Let $({\mathcal{X}},{\mathcal{L}},e)$ be a proper model of $(X,L)$. Let $s$ be a local section of $L^{{\text{\rm an}}}$ defined at a point $p\in X^{{\text{\rm an}}}$. Let $\mathcal{U}\subset{\mathcal{X}}$ be a trivializing open neighbourhood of ${\operatorname{red}}(p)$ and $\sigma$ a generator of ${\mathcal{L}}|_{{\mathcal{U}}}$. Let $U={\mathcal{U}}\cap X$ and $\lambda\in{\mathcal{O}}_{U^{{\text{\rm an}}}}$ such that $s^{\otimes e}=\lambda\sigma$ on $U^{{\text{\rm an}}}$. Then, the metric induced by the proper model $({\mathcal{X}},{\mathcal{L}},e)$ on $L^{{\text{\rm an}}}$,, denoted $\|\cdot\|_{{\mathcal{X}},{\mathcal{L}},e}$, is given by

$$\|s(p)\|_{{\mathcal{X}},{\mathcal{L}},e}=|\lambda(p)|^{1/e}.$$

This definition does neither depend on the choice of the open set ${\mathcal{U}}$ nor of the section $\sigma$, and it gives a metric on $L^{{\text{\rm an}}}$. The metrics on $L^{{\text{\rm an}}}$ obtained in this way are called algebraic, and a pair ${\overline{L}}:=(L,\|\cdot\|_{{\mathcal{X}},{\mathcal{L}},e})$ is called an algebraic metrized line bundle.