Let $X$ be a proper scheme over $K$ and $\overline{L}$ a line bundle on $X$ with an algebraic metric $\|\cdot\|_{\overline{L}}$. Let $\|\cdot\|$ be a piecewise $\mathbb{Q}$-linear metric on $\mathcal{O}_{X^{\textup{an}}}$ such that $\|\cdot\|_{\overline{L}}\otimes\|\cdot\|$ is a semipositive piecewise $\mathbb{Q}$-linear metric. Let $\mathscr{X}$ be an algebraic model of $X$ such that $\overline{L}$ has a model $\mathscr{L}$ on $\mathscr{X}$ and set $\varphi:=-\log\|1\|$. Then there is a sequence $(\mathfrak{a}_{n})_{n\in\mathbb{N}}$ of vertical coherent fractional ideals on $\mathscr{X}$ and a sequence $(d_{n})_{n\in\mathbb{N}}$ of positive integers such that $\frac{1}{d_{n}}\log|\mathfrak{a}_{n}|$ converges uniformly to $\varphi$.