Abstract

The previously developed spectral transparency method is improved in two ways: (a) A calculation scheme is developed that enables the consideration of light dispersion within a substance. We abandon the assumption that m(ν) = const and develop a method of calculating N(r) from γ*(ν) provided that the function m(ν) is specified. (b) A new scheme of processing data on γ*(ν) is proposed that significantly reduces the oscillation in the answer and requires less initial experimental data.

© 1980 Optical Society of America

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References

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  1. I. L. Zelmanovich, K. S. Shifrin, Light Scattering Tables (Gidrometeoizadat, Moscow, 1971), Vol. 4.
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. K. S. Shifrin, A. Ya. Perelman, Dokl. Akad. Nauk SSSR 151, 326 (1963).
  4. K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 15, 533, 667, 803 (1963);Opt. Spektrosk. 16, 117 (1964); [Opt. Spectrosc. 15, 285, 362, 434 (1963); 16, 61 (1964)];Geofis. Pura Appl. 58, 208 (1964).
  5. K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 143 (1966) [Opt. Spectrosc. 20, 75 (1966)].
  6. K. S. Shifrin, A. Ya. Perelman, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 964 (1965).
  7. K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 692 (1966) [Opt. Spectrosc. 20, 386 (1966)].
  8. A. S. Lagunov, L. P. Baivel, V. K. Litvinov, V. K. Kariazova, Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].
  9. It is essential that the kernel of the equation depend on the product of the variables y and r. All the details that refer to inverting Eq. (5) are given in Ref. 4.
  10. The calculations have been performed by S. Todorova, a probationer from Bulgaria, by means of the tables cited in Ref. 1. In these tables the dependence of m(ν) is given numerically. If m(ν) is known analytically, the calculations will be significantly simplified.
  11. A. Ya. Perelman, K. S. Shifrin, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 15, 66 (1979).
  12. K. S. Shifrin, Light Scattering in Turbid Media (GTTI, Moscow, 1951; NASA, Washington, D.C., 1968).
  13. K. S. Shifrin, A. Ya. Perelman, V. M. Volgin, Opt. Spektrosk.47, 1147 (1979) [Opt. Spectrosc. 47, in press (1979)].

1979

A. Ya. Perelman, K. S. Shifrin, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 15, 66 (1979).

1977

A. S. Lagunov, L. P. Baivel, V. K. Litvinov, V. K. Kariazova, Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].

1966

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 692 (1966) [Opt. Spectrosc. 20, 386 (1966)].

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 143 (1966) [Opt. Spectrosc. 20, 75 (1966)].

1965

K. S. Shifrin, A. Ya. Perelman, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 964 (1965).

1963

K. S. Shifrin, A. Ya. Perelman, Dokl. Akad. Nauk SSSR 151, 326 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 15, 533, 667, 803 (1963);Opt. Spektrosk. 16, 117 (1964); [Opt. Spectrosc. 15, 285, 362, 434 (1963); 16, 61 (1964)];Geofis. Pura Appl. 58, 208 (1964).

Baivel, L. P.

A. S. Lagunov, L. P. Baivel, V. K. Litvinov, V. K. Kariazova, Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].

Kariazova, V. K.

A. S. Lagunov, L. P. Baivel, V. K. Litvinov, V. K. Kariazova, Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].

Lagunov, A. S.

A. S. Lagunov, L. P. Baivel, V. K. Litvinov, V. K. Kariazova, Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].

Litvinov, V. K.

A. S. Lagunov, L. P. Baivel, V. K. Litvinov, V. K. Kariazova, Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].

Perelman, A. Ya.

A. Ya. Perelman, K. S. Shifrin, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 15, 66 (1979).

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 692 (1966) [Opt. Spectrosc. 20, 386 (1966)].

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 143 (1966) [Opt. Spectrosc. 20, 75 (1966)].

K. S. Shifrin, A. Ya. Perelman, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 964 (1965).

K. S. Shifrin, A. Ya. Perelman, Dokl. Akad. Nauk SSSR 151, 326 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 15, 533, 667, 803 (1963);Opt. Spektrosk. 16, 117 (1964); [Opt. Spectrosc. 15, 285, 362, 434 (1963); 16, 61 (1964)];Geofis. Pura Appl. 58, 208 (1964).

K. S. Shifrin, A. Ya. Perelman, V. M. Volgin, Opt. Spektrosk.47, 1147 (1979) [Opt. Spectrosc. 47, in press (1979)].

Shifrin, K. S.

A. Ya. Perelman, K. S. Shifrin, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 15, 66 (1979).

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 692 (1966) [Opt. Spectrosc. 20, 386 (1966)].

