Abstract

An approximate method [ J. Opt. Soc. Am. 72, 756 ( 1982)] for inferring the albedo of single scattering and Legendre moments of the angular scattering function from the asymptotic die-away of the backscattered radiance following a pulse on an optically thick slab target is numerically shown to be valid if the radiance is accurately known at a sufficient number of azimuthal angles for a fixed polar angle. The sensitivity of the estimated parameters to simulated random errors in the radiance is also examined, and it is inferred that the albedo of single scattering and the asymmetry factor, and perhaps a third parameter, might be obtainable by such a technique from measurements at only three azimuthal angles.

© 1985 Optical Society of America

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References

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  1. N. J. McCormick, “Remote characterization of a thick slab target with a pulsed laser,” J. Opt. Soc. Am. 72, 756–759 (1982).
    [CrossRef]
  2. N. J. McCormick, “Inverse methods for remote determination of properties of optically thick atmospheres,” Appl. Opt. 22, 2556–2558 (1983).
    [CrossRef] [PubMed]
  3. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  4. H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980).
  5. B. D. Ganapol, W. L. Filippone, “Time-dependent emergent radiance from an anisotropically-scattering semi-infinite atmosphere.” J. Quant. Spectrosc. Radiat. Transfer 27, 15–21 (1982).
    [CrossRef]
  6. The generalization of Ref. 5 to the azimuthally dependent problem is clearly inferred by noting the similarity between Eq. (12c) of Ref. 5 and Eq. (112) of Ref. 7.
  7. S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere XVII,” Astrophys. J. 105, 441–460 (1947).
    [CrossRef]
  8. S. Pahor, M. Gros, “Optical properties of thick fog layers,” Tellus 22, 321 (1970).
    [CrossRef]

1983 (1)

1982 (2)

B. D. Ganapol, W. L. Filippone, “Time-dependent emergent radiance from an anisotropically-scattering semi-infinite atmosphere.” J. Quant. Spectrosc. Radiat. Transfer 27, 15–21 (1982).
[CrossRef]

N. J. McCormick, “Remote characterization of a thick slab target with a pulsed laser,” J. Opt. Soc. Am. 72, 756–759 (1982).
[CrossRef]

1970 (1)

S. Pahor, M. Gros, “Optical properties of thick fog layers,” Tellus 22, 321 (1970).
[CrossRef]

1947 (1)

S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere XVII,” Astrophys. J. 105, 441–460 (1947).
[CrossRef]

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chandrasekhar, S.

S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere XVII,” Astrophys. J. 105, 441–460 (1947).
[CrossRef]

Filippone, W. L.

B. D. Ganapol, W. L. Filippone, “Time-dependent emergent radiance from an anisotropically-scattering semi-infinite atmosphere.” J. Quant. Spectrosc. Radiat. Transfer 27, 15–21 (1982).
[CrossRef]

Ganapol, B. D.

B. D. Ganapol, W. L. Filippone, “Time-dependent emergent radiance from an anisotropically-scattering semi-infinite atmosphere.” J. Quant. Spectrosc. Radiat. Transfer 27, 15–21 (1982).
[CrossRef]

Gros, M.

S. Pahor, M. Gros, “Optical properties of thick fog layers,” Tellus 22, 321 (1970).
[CrossRef]

McCormick, N. J.

Pahor, S.

S. Pahor, M. Gros, “Optical properties of thick fog layers,” Tellus 22, 321 (1970).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980).

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (1)

Astrophys. J. (1)

S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere XVII,” Astrophys. J. 105, 441–460 (1947).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

B. D. Ganapol, W. L. Filippone, “Time-dependent emergent radiance from an anisotropically-scattering semi-infinite atmosphere.” J. Quant. Spectrosc. Radiat. Transfer 27, 15–21 (1982).
[CrossRef]

Tellus (1)

S. Pahor, M. Gros, “Optical properties of thick fog layers,” Tellus 22, 321 (1970).
[CrossRef]

Other (3)

The generalization of Ref. 5 to the azimuthally dependent problem is clearly inferred by noting the similarity between Eq. (12c) of Ref. 5 and Eq. (112) of Ref. 7.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980).

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Figures (4)

Fig. 1
Fig. 1

Polar graph of I(0, −μ, ϕ; t) versus azimuthal angle ϕ for μ0 = 0.5 and (a) μ ≈ cos 45° and (b) μ ≈ cos 70° for different times t after the pulse.

Fig. 2
Fig. 2

Im(0, −μ; t) versus time t after the pulse for μ = 0.5 and μ ≈ cos 70° for different values of m. The values in parentheses represent the number of collisions necessary to obtain four-place accuracy in Eq. (10a).

