It is a fundamental observation by Zhang [Zha95b] that the non-Archimedean contribution of the Arakelov global height of a variety can be expressed in terms of a family of metrics. In particular, this global height only depends on the metrics and not on a particular choice of models, exhibiting the analogy between the Archimedean and non-Archimedean settings. The local heights were extended by Gubler [Gub02, Gub03] to non-necessarily discrete valuations and he also weakened the hypothesis of proper intersection.