{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T08:13:02Z","timestamp":1772611982319,"version":"3.50.1"},"reference-count":46,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T00:00:00Z","timestamp":1686528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42271330"],"award-info":[{"award-number":["42271330"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021YFB3901201"],"award-info":[{"award-number":["2021YFB3901201"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["42271330"],"award-info":[{"award-number":["42271330"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["2021YFB3901201"],"award-info":[{"award-number":["2021YFB3901201"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The leaf area index (LAI) is a crucial variable in climate, ecological, and land surface modeling. However, the estimation of the LAI from coarse-resolution remote sensing data can be affected by the spatial scaling bias, which arises from the nonlinearity of retrieval models and the heterogeneity of the land surface. This study provides an algorithm named Arithmetic Mean and Geometric Mean (AM\u2013GM) to correct the spatial scaling bias. It is established based on negative logarithmic functions and avoids second-order stationarity. In this algorithm, relationships are derived between the scaling bias of LAI and the arithmetic and geometric means of directional gap probability for two commonly used remote sensing models, the Beer\u2013Lambert law and a semi-empirical transfer function, respectively. According to the AM\u2013GM algorithm, the expression representing the model nonlinearity is derived and utilized for the analysis of LAI scaling bias. Furthermore, the AM\u2013GM algorithm is simplified by a linear relationship, which is constructed between two quantities related to the directional gap probability between two specific resolutions. Two scenes simulated by the LargE-Scale remote sensing data and image Simulation framework (LESS) model and three sites are used to evaluate the proposed algorithm and analyze the scaling bias of LAI. The validation results show that the AM\u2013GM algorithm provides accurate correction of LAI scaling bias. The analyses based on the AM\u2013GM algorithm demonstrate that the scaling bias of LAI increases with the increase in the LAI value, with stronger surface heterogeneity and coarser spatial resolution. The validation results of the simplified AM\u2013GM algorithm demonstrate that at the Sud-Ouest site, the absolute value of the bias for the estimated LAI decreases from 0.10, 0.22, 0.29, and 0.31 to 0.04, 0.01, 0.04, and 0.05 at 200 m, 500 m, 1000 m, and 1500 m resolutions, respectively. In conclusion, the proposed algorithm is effective in the analysis and correction of the scaling bias for coarse-resolution LAI.<\/jats:p>","DOI":"10.3390\/rs15123068","type":"journal-article","created":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T02:00:45Z","timestamp":1686621645000},"page":"3068","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["AM\u2013GM Algorithm for Evaluating, Analyzing, and Correcting the Spatial Scaling Bias of the Leaf Area Index"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4521-3988","authenticated-orcid":false,"given":"Jingyu","family":"Zhang","sequence":"first","affiliation":[{"name":"State Key Laboratory of Remote Sensing Science, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Beijing Engineering Research Center for Global Land Remote Sensing Products, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2070-3278","authenticated-orcid":false,"given":"Rui","family":"Sun","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Remote Sensing Science, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Beijing Engineering Research Center for Global Land Remote Sensing Products, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Faculty of Arts and Sciences, Beijing Normal University, Zhuhai 519087, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8245-6762","authenticated-orcid":false,"given":"Zhiqiang","family":"Xiao","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Remote Sensing Science, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Beijing Engineering Research Center for Global Land Remote Sensing Products, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"}]},{"given":"Liang","family":"Zhao","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Remote Sensing Science, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Beijing Engineering Research Center for Global Land Remote Sensing Products, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3923-6056","authenticated-orcid":false,"given":"Donghui","family":"Xie","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Remote Sensing Science, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"},{"name":"Beijing Engineering Research Center for Global Land Remote Sensing Products, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.rse.2012.12.027","article-title":"GEOV1: LAI and FAPAR essential climate variables and FCOVER global time series capitalizing over existing products. Part1: Principles of development and production","volume":"137","author":"Baret","year":"2013","journal-title":"Remote Sens. Environ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5596","DOI":"10.1109\/JSTARS.2021.3076075","article-title":"New Global MuSyQ GPP\/NPP Remote Sensing Products From 1981 to 2018","volume":"14","author":"Wang","year":"2021","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"106902","DOI":"10.1016\/j.compag.2022.106902","article-title":"Improving leaf area index estimation accuracy of wheat by involving leaf chlorophyll content information","volume":"196","author":"Chen","year":"2022","journal-title":"Comput. Electron. Agric."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"e2020GL091496","DOI":"10.1029\/2020GL091496","article-title":"Where Are Global Vegetation Greening and Browning Trends Significant?","volume":"48","author":"Mahecha","year":"2021","journal-title":"Geophys. Res. