{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T19:57:16Z","timestamp":1783627036695,"version":"3.55.0"},"reference-count":63,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,22]],"date-time":"2019-10-22T00:00:00Z","timestamp":1571702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IJGI"],"abstract":"<jats:p>The past several years have witnessed much progress in landscape ecology and fractal analysis. In landscape ecology, the gradient model of landscape patterns (i.e., landscape gradient) has emerged as a new operating paradigm, where most landscape metrics do not apply because they were developed for the patch mosaic model. In the fractal analysis, a new definition of fractal has been proposed, and various new fractal metrics have been developed. This technical note aims to provide an intersection of these two lines of advance, which will further present an opportunity to advance geo-informatics by considering the latest progress in both landscape ecology and fractal analysis. We first present an overview of the new definition of fractal and all the fractal metrics developed under this new definition. Since the chief obstacle to geographers and landscape ecologists in applying these metrics is the lack of readily accessible methods for their easy computation, we then develop an integrated tool to compute them on landscape gradients. The developed tool facilitates the computation of these new fractal metrics. A case study was carried out with real-life landscape gradients, namely a digital terrain model. These new fractal metrics and the developed tool can be expected to facilitate the fractal characterization of the patterns of gradient landscapes and the understanding of landscape dynamics from a new perspective.<\/jats:p>","DOI":"10.3390\/ijgi8100466","type":"journal-article","created":{"date-parts":[[2019,10,23]],"date-time":"2019-10-23T11:46:59Z","timestamp":1571831219000},"page":"466","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["FracL: A Tool for Characterizing the Fractality of Landscape Gradients from a New Perspective"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1714-779X","authenticated-orcid":false,"given":"Peichao","family":"Gao","sequence":"first","affiliation":[{"name":"State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China"},{"name":"Key Laboratory of Environmental Change and Natural Disaster, Beijing Normal University, Beijing 100875, China"},{"name":"Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Center for Geodata and Analysis, Beijing Normal University, Beijing 100875, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Samuel A.","family":"Cushman","sequence":"additional","affiliation":[{"name":"US Forest Service, Rocky Mountain Research Station, Flagstaff, AZ 86011, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gang","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Earth Sciences, Chengdu University of Technology, Chengdu 610000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8805-8914","authenticated-orcid":false,"given":"Sijing","family":"Ye","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9126-229X","authenticated-orcid":false,"given":"Shi","family":"Shen","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changxiu","family":"Cheng","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China"},{"name":"Key Laboratory of Environmental Change and Natural Disaster, Beijing Normal University, Beijing 100875, China"},{"name":"Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Center for Geodata and Analysis, Beijing Normal University, Beijing 100875, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1023\/B:LAND.0000021723.24247.ee","article-title":"Landscape metrics with ecotones: Pattern under uncertainty","volume":"19","author":"Arnot","year":"2004","journal-title":"Landsc. Ecol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s10980-012-9839-8","article-title":"Modeling landscape structure response across a gradient of land cover intensity","volume":"28","author":"Frazier","year":"2013","journal-title":"Landsc. Ecol."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Forman, R.T.T. (1995). Land Mosaics: The Ecology of Landscapes and Regions, Cambridge University Press.","DOI":"10.1017\/9781107050327"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wiens, J.A., and Moss, M.R. (2005). The gradient concept of landscape structure. Issues and Perspectives in Landscape Ecology, Cambridge University Press.","DOI":"10.1017\/CBO9780511614415"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cushman, S.A., and Huettmann, F. (2010). Landscape ecology: Past, present, and future. Spatial Complexity, Informatics, and Wildlife Conservation, Springer.","DOI":"10.1007\/978-4-431-87771-4"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s10980-009-9341-0","article-title":"Gradient modeling of conifer species using random forests","volume":"24","author":"Evans","year":"2009","journal-title":"Landsc. Ecol."