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We report a measurement of muon annual modulation in a 12-liter liquidscintillation detector with a live-time of more than 4 years at the SoudanUnderground Laboratory. Muon minimum ionization in the detector is identifiedby its observed pulse shape and large energy deposition. The measured muon ratein the detector is 28.69$\pm$2.09 muons per day with a modulation amplitude of2.66$\pm$ 1.0\% and a phase at Jul 22 $\pm$ 36.2 days. This annual modulationis correlated with the variation of the effective atmospheric temperature inthe stratosphere. The correlation coefficient, $\alpha_{T}$, is determined tobe $0.898 \pm 0.025$. This can be interpreted as a measurement of theatmospheric 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 fardetector. To further constrain the value of $K$/$\pi$ ratio, a Geant4simulation of the primary cosmic-ray protons with energy up to 100 TeV isimplemented to study the correlation of $K$/$\pi$ ratio and the muon annualmodulation 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 workat the production point 30 km above the Earth surface in the stratospherecannot induce muon annual modulation at the depth of Soudan.

Atmospheric Charged $K/\pi$ Ratio and Measurement of Muon Annual Modulation with a Liquid Scintillation Detector at Soudan