Abstract

A new method is proposed for the calibration of the sunphotometer. Well-known difficulties of the usual Langley-plot method when applied to unsteady turbidity conditions can be avoided by monitoring the circumsolar radiation. To realize this idea, an alternate of the Langley-plot method is developed, in which the logarithm of the sunphotometer reading is plotted against the ratio of intensity of singly scattered circumsolar radiation to that of direct solar radiation instead of the optical air mass in the usual Langley-plot method. Results of numerical simulations and field tests with a newly developed instrument show that the rms error of the calibration constant could be reduced to 1/5–1/10 of the usual method for wavelengths larger than 500 nm.

© 1986 Optical Society of America

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References

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  1. F. E. Volz, “Photometer snit selenphotoelement zur bestimmung der wellenlaugenabhangegkeit der dunsthulung,” Arch. Meteorol. Geophys. Bioklimatol. Ser. B 10, 100 (1959).
    [CrossRef]
  2. E. C. Flowers, R. A. McCormick, K. R. Kurfis, “Atmospheric Turbidity over the U.S.1961–1966,” J. Appl. Meteorol. 8, 955 (1969).
    [CrossRef]
  3. E. C. Flowers, R. A. McCormick, K. R. Kurfis, T. H. Bilton, “Atmospheric Turbidity Measurements with the Dual-Wavelength Sunphotometer. Observation and Measurement of Atmospheric Pollution,” Special Environ. Rep. 3, World Meteorological Organization, Geneva (1974), pp. 463–480.
  4. J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
    [CrossRef]
  5. G. E. Shaw, “Aerosols at Mauna Loa: Optical Properties,” J. Atmos. Sci. 36, 862 (1979).
    [CrossRef]
  6. J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Sahara Aerosol Transport by Means of Atmospheric Turbidity Measurements,” in Saharan Dust: Mobilization, Transport, Deposition, SCOPE Rep. 14, C. Morales, Ed. (Wiley, Chichester, England, (1979), pp. 171–186.
  7. N. S. Laulainen, B. J. Taylor, “The Precision and Accuracy of Volz Sunphotometry,” J. Appl. Meteorol. 13, 298 (1974).
    [CrossRef]
  8. P. B. Russell, G. E. Shaw, “Comments on ‘The Precision and Accuracy of Volz Sunphotometry,’ ” J. Appl. Meteorol. 14, 1206 (1975).
    [CrossRef]
  9. F. E. Volz, “Precision and Accuracy of Sunphotometry—A response,” J. Appl. Meteorol. 14, 1209 (1975).
    [CrossRef]
  10. E. Dutton, J. DeLuisi, “Results of a Sunphotometer Intercomparison held at Boulder, 19 October to 16 December 1981,” NOAA Tech. Memor. ERL ARL-114 (1982).
  11. G. E. Shaw, “Error Analysis of Multi-wavelength Sunphotometry,” Pure Appl. Geophys. 114, 1 (1976).
    [CrossRef]
  12. G. E. Shaw, “Sun Photometry,” Bull. Am. Meteorol. Soc. 64, 4 (1983).
    [CrossRef]
  13. N. T. O'Neill, J. R. Miller, “Combined Solar Aureole and Solar Beam Extinction Measurements. 1: Calibration Considerations,” Appl. Opt. 23, 3691 (1984).
    [CrossRef]
  14. S. Kawaguchi, T. Nakazawa, T. Nakajima, M. Tanaka, T. Yamanouchi, “Remote Sensing of the Antarctic Atmosphere Using a Sunphotometer,” Annual Report of Co-operative Research, 1981.Annual Report of Co-operative ResearchNational Institute of Polar Research (1982) p.101.
  15. C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithsonian Inst. 4 (1922).
  16. C. G. Abbot, L. B. Aldrich, F. E. Fowle, Ann. Astrophys. Obs. Smithsonian Inst. 5 (1923).
  17. D. Deirmendjian, “Use of Scattering Techniques in Cloud Mi-crophysics Research: I. The Aureole Method,” Rand Report R-590-PROct.1970,The Rand Corp., Santa Monica, CA.
  18. T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the Optical Properties of Aerosols from Aureole and Extinction Data,” Appl. Opt. 22, 2951 (1983).
    [CrossRef] [PubMed]
  19. M. Tanaka, T. Takamura, T. Nakajima, “Refractive Index and Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Clim. Appl. Meteorol. 22, 1253 (1983).
    [CrossRef]

1984 (1)

1983 (3)

M. Tanaka, T. Takamura, T. Nakajima, “Refractive Index and Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Clim. Appl. Meteorol. 22, 1253 (1983).
[CrossRef]

