raman scattering channel of the atmospheric laser sounding station,” Atmos. Oceanic Opt. 5, 384–385 (1992).

Wald, A.

H. B. Mann, A. Wald, “On the choice of number of class intervals in the application of the chi-squared test,” Ann. Math. Stat. 13, 306–317 (1942).
[CrossRef]

Wilks, S. S.

S. S. Wilks, Mathematical Statistics (Wiley, New York, 1962).

Ann. Math. Stat.

H. B. Mann, A. Wald, “On the choice of number of class intervals in the application of the chi-squared test,” Ann. Math. Stat. 13, 306–317 (1942).
[CrossRef]

Appl. Opt.

Atmos. Oceanic Opt.

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Measurements of the vertical temperature profile by means of the spontaneous Raman scattering channel of the atmospheric laser sounding station,” Atmos. Oceanic Opt. 5, 384–385 (1992).

S. N. Volkov, B. V. Kaul, “Method for determination of light backscattering and extinction coefficients in tropospheric aerosol layers using elastic- and Raman-backscatter-based lidar,” Atmos. Oceanic Opt. 7, 864–869 (1994).

S. N. Volkov, B. V. Kaul, V. A. Shapranov, D. I. Shelefontyk, “Some problems in smoothing lidar signals,” Atmos. Oceanic Opt. 13, 702–706 (2000).

C. R. Acad. Sci.

G. Darmois, “Sur les lois de probabilité à estimation exhaustive,” C. R. Acad. Sci. 200, 1265–1266 (1935).

Other

S. S. Wilks, Mathematical Statistics (Wiley, New York, 1962).

W. G. Cochran, Sampling Techniques (Wiley, New York, 1977).

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions [Dover, New York, 1974; (Vol. 55 of National Bureau of Standards Applied Mathematics Series, Washington, D.C., 1964)].

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (Elsevier, New York, 1968).

V. E. Derr, Remote Sensing of the Troposphere (U.S. Government Printing Office, Washington, D.C., 1972).

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

E. D. Hinkly, ed., Laser Monitoring of the Atmosphere (Springer-Verlag, Berlin, 1976).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Temperature profile and standard deviation in the free atmosphere before application of the OMLR.

Fig. 2
Fig. 2

Temperature profile and standard deviation in the free atmosphere after application of the OMLR.

Fig. 3
Fig. 3

Comparison interval during sliding reconstruction of the temperature profile.

Equations (31)

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Yxi=fxi+qxi  i=1,  , n; x1<x2<  <xn,
fxi=AN K xi-Δx/2xi+Δx/2 Gxx-2βπx×exp-2 0x εxdxdx, xi=Δxi+12i=1,  , n,
Y=f+q,
fi=j=1mcjaij  i=1, 2,  , n,
f=Ac,
Q2=Y-AcTD-1Y-Ac,
ĉ=ATD-1A-1ATD-1Y,
Q2χ1-α2n-m,
i=1n ωiφijφik=δjk=0jk1j=kj, k=1, 2,  m,
ĉj=i=1n ωiYiφij,
Q2=i=1n ωiYi2-j=1m ĉj2j=1, 2,  , m.
φik=uikj=1n ωjujk21/2; ui1=1, uik+1=xik-l=1kj=1n ωjxjkφjlφili=1, 2,  , n; k=1, 2, .
σˆi2fˆ=j=1mφij2  i=1, 2,  , n,
fˆi=j=1m ĉjφij±t1-αn-mσˆifˆ i=1, 2,  , n,
i=1n1γi=m,
γnmmn.
σˆ20=ˆσ2Y,
σˆ2k+1=σˆ2fˆk  k=0, 1, ,
σˆi2kσˆi2k+1F1-αν1, ν2  i=1, 2,  n; k=1, 2, ,
p=-Li-1/2Li-1/21γi+p=mi  i=1, 2,  , n,
rγi+p, Li, σi+p2=0 [i=1, 2,  n; p=-L-1/2,  , L-1/2)],
rγi+p, Li, σi+p2=γ-Liσˆ02σˆi2=0 i=1, 2,  , n; p=0,
βπxβπx, T=NFTT,
FTC1Jexp-C2JT,
YiTi=0.842Ri2-20.71Ri+341.3 i=1, 2,  , n,
Ri=Yi1/Yi2  i=1, 2,  , n,
σˆi2Tk=12TiYik2σˆik2Yk  i=1, 2,  , n,
σ20=Y.
rγi+p, Li, σi+p2=γ-LikFσˆ0k2Fσˆik2=0,
Fσˆik2=TiYik2σˆik2Yk  i=1, 2,  , n; k=1, 2; p=0.
Si=Yixi2  i=1, 2,  , n

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