Let $D$ be a divisor on $X$ with $h^{0}(X,{\mathcal{O}}_{X}(mD))\neq 0$ for some $m>0$. Define the asymptotic test ideal of exponent $\lambda\in\mathbb{R}_{\geq 0}$ associated with $X$ and $D$ as

$$\tau(\lambda\cdot\|D\|):=\tau({\mathfrak{a}}_{\bullet}^{\lambda})$$

where ${\mathfrak{a}}_{\bullet}$ denotes the graded sequence of base ideals for $D$, i.e. ${\mathfrak{a}}_{m}$ is the image of the natural map

$$H^{0}(X,{\mathcal{O}}(mD))\otimes_{k}{\mathcal{O}}_{X}(-mD)\to{\mathcal{O}}_{X}.$$

If $D$ is a $\mathbb{Q}$-divisor such that $h^{0}(X,{\mathcal{O}}_{X}(mD))\neq 0$ for some positive integer $m$ such that $mD$ is a usual divisor then we put $\tau(\lambda\cdot\|D\|):=\tau(\lambda/r\cdot\|rD\|)$ for some $r\in\mathbb{N}$ such that $rD$ has integral coefficients.