},{"key":"ref_7","unstructured":"McGarigal, K., Cushman, S.A., and Eduard, E. (1970, January 01). FRAGSTATS v4: Spatial Pattern Analysis Program for Categorical and Continuous Maps. Available online: http:\/\/www.umass.edu\/landeco\/research\/fragstats\/fragstats.html."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2049","DOI":"10.1007\/s10980-019-00889-6","article-title":"Describing and analyzing landscape patterns: Where are we now, and where are we going?","volume":"34","author":"Costanza","year":"2019","journal-title":"Landsc. Ecol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1007\/s10980-018-0709-x","article-title":"How has the state-of-the-art for quantification of landscape pattern advanced in the twenty-first century?","volume":"34","author":"Gustafson","year":"2019","journal-title":"Landsc. Ecol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s10980-009-9327-y","article-title":"Surface metrics: An alternative to patch metrics for the quantification of landscape structure","volume":"24","author":"McGarigal","year":"2009","journal-title":"Landsc. Ecol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1489","DOI":"10.1007\/s10980-018-0685-1","article-title":"Surface metrics for landscape ecology: A comparison of landscape models across ecoregions and scales","volume":"33","author":"Kedron","year":"2018","journal-title":"Landsc. Ecol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s10980-015-0248-7","article-title":"Surface metrics: Scaling relationships and downscaling behavior","volume":"31","author":"Frazier","year":"2016","journal-title":"Landsc. Ecol."},{"key":"ref_13","unstructured":"Smith, A.C. (2019, September 09). Bioxgeo\/geodiv: Methods for Calculating Gradient Surface Metrics. Available online: https:\/\/rdrr.io\/github\/bioXgeo\/geodiv\/."},{"key":"ref_14","first-page":"681","article-title":"Sur une courbe continue sans tangente, obtenue par une construction g\u00e9om\u00e9trique \u00e9l\u00e9mentaire [On a continuous curve without tangents constructible from elementary geometry]","volume":"1","author":"Koch","year":"1904","journal-title":"Arkiv f\u00f6r Matematik"},{"key":"ref_15","first-page":"629","article-title":"Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donn\u00e9e [On a cantorian curve which contains a continuous one-to-one image of every given curve]","volume":"162","author":"Sierpinski","year":"1916","journal-title":"Comptes Rendus"},{"key":"ref_16","first-page":"476","article-title":"Allgemeine R\u00e4ume und Cartesische R\u00e4ume., I","volume":"29","author":"Menger","year":"1927","journal-title":"Proc. Amst."},{"key":"ref_17","unstructured":"Mandelbrot, B.B. (1977). Fractals: Form, Chance, and Dimension, Freeman."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1126\/science.156.3775.636","article-title":"How long is the coast of Britain? Statistical self-similarity and fractional dimension","volume":"156","author":"Mandelbrot","year":"1967","journal-title":"Science"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1080\/00045608.2013.834239","article-title":"Ht-index for quantifying the fractal or scaling structure of geographic features","volume":"104","author":"Jiang","year":"2014","journal-title":"Ann. Assoc. Am. Geogr."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s10980-014-0105-0","article-title":"A review on the use of entropy in landscape ecology: Heterogeneity, unpredictability, scale dependence and their links with thermodynamics","volume":"30","author":"Vranken","year":"2015","journal-title":"Landsc. Ecol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.ecoinf.2008.07.001","article-title":"Three-dimensional metrics for the analysis of spatiotemporal data in ecology","volume":"3","author":"Parrott","year":"2008","journal-title":"Ecol. Inform."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1080\/00330124.2012.700499","article-title":"Head\/tail breaks: A new classification scheme for data with a heavy-tailed distribution","volume":"65","author":"Jiang","year":"2013","journal-title":"Prof. Geogr."},{"key":"ref_23","first-page":"203","article-title":"A smooth curve as a fractal under the third definition","volume":"53","author":"Ma","year":"2018","journal-title":"Cartogr. Int. J. Geogr. Inform. Geovis."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.socnet.2016.06.003","article-title":"Calling Dunbar\u2019s numbers","volume":"47","author":"Kaski","year":"2016","journal-title":"Soc. Netw."},{"key":"ref_25","first-page":"96","article-title":"Mapping block-level urban areas for all Chinese cities","volume":"106","author":"Long","year":"2016","journal-title":"Ann. Am. Assoc. Geogr."