Wasserstein metric-based Boltzmann entropy of a landscape mosaic: a clarification, correction, and evaluation of thermodynamic consistency

Context: Boltzmann entropy, also called thermodynamic entropy, has long been suggested and recently reemphasized as a basis for achieving a profound understanding of landscape dynamics with thermodynamic insights. The difficulty in practically applying this entropy lies in its computation with landscapes, and many solutions have attempted to address this. The latest solution for landscape mosaics is the Wasserstein metric-based method. Objectives: The first objective is to provide a clarification of and a correction to the Wasserstein metric-based method. The second is to evaluate the method in terms of thermodynamic consistency using different implementations. Methods: Two implementation methods, namely the von Neumann and the Moore neighborhood, were used, which led to two different Wasserstein metric-based entropies. Thermodynamic consistency, the fundamental property of entropy, was used as the evaluation principle. Three criteria (validity, reliability, and ability) were designed in terms of thermodynamic consistency, and corresponding indicators were proposed. Boltzmann entropies computed using all existing methods were used as benchmarks. Results: The three indicators of the five Boltzmann entropies (including two based on the Wasserstein metric and three using existing methods) against 100,000 landscapes were computed and investigated. The reasons for both the good and poor performance of the Wasserstein metric-based entropies were identified. Conclusions: The Wasserstein metric-based method can be safely used with the von Neumann neighborhood. Compared with the entropies produced by existing methods, the Wasserstein metric-based entropy has worse reliability but better ability (i.e., working range). The most reliable entropy was computed using the total edge-based method.