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###  $P(\nu_{\mu} \rightarrow \nu_{e})$ 

In figure \[fig:Pmue\_energy\_dist\]-(a) (upper panel) and (b) (lower panel), presented are the iso-contours of $P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ and $\Delta P (\nu_{\mu} \rightarrow \nu_{e}) \equiv P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ in $E - L$ space. Here, the superscript $(0)$ implies that it is calculated in zeroth-order in $W$ using (\[Schroedinger-eq-0th\]) with appropriate initial condition and final projection to flavor eigenstate. In most of the $E - L$ space $P(\nu_{\mu} \rightarrow \nu_{e})$ is small. However, we identify the two regions where $P(\nu_{\mu} \rightarrow \nu_{e})$ is relatively large, $\gsim 0.3$. One of them is at low energy, $E \lsim \text{a few hundred}$ MeV, and baseline $L \gsim$1000 km. The other one is a region $E \sim 10$ GeV and $L \sim$ 10000 km. The former may be understood as due to the solar MSW enhancement, and the latter as the atmospheric MSW enhancement [@Mikheev:1986gs; @Wolfenstein:1977ue]. Roughly speaking, the regions with relatively large $| \Delta P (\nu_{\mu} \rightarrow \nu_{e}) |$ overlap with these regions.

###  $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$ 

In figures \[fig:Pmutau\_energy\_dist\] and \[fig:Pmumu\_energy\_dist\], the same quantities (in each upper (a) and lower (b) panel) are presented but in $\nu_{\mu} \rightarrow \nu_{\tau}$ and $\nu_{\mu} \rightarrow \nu_{\mu}$ channels, respectively. In contrast to $\nu_{\mu} \rightarrow \nu_{e}$ channel, $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$ contours are globally “vacuum effect dominated”, apart from the solar MSW region, both in the standard (not shown) and the non-unitary cases. The first oscillation peak of $P(\nu_{\mu} \rightarrow \nu_{\tau})$ scales roughly as the vacuum oscillation peak does, $L / 10^3 \,\text{km} = 0.33 E / 1 \, \text{GeV}$. This f