  The production spectrum of high-energy muons as a function of depth in the
atmosphere is relevant for understanding properties of event rates in deep
detectors. For a given atmospheric profile, cascades of heavy nuclei develop at
higher altitude than proton showers, giving rise to larger separation of muons
at depth. For a given type of primary cosmic ray, seasonal variations in muon
rates reflect the fact that higher temperatures correspond to lower densities
and to a relative increase in the ratio of decay to re-interaction of the
parent mesons. In this paper, we present a generalization of the Elbert formula
that tracks meson decay to muons along the trajectory of the primary cosmic-ray
nucleus. The convolution of the production spectrum with a changing atmospheric
profile provides the dependence of event rates and sizes of muon bundles on
temperature and primary mass. We consider applications to IceCube and also to
multiple muon events in the compact underground detectors of MINOS and the NOvA
Near Detector.
