Abstract

Measurements of scattered sunlight often employ simple radiometers that are incapable of scanning azimuth ϕ and zenith angle θ. Instead, they measure integral moments of the radiance. Consider a set of N + 1 meters designed to measure moments Λ1 through ΛN+1. If each meter is judiciously designed, the Λj may be processed quite simply to deduce inherent properties of the scattering medium, namely, N moments of the volume scattering function. In the optimum design, each meter has an angular response that approximates a polynomial of degree N in cosθ. The theory of these meters is based on Boltzmann's equation of radiative transfer expressed in terms of spherical harmonics, a form in which this equation is particularly simple.

© 1983 Optical Society of America

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References

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  1. H. R. Gordon, O. B. Brown, M. M. Jacobs, Appl. Opt. 14, 417 (1975).
    [CrossRef] [PubMed]
  2. W. H. Wells, J. Opt. Soc. Am. 59, 686 (1969).
    [CrossRef]
  3. B. Davison, J. B. Sykes, Neutron Transport Theory (Clarendon, Oxford, 1957).
  4. W. H. Wells, J. J. Sidorowich, Ann. Phys. N.Y. 144, 203 (1982).
    [CrossRef]
  5. A. Gershun, J. Math. Phys. 18, 51 (1939).
  6. J. R. V. Zaneveld, “New Developments of the Theory of Radiative Transfer in the Oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, Eds. (Academic, New York, 1974), Chap. 6. See especially the last equation on p. 129; parentheses are missing on the left.
  7. H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).
  8. K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).
  9. H. R. Gordon, Appl. Opt. 19, 2092 (1980).
    [CrossRef] [PubMed]
  10. M. C. Wang, E. Guth, Phys. Rev. 84, 1092 (1951).
    [CrossRef]
  11. H. Hodara, “Experimental Results of Small-Angle Scattering,” Lecture 3.4 in AGARD Lect. Ser. 61 (1973).

1982

W. H. Wells, J. J. Sidorowich, Ann. Phys. N.Y. 144, 203 (1982).
[CrossRef]

1980

1979

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

1975

1973

H. Hodara, “Experimental Results of Small-Angle Scattering,” Lecture 3.4 in AGARD Lect. Ser. 61 (1973).

1969

1951

M. C. Wang, E. Guth, Phys. Rev. 84, 1092 (1951).
[CrossRef]

1939

A. Gershun, J. Math. Phys. 18, 51 (1939).

Baker, K. S.

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

Brown, O. B.

Davison, B.

B. Davison, J. B. Sykes, Neutron Transport Theory (Clarendon, Oxford, 1957).

Gershun, A.

A. Gershun, J. Math. Phys. 18, 51 (1939).

Gordon, H. R.

H. R. Gordon, Appl. Opt. 19, 2092 (1980).
[CrossRef] [PubMed]

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

H. R. Gordon, O. B. Brown, M. M. Jacobs, Appl. Opt. 14, 417 (1975).
[CrossRef] [PubMed]

Guth, E.

M. C. Wang, E. Guth, Phys. Rev. 84, 1092 (1951).
[CrossRef]

Hodara, H.

H. Hodara, “Experimental Results of Small-Angle Scattering,” Lecture 3.4 in AGARD Lect. Ser. 61 (1973).

Jacobs, M. M.

Sidorowich, J. J.

W. H. Wells, J. J. Sidorowich, Ann. Phys. N.Y. 144, 203 (1982).
[CrossRef]

Smith, R. C.

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

Sykes, J. B.

B. Davison, J. B. Sykes, Neutron Transport Theory (Clarendon, Oxford, 1957).

Wang, M. C.

M. C. Wang, E. Guth, Phys. Rev. 84, 1092 (1951).
[CrossRef]

Wells, W. H.

W. H. Wells, J. J. Sidorowich, Ann. Phys. N.Y. 144, 203 (1982).
[CrossRef]

W. H. Wells, J. Opt. Soc. Am. 59, 686 (1969).
[CrossRef]

Zaneveld, J. R. V.

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

J. R. V. Zaneveld, “New Developments of the Theory of Radiative Transfer in the Oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, Eds. (Academic, New York, 1974), Chap. 6. See especially the last equation on p. 129; parentheses are missing on the left.

