Competition between decay and re-interaction of charged pions and kaons depends
on the temperature/density profile of the upper atmosphere. The amplitude and phase of
the variations depend on the minimum muon energy required to reach the detector and on
muon multiplicity in the detector. Here we compare different methods for characterizing
the muon production profile and the corresponding effective temperature. A muon production
profile based on a parameterization of simulations of muons as a function of primary energy
is compared with approximate analytic solutions of the cascade equation integrated over
primary energy. In both cases, we compare two definitions of effective temperature.
We emphasize applications to compact underground detectors like MINOS and OPERA, while indicating how they relate to
extended detectors like IceCube.