We say that an element of $N^{1}({\mathscr{X}}/S)$ is ample if it is of the form $\sum_{i}a_{i}{\mathscr{L}}_{i}$ for some real numbers $a_{i}>0$ and some ample line bundles ${\mathscr{L}}_{i}$. A closed $(1,1)$-form $\theta$ is called ${\mathscr{X}}$-positive if $\theta_{\mathscr{X}}$ is ample. We say that a model metric ${\|\hskip 4.30554pt\|}$ of a line bundle $L$ is ${\mathscr{X}}$-positive if the same holds for the curvature form $c_{1}(L,{\|\hskip 4.30554pt\|})$. We say that an element $\theta\in N^{1}({\mathscr{X}}/S)$ is nef if $\theta\cdot C\geq 0$ for any closed curve $C\subset{\mathscr{X}}_{s}$. A closed $(1,1)$-form $\theta$ is said to be semipositive if it is determined by a nef class $\theta_{\mathscr{X}}\in N^{1}({\mathscr{X}}/S)$ on a model ${\mathscr{X}}$.

If $\theta$ is a closed $(1,1)$-form, we say that a model function $\varphi$ is $\theta$-plurisubharmonic (briefly $\theta$-psh) if $\theta+dd^{c}\varphi$ is semipositive. If $\theta$ is the closed $(1,1)$-form associated with some line bundle ${\mathscr{L}}$ on ${\mathscr{X}}$ and if $D$ is a vertical Cartier divisor on ${\mathscr{X}}$, then by definition $\varphi_{D}$ is a $\theta$-psh function if and only if ${\mathscr{L}}\otimes{\mathcal{O}}(D)$ is nef if and only if $\|\ \|_{{\mathscr{L}}\otimes{\mathcal{O}}(D)}$ is a semipositive metric.