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This study quantifies the main characteristics of a terrain-following,σ-coordinate through mathematical analyses of its covariant and contravariant basis vectors as well as the vertical coordinate of σ.A 3-D schematic of the σ-coordinate in a curvilinear coordinate system is provided in this study.The characteristics of the basis vectors were broken down into their “local vector characteristics” and “spatial distribution characteristics”,and the exact expressions of the covariant; in addition,the contravariant basis vectors of the σ-coordinate used to elucidate their detailed characteristics were properly solved.Through rewriting the expression of the vertical coordinate of σ,a mathematical expression of all the σ-coordinate surfaces was found,thereby quantifying the socalled terrain-following characteristics and lack of flexibility to adjust the slope variation of σ-coordinate surfaces for the classic definition of σ.Finally,an analysis on the range value of the vertical coordinate demonstrated that the general value range of σ could be obtained by eliminating the σ-coordinate surfaces below the Earth's surface.All these quantitative descriptions of the characteristics of σ-coordinate were the foundation for improving the σ-coordinate or creating a new one.