Abstract

A method of evaluating the aerosol size distribution from the spectral attenuation measurements is shown. The process consists of solving the simultaneous integral equations, and examples are given of solutions based on the attenuation measurements made by Knestrick et al. over the Chesapeake Bay. It is found that the evaluated individual size distributions do not necessarily follow the power law, although departures from it are mostly small. If the power law is to be adopted neglecting small departures, the evaluated results are in average expressed by r−3.57, where r is the radius of aerosols. In this study, the refractive index of aerosols is assumed to be 1.50, and some discussion is made of the effect of adopting a different refractive index value on estimation of the size distribution.

© 1969 Optical Society of America

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References

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  1. C. Junge, J. Meteorol. 12, 13 (1955).
    [CrossRef]
  2. J. A. Curcio, J. Opt. Soc. Amer. 51, 548 (1961).
    [CrossRef]
  3. R. G. Eldridge, J. Meteorol. 14, 55 (1957).
    [CrossRef]
  4. R. G. Eldridge, J. Atmos. Sci. 23, 605 (1966).
    [CrossRef]
  5. K. S. Shifrin, A. Ya. Perelman. Opt. Spectrosc. 15, 285 (1963).
  6. K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 362 (1963).
  7. K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 434 (1963).
  8. H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  9. D. L. Phillips, J. Ass. Comp. Mach. 9, 84 (1962).
    [CrossRef]
  10. S. Twomey, J. Ass. Comp. Mach. 10, 97 (1963).
    [CrossRef]
  11. S. Twomey, J. Franklin Inst. 279, 95 (1965).
    [CrossRef]
  12. D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
    [CrossRef]
  13. H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).
  14. D. Sinclair, in Proceedings of the U.S. Technical Conference on Air Pollution (McGraw-Hill Book Company, Inc., New York, 1952), Chap. 18.
  15. M. G. Gibbons, J. Opt. Soc. Amer. 48, 172 (1958).
    [CrossRef]
  16. F. E. Voltz, Ber. Deutsch. Wetterd. 2, No. 13, 47 (1954).
  17. R. Eiden, Appl. Opt. 5, 569 (1966).
    [CrossRef] [PubMed]
  18. R. Penndorf, Geophys. Res. Papers No. 45 [AFCRC–TR–56–204 (6)].
  19. G. L. Knestrick, T. H. Cosden, J. A. Curcio, J. Opt. Soc. Amer. 52, 1010 (1962).
    [CrossRef]
  20. A. Ångström, Geogr. Ann. 11, 156 (1929).
    [CrossRef]

1966 (3)

R. G. Eldridge, J. Atmos. Sci. 23, 605 (1966).
[CrossRef]

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

R. Eiden, Appl. Opt. 5, 569 (1966).
[CrossRef] [PubMed]

1965 (1)

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

1963 (4)

K. S. Shifrin, A. Ya. Perelman. Opt. Spectrosc. 15, 285 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 362 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 434 (1963).

S. Twomey, J. Ass. Comp. Mach. 10, 97 (1963).
[CrossRef]

1962 (2)

D. L. Phillips, J. Ass. Comp. Mach. 9, 84 (1962).
[CrossRef]

G. L. Knestrick, T. H. Cosden, J. A. Curcio, J. Opt. Soc. Amer. 52, 1010 (1962).
[CrossRef]

1961 (1)

J. A. Curcio, J. Opt. Soc. Amer. 51, 548 (1961).
[CrossRef]

1958 (1)

M. G. Gibbons, J. Opt. Soc. Amer. 48, 172 (1958).
[CrossRef]

1957 (1)

R. G. Eldridge, J. Meteorol. 14, 55 (1957).
[CrossRef]

1955 (1)

C. Junge, J. Meteorol. 12, 13 (1955).
[CrossRef]

1954 (1)

F. E. Voltz, Ber. Deutsch. Wetterd. 2, No. 13, 47 (1954).

1951 (1)

H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).

1929 (1)

A. Ångström, Geogr. Ann. 11, 156 (1929).
[CrossRef]

Ångström, A.

A. Ångström, Geogr. Ann. 11, 156 (1929).
[CrossRef]

Cosden, T. H.

G. L. Knestrick, T. H. Cosden, J. A. Curcio, J. Opt. Soc. Amer. 52, 1010 (1962).
[CrossRef]

Curcio, J. A.

G. L. Knestrick, T. H. Cosden, J. A. Curcio, J. Opt. Soc. Amer. 52, 1010 (1962).
[CrossRef]

J. A. Curcio, J. Opt. Soc. Amer. 51, 548 (1961).
[CrossRef]

Eiden, R.

Eldridge, R. G.

