Abstract

An analytical solution to the problem of inverting multispectral extinction measurements including absorption effects is constructed, via Laplace transform theory, under the assumption that the particulate extinction efficiency is given by the anomalous diffraction model of van de Hulst. A solution representation in terms of a finite series of generalized Laquerre polynomials is carried out to provide a framework for coping with the problem of limited extinction information. The solution is stabilized by means of the technique of constrained linear inversion.

© 1984 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. K. S. Shifrin, A. Y. Perelman, “Determination of the Particle Spectrum of a Dispersed System from Data on its Transparency,” Opt. Spectrosc. 15, 285 (1963).
  3. K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” PAGEOPH 58, 208 (1964).
    [CrossRef]
  4. M. A. Box, B. H. J. McKellar, “Analytic Inversion of Multispectral Extinction Data in the Anomalous Diffraction Approximation,” Opt. Lett. 3, 91, (1978).
    [CrossRef] [PubMed]
  5. 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).
  6. V. A. Punina, A. Y. Perelman, “Uber die Berechnung der Grössenverteilung von den Absorbierenden Kugelförmigen,” PAGEOPH 74, 92 (1969).
    [CrossRef]
  7. D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
    [CrossRef]
  8. S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
    [CrossRef]
  9. P. T. Walters, “Practical Applications of Inverting Spectral Turbidity Data to Provide Aerosol Size Distributions,” Appl. Opt. 19, 2353 (1980.
    [PubMed]
  10. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).
  11. A. L. Fymat, C. B. Smith, “Analytical Inversions in Remote Sensing of Particle Size Distributions. 4: Comparison of Fymat and Box-McKeller Solutions in the Anomalous Diffraction Approximation,” Appl. Opt. 18, 3595 (1979).
    [CrossRef] [PubMed]
  12. R. Piessens, M. Branders, “Numerical Inversion of the Laplace Transform Using Generalized Laguerre Polynomials,” Proc. Inst. Electr. Eng. 118, 1517 (1971).
    [CrossRef]
  13. Y. L. Luke, The Special Functions and Their Approximations, Vol. 2 (Academic, New York, 1969).
  14. S. Twomey, Introduction to the Mathematics of Inversion in Remove Sensing and Indirect Measurements (Elsevier, New York, 1977).
  15. G. Yamamoto, M. Tanaka, “Determination of Aerosol Size Distributions from Spectral Attenuation Measurements,” Appl. Opt. 8, 447 (1969
    [CrossRef] [PubMed]

1979 (1)

1978 (2)

1971 (1)

R. Piessens, M. Branders, “Numerical Inversion of the Laplace Transform Using Generalized Laguerre Polynomials,” Proc. Inst. Electr. Eng. 118, 1517 (1971).
[CrossRef]

1969 (2)

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

V. A. Punina, A. Y. Perelman, “Uber die Berechnung der Grössenverteilung von den Absorbierenden Kugelförmigen,” PAGEOPH 74, 92 (1969).
[CrossRef]

1964 (1)

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” PAGEOPH 58, 208 (1964).
[CrossRef]

1963 (2)

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

S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

1962 (1)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Box, M. A.

Branders, M.

R. Piessens, M. Branders, “Numerical Inversion of the Laplace Transform Using Generalized Laguerre Polynomials,” Proc. Inst. Electr. Eng. 118, 1517 (1971).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

Fymat, A. L.

Luke, Y. L.

Y. L. Luke, The Special Functions and Their Approximations, Vol. 2 (Academic, New York, 1969).

McKellar, B. H. J.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

Perelman, A. Y.

V. A. Punina, A. Y. Perelman, “Uber die Berechnung der Grössenverteilung von den Absorbierenden Kugelförmigen,” PAGEOPH 74, 92 (1969).
[CrossRef]

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” PAGEOPH 58, 208 (1964).
[CrossRef]

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

Phillips, D. L.

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Piessens, R.

