  Fix a subset $I\subseteq \mathbb R_{>0}$ such that $\gamma=\inf\{
\sum_{i}n_ib_i-1>0 \mid n_i\in \mathbb Z_{\geq 0}, b_i\in I \}>0$. We give a
explicit upper bound $\ell(\gamma)\in O(1/\gamma^2)$ as $\gamma\to 0$, such
that for any smooth surface $A$ of arbitrary characteristic with a closed point
0 and an $\mathbb R$-ideal $\mathfrak{a}$ with exponents in $I$, there always
exists a prime divisor $E$ over $A$ computing the minimal log discrepancy of
$(A,\mathfrak{a})$ at 0 and with its log discrepancy $k_E+1\leq \ell(\gamma)$.
Some examples indicate that our bound is optimal.
