For each $\xi\in\mathcal{X}_{0}$ let $I_{\xi}$ be the set of components $E_{j}$ passing through $\xi$. Arguing as above shows that there exists a unique way to define for each $s\in\sigma_{I_{\xi}}$ a valuation $\val_{\widehat{\mathcal{X}}_{\xi},s}$ on $\widehat{\mathcal{X}}_{\xi}:=\spec\widehat{\mathcal{O}}_{\mathcal{X},\xi}$, if we impose that:

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  $\val_{\widehat{\mathcal{X}}_{\xi},s}$ is centered at $\xi_{J}$ for $s\in\rel(\sigma_{J})\subset\sigma_{I_{\xi}}$;

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  $\val_{\widehat{\mathcal{X}}_{\xi},s}(E_{i})=s_{i}$ for each $i\in I_{\xi}$;

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  $s\mapsto\val_{\widehat{\mathcal{X}}_{\xi},s}(f)$ is continuous for each $f\in\widehat{\mathcal{O}}_{\mathcal{X},\xi}$.

Indeed, choose a regular system of parameters $(z_{i})_{i\in L}$ of $\mathcal{O}_{\mathcal{X},\xi}$ such that $z_{j}$ is a local equation of $E_{j}$ for $j\in I_{\xi}\subset L$, and a field of representatives of $\kappa(\xi)$ in $\mathcal{O}_{\mathcal{X},\xi}$. We then have an isomorphism $\widehat{\mathcal{O}}_{\mathcal{X},\xi}\simeq\kappa(\xi)[[t_{i},\,i\in L]]$ under which $\val_{\widehat{\mathcal{X}}_{\xi},s}$ corresponds to the monomial valuation taking value $s_{i}$ on $t_{i}$ for $j\in I_{\xi}$, and $0$ on $t_{i}$ for $i\in L\setminus J$. Note that $\val_{\mathcal{X},s}$ is then the image of $\val_{\widehat{\mathcal{X}}_{\xi},s}$ under the natural map $\widehat{\mathcal{X}}_{\xi}\to\mathcal{X}$.