R. G. Eldridge, J. Atmos. Sci. 23, 605 (1966).
[CrossRef]

R. G. Eldridge, J. Meteorol. 14, 55 (1957).
[CrossRef]

Fleming, H. E.

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

Gebbie, H. A.

H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).

Gibbons, M. G.

M. G. Gibbons, J. Opt. Soc. Amer. 48, 172 (1958).
[CrossRef]

Harding, W. R.

H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).

Hilsum, C.

H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).

Junge, C.

C. Junge, J. Meteorol. 12, 13 (1955).
[CrossRef]

Knestrick, G. L.

G. L. Knestrick, T. H. Cosden, J. A. Curcio, J. Opt. Soc. Amer. 52, 1010 (1962).
[CrossRef]

Penndorf, R.

R. Penndorf, Geophys. Res. Papers No. 45 [AFCRC–TR–56–204 (6)].

Perelman, A. Ya.

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 434 (1963).

K. S. Shifrin, A. Ya. Perelman. Opt. Spectrosc. 15, 285 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 362 (1963).

Phillips, D. L.

D. L. Phillips, J. Ass. Comp. Mach. 9, 84 (1962).
[CrossRef]

Pryce, A. W.

H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).

Roberts, V.

H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).

Shifrin, K. S.

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 434 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 362 (1963).

K. S. Shifrin, A. Ya. Perelman. Opt. Spectrosc. 15, 285 (1963).

Sinclair, D.

D. Sinclair, in Proceedings of the U.S. Technical Conference on Air Pollution (McGraw-Hill Book Company, Inc., New York, 1952), Chap. 18.

Twomey, S.

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

S. Twomey, J. Ass. Comp. Mach. 10, 97 (1963).
[CrossRef]

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

Voltz, F. E.

F. E. Voltz, Ber. Deutsch. Wetterd. 2, No. 13, 47 (1954).

Wark, D. Q.

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

Appl. Opt. (1)

Ber. Deutsch. Wetterd. (1)

F. E. Voltz, Ber. Deutsch. Wetterd. 2, No. 13, 47 (1954).

Geogr. Ann. (1)

A. Ångström, Geogr. Ann. 11, 156 (1929).
[CrossRef]

J. Ass. Comp. Mach. (2)

D. L. Phillips, J. Ass. Comp. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, J. Ass. Comp. Mach. 10, 97 (1963).
[CrossRef]

J. Atmos. Sci. (1)

R. G. Eldridge, J. Atmos. Sci. 23, 605 (1966).
[CrossRef]

J. Franklin Inst. (1)

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

J. Meteorol. (2)

C. Junge, J. Meteorol. 12, 13 (1955).
[CrossRef]

R. G. Eldridge, J. Meteorol. 14, 55 (1957).
[CrossRef]

J. Opt. Soc. Amer. (3)

M. G. Gibbons, J. Opt. Soc. Amer. 48, 172 (1958).
[CrossRef]

J. A. Curcio, J. Opt. Soc. Amer. 51, 548 (1961).
[CrossRef]

G. L. Knestrick, T. H. Cosden, J. A. Curcio, J. Opt. Soc. Amer. 52, 1010 (1962).
[CrossRef]

Mon. Weather Rev. (1)

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

Opt. Spectrosc. (3)

K. S. Shifrin, A. Ya. Perelman. Opt. Spectrosc. 15, 285 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 362 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 434 (1963).

Proc. Roy. Soc. London (1)

H. A. Gebbie, W. R. Harding, C. Hilsum, A. W. Pryce, V. Roberts, Proc. Roy. Soc. London A206, 87 (1951).

Other (3)

D. Sinclair, in Proceedings of the U.S. Technical Conference on Air Pollution (McGraw-Hill Book Company, Inc., New York, 1952), Chap. 18.

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

R. Penndorf, Geophys. Res. Papers No. 45 [AFCRC–TR–56–204 (6)].

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Figures (8)

Fig. 1
Fig. 1

Weighting function K(λ,x), vs x = log10r for various wavelengths.

Fig. 2
Fig. 2

(a) Comparison of the assumed model size distribution and estimated one. Solid line shows the model size distribution and white or black circle, estimated one. (b) Same as Fig. 2(a), but for different models.

Fig. 3
Fig. 3

Observed extinction coefficient and estimated size distribution. The curve marked by L (or R) should be referred to the ordinate scaled on the left (or right) side of the figure. Curves 1, 2, and 3 are based on observations on 13 January, 13:45, 11:10, and 21 April, respectively.

Fig. 4
Fig. 4

Same as Fig. 3, except that curves 1, 2, and 3 are based on observations on 13 January, 14:40, 29 April, 13:00 and 11:00, respectively.