T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the Optical Properties of Aerosols from Aureole and Extinction Data,” Appl. Opt. 22, 2951 (1983).
[CrossRef] [PubMed]

G. E. Shaw, “Sun Photometry,” Bull. Am. Meteorol. Soc. 64, 4 (1983).
[CrossRef]

1981 (1)

J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
[CrossRef]

1979 (1)

G. E. Shaw, “Aerosols at Mauna Loa: Optical Properties,” J. Atmos. Sci. 36, 862 (1979).
[CrossRef]

1976 (1)

G. E. Shaw, “Error Analysis of Multi-wavelength Sunphotometry,” Pure Appl. Geophys. 114, 1 (1976).
[CrossRef]

1975 (2)

P. B. Russell, G. E. Shaw, “Comments on ‘The Precision and Accuracy of Volz Sunphotometry,’ ” J. Appl. Meteorol. 14, 1206 (1975).
[CrossRef]

F. E. Volz, “Precision and Accuracy of Sunphotometry—A response,” J. Appl. Meteorol. 14, 1209 (1975).
[CrossRef]

1974 (1)

N. S. Laulainen, B. J. Taylor, “The Precision and Accuracy of Volz Sunphotometry,” J. Appl. Meteorol. 13, 298 (1974).
[CrossRef]

1969 (1)

E. C. Flowers, R. A. McCormick, K. R. Kurfis, “Atmospheric Turbidity over the U.S.1961–1966,” J. Appl. Meteorol. 8, 955 (1969).
[CrossRef]

1959 (1)

F. E. Volz, “Photometer snit selenphotoelement zur bestimmung der wellenlaugenabhangegkeit der dunsthulung,” Arch. Meteorol. Geophys. Bioklimatol. Ser. B 10, 100 (1959).
[CrossRef]

1923 (1)

C. G. Abbot, L. B. Aldrich, F. E. Fowle, Ann. Astrophys. Obs. Smithsonian Inst. 5 (1923).

1922 (1)

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithsonian Inst. 4 (1922).

Abbot, C. G.

C. G. Abbot, L. B. Aldrich, F. E. Fowle, Ann. Astrophys. Obs. Smithsonian Inst. 5 (1923).

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithsonian Inst. 4 (1922).

Aldrich, L. B.

C. G. Abbot, L. B. Aldrich, F. E. Fowle, Ann. Astrophys. Obs. Smithsonian Inst. 5 (1923).

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithsonian Inst. 4 (1922).

Berri, G. J.

J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
[CrossRef]

Bilton, T. H.

E. C. Flowers, R. A. McCormick, K. R. Kurfis, T. H. Bilton, “Atmospheric Turbidity Measurements with the Dual-Wavelength Sunphotometer. Observation and Measurement of Atmospheric Pollution,” Special Environ. Rep. 3, World Meteorological Organization, Geneva (1974), pp. 463–480.

Carlson, T. N.

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Sahara Aerosol Transport by Means of Atmospheric Turbidity Measurements,” in Saharan Dust: Mobilization, Transport, Deposition, SCOPE Rep. 14, C. Morales, Ed. (Wiley, Chichester, England, (1979), pp. 171–186.

Deirmendjian, D.

D. Deirmendjian, “Use of Scattering Techniques in Cloud Mi-crophysics Research: I. The Aureole Method,” Rand Report R-590-PROct.1970,The Rand Corp., Santa Monica, CA.

DeLuisi, J.

E. Dutton, J. DeLuisi, “Results of a Sunphotometer Intercomparison held at Boulder, 19 October to 16 December 1981,” NOAA Tech. Memor. ERL ARL-114 (1982).

Dutton, E.

E. Dutton, J. DeLuisi, “Results of a Sunphotometer Intercomparison held at Boulder, 19 October to 16 December 1981,” NOAA Tech. Memor. ERL ARL-114 (1982).

Flowers, E. C.

J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
[CrossRef]

E. C. Flowers, R. A. McCormick, K. R. Kurfis, “Atmospheric Turbidity over the U.S.1961–1966,” J. Appl. Meteorol. 8, 955 (1969).
[CrossRef]

E. C. Flowers, R. A. McCormick, K. R. Kurfis, T. H. Bilton, “Atmospheric Turbidity Measurements with the Dual-Wavelength Sunphotometer. Observation and Measurement of Atmospheric Pollution,” Special Environ. Rep. 3, World Meteorological Organization, Geneva (1974), pp. 463–480.

Fowle, F. E.