AGARD Lect. Ser. 61

H. Hodara, “Experimental Results of Small-Angle Scattering,” Lecture 3.4 in AGARD Lect. Ser. 61 (1973).

Ann. Phys. N.Y.

W. H. Wells, J. J. Sidorowich, Ann. Phys. N.Y. 144, 203 (1982).
[CrossRef]

Appl. Opt.

J. Math. Phys.

A. Gershun, J. Math. Phys. 18, 51 (1939).

J. Opt. Soc. Am.

Phys. Rev.

M. C. Wang, E. Guth, Phys. Rev. 84, 1092 (1951).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

Other

J. R. V. Zaneveld, “New Developments of the Theory of Radiative Transfer in the Oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, Eds. (Academic, New York, 1974), Chap. 6. See especially the last equation on p. 129; parentheses are missing on the left.

B. Davison, J. B. Sykes, Neutron Transport Theory (Clarendon, Oxford, 1957).

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Figures (7)

Fig. 1
Fig. 1

One configuration for a nonscanning compound radiometer.

Fig. 2
Fig. 2

Attenuation rates for spherical moments of radiance typical of seawater and clouds.

Fig. 3
Fig. 3

Response functions of the top two meters, i.e., R1(μ) and R2(μ) based on roots of the Legendre polynomial P4(μ).

Fig. 4
Fig. 4

Computed surfaces for the top two radiometers. They give response of Fig. 3 when used with a cosine detector.

Fig. 5
Fig. 5

Transmissometer (α meter).

Fig. 6
Fig. 6

Contrived response function for a nadir radiometer used to force decay of the R4n series.

Fig. 7
Fig. 7

Geometry for computing the reflecting surface of the radiometer ρ(z); vertex at z = 1.

Tables (3)

Tables Icon

Table I Matrix of Rjn for N = 3

Tables Icon

Table II Matrix of Qnj to Find Harmonics by Eq. (22)

Tables Icon

Table III Matrix of Qnj to Find Harmonics Using Eq. (23)

Equations (57)

