Abstract

Experimental measurements of the transmission properties of New England haze and fog are correlated with a modification of the single scatter theory. The modified single scatter theory is derived and its consequences are compared graphically with measured range dependent data and the angular distribution of scattered flux. The results may be summarized within the limits of 0.04 < σ <16 (km−1) and 0.2 < D < 10 (km) by

T=eσD+0.3σ0.5Deσ0.3D,

where T is the transmission function for radiant energy intercepted by a 2π detector, σ the atmospheric attenuation coefficient, and D the range from the isotropic source.

© 1962 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. S. Stewart and J. A. Curcio, J. Opt. Soc. Am. 42, 801 (1952).
    [Crossref]
  2. M. G. Gibbons, J. Opt. Soc. Am. 48, 550 (1958).
    [Crossref]
  3. M. G. Gibbons, J. Opt. Soc. Am. 49, 702 (1959).
    [Crossref]
  4. M. G. Gibbons, J. R. Nichols, F. I. Laughridge, and R. L. Rudkin, J. Opt. Soc. Am. 51, 633 (1961).
    [Crossref]
  5. R. G. Eldridge and J. C. Johnson, J. Opt. Soc. Am. 48, 463 (1958).
    [Crossref]
  6. B and J 4 × 5 press view cameras with Schneider-Kreuznach Xenar f/3.5, 105-mm lens in Synchro-Compur-P shutter, and Kodak Tri-X film in pack form.
  7. Equations (5) and (6) of Gibbons [M. G. Gibbons, J. Opt. Soc. Am. 48, 550 (1958)].
    [Crossref]
  8. C. Wiener, Abhandl. Naturforsch. 73, 1–240 (1907).
  9. R. O. Gumprecht, Neng-Lun Sun, Jin H. Chin, and C. M. Sliepcevich, J. Opt. Soc. Am. 42, 226–231 (1952).
    [Crossref]

1961 (1)

1959 (1)

1958 (3)

1952 (2)

1907 (1)

C. Wiener, Abhandl. Naturforsch. 73, 1–240 (1907).

Chin, Jin H.

Curcio, J. A.

Eldridge, R. G.

Gibbons, M. G.

Gumprecht, R. O.

Johnson, J. C.

Laughridge, F. I.

Nichols, J. R.

Rudkin, R. L.

Sliepcevich, C. M.

Stewart, H. S.

Sun, Neng-Lun

Wiener, C.

C. Wiener, Abhandl. Naturforsch. 73, 1–240 (1907).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

The measured direct (solid curves) and scattered (dashed curves) irradiance in light haze (A—□ B—Δ, and C—○) as a function of distance.

Fig. 2
Fig. 2

The measured direct (solid curves) and scattered (dashed curves) irradiance in moderate haze (D—○) and dense haze and smoke (E—□) as a function of distance.

Fig. 3
Fig. 3

The measured direct (solid curves) and scattered (dashed curves) irradiance in dense fog (F—□, G—○, and H—Δ) as a function of distance.

Fig. 4
Fig. 4

The geometry of single scatter theory.

Fig. 5
Fig. 5

Comparison of differential scattering cross section as a function of the forward scattering angle for Mie theory (— – —), Wiener plus diffraction function (– – – –), and the postulated function (——).

Fig. 6
Fig. 6

Schematic representation of the limits of the phase function as applied to modified single scatter theory.

Fig. 7
Fig. 7

The radiance functions F1, F2, and 1 2 ( e π sin Ψ ) as a function of the acceptance half-angle.

Fig. 8
Fig. 8

The irradiance function G1, 1 − eπsinΨ, and 1 − e as a function of the acceptance half-angle.

Fig. 9
Fig. 9

Experimental relative radiance n1D2, where n1 = N1/J0, as a function of the acceptance half-angle for all atmospheres. The solid curve represents the postulated radiance derived from modified single scatter theory. The numbers in parentheses indicate the number of data points included in the elongated circles.

Fig. 10
Fig. 10

Trend of the experimental relative radiance as a function of the acceptance half-angle (solid curves) for atmospheres of light haze. Each atmosphere is identified by the letters A, B, and C, at the terminus of the curve. The dashed curve represents the postulated radiance derived from modified single scatter theory.

Fig. 11
Fig. 11

Trend of the experimental relative radiance as a function of the acceptance half-angle (solid curves) for atmospheres of moderate haze D and dense haze and smoke E. Each atmosphere is identified by the letters at the terminus of the curves. The dashed curve represents the postulated radiance derived from modified single scatter theory.

