Abstract

The Phillips-Twomey and Backus-Gilbert inversion techniques are applied to determine the size distribution of water droplets in clouds from light scattering data at backward angles. The data are generated numerically from the Mie scattering functions and an assumed cloud model. The size distribution is recovered from these data using the two inversion techniques and is compared with the assumed model. It is found that the Phillips-Twomey technique gives better agreement between the assumed and recovered size distributions than the Backus-Gilbert technique. Also, it is more stable to random errors artificially introduced into the scattering data.

© 1976 Optical Society of America

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References

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  1. R. Frith, Q. J. R. Meteorol. Soc. 77, 441 (1951).
    [CrossRef]
  2. W. G. Durbin, Tellus 11, 202 (1959).
    [CrossRef]
  3. D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
    [CrossRef]
  4. S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
    [CrossRef]
  5. G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. Lond. A266, 123 (1970).
  6. B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
    [CrossRef]
  7. E. R. Westwater, A. Cohen, Appl. Opt. 12, 1340 (1973).
    [CrossRef] [PubMed]
  8. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [CrossRef]
  9. S. Twomey, H. B. Howell, Appl. Opt. 6, 2125 (1967).
    [CrossRef] [PubMed]
  10. J. V. Dave, Rep. 320-3237, IBM Scientific Center (1968).
  11. J. V. Dave, Appl. Opt. 8, 1161 (1969).
    [CrossRef] [PubMed]

1973 (1)

1971 (1)

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

1970 (1)

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. Lond. A266, 123 (1970).

1969 (1)

1967 (1)

1964 (1)

1963 (1)

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

1962 (1)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

1959 (1)

W. G. Durbin, Tellus 11, 202 (1959).
[CrossRef]

1951 (1)

R. Frith, Q. J. R. Meteorol. Soc. 77, 441 (1951).
[CrossRef]

Backus, G. E.

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. Lond. A266, 123 (1970).

Browning, S. R.

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

Cohen, A.

Dave, J. V.

J. V. Dave, Appl. Opt. 8, 1161 (1969).
[CrossRef] [PubMed]

J. V. Dave, Rep. 320-3237, IBM Scientific Center (1968).

Deirmendjian, D.

Durbin, W. G.

W. G. Durbin, Tellus 11, 202 (1959).
[CrossRef]

Frith, R.

R. Frith, Q. J. R. Meteorol. Soc. 77, 441 (1951).
[CrossRef]

Gilbert, J. F.

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. Lond. A266, 123 (1970).

Herman, B. M.

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

Howell, H. B.

Phillips, D. L.

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Reagan, J. A.

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

Twomey, S.

Westwater, E. R.

Appl. Opt. (4)

J. Assoc. Comput. Mach. (2)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

J. Atmos. Sci. (1)

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

Philos. Trans. R. Soc. Lond. (1)

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. Lond. A266, 123 (1970).

Q. J. R. Meteorol. Soc. (1)

R. Frith, Q. J. R. Meteorol. Soc. 77, 441 (1951).
[CrossRef]

Tellus (1)

W. G. Durbin, Tellus 11, 202 (1959).
[CrossRef]

Other (1)

J. V. Dave, Rep. 320-3237, IBM Scientific Center (1968).

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Figures (8)

Fig. 1
Fig. 1

Scattering data g(θ).

Fig. 2
Fig. 2

Recovered f(α) from the Phillips-Twomey inversion scheme for g(θ) with no random errors introduced.

Fig. 3
Fig. 3

Recovered f(α) from the Phillips-Twomey inversion scheme for g(θ) with 1% random errors introduced.

Fig. 4
Fig. 4

Recovered f(α) from the Phillips-Twomey inversion scheme for g(θ) with 5% random errors introduced.

Fig. 5
Fig. 5

Recovered f(α) from the Phillips-Twomey inversion scheme with trial value γ = 5 × 103 for g(θ) with 1% random errors introduced.

Fig. 6
Fig. 6

Recovered f(α) from the Phillips-Twomey inversion scheme with trial value γ = 5 × 104 for g(θ) with 1% random errors introduced.

Fig. 7
Fig. 7

Recovered f(α) from the Backus-Gilbert inversion scheme for g(θ) with no random errors introduced.

Fig. 8
Fig. 8

Recovered f(α) from the Backus-Gilbert inversion scheme for g(θ) with 1% random errors introduced.

Tables (2)

Tables Icon

Table I Spread s as a Function of the Size Parameter α for the Backus-Gilbert Inversion Scheme

Tables Icon

Table II Recovered f(α) from the Backus-Gilbert Inversion Scheme for g (θ) with 5% Random Errors Introduced

Equations (26)

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n ( r ) = c r 6 exp ( 1.5 r ) cm 3 μ m 1 ,
E m ( θ ) = λ 2 I 0 8 π 2 α 1 α 2 [ i 1 ( θ , α ) + i 2 ( θ , α ) ] f ( α ) d α ,
g ( θ ) = α 1 α 2 K ( θ , α ) f ( α ) d α ,
g ( θ ) = ( E m / I 0 ) ( 4 π 2 / λ 2 )
K ( θ , α ) = 1 2 [ i 1 ( θ , α ) + i 2 ( θ , α ) ] .
g = Bf ,
g j = k = 1 M w k K j k f k , j = 1 , . . . , N ,
f = B 1 g .
g j + j = k = 1 M w k K j k f k , j = 1 , . . . , N .
j = 1 N j 2 .
α 1 α 2 ( f t ) 2 d α = min f F α 1 α 2 ( f ) 2 d α .
j = 1 N j 2 = constant
f = ( B T B + γ H ) 1 B T g ,
H = [ 1 2 1 0 0 0 2 5 4 1 0 0 O 1 4 6 4 1 0 0 1 4 6 4 1 O 1 4 6 4 1 0 0 1 4 6 4 1 0 0 1 4 5 2 0 0 0 1 2 1 ] .
f ( α 0 ) ¯ = α 1 α 2 A ( α , α 0 ) f ( α ) d α
α 1 α 2 A ( α , α 0 ) d α = 1 ,
A ( α , α 0 ) = i = 1 N a i ( α 0 ) K i ( α ) .
f ( α 0 ) ¯ = i = 1 N a i ( α 0 ) α 1 α 2 K i ( α ) f ( α ) d α = i = 1 N a i ( α 0 ) g i .
s ( α 0 ) = 12 α 1 α 2 ( α α 0 ) 2 A 2 ( α , α 0 ) d α = a T Sa ,
S i j = 12 α 1 α 2 ( α α 0 ) 2 K i ( α ) K j ( α ) d α , i , j = 1 , . . . , N .
σ 2 = a T S a ,
Q ( α 0 ) = q s ( α 0 ) + ( 1 q ) w σ 2 ( α 0 ) ,
u i = α 1 α 2 K i ( α ) d α
w ( α 0 ) = q S ( α 0 ) + ( 1 q ) w S ( α 0 ) .
a ( α 0 ) = w 1 ( α 0 ) u u T w 1 ( α 0 ) u .
f ( α ) = 0.42 ( α / 6.3 ) 6 exp [ 1.5 ( α / 6.3 ) ]

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