Abstract: The linear arboricity of a graph $G$ is the minimum number of linear forests which partition the edges of $G$. In the present, it is proved that if a upper-embedded graph $G$ has $\Delta\geqslant 3\sqrt{4-3\varepsilon}$  then its linear arboricity is $\lceil\frac{\Delta}{2}\rceil$\,and if a secondary upper-embedded graph $G$ has $\Delta\geqslant 6\sqrt{1-\varepsilon}$ then its linear arboricity is $\lceil \frac{\Delta}{2}\rceil$, where $\varepsilon\leqslant0$. It improves the bound of the conclusion in [1]. As its application, the linear arboricity of a  triangulation graph on double torus is concluded

Key words: linear arboricity, surface, (secondary) upper-embedded graph, Euler , characteristic