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By using the integral method in the muon propagation through water, wecalculate the range fluctuation of high and ultra high energy muons. Manyauthors divide all radiative processes into two parts, namely, the continuouspart and radiative part in their Monte Carlo simulation in order to considerthe fluctuation in the both ranges and energies of the muons, while we treatall stochastic processes as exactly as possible, without the introduction ofthe continuous parts in all stochastic processes. The validity of our MonteCarlo method is checked by the corresponding analytical method which ismethodologically independent on the Monte Carlo procedure. Accompanied cascadeshowers are generated by the direct electron pair production, bremsstrahlungand photo-nuclear interaction. These showers are calculated by the exact MonteCarlo Method in one dimensional way. We report survival probabilities, rangedistributions and examples of individual muon behavior.

Fluctuation of TeV to EeV Energy Muons and Induced Muon Showers in Water