We call model of $X$ any projective and flat $\mathfrak{o}_{k}$-scheme $\mathscr{X}\to\operatorname{Spec}(\mathfrak{o}_{k})$ such that the generic fiber of $\mathscr{X}\to\operatorname{Spec}(\mathfrak{o}_{k})$ is $X$. We denote by $\mathscr{X}_{\circ}:=\mathscr{X}\otimes_{\mathfrak{o}_{k}}(\mathfrak{o}_{k}/\mathfrak{m}_{k})$ the central fiber of $\mathscr{X}\to\operatorname{Spec}(\mathfrak{o}_{k})$. By the valuative criterion of properness, for any point $x\in X^{\mathrm{an}}$, the canonical $k$-morphism $\operatorname{Spec}\hat{\kappa}(x)\rightarrow X$ extends in a unique way to an $\mathfrak{o}_{k}$-morphism of schemes $\mathscr{P}_{x}:\operatorname{Spec}\mathfrak{o}_{x}\rightarrow\mathscr{X}$. We denote by $r_{\mathscr{X}}(x)$ the image of $\mathfrak{m}_{x}\in\operatorname{Spec}\mathfrak{o}_{x}$ by the map $\mathscr{P}_{x}$. Thus we obtain a map $r_{\mathscr{X}}$ from $X^{\mathrm{an}}$ to $\mathscr{X}_{\circ}$, called the reduction map of $\mathscr{X}$.

Let $\mathscr{L}$ be an element of $\operatorname{Pic}(\mathscr{X})\otimes{\mathbb{Q}}$ such that $\left.{\mathscr{L}}\right|_{{X}}=L$ in $\operatorname{Pic}(X)\otimes{\mathbb{Q}}$. The ${\mathbb{Q}}$-invertible sheaf $\mathscr{L}$ yields a continuous metric $|\raisebox{1.72218pt}{.}|_{\mathscr{L}}$ as follows.

First we assume that $\mathscr{L}\in\operatorname{Pic}(\mathscr{X})$ and $\left.{\mathscr{L}}\right|_{{X}}=L$ in $\operatorname{Pic}(X)$. For any $x\in X^{\mathrm{an}}$, let $\omega_{x}$ be a local basis of $\mathscr{L}$ around $r_{\mathscr{X}}(x)$ and $\bar{\omega}_{x}$ the class of $\omega_{x}$ in ${L(x)}:=L\otimes_{\mathscr{O}_{X}}\hat{\kappa}(x)$. For $l\in L\otimes_{\mathscr{O}_{X}}\hat{\kappa}(x)$, if we set $l=a_{x}\bar{\omega}_{x}$ ($a_{x}\in\hat{\kappa}(x)$), then $|l|_{\mathscr{L}}(x):=|a_{x}|_{x}$. Here we set $h:=\{|\raisebox{1.72218pt}{.}|_{\mathscr{L}}(x)\}_{x\in X^{\mathrm{an}}}$. Note that $h$ is continuous because, for a local basis $\omega$ of $\mathscr{L}$ over an open set $\mathscr{U}$ of $\mathscr{X}$, $|\omega|_{\mathscr{L}}(x)=1$ for all $x\in r_{\mathscr{X}}^{-1}(\mathscr{U}_{\circ})$. Moreover,

(2)

$$|\raisebox{1.72218pt}{.}|_{h^{n}}(x)=|\raisebox{1.72218pt}{.}|_{\mathscr{L}^{n}}(x)$$

for all $n\geq 0$ and $x\in X^{\mathrm{an}}$. Indeed, if we set $l=a_{x}\bar{\omega}_{x}$ for $l\in{L(x)}$, then $l^{\otimes n}=a_{x}^{n}\bar{\omega}_{x}^{\otimes n}$. Thus

$$|l^{\otimes n}|_{h^{n}}(x)=(|l|_{h}(x))^{n}=|a_{x}|_{x}^{n}=|l^{\otimes n}|_{\mathscr{L}^{n}}(x).$$

In general, there are $\mathscr{M}\in\operatorname{Pic}(\mathscr{X})$ and a positive integer $m$ such that $\mathscr{L}^{\otimes m}=\mathscr{M}$ in $\operatorname{Pic}(\mathscr{X})\otimes{\mathbb{Q}}$ and $\left.{\mathscr{M}}\right|_{{X}}=L^{\otimes m}$ in $\operatorname{Pic}(X)$. Then

$$|\raisebox{1.72218pt}{.}|_{{\mathscr{L}}}(x):=(|\raisebox{1.72218pt}{.}|_{{\mathscr{M}}}(x))^{1/m}.$$

Note that the above definition does not depend on the choice of $\mathscr{M}$ and $m$. Indeed, let $\mathscr{M}^{\prime}$ and $m^{\prime}$ be another choice. As $\mathscr{M}^{\otimes m^{\prime}}=\mathscr{M^{\prime}}^{\otimes m}$ in $\operatorname{Pic}(\mathscr{X})\otimes{\mathbb{Q}}$, there is a positive integer $N$ such that $\mathscr{M}^{\otimes Nm^{\prime}}=\mathscr{M^{\prime}}^{\otimes Nm}$ in $\operatorname{Pic}({\mathscr{X}})$, so that, by using (2),

$$(|\raisebox{1.72218pt}{.}|_{{\mathscr{M}}}(x))^{Nm^{\prime}}=|\raisebox{1.72218pt}{.}|_{{\mathscr{M}}^{\otimes Nm^{\prime}}}(x)=|\raisebox{1.72218pt}{.}|_{{{\mathscr{M}}^{\prime}}^{\otimes Nm}}(x)=(|\raisebox{1.72218pt}{.}|_{{{\mathscr{M}}^{\prime}}}(x))^{Nm},$$

as desired.