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"321","DOI":"10.2307\/3565520","article-title":"Landscape patterns in a disturbed environment","volume":"48","author":"Krummel","year":"1987","journal-title":"Oikos"},{"key":"ref_27","first-page":"67","article-title":"Measuring the fractal geometry of landscapes","volume":"27","author":"Milne","year":"1988","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/BF02059856","article-title":"The VFractal: A new estimator for fractal dimension of animal movement paths","volume":"11","author":"Nams","year":"1996","journal-title":"Landsc. Ecol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.ecoinf.2011.08.005","article-title":"Multifractal pattern and process during a recent period of forest expansion in a temperate mountainous region of China","volume":"6","author":"Zhang","year":"2011","journal-title":"Ecol. Inform."},{"key":"ref_30","first-page":"1315","article-title":"Unified metrics for characterizing the fractal nature of geographic features","volume":"107","author":"Gao","year":"2017","journal-title":"Ann. Am. Assoc. Geogr."},{"key":"ref_31","unstructured":"Campagna, M., Montis, A.D., Isola, F., Lai, S., Pira, C., and Zoppi, C. (2012). Computing the image of the city. Planning Support Tools: Policy Analysis, Implementation and Evaluation (Proceedings of the 7th International Conference on Informatics and Urban and Regional Planning INPUT2012), FrancoAngeli."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1080\/23729333.2016.1278151","article-title":"Geospatial big data and cartography: Research challenges and opportunities for making maps that matter","volume":"3","author":"Robinson","year":"2017","journal-title":"Int. J. Cartogr."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"278","DOI":"10.26417\/ejes.v9i1.p278-290","article-title":"Spatial variability of soil aggregate stability in a disturbed river watershed","volume":"9","author":"Mainuri","year":"2017","journal-title":"Eur. J. Econ. Bus. Stud."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.physa.2014.06.060","article-title":"Urban chaos and replacement dynamics in nature and society","volume":"413","author":"Chen","year":"2014","journal-title":"Phys. A Stat.Mech. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"He, J., Zhang, H., Lan, T., Cao, W.W., and Wu, X. (2015, January 9). Exploring the hierarchical structure in road network. Proceedings of the International Conference on Intelligent Earth Observing and Applications, Guilin, Guangxi, China.","DOI":"10.1117\/12.2207827"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1111\/gean.12122","article-title":"A socio-geographic perspective on human activities in social media","volume":"49","author":"Ma","year":"2017","journal-title":"Geogr. Anal."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"McNutt, S.R., and Roman, D.C. (2015). Volcanic seismicity. The Encyclopedia of Volcanoes, Elsevier. [2nd ed.].","DOI":"10.1016\/B978-0-12-385938-9.00059-6"},{"key":"ref_38","first-page":"59","article-title":"Second light: The biggest stars","volume":"99","author":"Sage","year":"2005","journal-title":"J. R. Astron. Soc. Can."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1023\/B:LAND.0000021711.40074.ae","article-title":"Effects of changing scale on landscape pattern analysis: Scaling relations","volume":"19","author":"Wu","year":"2004","journal-title":"Landsc. Ecol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2092","DOI":"10.1016\/j.physa.2017.11.126","article-title":"Scaling relation of earthquake seismic data","volume":"492","author":"Liu","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1080\/00045608.2011.620505","article-title":"Fractality and self-similarity in the structure of road networks","volume":"102","author":"Zhang","year":"2012","journal-title":"Ann. Assoc. Am. Geogr."},{"key":"ref_42","first-page":"943","article-title":"Urban allometric scaling beneath structural fractality of road networks","volume":"109","author":"Lan","year":"2019","journal-title":"Ann. Assoc. Am. Geogr."},{"key":"ref_43","unstructured":"Huang, K.Y. (2015). Fractal or Scaling Analysis of Natural Cities Extracted from Open Geographic Data Sources. [Ph.D. Thesis, University of G\u00e4vle]."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TGRS.2018.2832193","article-title":"A Zipf\u2019s law-based method for mapping urban areas using NPP-VIIRS nighttime light data","volume":"10","author":"Wu","year":"2018","journal-title":"Remote Sens."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.landurbplan.2017.05.008","article-title":"Understanding uneven urban expansion with natural cities using open data","volume":"177","author":"Long","year":"2017","journal-title":"Landsc. Urban Plann."},{"key":"ref_46","first-page":"555","article-title":"Classification of PM2. 