  Observation of the interference between the atmospheric-scale and solar-scale
oscillations is one of the challenging and tantalizing goals of the ongoing and
upcoming neutrino experiments. An inevitable first step required for such
analyses is to establish the way of how the oscillation $S$ matrix can be
decomposed into the atmospheric and solar waves, the procedure dubbed as the
amplitude decomposition. In this paper, with use of the perturbative framework
proposed by Denton et al. (DMP), we establish the prescription for amplitude
decomposition which covers the whole kinematical region of the terrestrial
neutrino experiments. We analyze the limits to the atmospheric- and
solar-resonance regions to argue that the dynamical two modes of the DMP
decomposition can be interpreted as the matter-dressed atmospheric and solar
oscillations. The expressions of the oscillation probability, which are
decomposed into the non-interference and interference terms, are derived for
all the relevant flavor oscillation channels. Through construction of the DMP
decomposition, we reveal the nature of $\psi$ ($\theta_{12}$ in matter)
symmetry as due to the $S$ matrix rephasing invariance. A new picture of the
DMP perturbation theory emerged, a unified perturbative framework for neutrino
oscillation in earth matter.

High Energy Physics - Phenomenology