Abstract

An important step in any planned remote-sensing experiment is an analysis of the information content of the equations which will finally be inverted. In this paper we show the value of performing such an analysis using a recently developed analytic eigenfunction theory. So that we may fully utilize the analytic nature of this technique, we have applied it first to the anomalous diffraction approximation to the Mie theory extinction efficiency. Analytical expressions for the eigenfunctions and eigenvalues are derived. The effects of ill-conditioning, and their amelioration due to the inclusion of certain a priori knowledge, are then investigated.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Deepak, Ed., Inversion Methods in Atmospheric Remote Sounding (Academic, New York, 1977).
  2. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).
  3. 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 (1978).
    [CrossRef]
  4. G. E. Shaw, “Inversion of Optical Scattering and Spectral Extinction Measurements to Recover Aerosol Size Spectra,” Appl. Opt. 18, 988 (1979).
    [CrossRef] [PubMed]
  5. E. Thomalla, H. Quenzel, “Information Content of Aerosol Optical Properties with Respect to Their Size Distribution,” Appl. Opt. 21, 3170 (1982).
    [CrossRef] [PubMed]
  6. G. Yamamoto, M. Tanaka, “Determination of Aerosol Size Distribution from Spectral Attenuation Measurements,” Appl. Opt. 8, 447 (1969).
    [CrossRef] [PubMed]
  7. R. Rizzi, R. Guzzi, R. Legnani, “Aerosol Size Spectra from Spectral Extinction Data: the Use of a Linear Inversion Method,” Appl. Opt. 21, 1578 (1982).
    [CrossRef] [PubMed]
  8. H. Quenzel, “Determination of Size Distribution of Atmospheric Aerosol Particles from Spectral Solar Radiation Measurements,” J. Geophys. Res. 75, 2915 (1970).
    [CrossRef]
  9. G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of Atmospheric Extinctions Using Direct Solar Radiation Measurements Made with a Multiple Wavelength Radiometer,” J. Appl. Meteorol. 12, 374 (1973).
    [CrossRef]
  10. C. Tomasi, F. Prodi, M. Sentimenti, G. Cesari, “Multiwavelength Sun-Photometers for Accurate Measurements of Atmospheric Extinction in the Visible and Near-IR Spectral Range,” Appl. Opt. 22, 622 (1983).
    [CrossRef] [PubMed]
  11. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  12. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  13. G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, Amsterdam, 1976).
  14. E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976).
  15. F. Smithies, Integral Equations (Cambridge U.P., London, 1958).
  16. H. P. Baltes, Ed., Inverse Source Problems in Optics (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  17. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  18. A. Deepak, G. P. Box. “Representation of Aerosol Size Distribution Data by Analytic Models,” in Atmospheric Aerosols, Their Formation, Optical Properties, and Effects, A. Deepak, Ed. (Spectrum, Hampton, Va., 1982).
  19. O. N. Strand, “Theory and Methods Related to the Singular Function Expansion and Landweber’s Iteration for Integral Equations of the First Kind,” SIAM J. Numer. Anal. 11, 798 (1974).
    [CrossRef]
  20. C. D. Capps, R. L. Henning, G. M. Hess, “Analytic Inversion of Remote-Sensing Data,” Appl. Opt. 21, 3581 (1982).
    [CrossRef] [PubMed]
  21. Preliminary results are contained in M. A. Box, G. Viera, “Singular Function Theory of Remote Sensing Inversion,” presented at the IAMAP International Radiation Symposium, Perugia, Italy, Aug. 1984, to be published.
  22. 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 (1978).
    [CrossRef]
  23. G. F. Miller, “Fredholm Equations of the First Kind,” in Numerical Solution of Integral Equations, L. M. Delves, J. E. Walsh, Eds. (Clarendon, Oxford, 1974), pp. 178–188.
  24. M. A. Box, B. H. J. McKellar, “Determination of Moments of the Size Distribution Function in Scattering by Polydispersions,” Appl. Opt. 15, 2610 (1976).
    [CrossRef] [PubMed]
  25. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  26. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  27. M. A. Box, S. Y. Lo, B. H. J. McKellar, “Aerosol Optical Thicknesses Calculated Using the Anomalous Diffraction Approximation,” journal00, 000 (198x), in preparation.
  28. M. A. Box, B. H. J. McKellar, “Analytic Inversion of Multi-spectral Extinction Data in the Anomalous Diffraction Approximation,” Opt. Lett. 3, 91 (1978).
    [CrossRef] [PubMed]
  29. M. V. Barnhill, C. Russ, “Improving the Convergence of Solutions to the Fredholm Integral Equation of the First Kind,” J. Comput. Phys. 37, 268 (1980).
    [CrossRef]
  30. C. E. Junge, Air Chemistry and Radioactivity (Academic, New York, 1963).
  31. M. A. Box, B. H. J. McKellar, “Inversion of Multispectral Extinction Data—The Anomalous Diffraction Approximation,” presented at the Topical Meeting on Atmospheric Aerosols, Williamsburg, Va. (Dec. 1976), paper WA-10A (available as NASA CP-2004).
  32. A. L. Fymat, “Analytical Inversions in Remote Sensing of Particle Size Distributions. 1: Multispectral Extinctions in the Anomalous Diffraction Approximation,” Appl. Opt. 17, 1675 (1978).
  33. K. S. Shifrin, A. Ya. Perelman, “The Determination of the Spectrum of Particles in a Dispersed System from Data on its Transparency,” Opt. Spectrosc. 15, 285 (1963).
  34. A. Ya. Perelman, K. S. Shifrin, “Improvements to the Spectral Transparency Method for Determining Particle-Size Distribution,” Appl. Opt. 19, 1787 (1980).
    [CrossRef] [PubMed]
  35. M. A. Box, B. H. J. McKellar, “Relationship Between Two Analytic Inversion Formulae for Multispectral Extinction Data,” Appl. Opt. 18, 3599 (1979).
    [CrossRef] [PubMed]
  36. M. A. Box, B. H. J. McKellar, “Further Relations Between Analytic Inversion Formulas for Multispectral Extinction Data,” Appl. Opt. 20, 3829 (1981).
    [CrossRef] [PubMed]
  37. A. L. Fymat, C. B. Smith, “Analytical Inversions in Remote Sensing of Particle Size Distributions. 4: Comparison of Fymat and Box-McKellar Solutions in the Anomalous Diffraction Approximation,” Appl. Opt. 18, 3595 (1979).
    [CrossRef] [PubMed]
  38. J. G. McWhirter, “A Well-Conditioned Cubic b-Spline Model for Processing Laser Anemometry Data,” Opt. Acta 28, 1453 (1981).
    [CrossRef]
  39. M. Bertero, E. R. Pike, “Particle Size Distributions from Fraunhofer Diffraction I. An Analytic Eigenfunction Approach,” Opt. Acta 30, 1043 (1983).
    [CrossRef]
  40. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency Factors in Mie Scattering,” Phys. Rev. Lett. 45, 1490 (1980).
    [CrossRef]
  41. P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic Behaviour of Mie Scattering Amplitude,” J. Opt. Soc. Am. A (1986), to be published.
    [CrossRef]
  42. B. H. J. McKellar, M. A. Box, C. F. Bohren, “Sum Rules for Optical Scattering Amplitudes,” J. Opt. Soc. Am. 72, 535 (1982).
    [CrossRef]
  43. M. A. Box, P. Attard, G. Bryant, B. H. J. McKellar, “Sum Rules for Mie Scattering,” J. Opt. Soc. Am. A submitted.
  44. G. Viera, M. A. Box, “Information Content Analysis of Remote Sensing Experiments Using Eigenfunctions, Workshop on Advances in Remote Sensing Retrieval Methods, Williamsburg, Va., Oct–Nov. 1984, to be published.