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 143 (1966) [Opt. Spectrosc. 20, 75 (1966)].

K. S. Shifrin, A. Ya. Perelman, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 964 (1965).

K. S. Shifrin, A. Ya. Perelman, Dokl. Akad. Nauk SSSR 151, 326 (1963).

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 15, 533, 667, 803 (1963);Opt. Spektrosk. 16, 117 (1964); [Opt. Spectrosc. 15, 285, 362, 434 (1963); 16, 61 (1964)];Geofis. Pura Appl. 58, 208 (1964).

I. L. Zelmanovich, K. S. Shifrin, Light Scattering Tables (Gidrometeoizadat, Moscow, 1971), Vol. 4.

K. S. Shifrin, A. Ya. Perelman, V. M. Volgin, Opt. Spektrosk.47, 1147 (1979) [Opt. Spectrosc. 47, in press (1979)].

K. S. Shifrin, Light Scattering in Turbid Media (GTTI, Moscow, 1951; NASA, Washington, D.C., 1968).

van de Hulst, H. C.

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

Volgin, V. M.

K. S. Shifrin, A. Ya. Perelman, V. M. Volgin, Opt. Spektrosk.47, 1147 (1979) [Opt. Spectrosc. 47, in press (1979)].

Zelmanovich, I. L.

I. L. Zelmanovich, K. S. Shifrin, Light Scattering Tables (Gidrometeoizadat, Moscow, 1971), Vol. 4.

Dokl. Akad. Nauk SSSR

K. S. Shifrin, A. Ya. Perelman, Dokl. Akad. Nauk SSSR 151, 326 (1963).

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana

K. S. Shifrin, A. Ya. Perelman, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 964 (1965).

A. Ya. Perelman, K. S. Shifrin, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 15, 66 (1979).

Opt. Spektrosk.

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 692 (1966) [Opt. Spectrosc. 20, 386 (1966)].

A. S. Lagunov, L. P. Baivel, V. K. Litvinov, V. K. Kariazova, Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 15, 533, 667, 803 (1963);Opt. Spektrosk. 16, 117 (1964); [Opt. Spectrosc. 15, 285, 362, 434 (1963); 16, 61 (1964)];Geofis. Pura Appl. 58, 208 (1964).

K. S. Shifrin, A. Ya. Perelman, Opt. Spektrosk. 20, 143 (1966) [Opt. Spectrosc. 20, 75 (1966)].

Other

I. L. Zelmanovich, K. S. Shifrin, Light Scattering Tables (Gidrometeoizadat, Moscow, 1971), Vol. 4.

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

It is essential that the kernel of the equation depend on the product of the variables y and r. All the details that refer to inverting Eq. (5) are given in Ref. 4.

The calculations have been performed by S. Todorova, a probationer from Bulgaria, by means of the tables cited in Ref. 1. In these tables the dependence of m(ν) is given numerically. If m(ν) is known analytically, the calculations will be significantly simplified.

K. S. Shifrin, Light Scattering in Turbid Media (GTTI, Moscow, 1951; NASA, Washington, D.C., 1968).

K. S. Shifrin, A. Ya. Perelman, V. M. Volgin, Opt. Spektrosk.47, 1147 (1979) [Opt. Spectrosc. 47, in press (1979)].

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Figures (3)

Fig. 1
Fig. 1

Comparison of different inversion methods: (1) accurate distribution; (2) distribution obtained with constant m; (3) distribution obtained with variable m.

Fig. 2
Fig. 2

Curves of spectral transparency: (1) accurate curve; (2) curve corresponding to the nonoptimum pair (4,1); (3) curve corresponding to the optimum pair (2,2).

Fig. 3
Fig. 3

Results of inversion: (1) accurate distribution; (2) distribution corresponding to the nonoptimum pair (4,1); (3) distribution corresponding to the optimum pair (2,2).

Equations (85)

