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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 aexplicit upper bound $\ell(\gamma)\in O(1/\gamma^2)$ as $\gamma\to 0$, suchthat for any smooth surface $A$ of arbitrary characteristic with a closed point0 and an $\mathbb R$-ideal $\mathfrak{a}$ with exponents in $I$, there alwaysexists 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.

Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces