The tropical weight $m$ on an $n$-dimensional polyhedron $\sigma$ of ${\rm Trop}(Y)$ is defined in the following way. By density of the value group $\Gamma$ in ${\mathbb{R}}$, there is $\omega\in\Gamma^{r}\cap{\rm relint}(\sigma)$. We choose $t\in{\mathbb{G}}_{m}^{r}(K)$ with ${\rm trop}(t)=\omega$. Then the closure of $t^{-1}Y$ in $({{\mathbb{G}}_{m}^{r}})_{{K^{\circ}}}$ is a flat variety over ${K^{\circ}}$ whose special fibre is called the initial degeneration ${\rm in}_{\omega}(Y)$ of $Y$ at $\omega$. Note that ${\rm in}_{\omega}(Y)$ is a closed subscheme of $({\mathbb{G}}_{m}^{r})_{{\tilde{K}}}$. Let $m_{W}$ be the multiplicity of the irreducible component $W$ of ${\rm in}_{\omega}(Y)$. Then the tropical weight $m_{\sigma}$ is defined by $m_{\sigma}:=\sum_{W}m_{W}$, where $W$ ranges over all irreducible components of ${\rm in}_{\omega}(Y)$. One can show that the definition is independent of the choices of $\omega$ and $t$. It is a non-trivial fact from tropical geometry that $({\rm Trop}(Y),m)$ is a tropical cycle (see [Gu12], §13, for details).