5 for natural cities based on co-Kriging and head\/tail break algorithms","volume":"57","author":"Liu","year":"2017","journal-title":"J. Tsinghua Univ. Sci. Technol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"17801","DOI":"10.1007\/s11042-015-3095-2","article-title":"Assessment of lively street network based on geographic information system and space syntax","volume":"76","author":"Li","year":"2017","journal-title":"Multimed. Tools Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF00131538","article-title":"A hierarchical framework for the analysis of scale","volume":"3","author":"Johnson","year":"1989","journal-title":"Landsc. Ecol."},{"key":"ref_49","unstructured":"O\u2019Neill, R.V., Deangelis, D.L., Waide, J.B., Allen, T.F., and Allen, G.E. (1986). A Hierarchical Concept of Ecosystems, Princeton University Press."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1080\/00330124.2015.1099448","article-title":"CRG index: A more sensitive ht-index for enabling dynamic views of geographic features","volume":"68","author":"Gao","year":"2016","journal-title":"Prof. Geogr."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Gao, P.C., Liu, Z., Tian, K., and Liu, G. (2016). Characterizing traffic conditions from the perspective of spatial-temporal heterogeneity. ISPRS Int. J. Geo-Inform., 5.","DOI":"10.3390\/ijgi5030034"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/s41651-017-0009-z","article-title":"How complex is a fractal? Head\/tail breaks and fractional hierarchy","volume":"2","author":"Jiang","year":"2018","journal-title":"J. Geovis. Spatial Anal."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1007\/s12665-019-8357-2","article-title":"Spatial correlation analysis between topographic parameters for defining the geomorphometric diversity index: Application in the environmental protection area of the Serra da Esperan\u00e7a (state of Paran\u00e1, Brazil)","volume":"78","author":"Santos","year":"2019","journal-title":"Environ. Earth Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2183","DOI":"10.1007\/s10980-019-00814-x","article-title":"Computation of the Boltzmann entropy of a landscape: A review and a generalization","volume":"34","author":"Gao","year":"2019","journal-title":"Landsc. Ecol."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Cushman, S.A. (2018). Calculation of configurational entropy in complex landscapes. Entropy, 20.","DOI":"10.3390\/e20040298"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1023\/A:1008004423783","article-title":"Estimating regional forest productivity and water yield using an ecosystem model linked to a GIS","volume":"13","author":"Ollinger","year":"1998","journal-title":"Landsc. Ecol."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1007\/s10980-017-0537-4","article-title":"A multi-method approach to delineate and validate migratory corridors","volume":"32","author":"Bond","year":"2017","journal-title":"Landsc. Ecol."},{"key":"ref_58","first-page":"1133","article-title":"A hierarchy-based solution to calculate the configurational entropy of landscape gradients","volume":"32","author":"Gao","year":"2017","journal-title":"Landsc. Ecol."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1023\/B:LAND.0000030521.19856.cb","article-title":"Behavior of class-level landscape metrics across gradients of class aggregation and area","volume":"19","author":"Neel","year":"2004","journal-title":"Landsc. Ecol."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.ecolind.2007.12.002","article-title":"Parsimony in landscape metrics: Strength, universality, and consistency","volume":"8","author":"Cushman","year":"2008","journal-title":"Ecol. Indic."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1080\/13658816.2018.1427754","article-title":"Geographic space as a living structure for predicting human activities using big data","volume":"33","author":"Jiang","year":"2019","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_62","unstructured":"Alexander, C. (2002). The Nature of Order: An Essay on the Art of Building and The Nature of the Universe, The Center for Environmental Structure."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Jiang, B. (2019). A recursive definition of goodness of space for bridging the concepts of space and place for sustainability. Sustainability, 11.","DOI":"10.3390\/su11154091"}],"container-title":["ISPRS International Journal of Geo-Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2220-9964\/8\/10\/466\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:28:29Z","timestamp":1760189309000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2220-9964\/8\/10\/466"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,22]]},"references-count":63,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,10]]}},"alternative-id":["ijgi8100466"],"URL":"https:\/\/doi.org\/10.3390\/ijgi8100466","relation":{},"ISSN":["2220-9964"],"issn-type":[{"value":"2220-9964","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,22]]}}}