Fig. 5
Fig. 5

Same as Fig. 3, except that curves 1, 2, 3, and 4 are based on observations on 29 April, 29 September, 08:25 29 September, 08:10 and 08:50, 5 January, respectively.

Fig. 6
Fig. 6

Same as Fig. 3, except that curves 1, 2, 3, and 4 are based on observations on 27 April, 28 April, 24 April, and 27 October, respectively.

Fig. 7
Fig. 7

Same as Fig. 3, except that curves 1, 2, 3, and 4 are based on observations on 13 January, 11:35, 30 October, 18 September, and 28 October, respectively.

Fig. 8
Fig. 8

Volume concentration vs mean extinction coefficient. White circles are due to observations and estimations, while the straight lines represent the relation for aerosols obeying the power law, r−3.57 and r−4, respectively.

Tables (1)

Tables Icon

Table I Conversion Factors A and A′ for Various Values of m* and ν

Equations (31)

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k ( λ ) = 0 π r 2 Q [ 2 π r λ , m * ( λ ) ] n ( r ) d r , ( λ = λ 1 , λ 2 , , λ N ) ,
k ( λ ) + ( λ ) = x s x l K ( λ , x ) f ( x ) d x ,
x = log 10 r ,
K ( λ , x ) = π ln 10 × 10 - x Q [ 2 π 10 x λ , m * ( λ ) ] ,
f ( x ) = 10 4 x n ( 10 x ) .
λ 2 ( λ ) = const .
x s x l ( d 2 f s d x 2 ) 2 d x = min f F x s x l ( d 2 f d x 2 ) 2 d x ,
x s x l ( f t - f s ) 2 d x = min f F x s x l ( f t - f ) 2 d x ,
f = ( A T A + γ H ) - 1 A T k ,
f = ( A T A + γ I ) - 1 ( A T k + γ f t ) ,
A i j = x j x j + 1 K ( λ i , x ) d x , ( i , j = 1 , 2 , , N ) ,
H = [ 1 - 2 1 0 0 - 2 5 - 4 1 0 0 1 - 4 6 - 4 1 0 0 1 - 4 6 - 4 1 0 · · · 0 1 - 4 5 - 2 0 1 - 2 1 ] .
A i j = x j x j + 1 K ( λ i , x ) x d x + x j x 1 x j K ( λ i , x ) d x - x j + 1 × x 1 x j + 1 K ( λ i , x ) d x ,     ( i = 1 , 2 , , N j = 1 , 2 , , N - 2 )
A i , N - 1 = x N - 1 x N K ( λ i , x ) x d x + x N - 1 x 1 x N - 1 K ( λ i , x ) d x ,             ( i = 1 , 2 , , N )
A i , N = x 1 x N K ( λ i , x ) d x i ,     ( i = 1 , 2 , , N ) .
V = 4 3 π r 1 r 2 r 3 n ( r ) d r ,
0 k ( λ ) d λ = 3 2 π V 0 Q ( α ) α - 2 d α ,
V = 4 π 3 0 r 3 n ( r ) d r .
1 λ 2 - λ 1 λ 1 λ 2 k ( λ ) d λ ,
n ( r ) = c r - ν .
1 λ 2 - λ 1 λ 1 λ 2 k ( λ ) d λ / V = 3 ( 2 ) ν - 5 π ν - 3 λ 2 - λ 1 0 Q ( α , m * ) α 2 - ν d α λ 1 λ 2 λ 3 - ν d λ r 1 r 2 r 3 - ν d r .
k ( λ ) = 2 ν - 3 π ν - 2 λ 3 - ν C ( m * ) 0 Q ( α , m * ) α 2 - ν d α ,
C ( m * ) = C ( 1.50 ) 0 Q ( α , 1.50 ) α 2 - ν d α 0 Q ( α , m * ) α 2 - ν d α .
t = r ( m * - 1 ) ,
Q ( 2 π t λ ( m * - 1 ) , m * ) = Q * ( 2 π t λ ) ,
k ( λ ) = 0 π t 2 Q * ( 2 π t λ ) g ( t ) d t ,
g ( t ) = 1 ( m * - 1 ) 3 n ( t m * - 1 ) .
n ( 0.5 m * - 1 R ) = ( m * - 1 0.5 ) n ( R ) ,
k ( λ ) = 2 ν - 3 π ν - 2 ( m * - 1 ) ν - 3 C ( m * ) λ 3 - ν × 0 Q * ( σ ) σ 2 - ν d σ ,
σ = ( 2 π t / λ ) ( = ρ / 2 ) .
C ( m * ) = ( 0.5 m * - 1 ) ν - 3 C ( 1.50 )

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