Why ratio atmospheric ##\nu_\mu : \nu_e = 2:1##

[ChrisVer]

Why is the ratio of atmospheric neutrino fluxes expected to be:
$R= \frac{\nu_\mu + \bar{\nu}_\mu}{\nu_e +\bar{\nu}_e} =2$
for no Neutrino Oscillations?
An answer that I found gives:
$\pi \rightarrow \mu \nu_{\mu} \rightarrow e \nu_e \nu_\mu \nu_\mu$
But I think this is "wrong" in general, because the muons can reach the Earth before interacting... at least it should be a function of the Zenith angle $\theta$ and their energies (boost).

 

[mfb]

In general, the spectrum is dominated by low-energetic particles, and low-energetic muons decay in flight.
You also have some other particles producing muon pairs. That ratio of 2 is only an approximation.