## 4 Experiment and Analysis
### Results and Analysis
#### 4.2.2 Evaluation of Consistency with the Original Meaning of Entropy  
Following the previously reported method for verifying the consistency of the information entropy calculated in grayscale images with the original meaning of entropy [20], three criteria, including validity, reliability, and measurement ability, were used to quantify entropy performance. 1. Validity  
The most crucial criterion is validity, which determines whether the tool genuinely assesses the concept. In this evaluation experiment, terrain entropy's validity pertains to its ability to accurately reflect spatial disorder variation. The effective terrain entropy values are expected to show a logarithmic trend when processing test images in the mixed-iteration process. It is qualitatively defined as follows:  
\[V=\left\{ \begin{array}{ll}\textit{Yes},&if\ R^{2}\geq 50\%,a>0\\ \textit{No},&\textit{otherwise}\end{array} \right. \tag{6}\]  
Figure 7: Twelve real DEMs (\(30\times 30\)). (**a–d**) terrains from flat areas. (**e–h**) terrains from hilly areas. (**i–l**) terrains from mountainous areas. where \(a\) represents the coefficient of the function obtained by regressing the landscape entropy values and landscape indices using the logarithmic model \(y=\ln a+b\), and \(R^{2}\) is the coefficient of determination. Based on existing empirical experience [21], 50% was set. 2. Reliability  
Reliability indicates the consistency of a measurement. It encompasses two facets: The first is that a reliable measure provides the same result when the same entity is measured over and over. Second, a dependable metric for two similar objects should yield results that are nearly identical. In this context, if the terrain entropy calculation is trustworthy, the variation between its values in two successive mixed iterations should be minimal. To put it differently, when the dependable enhanced terrain entropy values for the test images are shown in a scatter plot, the lines linking each successive pair of points should be smooth. Reliability, denoted as \(R\), is quantitatively defined as follows:  
\[R=\frac{\left(\sum_{i=1}^{n-1}\left|v_{i+1}-v_{i}\right|\right)}{\left(max-min \right)} \tag{7}\]  
where \(v_{i}\) is the improved terrain entropy value of the unordered image configuration at the i-th iteration (where \(i=1,2,3,\ldots,n\)), N is the total number of iterations, and \(max\) and \(min\) are maximum and minimum values of all \(v_{i}\), respectively. It can be seen from Equation (7) that \(R\) is the ratio of the cumulative growth of the improved terrain entropy values from the first iteration to the last iteration to the range of all improved terrain entropy values in the iterations. The smaller this ratio, the smoother the scatter plot, and the more reliable the improved terrain entropy value. 3. Measurement ability  
The measurement ability refers to the scope of disorder that terrain entropy can identify. A terrain entropy with high measurement ability should encompass a wide range of configurations, from nearly ordered to fully disordered. In test images, a high-capability terrain entropy value converges slowly during mixed iterations, while a lower-capability one converges more rapidly. The measurement ability, denoted as \(A\), is defined as follows:  
\[\left\{ \begin{array}{l}A=\frac{S_{1}}{S_{2}}\\ S_{1}=\sum\limits_{i=1}^{n-1}\frac{1}{2}[(v_{i}-min)+(v_{i+1}-min)]\\ S_{2}=(n-1)\times(max-min)\end{array} \right. \tag{8}\]  
The meanings of \(v_{i}\), \(n\), \(max\), and \(min\) in Equation (8) are the same as those in Equation (7). A is the ratio of the areas of two shapes formed by the scatter plot of the improved terrain entropy values of a series of increasingly disordered configurations. The smaller this ratio, the slower the convergence of the improved entropy values during the mixed-iteration process. Therefore, a smaller \(A\) indicates stronger ability of the improved entropy. Elevation, relative elevation, and elevation probability terrain entropy values remain unchanged as the number of mixed iterations increases.