Abstract

It has recently been suggested that more satisfactory inversion results for the aerosol size distribution may be obtained if the scattered (aureole) data are first differentiated with respect to angle—the so-called differential-kernel method. Analytic eigenfunction theory provides an ideal framework for determining the relative information content of this method versus the standard approach. Our results, supported by the inversion of synthetic data sets, show the differential-kernel method to have significant advantages.

© 1990 Optical Society of America

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References

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  1. A. E. S. Green, A. Deepak, B. J. Lipofsky, “Interpretation of the Sun’s aureole based on atmospheric aerosol models,” Appl. Opt. 10, 1263–1279 (1971).
    [CrossRef] [PubMed]
  2. J. T. Twitty, “The inversion of aureole measurements to derive aerosol size distributions,” J. Atmos. Sci. 32, 584–591 (1975).
    [CrossRef]
  3. A. Deepak, ed., Inversion Methods in Atmospheric Remote Sounding (Academic, New York, 1977).
  4. D. Deirmendjian, “A survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
    [CrossRef]
  5. M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
    [CrossRef]
  6. E. Trakhovsky, E. P. Shettle, “Improved inversion procedure for the retrieval of aerosol size distributions using aureole measurements,” J. Opt. Soc. Am. A 2, 2054–2061 (1985).
    [CrossRef]
  7. M. T. Chahine, “Inversion problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
    [CrossRef]
  8. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).
  9. N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
    [CrossRef]
  10. G. Viera, M. A. Box, “Information content analysis of aerosol remote sensing experiments using singular function theory. 1: Extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).
    [CrossRef] [PubMed]
  11. G. Viera, M. A. Box, “Information content analysis of aerosol remote sensing experiments using singular function theory. 2: Scattering measurements,” Appl. Opt. 27, 3262–3274 (1988).
    [CrossRef] [PubMed]
  12. J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
    [CrossRef]
  13. M. Bertero, E. R. Pike, “Particle size distributions from Fraunhofer diffraction. I. An analytic eigenfunction approach,” Opt. Acta 30, 1043–1049 (1983).
    [CrossRef]
  14. G. Viera, M. A. Box, “Information content analysis of aerosol remote sensing experiments using an analytic eigenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).
    [CrossRef] [PubMed]
  15. F. Smithies, Integral Equations (Cambridge U. Press, London, 1958).
  16. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  18. M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, New York, 1969).
  19. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  20. E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976).
  21. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  22. E. P. Shettle, U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Bedford, Massachusetts 01731 (personal communication).
  23. P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic behavior of the Mie scattering amplitude,” J. Opt. Soc. Am. A 3, 256–258 (1986).
    [CrossRef]
  24. I. S. Gradshteyn, I. W. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).
  25. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

1988

1987

1986

1985

1983

M. Bertero, E. R. Pike, “Particle size distributions from Fraunhofer diffraction. I. An analytic eigenfunction approach,” Opt. Acta 30, 1043–1049 (1983).
[CrossRef]

1980

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

D. Deirmendjian, “A survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
[CrossRef]

1979

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

1978

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

1975

J. T. Twitty, “The inversion of aureole measurements to derive aerosol size distributions,” J. Atmos. Sci. 32, 584–591 (1975).
[CrossRef]

1971

1970

M. T. Chahine, “Inversion problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

Attard, P.

Bertero, M.

M. Bertero, E. R. Pike, “Particle size distributions from Fraunhofer diffraction. I. An analytic eigenfunction approach,” Opt. Acta 30, 1043–1049 (1983).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Box, M. A.

Bryant, G.

Byrne, D. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Chahine, M. T.

M. T. Chahine, “Inversion problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

Deepak, A.

Deirmendjian, D.

D. Deirmendjian, “A survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. W. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Green, A. E. S.

Herman, B. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Joseph, J. H.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, New York, 1969).

King, M. D.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Lipofsky, B. J.

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976).

McKellar, B. H. J.

McWhirter, J. G.

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Mekler, Y.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Pike, E. R.

M. Bertero, E. R. Pike, “Particle size distributions from Fraunhofer diffraction. I. An analytic eigenfunction approach,” Opt. Acta 30, 1043–1049 (1983).
[CrossRef]

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Reagan, J. A.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Ryzhik, I. W.

I. S. Gradshteyn, I. W. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Shettle, E. P.

E. Trakhovsky, E. P. Shettle, “Improved inversion procedure for the retrieval of aerosol size distributions using aureole measurements,” J. Opt. Soc. Am. A 2, 2054–2061 (1985).
[CrossRef]

E. P. Shettle, U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Bedford, Massachusetts 01731 (personal communication).

