We propose a new technique for eliminating flat edges from a Triangulated Irregular Network (TIN) in a morphologically consistent way. The algorithm is meant to be a preprocessing step for performing morphological computations on a terrain. Terrain morphology is rooted in Morse theory for smooth functions. Segmentation algorithms have been defined for TINs, mostly based on discrete versions of Morse theory, and under the assumption that the terrain model does not include flat edges. On the other hand, flat edges often occur in real data, and thus either they are eliminated through data perturbation, or the segmentation algorithms must be able to deal with them. In both cases, the resulting Morse segmentations are highly affected by the presence of flat edges. The new technique we propose provides a betster solution, as it preserves the set of maxima and minima of the original terrain, and improves consistency among the terrain decompositions produced by different segmentation algorithms.