Abstract

The dependence on bandwidth of the error in estimating both optical depth and extraterrestrial solar flux due to an assumption of Beer’s law applicability across finite bandwidths is determined by a numerical integration of Beer’s law across the bandwidth. It was found that for 0.1% accuracy, 100-Å bandwidths suffice for central wavelengths of 0.45 μm or greater; the maximum width yielding 0.1% accuracy decreases rapidly for shorter wavelengths. The accuracy to which solar elevation angle must be known to yield 0.1% accuracy is also examined and found to be a noteworthy though noncritical effect.

© 1982 Optical Society of America

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References

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  1. M. A. Box, Appl. Opt. 20, 2215 (1981).
    [CrossRef] [PubMed]
  2. M. P. Thekaekara, Appl. Opt. 13, 518 (1974).
    [CrossRef] [PubMed]

1981

1974

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Figures (9)

Fig. 1
Fig. 1

Error in estimation of solar intensity caused by assuming Beer’s law as a function of bandwidth for a Planckian source.

Fig. 2
Fig. 2

Same as Fig. 1 except for a source from the resolved spectrum.

Fig. 3
Fig. 3

Maximum bandwidth for 0.1% accuracy in estimation of solar intensity as a function of wavelength for a Planckian source.

Fig. 4
Fig. 4

Error in estimation of the optical depth at the central wavelength caused by assuming Beer’s law as a function of bandwidth for a Planckian source.

Fig. 5
Fig. 5

Same as Fig. 4 except for a source from the resolved spectrum.

Fig. 6
Fig. 6

Maximum bandwidth for 0.1% accuracy in estimation of the central wavelength optical depth as a function of wavelength for a Planckian source.

Fig. 7
Fig. 7

Same as Fig. 6 except for a source from the resolved spectrum.

Fig. 8
Fig. 8

Maximum uncertainty in the zenith angle for 0.1% accuracy in intensity measurements.

Fig. 9
Fig. 9

Maximum uncertainty in local time for 0.1% accuracy in intensity measurements for Tucson, Ariz., at various times of the year.

Equations (14)

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F λ ( τ λ ) = F λ ( 0 ) exp ( - m τ λ ) ,
F ( m ) = c 0 F λ ( 0 ) exp ( - m τ λ ) f ( λ - λ 0 ) d λ ,
F ( m ) = c i = 1 N F ( λ i ) exp [ - m τ ( λ i ) ] f ( λ i - λ 0 ) Δ λ .
f ( λ i - λ 0 ) = 1 2 π σ exp [ - ½ ( λ i - λ 0 ) 2 / σ 2 ] ,
τ ( λ i ) = τ R ( λ i ) + τ M ( λ i ) ,
τ R ( λ i ) = τ R a ( λ i λ a ) - 4 ,
τ M ( λ i ) = τ M a ( λ i λ a ) - 1 ,
F ( m ) = c 1 i = 1 N F ( λ i ) exp { - m [ 0.116 ( λ i 0.5 ) - 4 + 0.03 ( λ i 0.5 ) - 1 ] - 2 [ ( λ i - λ 0 ) 2 ] / B 2 ( 1.1775 ) 2 } ( 2.335 Δ λ / B ) ,
F ( 0 ) exact - F ( 0 ) calc F ( 0 ) exact , τ ( λ 0 ) - τ calc τ ( λ 0 ) .
F λ ( τ λ ) = F λ ( 0 ) exp ( - τ λ sec θ ) .
d F λ ( τ λ ) F λ ( τ λ ) = - τ λ d ( sec θ ) = - τ λ | sin θ d θ cos 2 θ | .
Δ θ max = 0.001 cos 2 θ sin θ
Δ t max = Δ θ max ( d θ / d t ) - 1 .
cos θ = sin ϕ sin δ + cos ϕ cos δ cos h ,

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