  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.