C. G. Abbot, L. B. Aldrich, F. E. Fowle, Ann. Astrophys. Obs. Smithsonian Inst. 5 (1923).

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithsonian Inst. 4 (1922).

Kawaguchi, S.

S. Kawaguchi, T. Nakazawa, T. Nakajima, M. Tanaka, T. Yamanouchi, “Remote Sensing of the Antarctic Atmosphere Using a Sunphotometer,” Annual Report of Co-operative Research, 1981.Annual Report of Co-operative ResearchNational Institute of Polar Research (1982) p.101.

Kurfis, K. R.

E. C. Flowers, R. A. McCormick, K. R. Kurfis, “Atmospheric Turbidity over the U.S.1961–1966,” J. Appl. Meteorol. 8, 955 (1969).
[CrossRef]

E. C. Flowers, R. A. McCormick, K. R. Kurfis, T. H. Bilton, “Atmospheric Turbidity Measurements with the Dual-Wavelength Sunphotometer. Observation and Measurement of Atmospheric Pollution,” Special Environ. Rep. 3, World Meteorological Organization, Geneva (1974), pp. 463–480.

Laulainen, N. S.

N. S. Laulainen, B. J. Taylor, “The Precision and Accuracy of Volz Sunphotometry,” J. Appl. Meteorol. 13, 298 (1974).
[CrossRef]

McCormick, R. A.

E. C. Flowers, R. A. McCormick, K. R. Kurfis, “Atmospheric Turbidity over the U.S.1961–1966,” J. Appl. Meteorol. 8, 955 (1969).
[CrossRef]

E. C. Flowers, R. A. McCormick, K. R. Kurfis, T. H. Bilton, “Atmospheric Turbidity Measurements with the Dual-Wavelength Sunphotometer. Observation and Measurement of Atmospheric Pollution,” Special Environ. Rep. 3, World Meteorological Organization, Geneva (1974), pp. 463–480.

Miller, J. R.

Nakajima, T.

M. Tanaka, T. Takamura, T. Nakajima, “Refractive Index and Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Clim. Appl. Meteorol. 22, 1253 (1983).
[CrossRef]

T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the Optical Properties of Aerosols from Aureole and Extinction Data,” Appl. Opt. 22, 2951 (1983).
[CrossRef] [PubMed]

S. Kawaguchi, T. Nakazawa, T. Nakajima, M. Tanaka, T. Yamanouchi, “Remote Sensing of the Antarctic Atmosphere Using a Sunphotometer,” Annual Report of Co-operative Research, 1981.Annual Report of Co-operative ResearchNational Institute of Polar Research (1982) p.101.

Nakazawa, T.

S. Kawaguchi, T. Nakazawa, T. Nakajima, M. Tanaka, T. Yamanouchi, “Remote Sensing of the Antarctic Atmosphere Using a Sunphotometer,” Annual Report of Co-operative Research, 1981.Annual Report of Co-operative ResearchNational Institute of Polar Research (1982) p.101.

Nees, R. T.

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Sahara Aerosol Transport by Means of Atmospheric Turbidity Measurements,” in Saharan Dust: Mobilization, Transport, Deposition, SCOPE Rep. 14, C. Morales, Ed. (Wiley, Chichester, England, (1979), pp. 171–186.

O'Neill, N. T.

Peterson, J. T.

J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
[CrossRef]

Prospero, J. M.

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Sahara Aerosol Transport by Means of Atmospheric Turbidity Measurements,” in Saharan Dust: Mobilization, Transport, Deposition, SCOPE Rep. 14, C. Morales, Ed. (Wiley, Chichester, England, (1979), pp. 171–186.

Reynolds, C. L.

J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
[CrossRef]

Rudisill, J. H.

J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
[CrossRef]

Russell, P. B.

P. B. Russell, G. E. Shaw, “Comments on ‘The Precision and Accuracy of Volz Sunphotometry,’ ” J. Appl. Meteorol. 14, 1206 (1975).
[CrossRef]

Savoie, D. L.

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Sahara Aerosol Transport by Means of Atmospheric Turbidity Measurements,” in Saharan Dust: Mobilization, Transport, Deposition, SCOPE Rep. 14, C. Morales, Ed. (Wiley, Chichester, England, (1979), pp. 171–186.

Shaw, G. E.

G. E. Shaw, “Sun Photometry,” Bull. Am. Meteorol. Soc. 64, 4 (1983).
[CrossRef]

G. E. Shaw, “Aerosols at Mauna Loa: Optical Properties,” J. Atmos. Sci. 36, 862 (1979).
[CrossRef]

G. E. Shaw, “Error Analysis of Multi-wavelength Sunphotometry,” Pure Appl. Geophys. 114, 1 (1976).
[CrossRef]

P. B. Russell, G. E. Shaw, “Comments on ‘The Precision and Accuracy of Volz Sunphotometry,’ ” J. Appl. Meteorol. 14, 1206 (1975).
[CrossRef]

Takamura, T.