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α = a + s , s σ d ω ,
A n = α S n , S n = P n ( cos η ) σ ( η ) d ω ,
A 0 = a , A = α .
σ ( η ) = n = 0 2 n + 1 4 π ( α A n ) P n ( cos η ) .
L ( z , θ , ϕ ) = n m = 0 2 n + 1 4 π ( 2 δ 0 m ) ( n m ) ! ( n + m ) ! L n m ( z ) P n m ( cos θ ) cos m ϕ ,
L n m = L ( θ , ϕ ) P n m ( cos θ ) cos m ϕ d ω ,
L 0 = L d ω ,
uplooking ( downwelling ) } L 1 up = 2 π 0 1 L μ d μ ,
downlooking ( upwelling ) } L 1 dn = 2 π 1 0 L μ d μ ,
L 1 = L 1 up L 1 dn .
( n + m ) L ˙ n 1 m + ( n m + 1 ) L ˙ n + 1 m + ( 2 n + 1 ) A n L n m = 0 ,
n L ˙ n 1 + ( n + 1 ) L ˙ n + 1 + ( 2 n + 1 ) A n L n = 0 ,
L ˙ 1 + A 0 L 0 = 0 , L ˙ 0 + 2 L ˙ 2 + 3 A 1 L 1 = 0 , 2 L ˙ 1 + 3 L ˙ 3 + 5 A 2 L 2 = 0 . }
a = ( L ˙ 1 dn L ˙ 1 up ) / L 0 .
A n = [ n L ˙ n 1 + ( n + 1 ) L ˙ n + 1 ] / ( 2 n + 1 ) L n .
Λ j = R j ( cos θ ) L ( θ , ϕ ) d ω = 2 π 1 1 R j ( μ ) L ¯ ( μ ) d μ ,
L ¯ ( μ ) = n = 0 2 n + 1 4 π L n P n ( μ ) .
Λ j = n = 0 ( n + 1 / 2 ) R j n L n ,
R j n = 1 1 R j ( μ ) P n ( μ ) d μ .
n = 0 N ( n + 1 / 2 ) R j n L n = Λ j X j ,
X j = n = N + 1 ( n + 1 / 2 ) R j n L n
L n ( z ) = j = 1 N + 1 Q n j ( Λ j X j ) ,
[ Q n j ] = [ ( n + 1 / 2 ) R j n ] 1 .
L n > N = H B n sec ζ P n ( cos ζ ) exp [ sec ζ 0 z A n ( z ) d z ] .
R j ( μ ) = i j ( μ μ i ) / ( μ j μ i )
= ( 1 μ j 2 ) ( N + 1 ) P N ( μ j ) P N + 1 ( μ ) ( μ μ j ) .
sinc ( x n π ) sin ( x n π ) x n π = ( 1 ) n sin x x n π .
μ 2 = { 0.7416 0.1156 ± μ = { 0.8611 0.3400 θ = { 30.56 70.12 .
R 1 , 4 = ( μ 2 0.1156 0.6260 ) ( μ ± 0.8611 ± 1.7223 ) , peaks near 30 ° 150 ° R 2 , 3 = ( μ 2 0.7416 0.6260 ) ( μ ± 0.3400 0.6800 ) , peaks near 70 ° 110 ° } .
R j n = { 2 ( 1 μ j 2 ) P n ( μ j ) [ ( N + 1 ) P N ( μ j ) ] 2 , n N , 0 , n > N ,
Q n j = P n ( μ j ) .
( n + 1 ) ( d d r + n + 2 r ) L n + 1 + n ( d d r n 1 r ) L n 1 + ( 2 n + 1 ) A n L n ( r ) = 0 .
n m ( ϕ ) = 1 1 P n m ( μ ) L ( μ , ϕ ) d μ ,
L n m = 0 2 π cos m ϕ n m ( ϕ ) d ϕ .
n m ( ϕ ) = j Q n j m [ Λ j m ( ϕ ) X j m ( ϕ ) ] ,
X j m ( ϕ ) = n = N + 1 ( n + ½ ) R j n m n m ( ϕ ) .
R 1 n = 2 δ 0 n
R 2 ( μ ) = { 4 μ , μ > 0 , 0 , μ < 0 , R 3 ( μ ) = { 0 , μ > 0 , 4 μ , μ < 0 ,
uplooking : R 2 n 4 = { 1 / 3 , n = 1 ( 1 ) ( n + 2 ) / 2 ( n 1 ) ! ! ( n 1 ) ( n + 2 ) n ! ! , n even , 0 , otherwise ,
downlooking : R 3 n = ( 1 ) n R 2 n .
R 4 n = 4 π δ ( μ 1 ) P n ( μ ) d ω 2 π = 2 ( ± 1 ) n .
nadir : R 4 n = 2 ( 1.0 , 0.81 , 0.53 , 0.24 , 0 , 0 , ) .
R j ( μ ) = P N + 1 ( μ ) P N + 1 ( μ j ) ( μ μ j ) .
r = 0 N ( 2 r + 1 ) P r ( μ ) P r ( ν ) = N + 1 μ ν [ P N + 1 ( μ ) P N ( ν ) P N ( μ ) P N + 1 ( ν ) ] .
P N + 1 ( μ ) μ μ j = r = 0 N ( 2 r + 1 ) P r ( μ j ) P r ( μ ) ( N + 1 ) P N ( μ j ) .
R j ( μ ) = ( 1 μ j 2 ) [ ( N + 1 ) P N ( μ j ) ] 2 r = 0 N ( 2 r + 1 ) P r ( μ j ) P r ( μ ) .
1 1 P n P r d μ = 2 δ r n 2 n + 1 .
p d ( ζ ) = P π arccos ζ 1 ν 2 π d ν = P sin 2 ζ ,
p ( μ ) = μ 1 μ R ( μ ) ( 2 π μ d μ ) .
P sin 2 ζ = p ( μ ) .
P = p ( μ 2 ) / sin 2 ζ m x .
I = ζ + η , R = θ η η = ½ ( θ ζ ) .
d z / d ρ = tan η .
tan ζ = ρ old + Δ ρ z old + Δ z ,
Δ z = Δ ρ d z / d ρ = Δ ρ tan η ,
Δ ρ = z old tan ζ ρ old 1 tan ζ tan η .
ρ new = ρ old + 2 Δ ρ , z new = z old + 2 Δ z = z old + 2 tan η Δ ρ . }

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