Do observations of sky illumination at noon confirm the angular dependence of Rayleigh scattering?

I wondered about the possibility of mathematically modeling the the magnitude of the blue component of sky illumination. As is well known, Rayleigh scattering of sunlight, primarily by Nitrogen ( $ N_2 $ ) molecules (due to their abundance in the earth atmosphere and high scattering cross section $\sigma_{scattering}$ ), is strongest for the shorther wavelengths ( $I_{scattering}\propto \frac{1}{\lambda^4}$ ), and this fact alone explains why the sky appear blue.

While thinking about this problem I wondered if, after applying some analytic procedures (which will be explained below), experimental observations of the brightness of blue component of the sky when the sun is at the zenith also confirm the angular dependence of Rayleigh's formula for scattered light power, i.e $P(\theta)\propto (1+\cos^2\theta)$. My main argument why this formula is relevant to the sky illumination problem is that most of the skylight is scattered only once by the atmosphere, because the product of the numerical volume density of scatterers $n$ by their effective cross section $\sigma_{scattering}$ and the thickness $H$ of the atmosphere is apparently low, so sunlight undergoes only one (Rayleigh) scattering. In other words, multiple scattering theory is irrelevant here.

However, this problem is more complex then it seems at first sight, since the amount of blue light arriving to the earth observer also depends on the length of the part of the line of sight that is inside earth atmosphere (these lines of sight represent different viewing directions for the observer), because higher such length means effectively thicker atmosphere (larger amount of air to scatter the blue wavelength). Perhaps the correct magnitude to measure is not the total blue light energy arriving from each direction but to calculate some modified version of it, like total blue light energy divided by the length of the atmospheric segment of the line of sight.

Under these assumptions (sun is at the zenith and only blue light is concerned) - was such an experiment ever conducted? this might be another striking confirmation of Rayleigh law, since many descriptions of it usually refer only to the fact that it explains the blue colour of the sky (while my suggestion might show that it also explains the brightness profile of the sky).