M. Tanaka, T. Takamura, T. Nakajima, “Refractive Index and Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Clim. Appl. Meteorol. 22, 1253 (1983).
[CrossRef]

Tanaka, M.

M. Tanaka, T. Takamura, T. Nakajima, “Refractive Index and Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Clim. Appl. Meteorol. 22, 1253 (1983).
[CrossRef]

T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the Optical Properties of Aerosols from Aureole and Extinction Data,” Appl. Opt. 22, 2951 (1983).
[CrossRef] [PubMed]

S. Kawaguchi, T. Nakazawa, T. Nakajima, M. Tanaka, T. Yamanouchi, “Remote Sensing of the Antarctic Atmosphere Using a Sunphotometer,” Annual Report of Co-operative Research, 1981.Annual Report of Co-operative ResearchNational Institute of Polar Research (1982) p.101.

Taylor, B. J.

N. S. Laulainen, B. J. Taylor, “The Precision and Accuracy of Volz Sunphotometry,” J. Appl. Meteorol. 13, 298 (1974).
[CrossRef]

Volz, F. E.

F. E. Volz, “Precision and Accuracy of Sunphotometry—A response,” J. Appl. Meteorol. 14, 1209 (1975).
[CrossRef]

F. E. Volz, “Photometer snit selenphotoelement zur bestimmung der wellenlaugenabhangegkeit der dunsthulung,” Arch. Meteorol. Geophys. Bioklimatol. Ser. B 10, 100 (1959).
[CrossRef]

Yamanouchi, T.

S. Kawaguchi, T. Nakazawa, T. Nakajima, M. Tanaka, T. Yamanouchi, “Remote Sensing of the Antarctic Atmosphere Using a Sunphotometer,” Annual Report of Co-operative Research, 1981.Annual Report of Co-operative ResearchNational Institute of Polar Research (1982) p.101.

Yamauchi, T.

Ann. Astrophys. Obs. Smithsonian Inst. (2)

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithsonian Inst. 4 (1922).

C. G. Abbot, L. B. Aldrich, F. E. Fowle, Ann. Astrophys. Obs. Smithsonian Inst. 5 (1923).

Appl. Opt. (2)

Arch. Meteorol. Geophys. Bioklimatol. Ser. B (1)

F. E. Volz, “Photometer snit selenphotoelement zur bestimmung der wellenlaugenabhangegkeit der dunsthulung,” Arch. Meteorol. Geophys. Bioklimatol. Ser. B 10, 100 (1959).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

G. E. Shaw, “Sun Photometry,” Bull. Am. Meteorol. Soc. 64, 4 (1983).
[CrossRef]

J. Appl. Meteorol. (5)

E. C. Flowers, R. A. McCormick, K. R. Kurfis, “Atmospheric Turbidity over the U.S.1961–1966,” J. Appl. Meteorol. 8, 955 (1969).
[CrossRef]

J. T. Peterson, E. C. Flowers, G. J. Berri, C. L. Reynolds, J. H. Rudisill, “Atmospheric Turbidity over Central North Carolina,” J. Appl. Meteorol. 20, 229 (1981).
[CrossRef]

N. S. Laulainen, B. J. Taylor, “The Precision and Accuracy of Volz Sunphotometry,” J. Appl. Meteorol. 13, 298 (1974).
[CrossRef]

P. B. Russell, G. E. Shaw, “Comments on ‘The Precision and Accuracy of Volz Sunphotometry,’ ” J. Appl. Meteorol. 14, 1206 (1975).
[CrossRef]

F. E. Volz, “Precision and Accuracy of Sunphotometry—A response,” J. Appl. Meteorol. 14, 1209 (1975).
[CrossRef]

J. Atmos. Sci. (1)

G. E. Shaw, “Aerosols at Mauna Loa: Optical Properties,” J. Atmos. Sci. 36, 862 (1979).
[CrossRef]

J. Clim. Appl. Meteorol. (1)

M. Tanaka, T. Takamura, T. Nakajima, “Refractive Index and Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Clim. Appl. Meteorol. 22, 1253 (1983).
[CrossRef]

Pure Appl. Geophys. (1)

G. E. Shaw, “Error Analysis of Multi-wavelength Sunphotometry,” Pure Appl. Geophys. 114, 1 (1976).
[CrossRef]

Other (5)

S. Kawaguchi, T. Nakazawa, T. Nakajima, M. Tanaka, T. Yamanouchi, “Remote Sensing of the Antarctic Atmosphere Using a Sunphotometer,” Annual Report of Co-operative Research, 1981.Annual Report of Co-operative ResearchNational Institute of Polar Research (1982) p.101.