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γ * ( ν ) = 0 K ( ν , r , m ) f ( r ) dr .
f ( r ) = π r 2 N ( r ) ,
K ( ν , r , m ) Q ( ρ ) = 2 4 sin ρ ρ + 4 1 cos ρ ρ 2 , ρ = yr ,
y = φ ( ν ) = 4 π [ m ( ν ) 1 ] ν .
γ ( ν ) = 0 Q ( yr ) f ( r ) dr .
γ ( ν ) γ * ( ν ) .
c 0 = lim ν γ * ( ν ) = lim ν γ ( ν ) = 2 0 f ( r ) dr ,
lim ν K ( ν , r , m ) = lim ν Q ( ρ ) = 2 .
m ( ν ) = m 0
f ( r ) = 1 2 π { 1 2 π i σ i σ + i ( p + 1 ) Γ c ( p ) [ 0 γ ( ν ) y p dy ] r p dp } , Γ c ( p ) = Γ ( p ) cos π p 2 ,
f ( r ) = f 1 ( r ) + f 2 ( r ) ,
f l ( r ) = 1 4 π 2 i σ i σ + i ( p + 1 ) Γ c ( p ) I l ( p ) r p dp ( 2 < σ < 0 ) .
I 1 ( p ) = 0 τ γ ( ν ) φ ( ν ) φ p ( ν ) d ν ( Rep < 3 ) ,
I 2 ( p ) = τ γ ( ν ) φ ( ν ) φ p ( ν ) d ν ( Rep > 1 ) .
f 1 ( r ) 1 2 π j = 1 n γ ( ν j ) φ ( ν j ) Δ ν j ω [ φ ( ν j ) r ] ,
ω ( a ) = 1 2 π i σ i σ + i ( p + 1 ) Γ c ( p ) a p dp ( 2 < σ < 0 ) .
ω ( a ) = a sin a + cos a 1 .
γ ( ν ) c 0 + c 1 ν 1 + . . . + c K ν K ( ν > τ ) .
γ ( ν ) c 0 + c 1 ν 1 + c 2 ν 2 ( ν > τ ) .
m ( ν ) = m 0 ( ν > τ ) .
I 2 ( p ) 4 π ( m 0 1 ) ( c 0 τ p 1 + c 1 p + c 2 τ 1 p + 1 ) y τ p ,
y τ = 4 π ( m 0 1 ) τ .
f 2 ( r ) 2 ( m 0 1 ) [ c 0 τ ω 0 ( y τ r ) + c 1 ω 1 ( y τ r ) + c 2 τ 1 ω 2 ( y τ r ) ] ,
ω α ( a ) = 1 2 π i σ i σ + i p + 1 p + α 1 Γ c ( p ) a p dp ( 2 < σ < 0 ) .
ω α ( a ) = cos a 1 + ( α 2 ) a α 1 0 a cos b 1 b α db ( α < 3 ) .
ω 0 ( a ) = cos a 2 sin a a + 1 ω 1 ( a ) = cos a 1 + 0 a 1 cos b b db ω 2 ( a ) = cos a 1 . } .
ω α ( a ) = O ( a 2 ) ( a 0 ) , ω α ( a ) = ω α ( a ) .
f ( r ) 2 { 1 4 π j = 1 n γ ( ν j ) φ ( ν j ) Δ ν j ω [ φ ( ν j ) r ] + ( m 0 1 ) [ c 0 τ ω 0 ( y τ r ) + c 1 ω 1 ( y τ r ) + c 2 τ 1 ω 2 ( y τ r ) ] } .
f ( r ) 2 ( m 0 1 ) { j = 1 n γ ( ν j ) Δ ν j ω [ 4 π ( m 0 1 ) ν j r ] + c 0 τ ω 0 ( y τ r ) + c 1 ω 1 ( y τ r ) + c 2 τ 1 ω 2 ( y τ r ) } .
f ( r ) = A r 4 exp ( 3 r ̅ 1 r ) ,
m ( ν ) 1 + m 1 ν 1 + m 2 ν 2 , m 2 0 ( ν > τ ) .
lim ν m ( ν ) = 1 .
γ ( ν ) l = 0 κ γ κ l φ l ( ν ) , φ ( ν ) 4 π ( m 1 + m 2 ν 1 ) ( ν > τ ) ,
γ κ l = 1 ( 4 π ) l j = l κ ( 1 ) j l ( j l ) m 1 j l m 2 j c j .
I 2 ( p ) l = 0 κ γ κ l φ p + l + 1 ( τ ) m 1 p + l + 1 p l 1 ( Rep < 1 ) ,
f 2 ( r ) 2 l = 0 κ γ κ l φ l + 1 ( τ ) ω l [ φ ( τ ) r ] + 2 l = 0 κ γ κ l m 1 l + 1 ω l ( m 1 r ) .
ω 1 ( a ) = 3 a sin a + ( 1 3 a 2 ) cos a + 1 2 + 3 a 2 ω 2 ( a ) = ( 4 a + 8 a 3 ) sin a + ( 1 8 a 2 ) cos a + 1 3 } .
