Let ${\overline{L}}_{i}$, $i=0,\dots,d-1$, be line bundles on $X$ equipped with algebraic metrics. For each $i$, choose a model $(\mathcal{X}_{i},\mathcal{L}_{i},e_{i})$ that realizes the metric of ${\overline{L}}_{i}$. We can assume without loss of generality that the models $\mathcal{X}_{i}$ agree with a common model $\mathcal{X}$. Let $Y$ be a $d$-dimensional subvariety of $X$ and $Y^{{\text{\rm an}}}$ its analytification. Let $\mathcal{Y}\subset\mathcal{X}$ be the closure of $Y$, ${\widetilde{\mathcal{Y}}}$ be its normalization, ${\widetilde{{\mathcal{Y}}}}_{o}$ its special fibre, and ${\widetilde{{\mathcal{Y}}}}_{o}^{(0)}$ the set of irreducible components of this special fibre. For each $V\in{\widetilde{{\mathcal{Y}}}}_{o}^{(0)}$, consider the point $\xi_{V}\in Y^{{\text{\rm an}}}$ defined by (2.15). Let $\delta_{\xi_{V}}$ be the Dirac delta measure on $X^{{\text{\rm an}}}$ supported on $\xi_{V}$. We define a discrete signed measure on $X^{{\text{\rm an}}}$ by

(2.29)

$$\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\land\delta_{Y}=\sum_{V\in{\widetilde{{\mathcal{Y}}}}_{o}^{(0)}}{\operatorname{ord}}_{V}(\varpi)\frac{\deg_{\mathcal{L}_{0},\dots,\mathcal{L}_{d-1}}(V)}{e_{0}\dots e_{d-1}}\delta_{\xi_{V}}.$$

This notion extends by linearity to the group of $d$-dimensional cycles of $X$.