R. Piessens, M. Branders, “Numerical Inversion of the Laplace Transform Using Generalized Laguerre Polynomials,” Proc. Inst. Electr. Eng. 118, 1517 (1971).
[CrossRef]

Punina, V. A.

V. A. Punina, A. Y. Perelman, “Uber die Berechnung der Grössenverteilung von den Absorbierenden Kugelförmigen,” PAGEOPH 74, 92 (1969).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” PAGEOPH 58, 208 (1964).
[CrossRef]

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

Smith, C. B.

Tanaka, M.

Twomey, S.

S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Walters, P. T.

P. T. Walters, “Practical Applications of Inverting Spectral Turbidity Data to Provide Aerosol Size Distributions,” Appl. Opt. 19, 2353 (1980.
[PubMed]

Yamamoto, G.

Appl. Opt. (4)

J. Assoc. Comput. Mach. (2)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

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

PAGEOPH (2)

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” PAGEOPH 58, 208 (1964).
[CrossRef]

V. A. Punina, A. Y. Perelman, “Uber die Berechnung der Grössenverteilung von den Absorbierenden Kugelförmigen,” PAGEOPH 74, 92 (1969).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

R. Piessens, M. Branders, “Numerical Inversion of the Laplace Transform Using Generalized Laguerre Polynomials,” Proc. Inst. Electr. Eng. 118, 1517 (1971).
[CrossRef]

Other (4)

Y. L. Luke, The Special Functions and Their Approximations, Vol. 2 (Academic, New York, 1969).

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

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (10)

Fig. 1
Fig. 1

Dependence of extinction efficiency amplitude on index of refraction.

Fig. 2
Fig. 2

Comparison of extinction efficiencies for m ˜ = (1.2,0).

Fig. 3
Fig. 3

Comparison of extinction efficiencies for m ˜ = (1.18,.08) (H2O at 10.6 μm).

Fig. 4
Fig. 4

Comparison of extinction coefficients for gamma distribution as a function of wave number.

Fig. 5
Fig. 5

Inversion of anomalous diffraction extinction data for three different indices of refraction. Inversions show solution similarity. Ten extinction values and ten terms in solution expansion were used.

Fig. 6
Fig. 6

Inversion of Mie extinction data for gamma distribution input form. Ten extinction values and ten terms in solution expansion were used.

Fig. 7
Fig. 7

Anomalous diffraction and Mie extinction coefficients as a function of wave number for quasi-bimodal input distribution.

Fig. 8
Fig. 8

Inversion of anomalous diffraction extinction data. Input distribution is plotted as a continuous line. Ten extinction values and ten terms in solution expansion were used.

Fig. 9
Fig. 9

Inversion of Mie extinction data. Input distribution is plotted as a continuous line. Ten extinction values and ten terms in solution expansion were used.

Fig. 10
Fig. 10

Plots of f(r) using the series coefficients given in Table I. The graph for b/c = 1 is the complete reconstructed profile for the gamma distribution with M0 = 1, 〈r〉 = 20, and L = 4 (b = 0.25). The plots for b/c ≠ 1 are truncated approximations to f(r) for the two cases: (1) c = 0.5b, and (2) c = 2b. The curves for b/c = 0.5 and 1 coincide very closely except near r = 40, where the former curve begins to fall slightly below that for b = c (in the figure this shows up as a slight broadening of the continuous curve near r = 40). To extend the truncated approximations to greater ranges with accuracy requires the inclusion of more terms in the expansions.

Tables (1)

Tables Icon

Table I Coefficients Aj and Aj/j!, Calculated from Eqs.(B3) and (B4) for j = 0,1, …, 20, and for b/c = 0.5,1,2

Equations (66)