1983 (2)

1982 (4)

1981 (2)

M. A. Box, B. H. J. McKellar, “Further Relations Between Analytic Inversion Formulas for Multispectral Extinction Data,” Appl. Opt. 20, 3829 (1981).
[CrossRef] [PubMed]

J. G. McWhirter, “A Well-Conditioned Cubic b-Spline Model for Processing Laser Anemometry Data,” Opt. Acta 28, 1453 (1981).
[CrossRef]

1980 (3)

M. V. Barnhill, C. Russ, “Improving the Convergence of Solutions to the Fredholm Integral Equation of the First Kind,” J. Comput. Phys. 37, 268 (1980).
[CrossRef]

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

A. Ya. Perelman, K. S. Shifrin, “Improvements to the Spectral Transparency Method for Determining Particle-Size Distribution,” Appl. Opt. 19, 1787 (1980).
[CrossRef] [PubMed]

1979 (3)

1978 (4)

M. A. Box, B. H. J. McKellar, “Analytic Inversion of Multi-spectral Extinction Data in the Anomalous Diffraction Approximation,” Opt. Lett. 3, 91 (1978).
[CrossRef] [PubMed]

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 (1978).
[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 (1978).
[CrossRef]

A. L. Fymat, “Analytical Inversions in Remote Sensing of Particle Size Distributions. 1: Multispectral Extinctions in the Anomalous Diffraction Approximation,” Appl. Opt. 17, 1675 (1978).

