Abstract

Indirect measurement of parameters related to the drop size distribution in fog and cloud is discussed. Analysis of the kernel functions in the integral equation relating physical measurements (of transmission, forward scattering, and back scattering, respectively) to the drop size distribution reveals the number of useful independent inferences which can be made, with a specified error in measurement and from a given number of observations.

© 1967 Optical Society of America

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References

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  1. T. L. Gilbert, A Study of the Particle Size Distribution of Aerosols from Light Scattering Measurements, Sci. Rept. No. 1, Air Force Contract No. AF19(604)-1428, Armour Research Foundation, Chicago, Ill. (13Nov.1956).
  2. K. S. Shifrin, A. Y. Perelman, Geofis. Pura Appl. 58, 208 (1964).
    [CrossRef]
  3. K. S. Shifrin, V. I. Golikov, Tr. Glazr. Geofiz. Obs. 152, 3 (1964).
  4. R. G. Eldridge, J. Meteorol. 14, 55 (1957).
    [CrossRef]
  5. J. R. Hodkinson, in Aerosol Science, C. N. Davies, Ed. (Academic Press Inc., New York, 1966), pp. 278–355.
  6. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [CrossRef]

1964

K. S. Shifrin, A. Y. Perelman, Geofis. Pura Appl. 58, 208 (1964).
[CrossRef]

K. S. Shifrin, V. I. Golikov, Tr. Glazr. Geofiz. Obs. 152, 3 (1964).

D. Deirmendjian, Appl. Opt. 3, 187 (1964).
[CrossRef]

1957

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

Deirmendjian, D.

Eldridge, R. G.

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

Gilbert, T. L.

T. L. Gilbert, A Study of the Particle Size Distribution of Aerosols from Light Scattering Measurements, Sci. Rept. No. 1, Air Force Contract No. AF19(604)-1428, Armour Research Foundation, Chicago, Ill. (13Nov.1956).

Golikov, V. I.

K. S. Shifrin, V. I. Golikov, Tr. Glazr. Geofiz. Obs. 152, 3 (1964).

Hodkinson, J. R.

J. R. Hodkinson, in Aerosol Science, C. N. Davies, Ed. (Academic Press Inc., New York, 1966), pp. 278–355.

Perelman, A. Y.

K. S. Shifrin, A. Y. Perelman, Geofis. Pura Appl. 58, 208 (1964).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, A. Y. Perelman, Geofis. Pura Appl. 58, 208 (1964).
[CrossRef]

K. S. Shifrin, V. I. Golikov, Tr. Glazr. Geofiz. Obs. 152, 3 (1964).

Appl. Opt.

Geofis. Pura Appl.

K. S. Shifrin, A. Y. Perelman, Geofis. Pura Appl. 58, 208 (1964).
[CrossRef]

J. Meteorol.

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

Tr. Glazr. Geofiz. Obs.

K. S. Shifrin, V. I. Golikov, Tr. Glazr. Geofiz. Obs. 152, 3 (1964).

Other

J. R. Hodkinson, in Aerosol Science, C. N. Davies, Ed. (Academic Press Inc., New York, 1966), pp. 278–355.

T. L. Gilbert, A Study of the Particle Size Distribution of Aerosols from Light Scattering Measurements, Sci. Rept. No. 1, Air Force Contract No. AF19(604)-1428, Armour Research Foundation, Chicago, Ill. (13Nov.1956).

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Figures (10)

Fig. 1
Fig. 1

Schematic arrangement for measuring scattered light at several angles. (For back scatter measurements the corner reflector would be omitted.)

Fig. 2
Fig. 2

The normalized eigenvalues of the covariance matrix of transmission kernels, for indicated values of N (number of measurements).

Fig. 3
Fig. 3

Orthogonal (normalized) functions constructed from the transmission kernels.

Fig. 4
Fig. 4

The number of useful independent inferences vs the number of measurements (for indicated values of error in measurement) of transmission.

Fig. 5
Fig. 5

The normalized eigenvalues of the covariance matrix of forward scattering kernels, for indicated values of N (number of measurements).

Fig. 6
Fig. 6

Orthogonal (normalized) functions constructed from the forward scattering kernels.

Fig. 7
Fig. 7

The number of useful independent inferences vs the number of measurements (for indicated values of error in measurement) of forward scattering.

Fig. 8
Fig. 8

The normalized eigenvalues of the covariance matrix of back scattering kernels, for indicated values of N (number of measurements).

Fig. 9
Fig. 9

Orthogonal (normalized) functions constructed from the back scattering kernels.

Fig. 10
Fig. 10

The number of useful independent inferences vs the number of measurements (for indicated values of error in measurement) of back scattering.

Tables (1)

Tables Icon

Table I Relative Error in Approximating the Weighting Functions Which Give Number, Mean Radius, Etc. (Error Given in Per Cent)

Equations (21)

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I ( l ) / I 0 = exp [ - π r 2 κ ( 2 π r / λ ) n ( r ) d r ] ] .
I ( l ) ¯ / I 0 = exp [ - π r 2 κ ( 2 π r / λ ) n ( r ) d r ] av ,
g ( y ) = K ( x , y ) f ( x ) d x + ( y ) ,
g m = k m ( x ) f ( x ) d x + m .
k m ( r ) = [ i 1 ( θ m , 2 π r / λ m ) + i 2 ( θ m , 2 π r / λ m ) ] λ 2 m / 8 π 2 .
Q = a b [ ξ 1 k 1 ( r ) + ξ 2 k 2 ( r ) + ξ N k N ( r ) ] 2 d r .
C = ( k 1 2 ( r ) d r k 1 ( r ) k 2 ( r ) d r k 1 ( r ) k 2 ( r ) d r k 2 2 ( r ) d r ) .
( ϕ 1 ( r ) ϕ 2 ( r ) ϕ N ( r ) ) = U * ( k 1 ( r ) k 2 ( r ) k N ( r ) ) ,
ϕ ( r ) = U * k ( r )
ϕ ( r ) ϕ * ( r ) = U * k ( r ) k * ( r ) U ,
ϕ ( r ) ϕ * ( r ) d r = U * C U
f ( r ) = y 1 ϕ 1 ( r ) + y 2 ϕ 2 ( r ) + y N ϕ N ( r ) + ρ ( r ) .
g i = k i ( r ) m y m ϕ m ( r ) d r ,
g = [ k ( r ) ϕ * ( r ) d r ] y .
g = [ k ( r ) k * ( r ) d r ] U y = C U y ,
g = U Λ y
y = Λ - 1 U * g .
y i = λ i - 1 j u j i g j             ( i = 1 , 2 , N ) ,
y = Λ - 1 U * ( g + ɛ ) = Λ - 1 U * ( U Λ x + ɛ ) = y + Λ - 1 U * ɛ .
w ( r ) a 1 k 1 ( r ) + a 2 k 2 ( r ) + a N k N ( r ) ,
w ( r ) f ( r ) d r a 1 k 1 ( r ) f ( r ) d r + + a N k N ( r ) f ( r ) d r .

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