Fig. 12
Fig. 12

Trend of the experimental relative radiance as a function of the acceptance half-angle (solid curve) for atmosphere of dense fog. Each atmosphere is identified by the letters F, G, and H at the terminus of the curve. The dashed curve represents the postulated radiance derived from modified single scatter theory.

Fig. 13
Fig. 13

Relative scattered irradiance as a function of slant range for atmospheres of light and moderate haze, dense haze and smoke, and dense fog.

Fig. 14
Fig. 14

Correlation of the constant K with the experimental atmospheric attenuation coefficients.

Fig. 15
Fig. 15

Correlation of the experimental scattering attenuation coefficient with σ0.3.

Fig. 16
Fig. 16

Correlation of the constant with the experimental values of the zero range intercept of the relative irradiance.

Fig. 17
Fig. 17

Correlation of the values of the experimental and computed [Eq. (23)] irradiance.

Fig. 18
Fig. 18

The transmission function [Eq. (24)] plotted as a function of the slant range (solid curves) for various atmospheric opacities. The contribution from the direct flux [first term in Eq. (24)] is plotted for comparison (dashed curves). The number adjacent to each curve gives the value of σ corresponding to the curve. The difference between the solid curve and the dashed curve for a given value of σ represents the “buildup” caused by scattered-in flux [second term of Eq. (24)].

Tables (1)

Tables Icon

Table I Atmospheric parameters.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

T = e σ D + 0.3 σ 0.5 D e σ 0.3 D ,
H dir = E dir A dir S c F
H si = E si A si S c F ,
σ = ( 1 / D ) ln ( 1 / h dir D 2 ) .
H si = Ω N cos Ψ d Ω .
H si = g J 0 k σ 0 υ B ( Φ ) e ( r 1 + r 2 ) σ cos Ψ d υ r 1 2 r 2 2 .
H s i = g J 0 k σ D 0 V B ( Φ ) e ( R 1 + R 2 ) D σ cos Ψ d V R 1 2 R 2 2 .
d V = r 2 2 d Ω d r 2 / D 3 = R 2 2 d R 2 d Ω .
H si = Ω [ g J 0 k σ D 0 R 2 B ( Φ ) e ( R 1 + R 2 ) D σ R 1 2 d R 2 ] cos Ψ d Ω .
N = g J 0 k σ D 0 R 2 B ( Φ ) e ( R 1 + R 2 ) D σ R 1 2 d R 2
Φ = sin 1 { sin Ψ / [ ( cos Ψ R 2 ) 2 + sin 2 Ψ ] 1 2 } ,
R 1 = [ ( cos Ψ R 2 ) 2 + sin 2 Ψ ] 1 2 ,
N = g J 0 k σ D 0 R 2 B ( R 2 , Ψ ) × exp ( { R 2 + [ ( cos Ψ R 2 ) + sin 2 Ψ ] 1 2 } D σ ) ( cos Ψ R 2 ) 2 + sin 2 Ψ d R 2.
H = J 0 ( e σ D / D 2 ) + H si .
B 1 ( Φ ) = 3.35 cos 6 Φ 2.65 cos 4 Φ + 0.60 cos 2 Φ ( 0 Φ 1 2 π ) , B 1 ( Φ ) = 0 ( 1 2 π Φ π ) ,
B 2 ( Φ ) = 0.82 ( α 2 / 4 π ) [ 1 ( α sin Φ / 3.83 ) ] ( 0 α sin Φ / 3.83 1 ) , B 2 ( Φ ) = 0 ( α sin Φ 3.83 ) ,
R 1 + R 2 = R 2 + [ ( cos Ψ R 2 ) 2 + sin 2 Ψ ] 1 2
N = g J 0 k σ D e K σ D R 2 0 B ( R 2 , Ψ ) d R 2 ( cos Ψ R 2 ) 2 + sin 2 Ψ
N 1 = g k J 0 σ D e K σ D sin Ψ F 1 ( Ψ ) ( 0 < Ψ < 1 2 π ) .
H 1 = g k J 0 ( σ / D ) e K σ D G 1 ( Ψ ) ( 0 < Ψ < 1 2 π ) ,
h si = H si / J 0 = C ( σ / D ) e K σ D ,
K σ D m = 1.
K = σ 0.7 .
C = 0.3 σ 0.5 .
h si = ( 0.3 σ 0.5 / D ) e σ 0.3 D .
h = h dir + h si = e σ D D 2 + 0.3 σ 0.5 D e σ 0.3 D