Let $\pi\colon X\to{\mathbb{D}}^{*}$ be a proper submersion that is meromorphic at $0\in{\mathbb{D}}$, and let $\psi$ be a continuous metric on $K_{X/{\mathbb{D}}^{*}}$ with analytic singularities. Then there exists a positive measure $\mu_{0}$ on $X^{\mathrm{hyb}}_{0}$ such that if $\mu_{t}:=\frac{\lambda(t)^{d}e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi)^{d}}$, then $\lim_{t\to 0}\mu_{t}=\mu_{0}$ in the sense of weak convergence of measures on $X^{\mathrm{hyb}}$. Further, there exists a snc model ${\mathcal{X}}\to{\mathbb{D}}$ and a ${\mathbb{Q}}$-line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$ such that $\psi$ extends to a smooth metric on ${\mathcal{L}}$, and

$$\mu_{0}:=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}(\psi)\right)b_{\sigma}^{-1}\lambda_{\sigma},$$

where $\sigma$ ranges over the $d$-dimensional faces of $\Delta({\mathcal{L}})$. Here $\lambda_{\sigma}$ denotes normalized Lebesgue measure on $\sigma$ and $b_{\sigma}=\gcd_{i\in J}b_{i}$, where ${\mathcal{X}}_{0}=\sum_{i}b_{i}E_{i}$ and $E_{i}$, $i\in J$ are the divisors defining $\sigma$.