Abstract
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.
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References
Batty M (2010) Space, scale, and scaling in entropy maximizing. Geogr Anal 42(4):395–421
Boltzmann L (1872) Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen [Further studies on the thermal equilibrium of gas molecules]. Sitzungsberichte Akademie der Wissenschaften 66:275–370
Chen YG (2012) The rank-size scaling law and entropy-maximizing principle. Phys A 391(3):767–778
Chen YG, Wang JJ (2016) Describing urban evolution with the fractal parameters based on area-perimeter allometry. Discret Dyn Nat Soc 2016:4863907
Childress WM, Rykiel EJ, Forsythe W, Li BL, Wu H-i (1996) Transition rule complexity in grid-based automata models. Landsc Ecol 11(5):257–266
Clausius R (1850) Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen [On the moving force of heat, and the laws regarding the nature of heat itself which are deducible therefrom]. Ann Phys 79(4):368–397
Costanza JK, Riitters K, Vogt P, Wickham J (2019) Describing and analyzing landscape patterns: where are we now, and where are we going? Landsc Ecol 34(9):2049–2055
Cushman SA (2015) Thermodynamics in landscape ecology: the importance of integrating measurement and modeling of landscape entropy. Landsc Ecol 30(1):7–10
Cushman SA (2016) Calculating the configurational entropy of a landscape mosaic. Landsc Ecol 31(3):481–489
Cushman SA (2018) Calculation of configurational entropy in complex landscapes. Entropy 20(4):298
Cushman SA, Evans JS, McGarigal K (2010) Landscape ecology: past, present, and future. In: Cushman SA and Huettmann F (eds) Spatial complexity, informatics, and wildlife conservation. Springer, Tokyo, Japan
Denbigh K (1981) How subjective is entropy? Chem Br 17:168–185
Dirac PAM (1927) The physical interpretation of the quantum dynamics. Proc R Soc Lond 113(765):621–641
Forman RTT (1995) Land mosaics: the ecology of landscapes and regions. Cambridge University Press, Cambridge
Forman RTT, Godron M (1986) Landscape ecology. Wiley, New York
Frazier AE (2019) Emerging trajectories for spatial pattern analysis in landscape ecology. Landsc Ecol 34(9):2073–2082
Gao PC, Li ZL (2019a) Aggregation-based method for computing absolute Boltzmann entropy of landscape gradient with full thermodynamic consistency. Landsc Ecol 34(8):1837–1847
Gao PC, Li ZL (2019b) Computation of the Boltzmann entropy of a landscape: a review and a generalization. Landsc Ecol 34(9):2183–2196
Gao PC, Cushman SA, Liu G, Ye SJ, Shen S, Cheng CX (2019) FracL: a tool for characterizing the fractality of landscape gradients from a new perspective. ISPRS Int J Geo-Information 8(10):466
Gao PC, Li ZL, Zhang H (2018) Thermodynamics-based evaluation of various improved Shannon entropies for configurational information of gray-level images. Entropy 20(1):19
Gao PC, Zhang H, Li ZL (2017) A hierarchy-based solution to calculate the configurational entropy of landscape gradients. Landsc Ecol 32(6):1133–1146
Gould H, Tobochnik J (2010) Statistical and thermal physics: with computer applications. Princeton University Press, Princeton
Gustafson EJ (1998) Quantifying landscape spatial pattern: what is the state of the art? Ecosystems 1(2):143–156
Gustafson EJ (2019) How has the state-of-the-art for quantification of landscape pattern advanced in the twenty-first century? Landsc Ecol 34(9):2065–2072
Huettner DA (1976) Net energy analysis: an economic assessment. Science 192(4235):101–104
Kedron P, Zhao Y, Frazier AE (2019) Three dimensional (3D) spatial metrics for objects. Landsc Ecol 34(9):2123–2132
Li HB, Wu JG (2004) Use and misuse of landscape indices. Landsc Ecol 19(4):389–399
Longo G, Miquel P-A, Sonnenschein C, Soto AM (2012) Is information a proper observable for biological organization? Prog Biophys Mol Biol 109(3):108–114
Ma BR, Tian GJ, Kong LQ, Liu XJ (2018) How China’s linked urban–rural construction land policy impacts rural landscape patterns: a simulation study in Tianjin China. Landsc Ecol 33(8):1417–1434
McGarigal K, Cushman SA (2005) The gradient concept of landscape structure. In: Wiens JA and Moss MR (eds) Issues and perspectives in landscape ecology. Cambridge University Press, Cambridge
