Let $Z$ be a compact analytic space over $K$. An analytic moment map on $Z$ is an analytic morphism $\varphi:Z\rightarrow{T^{\rm an}}$ for a split torus $T={\mathbb{G}}_{m}^{r}$ over $K$ as before. Let $M$ be the character group of $T$, then we have $T={\rm Spec}(K[M])$. The map ${\varphi_{\rm trop}}:={\rm trop}\circ\varphi:Z\rightarrow N_{\mathbb{R}}$ is called the tropicalization map of $\varphi$ and we may use the coordinates on $T$ to identify $N_{\mathbb{R}}$ with ${\mathbb{R}}^{r}$.