  Let $S$ be a minimal surface of general type with $p_{g}(S)=0$ and
$K^{2}_{S}=4$. Assume the bicanonical map $\varphi$ of $S$ is a morphism of
degree $4$ such that the image of $\varphi$ is smooth. Then we prove that the
surface $S$ is a Burniat surface with $K^{2}=4$ and of non nodal type.
