  We report a measurement of muon annual modulation in a 12-liter liquid
scintillation detector with a live-time of more than 4 years at the Soudan
Underground Laboratory. Muon minimum ionization in the detector is identified
by its observed pulse shape and large energy deposition. The measured muon rate
in the detector is 28.69$\pm$2.09 muons per day with a modulation amplitude of
2.66$\pm$ 1.0\% and a phase at Jul 22 $\pm$ 36.2 days. This annual modulation
is correlated with the variation of the effective atmospheric temperature in
the stratosphere. The correlation coefficient, $\alpha_{T}$, is determined to
be $0.898 \pm 0.025$. This can be interpreted as a measurement of the
atmospheric charged kaon to pion ($K$/$\pi$) ratio of 0.094$^{+0.044}_{-0.061}$
for $E_{p} > $ 7 TeV, consistent with the measurement from the MINOS far
detector. To further constrain the value of $K$/$\pi$ ratio, a Geant4
simulation of the primary cosmic-ray protons with energy up to 100 TeV is
implemented to study the correlation of $K$/$\pi$ ratio and the muon annual
modulation for muon energy greater than 0.5 TeV. We find out that a charged
$K$/$\pi$ ratio of 0.1598, greater than the upper bound (0.138) from this work
at the production point 30 km above the Earth surface in the stratosphere
cannot induce muon annual modulation at the depth of Soudan.
