The Götz inversion of intensity-ratio in zenith-scattered sunlight

Abstract

Gotz, Meetham, and Dobson have recently studied the distribution of ozone in the earth’s atmosphere by a new method, comparing the intensities, at two wave-lengths, of ultra-violet scattered zenith-light. Regener in a much more direct manner has found strikingly similar results, using a spectrograph carried high into the stratosphere by a balloon. The zenith-light method has great importance for the study of the ozone-distribution, and may be applicable, in modified form, to other atmospheric problems, possibly including problems of radio waves scattered and absorbed in the ionosphere. But it is laborious and in some respects difficult to determine the ozone-distribution from the zenith-light measurements. It seems desirable, and the need has been urged upon me by Dr. Dobson, to examine further some of the mathematical questions involved in the method. This is attempted in the present paper, subject to the limitation that only primary scattered light is considered. The effect of secondary scattered light remains to be discussed in a further paper. The interpretation of their work by Gotz, Meetham, and Dobson has been questioned by Pekeris. His criticism is here shown to be invalid (7.2, 7.21). The plan of this paper is to consider certain special distributions of ozone for which the mathematical analysis can be carried to an advanced stage, though in some cases numerical integration must be resorted to at the final stage. In most of the cases considered, but not all, the air density is supposed to vary exponentially with the height. Complete numerical results have been calculated in a number of cases, mainly referring to an atmosphere on a flat earth ; but some cases relating to a spherical earth have been worked out numerically, and these suffice to show, in a general way, how the curvature of the earth affects the results.