Fig. 3
Fig. 3

Time derivative of Ym(0, −μ; t) versus time t after the pulse for μ = 0.5 and μ ≈ cos 70° for different values of m. {The derivatives were estimated using the approximation [Ym(t + Δt) − Ym(t)]/Δt for Δt = 1.}

Fig. 4
Fig. 4

Statistical fluctuation of Ym(0, −μ; t) versus time for μ0 = 0.5 and μ ≈ cos 70° for different values of m, random error parameter σc = 0.01, and (a) J = 9 or (b) J = 2. The shaded envelopes represent plus and minus one standard deviation about the unperturbed values for 20 simulated measurements at each azimuthal angle.

Tables (4)

Tables Icon

Table 1 Tm() versus m and for μ ≈ cos 45° with J = 9 and No Random Errors

Tables Icon

Table 2 Tm() versus m and for μ ≈ cos 70° with J = 9 and No Random Errors

Tables Icon

Table 3 Tm*(σ) for m = 0, 1, and 2 versus Random Error Parameters σ and σc for μ ≈ cos 70° with J = 9a

Tables Icon

Table 4 Tm*(σ) for m = 0, 1, and 2 versus Random Error Parameters σ and σc for μ ≈ cos 70° with J = 2a

Equations (20)

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υ 1 t I ( x , μ , ϕ ; t ) + μ x I ( x , μ , ϕ ; t ) + I ( x , μ , ϕ ; t ) = 0 2 π d ϕ 1 1 d μ p ( cos θ ) I ( x , μ , ϕ ; t ) .
p ( cos θ ) = ( 4 π ) 1 n = 0 N m = n N f n m P n m ( μ ) P n m ( μ ) cos m ( ϕ ϕ ) ,
f n m = f n ( 2 n + 1 ) ( n m ) ! / ( n + m ) ! .
I ( 0 , μ , ϕ , t ) = δ ( μ μ 0 ) δ ( ϕ ) δ ( t ) , 0 μ 1 , 0 ϕ 2 π .
I ( 0 , μ , ϕ j ; t ) = m = 0 N ( 2 δ m 0 ) I m ( 0 , μ ; t ) cos m ϕ j , 0 μ 1 ,
I m ( 0 , μ ; t ) = ( 2 π ) 1 0 2 π I ( 0 , μ , ϕ ; t ) cos m ϕ d ϕ , m = 0 to N
I m ( 0 , μ ; t ) = j = 0 J W j m I ( 0 , μ , ϕ j ; t ) , m = 0 to J .
j = 0 J W j m cos m ϕ j = δ m m / ( 2 δ m 0 ) , m = 0 to J , m = 0 to J .
I m ( 0 , μ ; t ) K m ( μ , μ 0 ) t 3 / 2 exp [ ( 1 f m ) υ t ] ,
( 1 f m ) [ υ ( t 2 t 1 ) ] 1 ln [ ( t 1 / t 2 ) 3 / 2 I m ( 0 , μ ; t 1 ) I m ( 0 , μ ; t 2 ) ] .
Y m ( 0 , μ ; t ) = t + υ 1 ln [ t 3 / 2 I m ( 0 , μ ; t ) ]
| Y m ( 0 , μ ; T m ) f m | / f m .
I m ( 0 , μ ; t ) = e υ t t k = 0 ( f 0 υ t ) k ( k 1 ) ! I k m ( 0 , μ ) , m = 0 to N .
I k m ( 0 , μ ) = μ 0 2 1 μ + μ 0 n = m N ( 1 ) n m f n m f 0 × k = 0 k 1 Φ k , n m ( μ ) Φ k 1 k , n m ( μ 0 ) ,
Φ 0 , n m ( μ ) = P n m ( μ ) , Φ k , n m ( μ ) = ( 1 ) n μ 2 k = 0 k 1 0 1 d μ P n m ( μ ) μ + μ × j = m N ( 1 ) j f j m f 0 Φ k , j m ( μ ) Φ k 1 k , j m ( μ )
ϕ j = ( π / 2 ) ( 2 j + 1 ) / ( J + 1 ) , j = 0 to J .
p ( z ) = ( 2 π ) 1 / 2 exp ( z 2 / 2 ) [ erf ( 2 1 / 2 ) ] 1 , 1 z 1 .
I p ( 0 , μ , ϕ j ; t ) = I ( 0 , μ , ϕ j ; t ) [ 1 + σ c z ( ξ ) ] ,
z ( ξ ) = [ 1 ( 2 π ) 1 / 2 e 1 / 2 erf ( 2 1 / 2 ) ] 1 / 2 sign ( ξ ) erf 1 [ | ξ | erf ( 2 1 / 2 ) ] .
σ T 2 = σ 2 + 2

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