  Observation of the interference between the atmospheric and solar oscillation
waves with the correct magnitude would provide another manifestation of the
three-generation structure of leptons. As a prerequisite for such analyses we
develop a method for decomposing the oscillation $S$ matrix into the
atmospheric and solar amplitudes. Though the similar method was recently
proposed successfully in vacuum, once an extension into the matter environment
is attempted, it poses highly nontrivial problems. Even for an infinitesimal
matter potential, inherent mixture of the atmospheric and solar oscillation
waves occurs, rendering a simple extension of the vacuum definition untenable.
We utilize general kinematic structure as well as analyses of the five
perturbative frameworks, in which the nature of matter-dressed atmospheric and
solar oscillations are known, to understand the origin of the trouble, how to
deal with the difficulty, and to grasp the principle of decomposition. Then, we
derive the amplitude decomposition formulas in these frameworks, and discuss
properties of the decomposed probabilities. We mostly discuss the $\nu_{\mu}
\rightarrow \nu_{e}$ channel, but a comparison with the $\nu_{\mu} \rightarrow
\nu_{\tau}$ channel reveals an interesting difference.
