Abstract

A methodology for investigating the variation of particle size distributions retrieved from constrained linear inversion of optical scattering data is presented. By plotting the expected inversion error vs angular scanning parameters (for typical size distribution vectors) one can determine sets of optimum angles based on a minimum error criteria at each particle size. An expression for the expected inversion error at each radius knot is derived. In addition a formulation for the Fredholm quadrature matrix in terms of Legendre coefficients and polynomials is introduced. This method of computation is advantageous when a large number of angles are to be investigated. The derived results are applied to the special case of a Junge Continental Aerosol.

© 1982 Optical Society of America

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References

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    [PubMed]
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    [CrossRef]
  10. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).
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    [CrossRef]
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    [CrossRef] [PubMed]

1975 (1)

J. T. Twitty, J. Atmos. Sci. 32, 584 (1975).
[CrossRef]

1974 (1)

1973 (1)

1971 (2)

1970 (1)

1968 (1)

G. Hanel, Tellus 20, 371 (1968).
[CrossRef]

1967 (1)

1963 (1)

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

1957 (1)

1955 (1)

Allen, E.

Chu, C.-M.

Churchill, S. W.

Clark, G. E.

Cohen, A.

Dave, J. V.

Deepak, A.

Green, A. E. S.

Hanel, G.

G. Hanel, Tellus 20, 371 (1968).
[CrossRef]

Howell, H. B.

Kipofsky, B. J.

Twitty, J. T.

J. T. Twitty, J. Atmos. Sci. 32, 584 (1975).
[CrossRef]

Twomey, S.

S. Twomey, H. B. Howell, Appl. Opt. 6, 2125 (1967).
[CrossRef] [PubMed]

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

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

Westwater, E. R.

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Figures (2)

Fig. 1
Fig. 1

Inversion errors as a function of the largest scanning angle and radius knot number (r = 0.025, γ = 1.5 r 2 λ 1). The smallest angle is fixed at 1.5°. N = 23 angular measurements.

Fig. 2
Fig. 2

Inversion errors as a function of the smallest scanning angle and radius knot number (r = 0.025, γ = 1.5 r 2 λ 1). The largest angle is fixed at 10°.

Tables (2)

Tables Icon

Table I Comparisons of aij Values

Tables Icon

Table II Second Component of Inversion Error (%) for rms Error of 1%

Equations (28)

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B λ p ( cos θ ) = x A x B [ ln 10 π λ 2 K ( α , θ , m ) 10 - 3 x ] [ n ( x ) 10 4 x ] d x ,
G ˜ t = A ˜ F ˜ t ,
a i j = x j - 1 x j ( x - x j - 1 x j - x j - 1 ) K ( α , θ i ) d x + x j x j + 1 ( x j + 1 - x x j + 1 - x j ) K ( α , θ i ) d x
a i 1 = x 1 x 2 ( x 2 - x x 2 - x 1 ) K ( α , θ i ) d x , a i M = x M - 1 x M ( x - x M - 1 x M - x M - 1 ) K ( α , θ i ) d x .
K ( α , θ i ) = l = 0 L C l ( α ) 4 P l ( cos θ i ) ,
γ l j = x j - 1 x j ( x - x j - 1 x j - x j - 1 ) C l 10 - 3 x 4 d x + x j x j + 1 ( x j + 1 - x x j + 1 - x j ) C l 10 - 3 x 4 d x
γ l 1 = x 1 x 2 ( x 2 - x x 2 - x 1 ) C l 10 - 3 x 4 d x , γ l M = x M - 1 x M ( x - x M - 1 x M - x M - 1 ) C l 10 - 3 x 4 d x ,
A ˜ = ln 10 λ 2 π P ˜ ( cos θ ) T Γ ˜ λ .
γ l j = γ l 2 j - 1 + ( γ l 2 j - 2 + γ l 2 j ) / 2 ,
Γ ˜ λ = ( 0.5 / λ ) 3 Γ ˜ 0.5 .
A ˜ T A ˜ F ˜ t = A ˜ T G ˜ t .
A ˜ T A ˜ = P ˜ u e Λ ˜ ( P ˜ u e ) - 1 ,
F ˜ = P ˜ u e ( Λ ˜ + γ I ˜ ) - 1 ( P ˜ u e ) T A ˜ T ( G ˜ t + E ˜ ) ,
F ˜ - F ˜ t = P ˜ u e ( Λ ˜ + γ I ˜ ) - 1 ( P ˜ u e ) T A ˜ T G ˜ t - P ˜ u e [ F ˜ t ] u + P ˜ u e ( Λ ˜ + γ I ˜ ) - 1 ( P ˜ u e ) T A ˜ T E ˜ ,
F ˜ - F ˜ t = P ˜ u e ( Λ ˜ + γ I ˜ ) - 1 Λ ˜ [ F ˜ t ] u - P ˜ u e [ F ˜ t ] u + P ˜ u e ( Λ ˜ + γ I ˜ ) - 1 ( p ˜ u e ) T A ˜ T E ˜ = - γ P ˜ u e ( Λ ˜ + γ I ˜ ) - 1 [ F ˜ t ] u + P ˜ u e ( Λ ˜ + γ I ˜ ) - 1 ( P ˜ u e ) T A ˜ T E ˜ .
( F ˜ - F ˜ t ) i = - F ˜ t · e ˜ i * + E ˜ · C ˜ i * γ ,
e ˜ j * = γ j = 1 M p i j u ˜ j λ j + γ ,
C ˜ i * = γ j = 1 M p i j A ˜ u ˜ j λ j + γ = A ˜ e ˜ i * ,
E ˜ = ( r 1 0 · · 0 r N ) G ˜ t = E ˜ r G ˜ t ,
( F ˜ - F ˜ t ) i 2 = ( F ˜ t · e ˜ i * ) 2 + r 2 G ˜ t 2 · C ˜ i * 2 γ 2 .
Δ f i 2 2 ~ r 2 k = 1 N g t k 2 j = 1 M p i j 2 s j k 2 ( λ j + γ ) 2 ,
k = 1 N s j k 2 = λ j and approximating k = 1 N g t k 2 s j k 2 by G ˜ t 2 N λ j
Δ f i 2 2 ~ r 2 G ˜ t 2 / N γ j = 1 M p i j 2 λ / γ ( λ j + 1 γ ) 2 .
γ i r 2 G ˜ t 2 / M f t i 2 .
j = 1 M Σ p i j 2 = 1 ) .             Thus · e ˜ i = j = 1 M p i j u ˜ j
F ˜ t · e ˜ i * = F ˜ t · [ γ j = 1 J p i j u ˜ j / ( λ j + γ ) + j = J + 1 M p i j u ˜ j ] ,
F ˜ t · e ˜ i * f t i 1 - j = 1 J λ j λ j + γ p i j F ˜ t · u ˜ j f t i = 1 + F ˜ t · f t i ( e ˜ i * - e ˜ i ) ,
C ˜ i - α C ˜ k 2 = j = 1 M ( p i j - α p k j ) 2 λ j ,

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