Log del Pezzo surfaces with not small fractional indices

For a log del Pezzo surface $S$, the fractional index $r(S)\in\mathbb{Q}_{>0}$ is the maximum of $r$ with which $-K_S$ can be written as $r$ times some Cartier divisor. We classify all the log del Pezzo surfaces $S$ with $r(S)\geq 1/2$, after the technique of Nakayama.