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τ ( k ) = π 0 r 2 Q ( m ˜ , k , r ) f ( r ) d r .
Q = 2 + 4 Re { K ( w ˜ ) } ,
K ( w ˜ ) = exp ( - w ˜ ) w ˜ + exp ( - w ˜ ) - 1 w ˜ 2 , w ˜ = ρ ( i + tan β ) = k ˜ r , }
ρ = 2 ( n - 1 ) k r ,
tan β = n / ( n - 1 ) .
Q ( ρ , β ) = 2 - 4 exp ( - ρ tan β ) ( cos β ρ ) [ sin ( ρ - β ) + ( cos β ρ ) cos ( ρ - 2 β ) ] + 4 ( cos β ρ ) cos 2 β .
Q ( ρ ) = 2 - 4 sin ρ ρ + 4 ( 1 - cos ρ ) ρ 2 .
T ( k ˜ ) 4 π 0 r 2 f ( r ) K ( k ˜ r ) d r .
τ ( k ) = Re { T ( k ˜ ) } - T ( 0 ) .
f ( r ) = 1 4 π d [ F ( r ) / r ] d r ,
T ( k ˜ ) = 0 F ( r ) exp ( - k ˜ r ) d r ,
F ( r ) = 1 2 π i C T ( k ˜ ) exp ( k ˜ r ) d k ˜ ,
k ^ r = i ρ = i k r ,
k = 2 ( n - 1 ) k .
Re { T ( k ˜ ) } = τ ( k ) - 2 π M 2 = 0 F ( r ) cos ( k r ) d r .
F ( r ) = 2 π 0 [ τ ( k ) - 2 π M 2 ] cos ( k r ) d k .
f ( r ) = - 1 2 π 2 r 2 0 [ τ ( k ) - 2 π M 2 ] ( k r sin k r + cos k r ) d k ,
δ F ( r ) = 2 π 0 δ τ ( k ) cos ( k r ) d k .
δ F ( r ) = 2 π r [ sin ( k 2 r ) - sin ( k 1 r ) ] ,
δ f ( r ) = - 2 π r [ 2 sin ( k r ) - ( k r ) cos ( k r ) ] | k 2 r k 1 r ,
F ( r ) = r exp ( - b r ) n = 0 n ! a n ( n + 1 ) ! L n 1 ( b r ) ,
L n 1 ( x ) = m = 0 n ( - 1 ) m ( n + 1 ) ! x m ( n - m ) ! ( M + 1 ) ! m ! .
T ( k ˜ ) = n = 0 k ˜ n a n ( k ˜ + b ) n + 2 .
c n = ( - 1 ) n + 1 4 π ( n + 1 ) ! m = n a m m ! ( m - n ) ! .
T ( k ˜ ) = - 4 π b 2 n = 0 ( n + 1 ) c n ( k ˜ b - 1 + 1 ) n + 2 .
F ( r ) = - 4 π r exp ( - b r ) n = 0 c n ( b r ) n n ! .
f ( r ) = b exp ( - b r ) n = 0 ( c n - c n + 1 ) ( b r ) n n ! .
F ( r ) = 4 π r | 0 r f ( r ) d r - M 0 | ,
lim r - > 0 F / r = - 4 π M 0 .
f ( r ) = b exp ( - b r ) n = 1 N ( c n - c n + 1 ) ( b r ) n n ! ,
f ( r ) = M 0 J ( x ) r = M 0 ( L + 1 ) ( L + 1 ) r L ! x L exp [ - ( L + 1 ) x ] ,
j = 1 J A i j c j = τ ( k i ) ,
Re { T ( k ˜ i ) } - T ( 0 ) = - 4 π M 0 b 2 j = 0 J ( j + 1 ) c j × [ Re { 1 / ( k ˜ i b - 1 + 1 ) j + 2 } - 1 ] .
cot ϕ = 1 ρ + tan β ,
ρ = 2 ( n - 1 ) k / b
A i j = cos 2 ( β + ϕ i ) cos 2 ϕ i cos 2 β + 2 cos 3 ( β + ϕ i ) cos 3 ϕ i cos 3 β - 3 ,
