Improvement of Terrain Entropy Calculation for Grid Digital Elevation Models Considering Spatial Structural Features

Existing methods for calculating terrain entropy in grid digital elevation models (DEMs) often face computational anomalies in specific topographies within small windows. To address this issue, an improved method was developed based on the Euclidean distance approach. This method was inspired by Claramunt’s technique of weighting information entropy by the average distance between points with the same value and different values. Specifically, vectors were formed between grid points and categorized by value consistency and relative positions. Those formed between points of different values were classified by the value of the starting point as well as parallel and adjacent relationships. This comprehensive grouping strategy was integrated into distance calculations, becoming a new probability operator that accurately reflects terrain spatial characteristics. Experimental verification confirms that the method proposed aligns with the fundamental concept of entropy, yielding a regression equation of y=0.011ln⁡x+0.463 with a coefficient of determination of 94.73%, a reliability of 44.015, and a measurement ability of 0.757. For the mixed iterative images with gradually increasing spatial disorder, their entropy values should follow a logarithmic trend. Therefore, a logarithmic function is used for fitting. A determination coefficient greater than 50% indicates that the method adheres to the original definition of entropy and is effective in capturing the increasing spatial disorder of the grid DEM. A lower reliability value suggests smoother data computation between the two iterations. A lower measurement ability value indicates slower convergence for grid DEMs with gradually increasing spatial disorder. The improved method was also tested on simulated and real DEMs, and the results showed a strong correlation between calculated terrain entropy values and terrain complexity. By effectively capturing spatial information changes, this approach overcomes the shortcoming of computational anomalies and demonstrates high reliability in terrain entropy calculation in grid DEMs.