Let $\mathcal{X}$ be a model of $X$. A vertical fractional ideal sheaf $\mathfrak{a}$ is a finitely generated $\mathcal{O}_{\mathcal{X}}$-submodule of the function field of $\mathcal{X}$ such that $\mathfrak{a}|_{X}=\mathcal{O}_{X}$. Then $\mathfrak{a}$ defines a continuous function $\log|\mathfrak{a}|\in C^{0}(X)$ by setting

$$\log|\mathfrak{a}|(x):=\max\left\{\log|f|_{x}\mid f\in\mathfrak{a}_{c_{\mathcal{X}}(x)}\right\}.$$

Note that each vertical Cartier divisor $D\in\Div_{0}(\mathcal{X})$ defines a vertical fractional ideal sheaf $\mathcal{O}_{\mathcal{X}}(D)$, hence a continuous function $f_{D}:=\log|\mathcal{O}_{\mathcal{X}}(D)|$. Note that $f_{\mathcal{X}_{0}}$ is the constant function $1$ since $\log|t|^{-1}=1$. The map $D\mapsto f_{D}$ extends by linearity to $\Div_{0}(\mathcal{X})_{\mathbf{R}}\to C^{0}(X)$.