Smithies, F.

F. Smithies, Integral Equations (Cambridge U. Press, London, 1958).

Trakhovsky, E.

Twitty, J. T.

J. T. Twitty, “The inversion of aureole measurements to derive aerosol size distributions,” J. Atmos. Sci. 32, 584–591 (1975).
[CrossRef]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Viera, G.

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Wolfson, N.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

Appl. Opt.

J. Appl. Meteorol.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

J. Atmos. Sci.

M. T. Chahine, “Inversion problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

J. T. Twitty, “The inversion of aureole measurements to derive aerosol size distributions,” J. Atmos. Sci. 32, 584–591 (1975).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Opt. Acta

M. Bertero, E. R. Pike, “Particle size distributions from Fraunhofer diffraction. I. An analytic eigenfunction approach,” Opt. Acta 30, 1043–1049 (1983).
[CrossRef]

Phys. Rev. Lett.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Rev. Geophys. Space Phys.

D. Deirmendjian, “A survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
[CrossRef]

Other

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).

A. Deepak, ed., Inversion Methods in Atmospheric Remote Sounding (Academic, New York, 1977).

F. Smithies, Integral Equations (Cambridge U. Press, London, 1958).

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, New York, 1969).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976).

E. P. Shettle, U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Bedford, Massachusetts 01731 (personal communication).

I. S. Gradshteyn, I. W. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

Normalized eigenvalues for the normal and differential kernels for three values of α.

Fig. 2
Fig. 2

Retrieved distribution for the normal-kernel method for two different cutoffs, ωm.

Fig. 3
Fig. 3

Retrieved distribution for the differential-kernel method for two different error levels.

Equations (27)

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β ( θ ) = k 2 n ( r ) i ( r , λ , m , θ ) d r ,
2 S ( 0 ° ) x 2 1 2 Q ext 1 2 ( 2 + 2 x 2 / 3 ) ,
S ( θ ) = x 2 J 1 ( u ) / u ,
S ( θ ) = i ( m 1 ) ( 2 π ) 1 / 2 x 3 J 3 / 2 ( u ) / u 3 / 2 ,
S ( θ ) = x ν J v ( x θ ) / θ v .
i ( θ ) = x 2 v J ν 2 ( x θ ) / θ 2 v .
i = x 2 v + 1 J v ( x θ ) J v + 1 ( x θ ) / θ 2 v .
β ( θ ) = k 2 0 x 2 v J ν 2 ( x θ ) θ 2 v n ( r ) d r ,
g ( y ) = 0 K ( y r ) f ( r ) d r ,
g ( y ) = k 2 4 v y α + 2 v β ( θ ) ,
K ( y r ) = ( y r ) α J ν 2 ( y r ) ,
f ( r ) = r 2 v α n ( r ) .
β ( θ ) = k 2 0 x 2 v + 1 J v ( x θ ) J v + 1 ( x θ ) n ( r ) d r ,
g ( y ) = k 4 v y α + 2 v + 1 β ( θ ) ,
K ( y r ) = ( y r ) α J v ( y r ) J v + 1 ( y r ) ,
f ( r ) = r 2 v α + 1 n ( r ) .
f ( r ) = r 1 / 2 Re 0 ω m G ( ω ) r i ω χ ( ω ) / λ ( ω ) d ω ,
G ( ω ) = χ ( ω ) 0 g ( y ) y 1 / 2 i ω d y ,
χ ( ω ) = ( K / π λ ) 1 / 2 ,
λ ( ω ) = | K ( ½ + i ω ) | ,
K ( 1 2 + i ω ) = 0 t 1 / 2 + i ω K ( t ) d t
2 v ½ < α < ½ ,
2 v 3 / 2 < α < ½ .
K ( 1 2 + i ω ) = 2 α 1 / 2 + i ω Γ ( 1 / 2 α i ω ) Γ ( 1 / 4 + α / 2 + v + i ω / 2 ) Γ 2 ( 3 / 4 α / 2 i ω / 2 ) Γ ( 3 / 4 α / 2 + v i ω / 2 ) ,
K ( 1 / 2 + i ω ) = 2 α 1 / 2 + i ω Γ ( 1 / 2 α i ω ) Γ ( 3 / 4 + α / 2 + v + i ω / 2 ) ( 1 / 4 α / 2 i ω / 2 ) Γ 2 ( 1 / 4 α / 2 i ω / 2 ) Γ ( 5 / 4 α / 2 + v i ω / 2 ) ,
λ ( ω ) ω α 1 ,
n ( r ) = A r 2 e b r ,

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