## 1 Introduction  
Digital terrain analysis (DTA) is a critical field in geospatial sciences, involving the study and interpretation of terrain features using digital elevation models (DEMs). DEMs are grid-based representations of terrain that store elevation data of each cell in a grid for the analysis of topographic features, such as slope, aspect, curvature, and landform classification. Terrain entropy, as a key indicator in DTA, is used to quantitatively assess the complexity and diversity of terrain [1]. Its theoretical foundation is the concept of entropy in information theory, which was introduced by an American mathematician called [PERSON] [2] for the measurement of the randomness or uncertainty of a system [3; 4]. As scholar [PERSON] proposed the use of grid DEM data for the calculation of entropy in 1997 [5], many scholars have used terrain entropy to quantify the complex structure of terrain [6; 7; 8]. In grid DEM studies, terrain entropy is used for DEM quality inspection and accuracy evaluation [9; 10], as well as providing a basis for determining the appropriate resolution for grid DEMs [10]. Additionally, terrain entropy has been applied in temporal analysis of land use structures [11] and in the analysis of terrain characteristics of river basins [12]. The methods for calculating the terrain entropy of grid DEMs can be broadly categorized into three types, namely elevation-based, terrain-factor-based, and multi-scale analysis methods. The most fundamental method is elevation-based, which divides the individual elevation value by the total sum of all elevation values in a region and takes the result as the probability Pi in [PERSON]'s entropy equation [13]. To simplify the computation, some researchers have used a Taylor series expansion of \(\log\text{P}_{\text{i}}\) instead of higher-order terms [14], greatly reducing the computational load while not impairing accuracy. The terrain-factor-based terrain entropy calculation method improves the matching of terrain complexity and entropy in some areas. This method involves first the extraction of terrain factors such as slope, aspect, and curvature from DEM data, then the individual reclassification of each terrain factor, and finally, the calculation of the percentage of each level relative to the total grid cells, which gives the probability of occurrence for each terrain factor class, as \(\text{P}_{\text{i}}\) in [PERSON]'s entropy equation, to obtain the terrain entropy for each factor [15; 16]. This approach allows for a more comprehensive description of terrain complexity. In the multi-scale analysis method, slope terrain entropy is calculated in each of the fixed small windows (e.g., 3 \(\times\) 3 or 5 \(\times\) 5) [17; 18], so terrain characteristics can be obtained at different scales in grid DEMs. This method provides accurate terrain information, thus enhancing the precision and reliability of terrain analysis [19]. The terrain entropy calculation method should be selected and modified according to specific research needs so that an effective analysis and description of geomorphological features can be achieved [20]. Although these methods provide a foundation for quantifying terrain complexity, their performance is not ideal when applied to certain special terrain types. For example, in the calculation of elevation-based terrain entropy, when the elevation values across a grid DEM calculation area are constant (but not zero), the terrain complexity should theoretically be zero. However, when directly calculating elevation-based terrain entropy, since the elevation values are non-zero, the probability calculated by dividing the elevation value at a specific point by the sum of all elevation values in the calculation area is also non-zero. As a result, the computed terrain entropy value will not be zero. Using relative elevation values for terrain entropy calculation can avoid this issue, but it still fundamentally relies on the direct use of elevation values.