We use the $\sigma$-Ponomarev-system $(f,M,X,\{{\mathcal{P}}_n\})$ to give a consistent method to construct an $s$-mapping (msss-mapping, mssc-mapping) $f$ with covering-properties onto a space $X$ from a metric space $M$. As applications, we systematically get characterizations of $s$-images (msss-images, mssc-images) of metric spaces.