D. Deirmendjian, “Use of Scattering Techniques in Cloud Mi-crophysics Research: I. The Aureole Method,” Rand Report R-590-PROct.1970,The Rand Corp., Santa Monica, CA.

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Sahara Aerosol Transport by Means of Atmospheric Turbidity Measurements,” in Saharan Dust: Mobilization, Transport, Deposition, SCOPE Rep. 14, C. Morales, Ed. (Wiley, Chichester, England, (1979), pp. 171–186.

E. C. Flowers, R. A. McCormick, K. R. Kurfis, T. H. Bilton, “Atmospheric Turbidity Measurements with the Dual-Wavelength Sunphotometer. Observation and Measurement of Atmospheric Pollution,” Special Environ. Rep. 3, World Meteorological Organization, Geneva (1974), pp. 463–480.

E. Dutton, J. DeLuisi, “Results of a Sunphotometer Intercomparison held at Boulder, 19 October to 16 December 1981,” NOAA Tech. Memor. ERL ARL-114 (1982).

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Figures (6)

Fig. 1
Fig. 1

Normalized phase functions of aerosols (p = 3.5 ∼ 5) and air molecules (dotted line). The parameter p is the exponent of the size distribution function of aerosols in Eq. (10). The index of refraction of aerosols is assumed to be m = 1.5 0.01 i

Fig. 2
Fig. 2

Single-scattering ratio R as a function of the optical air mass m for λ = 500 nm, τm = 0.139, Θ = 20°, m = 1.5 0.01 i, and various values of τ0, p, and A. The upper panel shows the dependence of R on τa and p at A = 0.1 and the lower panel the dependence of R on τa and A at p = 4.0.

Fig. 3
Fig. 3

Iteration scheme to determine the calibration constant from measurements of direct-solar and circumsolar radiations.

Fig. 4
Fig. 4

Absolute values of dF0 (= | F0F0* |) vs F0*. The solid line corresponds to λ = 500 nm and τa = 0.05 and the dashed line to λ = 369 nm and τa = 0.068.

Fig. 5
Fig. 5

Comparisons of the Langley-plot method (left panel) and the present method (right panel) at λ = 500 nm for turbidity conditions given in Eq. (9) with the optical thickness of aerosols at noon τa0 = 0.2 and Shaw's parabolic drift parameter α = 0, ●; 0.011, △; and −0.011, ○.

Fig. 6
Fig. 6

Comparisons of the Langley-plot method (left panel) and the present method (right panel) at λ = 500 nm for measured data of 15 Oct. 1981, ○; 10 Nov. 1981, ●; 16 Feb. 1982, △; and 18 Feb. 1982, ▲.

Tables (5)

Tables Icon

Table I Comparison of Simulated Calibration Between the Present and Langley-plot Methods for Turbidity Conditions Varying Parabolically with Time*

Tables Icon

Table II Comparison of Simulated Calibration Between the Present and Langley-plot Methods for Size Distributions Differing from Assumed One*

Tables Icon

Table III Results of Simulated Calibration for the Complex Index of Refraction Differing from the Assumed Value of 1.5–0.01/

Tables Icon

Table IV Specifications of the Aureolemeter

Tables Icon

Table V Calibration of the Aureolemeter by the Present and Langley-plot Methods*

Equations (12)

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

ln F = ln F 0 m τ ,
F a 1 ( μ 0 , ϕ ) = m τ ω 0 P ( cos Θ ) F 0 exp ( m τ ) Δ Ω ,
τ = τ a + τ m ,
ω 0 = ( ω 0 τ a + ω 0 m τ m ) / τ ,
P ( cos Θ ) = [ ω 0 a τ a P a ( cos Θ ) + ω 0 m τ m P m ( cos Θ ) ] / ω 0 τ ,
2 π 1 1 P ( x ) d x = 1 .
ln F = ln F 0 τ * ,
τ * = m τ = F a 1 / [ F Δ Ω ω 0 P ( cos Θ ) ] .
F a 1 = R ( m , τ a , τ m , m , A , Θ ) F a ,
| F 0 ( i ) F 0 * | | d F 0 ( i ) | <
n ( r ) = { C 10 p for r < 0.1 μ m , C r p for r > 0.1 μ m ,
τ a ( t ) = τ a 0 ( 1 + α t 2 ) ,

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