Abstract
In this paper the functions involved in the emission and reception of secondary scattered (rescattered) sunlight are discussed. Index-sum approximations of the form
are obtained for the integrals Ejn(x), defined by
The factors βnr are determined numerically by Prony’s method of interpolation by exponentials, and the coefficients Bnr by the method of least squares and Seidel’s method of iteration. The functions Ejn(x) occur as factors in the integrands of the integrals
involved in the emission of secondary scattered light. By means of the index-sum approximations for the integrals Ejn(x), approximate formulas for the functions Ln are obtained. Numerical tables are given for Ejn and Ln, showing the high degree of accuracy in our approximations. Likewise, the functions Ln occur as factors in the integrands of the integrals Lun and Ldn involved in the reception of secondary scattered light. These integrals are of the type
Hence, approximate formulas for such integrals can be obtained in which the degree of accuracy is higher than that obtained for the functions Ln.
According to the approximate formulas given in this paper, an estimate can be made for secondary scattered sunlight in which the error is less than one percent of the exact value.
The formulas apply to all solar altitudes and to a wide range of wave-lengths in the solar spectrum, in contrast with previously obtained approximate formulas which apply to weakly attenuated radiations and high solar altitudes.
© 1949 Optical Society of America
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