Terrain derivatives exhibit surface morphology in various aspects. However, existing spatial change calculation methods for terrain derivatives are based on a mathematical scalar operating system, which may disregard the directional property of the original data to a certain extent. This situation is particularly true in second-order terrain derivatives, in which original data can be terrain derivatives with clear directional properties, such as slope or aspect. Thus, this study proposes a mathematical vector operation method for the calculation of second-order terrain derivatives. Given the examples of the first-order terrain derivatives of slope and aspect, their second-order terrain derivatives are calculated using the proposed vector method. Directional properties are considered and vectorized using the following steps: rotation-type judgment, standardization of initial direction, and vector representation. The proposed vector method is applied to one mathematical Gaussian surface and three different ground landform areas using digital elevation models (DEMs) with 5 and 1 m resolutions. Comparison analysis results between the vector and scalar methods show that the former achieves more reasonable and accurate second-order terrain derivatives than the latter. Moreover, the vector method avoids overexpression or even exaggeration errors. This vector operation concept and its expanded methods can be applied in calculating other terrain derivatives in geomorphometry.

In recent years, applications and analyses based on slope units have become increasingly widespread. Compared with grid units, slope units can better represent terrain features and boundaries and allow a more complete view of the morphology of the Earth’s surface. Maps based on slope units also offer significant improvements for disaster prediction and the analysis of slope land resources. Therefore, we need a reasonable method of slope unit classification. Although some methods have been proposed for slope unit classification, they have been too focused on morphological variations and have not fully considered the importance of geomorphology, and the geomorphological and physical significance of slope partitioning remain unclear. Therefore, we propose a novel slope unit classification method by combining terrain feature lines (CTFL) derived from the meaning of geomorphology ontology that use several terrain feature lines, such as geomorphic water division lines, valley shoulder lines, slope toe lines, and shady/sunny slope boundary lines, to classify slopes. The Jiuyuangou and Lushan study areas were selected to test the CTFL method. Compared with the traditional hydrological method, the CTFL method can effectively overcome topographic abruptness and distortions, improve the uniformity of slope and aspect within individual units, and increase the accuracy of slope unit applications and analyses. This work fully considers the importance of geomorphology and is conducive to future studies of slope unit division.