1976 (1)

1974 (1)

O. N. Strand, “Theory and Methods Related to the Singular Function Expansion and Landweber’s Iteration for Integral Equations of the First Kind,” SIAM J. Numer. Anal. 11, 798 (1974).
[CrossRef]

1973 (1)

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of Atmospheric Extinctions Using Direct Solar Radiation Measurements Made with a Multiple Wavelength Radiometer,” J. Appl. Meteorol. 12, 374 (1973).
[CrossRef]

1970 (1)

H. Quenzel, “Determination of Size Distribution of Atmospheric Aerosol Particles from Spectral Solar Radiation Measurements,” J. Geophys. Res. 75, 2915 (1970).
[CrossRef]

1969 (1)

1963 (1)

K. S. Shifrin, A. Ya. Perelman, “The Determination of the Spectrum of Particles in a Dispersed System from Data on its Transparency,” Opt. Spectrosc. 15, 285 (1963).

Attard, P.

P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic Behaviour of Mie Scattering Amplitude,” J. Opt. Soc. Am. A (1986), to be published.
[CrossRef]

M. A. Box, P. Attard, G. Bryant, B. H. J. McKellar, “Sum Rules for Mie Scattering,” J. Opt. Soc. Am. A submitted.

Barnhill, M. V.

M. V. Barnhill, C. Russ, “Improving the Convergence of Solutions to the Fredholm Integral Equation of the First Kind,” J. Comput. Phys. 37, 268 (1980).
[CrossRef]

Bertero, M.

M. Bertero, E. R. Pike, “Particle Size Distributions from Fraunhofer Diffraction I. An Analytic Eigenfunction Approach,” Opt. Acta 30, 1043 (1983).
[CrossRef]

Bohren, C. F.

B. H. J. McKellar, M. A. Box, C. F. Bohren, “Sum Rules for Optical Scattering Amplitudes,” J. Opt. Soc. Am. 72, 535 (1982).
[CrossRef]

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

Box, G. P.

A. Deepak, G. P. Box. “Representation of Aerosol Size Distribution Data by Analytic Models,” in Atmospheric Aerosols, Their Formation, Optical Properties, and Effects, A. Deepak, Ed. (Spectrum, Hampton, Va., 1982).

Box, M. A.

B. H. J. McKellar, M. A. Box, C. F. Bohren, “Sum Rules for Optical Scattering Amplitudes,” J. Opt. Soc. Am. 72, 535 (1982).
[CrossRef]

M. A. Box, B. H. J. McKellar, “Further Relations Between Analytic Inversion Formulas for Multispectral Extinction Data,” Appl. Opt. 20, 3829 (1981).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Relationship Between Two Analytic Inversion Formulae for Multispectral Extinction Data,” Appl. Opt. 18, 3599 (1979).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Analytic Inversion of Multi-spectral Extinction Data in the Anomalous Diffraction Approximation,” Opt. Lett. 3, 91 (1978).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Determination of Moments of the Size Distribution Function in Scattering by Polydispersions,” Appl. Opt. 15, 2610 (1976).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Inversion of Multispectral Extinction Data—The Anomalous Diffraction Approximation,” presented at the Topical Meeting on Atmospheric Aerosols, Williamsburg, Va. (Dec. 1976), paper WA-10A (available as NASA CP-2004).

P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic Behaviour of Mie Scattering Amplitude,” J. Opt. Soc. Am. A (1986), to be published.
[CrossRef]

G. Viera, M. A. Box, “Information Content Analysis of Remote Sensing Experiments Using Eigenfunctions, Workshop on Advances in Remote Sensing Retrieval Methods, Williamsburg, Va., Oct–Nov. 1984, to be published.

M. A. Box, S. Y. Lo, B. H. J. McKellar, “Aerosol Optical Thicknesses Calculated Using the Anomalous Diffraction Approximation,” journal00, 000 (198x), in preparation.

