How do we know that the muons created by cosmic rays in the upper atmosphere are the same muons reaching our detectors at the surface? I was reading the summary of most significant experiments done so far verifying muon's Special Relativity time dilation and lifetime extension here:
https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation
However it is not clear to me:
How do we know that the muons created by cosmic rays in the upper atmosphere are the same muons reaching our detectors at the surface?
 A: Here is a calculation , and here you can read about the air showers and how they are modeled .
The basic assumption in the calculation is that very high energy cosmic rays hit the atmosphere at a specific distance . The calculation is done because of the paradox of the muons reaching the earth, whereas their half life measured on the surface would have them decay before reaching the surface.
From the studies in accelerators one can generate the primary interaction, have Monte Carlo  events and compare the distribution  with the observed data from the showers. For the muons the important point is that they are decay products of pions, and pions decay faster by two orders of magnitude then muons.
As you can see here there is a mutual interaction between cosmic ray experiments, and particle physics running ones, as it seems there are questions about the density of muons. The link gives also an overview of the cosmic experiments. I just learned that they can see the shower starting on the top of the atmosphere! (search for telescope) So if specific air showers can be correlated with telescope detection, this makes the muon timing and modeling more strict.
A: There's no such thing as "the same muons" or "different muons". All muons are not just identical, they are indistinguishable. If there is a process:
$$ \mu + X \rightarrow X^* \rightarrow \mu + X$$
in which a muon and $X$ combine into an intermediate state which decays to a new muon, then two things happen:
1: the final state muon is not going to live longer than the initial state muon. There are plenty of PSE answers explaining that decay is memoryless.
2: that reaction will occur along with some kind of exchange, where them muon scatters off of $X$, meaning the final state muon is now entangled...but not really b/c it's identical particles.
So: there are no "new" or "old" muons. The question is basically nonsense. Relativity is correct.