A i j = ( j + 1 ) { cos j + 2 ( β + ϕ i ) cos [ ( j + 2 ) ϕ i ] cos j + 2 β - 1 } ,
A c = τ .
c = ( A T A ) - 1 A T τ .
c = ( A T A + s I ) - 1 ( A T τ + s c 0 ) ,
τ Γ ( k , r , L ) = 4 π M 0 r 2 { ( L + 2 ) 2 ( L + 1 ) + ( cos β ρ ) 2 cos 2 β + ( cos β ρ ) [ cos ( β + ϕ ) cos β ] L + 2 sin [ β - ( L + 2 ) ϕ ] - ( cos β ρ ) 2 [ cos ( β + ϕ ) cos β ] L + 1 cos [ 2 β - ( L + 1 ) ϕ ] } ,
τ i = τ i / τ i max .
τ r ( k i , r , L ) = τ r ( k i , r , L ) / τ r ( k i max r , L ) .
= τ - τ r I τ
H m = 1.1 + ( n - 1.2 ) / 3
τ ( k ) = π H m 0 r 2 Q a . d . ( ρ ) f ( r ) d r ,
= π H m [ 2 ( n - 1 ) k ] 3 0 ρ 2 Q a . d . ( ρ ) f [ ρ 2 ( n - 1 ) k ] d ρ ,
H 1 ( n 1 - 1 ) 3 f 1 [ ρ 2 ( n 1 - 1 ) k ] = H 2 ( n 2 - 1 ) 3 f 2 [ ρ 2 ( n 2 - 1 ) k ] ,
f 2 ( r ) = H 1 H 2 ( n 2 - 1 n 1 - 1 ) 3 f 1 [ ( n 2 - 1 n 1 - 1 ) r ] .
M n 0 r n f ( r ) d r .
M n , 2 = H 1 H 2 ( n 1 - 1 n 2 - 1 ) n - 2 M n , 1.
r 2 = ( n 1 - 1 n 2 - 1 ) r 1 .
T ( k ˜ ) = 4 π 0 r 2 f ( r ) K ( k ˜ r ) d r .
F ( r ) = 1 2 π i C T ( k ˜ ) exp ( k ˜ r ) d k ˜
F ( r ) = 4 π 0 r 2 f ( r ) { 1 2 π i C K ( k ˜ r ) exp ( k ˜ r ) d k ˜ } d r .
1 2 π i C K ( k ˜ r ) exp ( k ˜ r ) d k ˜ = { - r / r 2 , 0 r r , 0 , r > r .
F ( r ) = - 4 π M 0 r exp ( - b r ) m = 0 L ( b r ) m m ! ,
F ( r ) = - 4 π r exp ( - c r ) j = 0 A j ( c r ) j j ! .
A = M 0 j ! { i = 0 j H ( i , j - i ) , i = 0 L H ( j - L + i , L - i ) ,             j L , j > L ,
H ( i , j ) = ( 1 - b / c ) i ( b / c ) j i ! j ! .
T ( k ˜ ) = - 4 π M 0 y ˜ 2 b 2 m = 0 L ( m + 1 ) y ˜ m ,
y ˜ = 1 ( k ˜ b - 1 + 1 ) = cos ( ϕ + β ) cos β exp ( - i ϕ ) ,
m = 0 L ( m + 1 ) y ˜ m = d d y ˜ m = 0 L y ˜ m + 1 = d d y ˜ [ y ˜ ( 1 - y ˜ L + 1 ) ( 1 - y ˜ ) ] = ( 1 - y ˜ L + 1 ) ( 1 - y ˜ ) 2 - ( L + 1 ) y ˜ L + 1 ( 1 - y ˜ ) .
T ( 0 ) = - 4 π M 0 b 2 m = 0 L ( m + 1 ) = - 4 π M 0 r 2 { ( L + 2 ) 2 ( L + 1 ) } ,
τ Γ ( k ) = 4 π M 0 r 2 Re { ( L + 2 ) 2 ( L + 1 ) - x ˜ - 2 + x ˜ - 1 y ˜ L + 2 + x ˜ - 2 y ˜ L + 1 } ,
x ˜ = k ˜ r = ρ cos β exp { i ( π / 2 - β ) } ,

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