M. A. Box, P. Attard, G. Bryant, B. H. J. McKellar, “Sum Rules for Mie Scattering,” J. Opt. Soc. Am. A submitted.

Preliminary results are contained in M. A. Box, G. Viera, “Singular Function Theory of Remote Sensing Inversion,” presented at the IAMAP International Radiation Symposium, Perugia, Italy, Aug. 1984, to be published.

Bryant, G.

M. A. Box, P. Attard, G. Bryant, B. H. J. McKellar, “Sum Rules for Mie Scattering,” J. Opt. Soc. Am. A submitted.

P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic Behaviour of Mie Scattering Amplitude,” J. Opt. Soc. Am. A (1986), to be published.
[CrossRef]

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 (1978).
[CrossRef]

Capps, C. D.

Cesari, G.

Deepak, A.

A. Deepak, G. P. Box. “Representation of Aerosol Size Distribution Data by Analytic Models,” in Atmospheric Aerosols, Their Formation, Optical Properties, and Effects, A. Deepak, Ed. (Spectrum, Hampton, Va., 1982).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Fymat, A. L.

Guzzi, R.

Henning, R. L.

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 (1978).
[CrossRef]

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of Atmospheric Extinctions Using Direct Solar Radiation Measurements Made with a Multiple Wavelength Radiometer,” J. Appl. Meteorol. 12, 374 (1973).
[CrossRef]

Hess, G. M.

Huffman, D. R.

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

Junge, C. E.

C. E. Junge, Air Chemistry and Radioactivity (Academic, New York, 1963).

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 (1978).
[CrossRef]

Legnani, R.

Liou, K. N.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

Lo, S. Y.

M. A. Box, S. Y. Lo, B. H. J. McKellar, “Aerosol Optical Thicknesses Calculated Using the Anomalous Diffraction Approximation,” journal00, 000 (198x), in preparation.

McCartney, E. J.

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

McKellar, B. H. J.

B. H. J. McKellar, M. A. Box, C. F. Bohren, “Sum Rules for Optical Scattering Amplitudes,” J. Opt. Soc. Am. 72, 535 (1982).
[CrossRef]

M. A. Box, B. H. J. McKellar, “Further Relations Between Analytic Inversion Formulas for Multispectral Extinction Data,” Appl. Opt. 20, 3829 (1981).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Relationship Between Two Analytic Inversion Formulae for Multispectral Extinction Data,” Appl. Opt. 18, 3599 (1979).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Analytic Inversion of Multi-spectral Extinction Data in the Anomalous Diffraction Approximation,” Opt. Lett. 3, 91 (1978).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Determination of Moments of the Size Distribution Function in Scattering by Polydispersions,” Appl. Opt. 15, 2610 (1976).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Inversion of Multispectral Extinction Data—The Anomalous Diffraction Approximation,” presented at the Topical Meeting on Atmospheric Aerosols, Williamsburg, Va. (Dec. 1976), paper WA-10A (available as NASA CP-2004).

M. A. Box, S. Y. Lo, B. H. J. McKellar, “Aerosol Optical Thicknesses Calculated Using the Anomalous Diffraction Approximation,” journal00, 000 (198x), in preparation.

P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic Behaviour of Mie Scattering Amplitude,” J. Opt. Soc. Am. A (1986), to be published.
[CrossRef]

M. A. Box, P. Attard, G. Bryant, B. H. J. McKellar, “Sum Rules for Mie Scattering,” J. Opt. Soc. Am. A submitted.

McWhirter, J. G.

J. G. McWhirter, “A Well-Conditioned Cubic b-Spline Model for Processing Laser Anemometry Data,” Opt. Acta 28, 1453 (1981).
[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 (1978).
[CrossRef]

Miller, G. F.

G. F. Miller, “Fredholm Equations of the First Kind,” in Numerical Solution of Integral Equations, L. M. Delves, J. E. Walsh, Eds. (Clarendon, Oxford, 1974), pp. 178–188.

Nussenzveig, H. M.

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

Paltridge, G. W.

G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, Amsterdam, 1976).

Perelman, A. Ya.

A. Ya. Perelman, K. S. Shifrin, “Improvements to the Spectral Transparency Method for Determining Particle-Size Distribution,” Appl. Opt. 19, 1787 (1980).
[CrossRef] [PubMed]

K. S. Shifrin, A. Ya. Perelman, “The Determination of the Spectrum of Particles in a Dispersed System from Data on its Transparency,” Opt. Spectrosc. 15, 285 (1963).

Pike, E. R.