γ 20 = m 2 2 c 0 m 1 m 2 c 1 + m 1 2 c 2 m 2 2 , γ 21 = m 2 c 1 2 m 1 c 2 4 π m 2 2 , γ 22 = c 2 16 π 2 m 2 2 .
f 2 ( r ) 2 m 2 [ c 0 τ ω 0 ( β r ) + c 1 τ 2 ω 1 ( β r ) + c 2 τ 3 ω 2 ( β r ) ] , β = 4 π m 2 τ ,
m ( ν ) 1 + m 2 ν 1 ( ν > τ ) .
f 2 ( r ) c 0 2 π φ ( τ ) ω 0 [ φ ( τ ) r ] .
ν = ψ ( y ) ,
m ( ν ) 1 m ( ν ) ν .
Q ( ρ ) = 4 ρ 3 κ ( ρ ) , κ ( ρ ) = ρ 2 sin ρ , Q ( 0 ) = κ ( 0 ) = 0 .
Q ( ρ ) = 4 0 ρ u 3 du 0 u t 2 sin tdt .
Q ( ρ ) = 2 ρ 0 1 ( 1 t 2 ) sin ρ tdt .
0 Q ( yr ) f ( r ) dr = g ( y ) ,
g ( y ) = γ [ ψ ( y ) ] .
0 f ( r ) cos yrdr = h ( y ) ,
h ( y ) = 2 c 0 y g ( y ) 2 g ( y ) 4 ,
g ( y ) = y 0 1 ( 1 t 2 ) z ( yt ) dt ,
z ( t ) = 2 0 rf ( r ) sin trdr .
0 y z ( t ) dt = y g ( y ) + 2 g ( y ) 2 .
0 y z ( t ) dt = 2 0 f ( r ) ( 1 cos yr ) dr .
f ( r ) = 1 π 0 h ( y ) cos rydy ,
c 0 = g ( ) = 2 0 f ( r ) dr .
h ( y ) = h ( y ) , lim y h ( y ) = 0
f ( r ) = 2 Im Rey > 0 res [ h ( y ) exp ( iyr ) ] ,
g ( y ) = 4 + 4 1 + y 2 8 ( 1 + y 2 ) 2 .
h ( y ) = 8 ( 1 + y 2 ) 3 6 ( 1 + y 2 ) 2 ,
f ( r ) = r 2 exp ( r ) .
0 Q ( yr ) r 2 exp ( r ) dr = 4 + 4 1 + y 2 8 ( 1 + y 2 ) 2
g * ( y ) = γ * [ ψ ( y ) ] .
g ( y ) g * ( y ) ,
f ( r ) = r μ + 2 exp ( γ r ) ( μ 0 , γ > 0 ) .
( 1 + y 2 μ 2 ) 1 ( 0 y < ) .
π n c ( y ) = P n ( 2 c 2 c 2 + y 2 1 ) ,
F ( y ) = n = 0 a n π n c ( y ) ,
a n = 2 ( 2 n + 1 ) c 2 0 F ( y ) π n c ( y ) y ( c 2 + y 2 ) 2 dy .
g ( y ) g n c ( y ) = l = 0 n a l π l c ( y ) ( 0 y < )
g ( y ) = g ( ) + O ( y 2 ) ( y ) ,
g ( y ) = O ( y 2 ) ( y 0 ) .
l = 0 n a l = 0 .
δ n c = max y 0 | g n c ( y ) g ( y ) | g m , g m = max y 0 g ( y ) .
n = 1 ( 1 ) 10 , c = 0.5 ( 0.5 ) 6.0 .
g n c ( y ) = l = 0 n b l ( 1 + y 2 c 2 ) l ,
h n c ( y ) = 1 2 [ n b n ( 1 + y 2 c 2 ) n 1 + l = 2 n ( l 1 ) ( b l 1 b l ) ( 1 + y 2 c 2 ) l ] ,
f ( r ) f n c ( r ) = 1 π [ n b n φ n + 1 c ( r ) + l = 1 n ( l 1 ) ( b l 1 b l ) φ l c ( r ) ] ,
φ l c ( r ) = 0 ( 1 + y 2 c 2 ) l cos rydy .
f n c ( r ) = c t n ( cr ) exp ( cr )
lim r f n c ( r ) = 0
f 1 c ( r ) = c 2 ( cr + 1 ) exp ( cr ) π a 1 f 2 c ( r ) = f 1 c ( r ) 3 c 4 ( c 2 r 2 cr 1 ) exp ( cr ) π a 2 } .
n c ( α ) = max r α | f n c ( r ) f ( r ) | f M , f M = max r 0 f ( r ) .
f ( r ) = π N γ r 2 exp ( γ r ) .
g ( y ) = 4 π N γ 2 [ 1 + ( 1 + y 2 γ 2 ) 1 2 ( 1 + y 2 γ 2 ) 2 ] .

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