McGarigal K, Cushman SA, Eduard E (2012) FRAGSTATS v4: spatial pattern analysis program for categorical and continuous maps. Available from http://www.umass.edu/landeco/research/fragstats/fragstats.html. Accessed 17 Jan 2016
Moore EF (1962) Machine models of self-reproduction. In: Proceedings of Symposia in Applied Mathematics, New York. vol 14. American Mathematical Society, pp 17–33
Naveh Z, Lieberman AS (1990) Landscape ecology: theory and application. Springer, New York
Nowosad J, Stepinski TF (2019) Information theory as a consistent framework for quantification and classification of landscape patterns. Landsc Ecol 34(9):2091–2101
O’Neill RV, Johnson AR, King AW (1989) A hierarchical framework for the analysis of scale. Landsc Ecol 3(3):193–205
Pelorosso R, Gobattoni F, Leone A (2017) The low-entropy city: a thermodynamic approach to reconnect urban systems with nature. Landsc Urban Plann 168:22–30
Prigogine I (1967) Introduction to thermodynamics of irreversible processes. Interscience Publishers, New York
Prigogine I, Nicolis G, Babloyantz A (1972) Thermodynamics of evolution. Phys Today 25(11):23–28
Riitters KH, Vogt P, Soille P, Estreguil C (2009) Landscape patterns from mathematical morphology on maps with contagion. Landsc Ecol 24(5):699–709
Rubner Y, Tomasi C, Guibas LJ (2000) The earth mover’s distance as a metric for image retrieval. Int J Comput Vis 40(2):99–121
Shannon CE (1948) A mathematical theory of communication. Bell Syst Techn J 27(3):379–423
Söndgerath D, Schröder B (2002) Population dynamics and habitat connectivity affecting the spatial spread of populations–a simulation study. Landsc Ecol 17(1):57–70
Teodorescu PP, Kecs WW, Toma A (2013) Distribution theory: with applications in engineering and physics. Wiley-VCH, Weinheim
Tischendorf L (2001) Can landscape indices predict ecological processes consistently? Landsc Ecol 16(3):235–254
Turner MG (2005) Landscape ecology: What is the state of the science? Annu Rev Ecol Evol Syst 36:319–344
Vaserstein LN (1969) Markov processes over denumerable products of spaces, describing large systems of automata. Problemy Peredachi Informatsii 5(3):64–72
Villani C (2008) Optimal transport: old and new. Springer, Berlin
Volkenstein MV (2009) Entropy and information. Birkhäuser, Basel
von Neumann J (1966) Theory of self-reproducing automata. University of Illinois Press, Champaign
Vranken I, Baudry J, Aubinet M, Visser M, Bogaert J (2015) A review on the use of entropy in landscape ecology: heterogeneity, unpredictability, scale dependence and their links with thermodynamics. Landsc Ecol 30(1):51–65
Wang G, Li JR, Ravi S (2019) A combined grazing and fire management may reverse woody shrub encroachment in desert grasslands. Landsc Ecol 34(8):2017–2031
Wilson A (2010) Entropy in urban and regional modelling: retrospect and prospect. Geogr Anal 42(4):364–394
Wu JG (2013) Key concepts and research topics in landscape ecology revisited: 30 years after the Allerton Park workshop. Landsc Ecol 28(1):1–11
Wu JG, Loucks OL (1995) From balance of nature to hierarchical patch dynamics: a paradigm shift in ecology. Q Rev Biol 70(4):439–466
Zhao Y, Zhang XC (2019) Calculating spatial configurational entropy of a landscape mosaic based on the Wasserstein metric. Landsc Ecol 34(8):1849–1858
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant No. 41901316), the Research Grants Council of Hong Kong (Grant No. PolyU 152219/18E), the Key Research and Development Program of Chengdu (2019-YF05-02119-SN), the Program of Science and Technology of Sichuan Province (2020YJ0325), the Shanghai Philosophy and Social Science Project (2020BGL034), the State Key Laboratory of Earth Surface Processes and Resource Ecology (Grant No. 2020-KF-03), and the Fundamental Research Funds for the Central Universities (Grant No. 2019NTST02).
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Gao, P., Zhang, H. & Wu, Z. Wasserstein metric-based Boltzmann entropy of a landscape mosaic: a clarification, correction, and evaluation of thermodynamic consistency. Landscape Ecol 36, 815–827 (2021). https://doi.org/10.1007/s10980-020-01177-4
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DOI: https://doi.org/10.1007/s10980-020-01177-4