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 $ u_{mu} rightarrow u_{e}$ channel, but a comparison with the $ u_{mu} rightarrow u_{tau}$ channel reveals an interesting difference.
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