  Fujino and Tanaka established the minimal model theory for $\mathbb
Q$-factorial log surfaces in characteristic $0$ and $p$, respectively. We prove
that every intermediate surface has only log terminal singularities if we run
the minimal model program starting with a pair consisting of a smooth surface
and a boundary $\mathbb R$-divisor. We further show that such a property does
not hold if the initial surface is singular.