M. Bertero, E. R. Pike, “Particle Size Distributions from Fraunhofer Diffraction I. An Analytic Eigenfunction Approach,” Opt. Acta 30, 1043 (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 (1978).
[CrossRef]

Platt, C. M. R.

G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, Amsterdam, 1976).

Prodi, F.

Quenzel, H.

E. Thomalla, H. Quenzel, “Information Content of Aerosol Optical Properties with Respect to Their Size Distribution,” Appl. Opt. 21, 3170 (1982).
[CrossRef] [PubMed]

H. Quenzel, “Determination of Size Distribution of Atmospheric Aerosol Particles from Spectral Solar Radiation Measurements,” J. Geophys. Res. 75, 2915 (1970).
[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 (1978).
[CrossRef]

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of Atmospheric Extinctions Using Direct Solar Radiation Measurements Made with a Multiple Wavelength Radiometer,” J. Appl. Meteorol. 12, 374 (1973).
[CrossRef]

Rizzi, R.

Russ, C.

M. V. Barnhill, C. Russ, “Improving the Convergence of Solutions to the Fredholm Integral Equation of the First Kind,” J. Comput. Phys. 37, 268 (1980).
[CrossRef]

Sentimenti, M.

Shaw, G. E.

G. E. Shaw, “Inversion of Optical Scattering and Spectral Extinction Measurements to Recover Aerosol Size Spectra,” Appl. Opt. 18, 988 (1979).
[CrossRef] [PubMed]

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of Atmospheric Extinctions Using Direct Solar Radiation Measurements Made with a Multiple Wavelength Radiometer,” J. Appl. Meteorol. 12, 374 (1973).
[CrossRef]

Shifrin, K. S.

A. Ya. Perelman, K. S. Shifrin, “Improvements to the Spectral Transparency Method for Determining Particle-Size Distribution,” Appl. Opt. 19, 1787 (1980).
[CrossRef] [PubMed]

K. S. Shifrin, A. Ya. Perelman, “The Determination of the Spectrum of Particles in a Dispersed System from Data on its Transparency,” Opt. Spectrosc. 15, 285 (1963).

Smith, C. B.

Smithies, F.

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

Strand, O. N.

O. N. Strand, “Theory and Methods Related to the Singular Function Expansion and Landweber’s Iteration for Integral Equations of the First Kind,” SIAM J. Numer. Anal. 11, 798 (1974).
[CrossRef]

Tanaka, M.

Thomalla, E.

Tomasi, C.

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (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.

Preliminary results are contained in M. A. Box, G. Viera, “Singular Function Theory of Remote Sensing Inversion,” presented at the IAMAP International Radiation Symposium, Perugia, Italy, Aug. 1984, to be published.

G. Viera, M. A. Box, “Information Content Analysis of Remote Sensing Experiments Using Eigenfunctions, Workshop on Advances in Remote Sensing Retrieval Methods, Williamsburg, Va., Oct–Nov. 1984, to be published.

Wiscombe, W. J.

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

Yamamoto, G.

Appl. Opt. (12)

G. Yamamoto, M. Tanaka, “Determination of Aerosol Size Distribution from Spectral Attenuation Measurements,” Appl. Opt. 8, 447 (1969).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Determination of Moments of the Size Distribution Function in Scattering by Polydispersions,” Appl. Opt. 15, 2610 (1976).
[CrossRef] [PubMed]

G. E. Shaw, “Inversion of Optical Scattering and Spectral Extinction Measurements to Recover Aerosol Size Spectra,” Appl. Opt. 18, 988 (1979).
[CrossRef] [PubMed]

A. L. Fymat, C. B. Smith, “Analytical Inversions in Remote Sensing of Particle Size Distributions. 4: Comparison of Fymat and Box-McKellar Solutions in the Anomalous Diffraction Approximation,” Appl. Opt. 18, 3595 (1979).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Relationship Between Two Analytic Inversion Formulae for Multispectral Extinction Data,” Appl. Opt. 18, 3599 (1979).
[CrossRef] [PubMed]

A. Ya. Perelman, K. S. Shifrin, “Improvements to the Spectral Transparency Method for Determining Particle-Size Distribution,” Appl. Opt. 19, 1787 (1980).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Further Relations Between Analytic Inversion Formulas for Multispectral Extinction Data,” Appl. Opt. 20, 3829 (1981).
[CrossRef] [PubMed]

R. Rizzi, R. Guzzi, R. Legnani, “Aerosol Size Spectra from Spectral Extinction Data: the Use of a Linear Inversion Method,” Appl. Opt. 21, 1578 (1982).
[CrossRef] [PubMed]

