An automatic geostatistical method of analyzing spatial characteristics of topography is presented. Anisotropy in spatial continuity is modeled by estimating the range of the directional semivariograms. Spatial characteristics of topography are defined by orientation of the semi-major axis, length of the semi-major axis and length of the semi-minor axis of the anisotropy model ellipse. Orientation of the ellipse gives the average orientation of the linear features. Aspect ratio of the ellipse estimates the degree of lineation of features. Area of the ellipse determines the roughness of the topography. The automatic geostatistical topography characterization method was applied to land topography from Western US and sea-floor topography from the Ross sea, Antarctica. The method was able to characterize the spatial patterns of topography from Western US. The method was used to separate highly lineated regions on the Antarctic sea floor from areas with less prominent or no lineations.