## Determination of the aerosol size distribution by analytic inversion of the extinction spectrum in the complex anomalous diffraction approximation

Applied Optics, Vol. 39, Issue 24, pp. 4214-4231 (2000)

http://dx.doi.org/10.1364/AO.39.004214

Enhanced HTML Acrobat PDF (188 KB)

### Abstract

A new derivation is presented for the analytical inversion of aerosol spectral extinction data to size distributions. It is based on the complex analytic extension of the anomalous diffraction approximation (ADA). We derive inverse formulas that are applicable to homogeneous nonabsorbing and absorbing spherical particles. Our method simplifies, generalizes, and unifies a number of results obtained previously in the literature. In particular, we clarify the connection between the ADA transform and the Fourier and Laplace transforms. Also, the effect of the particle refractive-index dispersion on the inversion is examined. It is shown that, when Lorentz’s model is used for this dispersion, the continuous ADA inverse transform is mathematically well posed, whereas with a constant refractive index it is ill posed. Further, a condition is given, in terms of Lorentz parameters, for which the continuous inverse operator does not amplify the error.

© 2000 Optical Society of America

**OCIS Codes**

(010.1110) Atmospheric and oceanic optics : Aerosols

(010.1310) Atmospheric and oceanic optics : Atmospheric scattering

(290.2200) Scattering : Extinction

(290.3200) Scattering : Inverse scattering

**History**

Original Manuscript: October 5, 1999

Revised Manuscript: March 31, 2000

Published: August 20, 2000

**Citation**

Ghislain Franssens, Martine De Mazière, and Dominique Fonteyn, "Determination of the aerosol size distribution by analytic inversion of the extinction spectrum in the complex anomalous diffraction approximation," Appl. Opt. **39**, 4214-4231 (2000)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-39-24-4214

Sort: Year | Journal | Reset

### References

- M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), pp. 633–664.
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
- D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
- C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
- P. T. Walters, “Practical applications of inverting spectral turbidity data to provide aerosol size distributions,” Appl. Opt. 19, 2353–2365 (1980). [CrossRef] [PubMed]
- R. A. Dobbins, G. S. Jizmagian, “Optical scattering cross sections for polydispersions of dielectric spheres,” J. Opt. Soc. Am. 56, 1345–1350 (1966). [CrossRef]
- A. Tarantola, B. Valette, “Generalized nonlinear inverse problems solved using the least squares criterion,” Rev. Geophys. Space Phys. 20, 219–232 (1982). [CrossRef]
- D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962). [CrossRef]
- S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975). [CrossRef]
- K. S. Shifrin, A. Y. Perelman, “The determination of the spectrum of particles in a dispersed system from data on its transparency. I. The fundamental equation for the determination of the spectrum of the particles,” Opt. Spectrosc. (USSR) 15, 285–289 (1963).
- A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 1. Multispectral extinctions in the anomalous diffraction approximation,” Appl. Opt. 17, 1675–1676 (1978).
- M. A. Box, B. H. McKellar, “Analytic inversion of multispectral extinction data in the anomalous diffraction approximation,” Opt. Lett. 3, 91–93 (1978). [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, 3595–3598 (1979). [CrossRef] [PubMed]
- M. A. Box, B. H. McKellar, “Relationship between two analytic inversion formulae for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979). [CrossRef] [PubMed]
- M. A. Box, B. H. McKellar, “Further relations between analytic inversion formulas for multispectral extinction data,” Appl. Opt. 20, 3829–3831 (1981). [CrossRef] [PubMed]
- K. S. Shifrin, A. Y. Perelman, V. M. Volgin, “Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient,” Opt. Spectrosc. (USSR) 51, 534–538 (1981).
- J. Wang, F. R. Hallett, “Spherical particle size determination by analytical inversion of the UV–visible–NIR extinction spectrum,” Appl. Opt. 35, 193–197 (1996). [CrossRef] [PubMed]
- E. O. Brigham, The Fast Fourier Transform and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).
- K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993). [CrossRef]
- C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982). [CrossRef] [PubMed]
- J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984). [CrossRef] [PubMed]
- G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction theory,” Appl. Opt. 24, 4525–4533 (1985). [CrossRef]
- M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247–258 (1986). [CrossRef]
- P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
- R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1999).
- B. Davies, Integral Transforms and Their Applications (Springer-Verlag, Berlin, 1978). [CrossRef]
- M. A. Box, B. H. McKellar, “Determination of moments of the size distribution function in scattering by polydispersions,” Appl. Opt. 15, 2610 (1976). [CrossRef] [PubMed]
- A. L. Fymat, “Determination of moments of the size distribution function in scattering by polydispersions: a comment,” Appl. Opt. 17, 3516–3517 (1978). [CrossRef] [PubMed]
- K. Aki, P. Richards, Quantitative Seismology (Freeman, San Francisco, Calif., 1980), Vol. I, pp. 170–177.
- M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
- I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.