E. Thomalla, H. Quenzel, “Information Content of Aerosol Optical Properties with Respect to Their Size Distribution,” Appl. Opt. 21, 3170 (1982).
[CrossRef] [PubMed]

C. D. Capps, R. L. Henning, G. M. Hess, “Analytic Inversion of Remote-Sensing Data,” Appl. Opt. 21, 3581 (1982).
[CrossRef] [PubMed]

C. Tomasi, F. Prodi, M. Sentimenti, G. Cesari, “Multiwavelength Sun-Photometers for Accurate Measurements of Atmospheric Extinction in the Visible and Near-IR Spectral Range,” Appl. Opt. 22, 622 (1983).
[CrossRef] [PubMed]

A. L. Fymat, “Analytical Inversions in Remote Sensing of Particle Size Distributions. 1: Multispectral Extinctions in the Anomalous Diffraction Approximation,” Appl. Opt. 17, 1675 (1978).

J. Appl. Meteorol. (1)

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of Atmospheric Extinctions Using Direct Solar Radiation Measurements Made with a Multiple Wavelength Radiometer,” J. Appl. Meteorol. 12, 374 (1973).
[CrossRef]

J. Atmos. Sci. (1)

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 (1978).
[CrossRef]

J. Comput. Phys. (1)

M. V. Barnhill, C. Russ, “Improving the Convergence of Solutions to the Fredholm Integral Equation of the First Kind,” J. Comput. Phys. 37, 268 (1980).
[CrossRef]

J. Geophys. Res. (1)

H. Quenzel, “Determination of Size Distribution of Atmospheric Aerosol Particles from Spectral Solar Radiation Measurements,” J. Geophys. Res. 75, 2915 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

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 (1978).
[CrossRef]

Opt. Acta (2)

J. G. McWhirter, “A Well-Conditioned Cubic b-Spline Model for Processing Laser Anemometry Data,” Opt. Acta 28, 1453 (1981).
[CrossRef]

M. Bertero, E. R. Pike, “Particle Size Distributions from Fraunhofer Diffraction I. An Analytic Eigenfunction Approach,” Opt. Acta 30, 1043 (1983).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

K. S. Shifrin, A. Ya. Perelman, “The Determination of the Spectrum of Particles in a Dispersed System from Data on its Transparency,” Opt. Spectrosc. 15, 285 (1963).

Phys. Rev. Lett. (1)

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

SIAM J. Numer. Anal. (1)

O. N. Strand, “Theory and Methods Related to the Singular Function Expansion and Landweber’s Iteration for Integral Equations of the First Kind,” SIAM J. Numer. Anal. 11, 798 (1974).
[CrossRef]

Other (20)

Preliminary results are contained in M. A. Box, G. Viera, “Singular Function Theory of Remote Sensing Inversion,” presented at the IAMAP International Radiation Symposium, Perugia, Italy, Aug. 1984, to be published.

C. E. Junge, Air Chemistry and Radioactivity (Academic, New York, 1963).

M. A. Box, B. H. J. McKellar, “Inversion of Multispectral Extinction Data—The Anomalous Diffraction Approximation,” presented at the Topical Meeting on Atmospheric Aerosols, Williamsburg, Va. (Dec. 1976), paper WA-10A (available as NASA CP-2004).

P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic Behaviour of Mie Scattering Amplitude,” J. Opt. Soc. Am. A (1986), to be published.
[CrossRef]

M. A. Box, P. Attard, G. Bryant, B. H. J. McKellar, “Sum Rules for Mie Scattering,” J. Opt. Soc. Am. A submitted.

G. Viera, M. A. Box, “Information Content Analysis of Remote Sensing Experiments Using Eigenfunctions, Workshop on Advances in Remote Sensing Retrieval Methods, Williamsburg, Va., Oct–Nov. 1984, to be published.

G. F. Miller, “Fredholm Equations of the First Kind,” in Numerical Solution of Integral Equations, L. M. Delves, J. E. Walsh, Eds. (Clarendon, Oxford, 1974), pp. 178–188.

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

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

M. A. Box, S. Y. Lo, B. H. J. McKellar, “Aerosol Optical Thicknesses Calculated Using the Anomalous Diffraction Approximation,” journal00, 000 (198x), in preparation.

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

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

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

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

G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, Amsterdam, 1976).

