A fundamental theory based on the Monte Carlo \textit{Time Sequential   Procedure} for the range fluctuations of high energy muons
  Lipari and Stanev developed a method for range fluctuation of high energy muons, stressing the importance of accounting for the fluctuations of the energy loss in radiative processes in 1991 and, now, their method has become the basement for the energy determination of high energy muons through the measurement of the Cherenkov light yields due to those muons in KM3 physics. Once, Takahashi et al. developed a method for the investigation on the depth intensity relation of high energy muons in which all the stochastic processes concerned are taken into account exactly (1983). Now, we make the method by Takahashi et al. revival for the same purpose of the application to the analysis of future KM3 physics. In the present paper, our concern is restricted to the introduction to the fundamental of our method and some subsequent results thereby in which the real simulated behaviors of high energy muons from $10^{12}$eV to $10^{18}$eV, the survival probabilities of high energy and so on are included. The discussion around the practical application of our method to the KM3 physics is entrusted in the subsequent papers. As far as the survival probability of high energy muons is concerned, our method gives nearly the same results to Lipari and Stanev's in some regions and gives the deviated results from theirs in another ones. Thus, we examine the application limit of their method and clarify the reason why, comparing with our method. The most distinct difference in the both methods may become apparent in the treatment on the Cherenkov light yields spectrum by which one may estimate the energies of the muons concerned. We will mention to them in subsequent papers. 
Yoshihide Okumura, Nobusuke Takahashi, Akeo Misaki
hep-ph astro-ph.HE hep-ex