Pluri-cotangent maps of surfaces of general type

Let $X$ be a compact, complex surface of general type whose cotangent bundle ${\mathrm{\Omega }}_X$ is strongly semi-ample. We study the pluri-cotangent maps of $X$, namely the morphisms ${\psi }_n\text{\hspace{0.17em}⁣}:\mathbb{P}({\mathrm{\Omega }}_X)\to \mathbb{P}(H^0(X,S^n{\mathrm{\Omega }}_X))$ defined by the vector space of global sections $H^0(X,S^n{\mathrm{\Omega }}_X)$.