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

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

H. P. Baltes, Ed., Inverse Source Problems in Optics (Springer-Verlag, Berlin, 1978).
[CrossRef]

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

A. Deepak, G. P. Box. “Representation of Aerosol Size Distribution Data by Analytic Models,” in Atmospheric Aerosols, Their Formation, Optical Properties, and Effects, A. Deepak, Ed. (Spectrum, Hampton, Va., 1982).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Normalized eigenvalues.

Fig. 2
Fig. 2

Coefficients gω+ for the haze H size distribution.

Fig. 3
Fig. 3

Retrieved volume distribution for the haze H size distribution.

Fig. 4
Fig. 4

Coefficients gω+ for the power-law size distribution.

Fig. 5
Fig. 5

Retrieved volume distribution for the power-law size distribution.

Fig. 6
Fig. 6

Retrieved area distribution for haze H, incorporating a priori knowledge of A.

Fig. 7
Fig. 7

Retrieved size distribution for haze H, incorporating a priori knowledge of both A and N.

Equations (78)

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

τ ( λ ) = 0 π r 2 Q ( r , λ , m ) n ( r ) d r ,
g ( x ) = a b K ( x , y ) f ( y ) d y c < x < d ,
a b K ( x , y ) V i ( y ) d y = κ i U i ( x ) ,
a d K ( x , y ) U i ( x ) d x = κ i V i ( y ) ,
f m ( y ) = i = 1 f i V i ( y ) ,
f i = g i κ i = a b f ( y ) V i ( y ) d y ,
g i = c d g ( x ) U i ( x ) d x .
a b K ( x , y ) f 0 ( y ) d y = 0 ,
f = f m + f 0 .
δ f i = κ i 1 c d ( x ) U i ( x ) d x ,
f N ( y ) = i = 1 N g i κ i V i ( y )
f ( y ) = f N ( y ) + f R ( y ) + f 0 ( y ) ,
g ( x ) = 0 K ( x y ) f ( y ) d y 0 < y < ,
0 | K ( t ) | t 1 / 2 d t < .
0 K ( x y ) ψ ω ( x ) d x = λ ω ψ ω ( y ) .
ψ ω + ( y ) = R e [ ϕ ω ( y ) ] ,
ψ ω ( y ) = Im [ ϕ ω ( y ) ] ,
λ ω ± = ± | K ˜ ( 1 2 + i ω ) | .
ϕ ω ( y ) = y 1 / 2 i ω K ˜ ( 1 2 + i ω ) ( π | K ˜ ( 1 2 + i ω ) | ) ,
K ˜ ( s ) = 0 x s 1 K ( x ) d x
ψ ω ± ( y ) = ± ψ ω ± ( y ) ,
f ( y ) = 0 f ω + ψ ω + ( y ) d ω + 0 f ω ψ ω ( y ) d ω ,
f ω ± = g ω ± / λ ω ± ,
g ω ± = 0 g ( x ) ψ ω ± ( x ) d x .
f ( y ) = f ω m ( y ) + f R ( y ) ,
f ω m ( y ) = 0 ω m f ω + ψ ω + ( y ) d ω + 0 ω m f ω ψ ω ( y ) d ω .
Q ( ρ ) = 2 + 4 ρ 2 4 sin ρ ρ 4 cos ρ ρ 2
= 4 ( π / 2 ρ ) 1 / 2 H 3 / 2 ( ρ ) ,
g ( κ ) = 0 K ( κ r ) f ( r ) d r 0 < κ < ,
g ( κ ) = ( 2 π ) 3 / 2 κ α + 1 / 2 τ ( κ ) ,
