When $X$ is regular and the metrics are smooth (in the Archimedean case) or algebraic (in the non-Archimedean case), the local heights of Definition 2.39 agree with the local heights that can be derived using the Gillet-Soulé arithmetic intersection product. In particular, in the Archimedean case, this local height agrees with the Archimedean contribution of the Arakelov global height introduced by Bost, Gillet and Soulé in [BGS94]. In the non-Archimedean case, the local height can be interpreted in terms of an intersection product. Assume that $Y$ is prime and choose models $(\mathcal{X}_{i},\mathcal{L}_{i},e_{i})$ of $(X,L_{i})$ that realize the algebraic metrics of ${\overline{L}}_{i}$. Without loss of generality, we may assume that all the models $\mathcal{X}_{i}$ agree with a common model $\mathcal{X}$. The sections $s^{\otimes e_{i}}_{i}$ can be seen as rational sections of $\mathcal{L}_{i}$ over $\mathcal{X}$. With the notations in Definition 2.28, the equation (2.15) implies that

$$\log\|s_{d}(\xi_{V})\|=\frac{\log|\varpi|{\operatorname{ord}}_{V}(s_{d}^{\otimes e_{d}})}{e_{d}{\operatorname{ord}}_{v}(\varpi)}.$$

Therefore, in this case the equation in Definition 2.39(2) can be written as

(2.43)

$$\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})=\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1}}(Y\cdot\operatorname{div}(s_{d});s_{0},\dots,s_{d-1})\\ -\frac{\log|\varpi|}{e_{0}\dots e_{d}}\sum_{V\in{\widetilde{\mathcal{Y}}}_{0}^{(0)}}{\operatorname{ord}}_{V}(s_{d}^{\otimes e_{d}})\deg_{\mathcal{L}_{0},\dots,\mathcal{L}_{d-1}}(V).$$