K ( κ r ) = ( κ r ) α H 3 / 2 ( κ r ) ,
f ( r ) = r 3 / 2 α n ( r ) ,
0 t μ ν 1 H ν ( t ) d t = 2 μ ν 1 Γ ( 1 2 μ ) tan ( 1 2 π μ ) Γ ( ν + 1 1 2 μ ) ,
K ˜ ( 1 2 + i ω ) = 2 μ 5 / 2 Γ ( 1 2 μ ) tan ( 1 2 π μ ) Γ ( 5 / 2 1 2 μ ) = ( 2 π ) 1 / 2 ( μ 2 ) Γ ( μ 3 ) sin 1 2 π μ ,
| sin 1 2 π ( α + 2 + i ω ) | exp ( 1 2 π ω ) ,
| Γ ( α 1 + i ω ) | | ω | α 3 / 2 exp ( 1 2 π ω ) ,
λ ω ± ~ ω α 1 / 2 , 3 < α < 1 .
n ( r ) = 1 2 a b 3 r 2 exp ( b r ) .
τ ( κ ) = t = 4 π a b 2 ( 45 t 2 + 40 t 4 + 25 t 6 + 6 t 8 ) ( 1 + t 2 ) 4 , κ / b }
g ( κ ) = ( 2 π ) 3 / 2 τ ( κ ) / κ ,
( Δ g ω ) 2 = ψ ω 2 ( t ) 2 ( t ) d t ,
n ( r ) = A r ν , r min < r < r max .
f ω ± ~ ω 5 exp ( 1 2 π ω ) .
g ω ± ~ ω 3 exp ( 1 2 π ω ) .
f ω ± ~ ω 1 ,
g ω ± ~ ω 3 .
n ( r ) = ( 2 π ) 3 / 2 0 ( κ r ) 1 / 2 H 3 / 2 ( κ r ) κ 2 [ τ ( κ ) 2 A 4 π N κ 2 ] d κ
A = π 0 r 2 n ( r ) d r ,
N = 0 n ( r ) d r
n ( r ) = ( 2 π ) 3 / 2 0 ( κ r ) 1 / 2 Y 3 / 2 ( κ r ) κ 2 [ τ ( κ ) 2 A ] d κ ,
g ( κ ) = 0 K ( κ r ) f ( r ) d r ,
g ( κ ) = ( 2 π ) 3 / 2 κ α + 3 / 2 [ τ ( κ ) 2 A ] ,
K ( κ r ) = ( κ r ) α H 1 / 2 ( κ r ) ,
f ( r ) = r 1 / 2 α d d r [ r 2 n ( r ) ] .
K ˜ ( 1 2 + i ω ) = ( 2 / π ) 1 / 2 Γ ( α + i ω ) cos 1 2 π ( α + i ω ) ,
g ( κ ) = 0 K ( κ r ) f ( r ) d r ,
g ( κ ) = ( 2 π ) 3 / 2 κ α + 3 / 2 [ τ ( κ ) 2 A ] ,
K ( κ r ) = ( 2 / π ) 1 / 2 ( κ r ) α 1 / 2 ( 1 κ r sin κ r cos κ r ) = ( κ r ) α [ Η 1 / 2 ( κ r ) κ r J 1 / 2 ( κ r ) ] ,
f ( r ) = r 1 / 2 α n ( r ) ,
K ˜ ( 1 2 + i ω ) = ( 2 / π ) 1 / 2 ( α + 1 + i ω ) Γ ( α + i ω ) cos 1 2 π ( α + i ω ) ,
λ ω ± ~ ω α + 1 / 2 .
K ( κ r ) = ( 2 / π ) 1 / 2 ( κ r ) α 1 / 2 ( 1 cos κ r ) ,
g ( κ ) = 0 K ( κ r ) f ( r ) d r ,
g ( κ ) = ( 2 π ) 3 / 2 κ α + 3 / 2 [ 4 π N κ 2 + 2 A τ ( κ ) ] ,
K ( κ r ) = ( 2 / π ) 1 / 2 ( κ r ) α 1 / 2 cos κ r ,
f ( r ) = r 1 / 2 α d d r [ r 2 n ( r ) ] .
K ˜ ( 1 2 + i ω ) = ( 2 / π ) 1 / 2 Γ ( α + i ω ) cos 1 2 π ( α + i ω ) ,
λ ω ± = ± 1 .
f ω ± = 0 r 2 n ( r ) d d r [ r 1 ψ ω ± ( r ) ] d r .
f ω ± = 3 2 n ω ± ω n ω ,
n ω ± = ± ( 3 2 g ± ω g ) [ λ ω ± ( ω 2 + 9 4 ) ] 1 .
n ω ± = ( 3 2 g ± ω g ) / ( ω 2 + 9 4 ) .
n ω ± ~ ± g ω ± / ω ,
y i + 1 = y i exp ( π / ω ) .
n = ω max π ln ( y max y min ) .
Q Mie ( x ) ~ 2 + 1.9923861 x 2 / 3 + O ( x 1 ) + ripple ,
τ ( k ) ~ 2 A + 1.9923861 π M 4 / 3 k 2 / 3 + O ( k 1 ) ,
K 1 0 K τ ( k ) d k ~ 2 A + 5.9771583 π M 4 / 3 K 2 / 3 + O ( K 1 ) ,

Metrics