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1038\/s41893-019-0220-7","article-title":"China and India lead in greening of the world through land-use management","volume":"2","author":"Chen","year":"2019","journal-title":"Nat. Sustain."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yu, T., Zhang, Q., and Sun, R. (2021). Comparison of Machine Learning Methods to Up-Scale Gross Primary Production. Remote Sens., 13.","DOI":"10.3390\/rs13132448"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4259","DOI":"10.1038\/s41467-019-12257-8","article-title":"Vegetation structural change since 1981 significantly enhanced the terrestrial carbon sink","volume":"10","author":"Chen","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.rse.2006.07.013","article-title":"Influence of the spatial heterogeneity on the non-linear estimation of Leaf Area Index from moderate resolution remote sensing data","volume":"105","author":"Garrigues","year":"2006","journal-title":"Remote Sens. Environ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1080\/17538947.2021.2019844","article-title":"Simultaneous retrieval of land surface temperature and emissivity from the FengYun-4A advanced geosynchronous radiation imager","volume":"15","author":"Liu","year":"2022","journal-title":"Int. J. Digit. Earth"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.3390\/s90301768","article-title":"Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling","volume":"9","author":"Wu","year":"2009","journal-title":"Sensors"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zhang, J., Wang, J., Sun, R., Zhou, H., and Zhang, H. (2020). A Model-Downscaling Method for Fine-Resolution LAI Estimation. Remote Sens., 12.","DOI":"10.3390\/rs12244147"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2219","DOI":"10.1109\/TGRS.2006.872100","article-title":"Algorithm for global leaf area index retrieval using satellite imagery","volume":"44","author":"Deng","year":"2006","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1080\/02757250009532420","article-title":"Numerical experiments on the spatial scaling of land surface albedo and leaf area index","volume":"19","author":"Liang","year":"2000","journal-title":"Remote Sens. Rev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5383","DOI":"10.1080\/01431160903130978","article-title":"Scale transformation of Leaf Area Index product retrieved from multiresolution remotely sensed data: Analysis and case studies","volume":"30","author":"Tao","year":"2009","journal-title":"Int. J. Remote Sens."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1109\/LGRS.2014.2341925","article-title":"Improving Leaf Area Index Retrieval Over Heterogeneous Surface by Integrating Textural and Contextual Information: A Case Study in the Heihe River Basin","volume":"12","author":"Yin","year":"2015","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"111700","DOI":"10.1016\/j.rse.2020.111700","article-title":"Improving leaf area index retrieval over heterogeneous surface mixed with water","volume":"240","author":"Xu","year":"2020","journal-title":"Remote Sens. Environ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1029\/2018RG000608","article-title":"An Overview of Global Leaf Area Index (LAI): Methods, Products, Validation, and Applications","volume":"57","author":"Fang","year":"2019","journal-title":"Rev. Geophys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"e2018MS001453","DOI":"10.1029\/2018MS001453","article-title":"Perspectives on the Future of Land Surface Models and the Challenges of Representing Complex Terrestrial Systems","volume":"12","author":"Fisher","year":"2020","journal-title":"J. Adv. Model. Earth Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"83851","DOI":"10.1109\/ACCESS.2021.3087411","article-title":"Quantitative Representation of Spatial Heterogeneity in the LAI Scaling Transfer Process","volume":"9","author":"Zhao","year":"2021","journal-title":"IEEE Access"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1016\/j.jenvman.2006.08.016","article-title":"Spatial scaling between leaf area index maps of different resolutions","volume":"85","author":"Jin","year":"2007","journal-title":"J. Environ. Manag."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0034-4257(92)90008-8","article-title":"Change of scale in models of remote sensing: A general method for spatialization of models","volume":"40","author":"Raffy","year":"1992","journal-title":"Remote Sens. Environ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1109\/36.581996","article-title":"A framework for analyzing and designing scale invariant remote sensing algorithms","volume":"35","author":"Hu","year":"1997","journal-title":"IEEE Trans. Geosci. Remote. Sens."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3068","DOI":"10.1109\/JSTARS.2014.2346251","article-title":"Analyzing the Spatial Scaling Bias of Rice Leaf Area Index From Hyperspectral Data Using Wavelet-Fractal Technique","volume":"8","author":"Jiang","year":"2015","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1472","DOI":"10.1109\/JSTARS.2018.2799955","article-title":"Scaling Correction of Remotely Sensed Leaf Area Index for Farmland Landscape Pattern With Multitype Spatial Heterogeneities Using Fractal Dimension and Contextural Parameters","volume":"11","author":"Wu","year":"2018","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Wu, L., Qin, Q., Liu, X., Ren, H., Wang, J., Zheng, X., Ye, X., and Sun, Y. (2016). Spatial Up-Scaling Correction for Leaf Area Index Based on the Fractal Theory. Remote Sens., 8.","DOI":"10.3390\/rs8030197"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2308","DOI":"10.1109\/JSTARS.2019.2906053","article-title":"An Improved Computational Geometry Method for Obtaining Accurate Remotely Sensed Products via Convex Hulls With Dynamic Weights: A Case Study With Leaf Area Index","volume":"12","author":"Chen","year":"2019","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_27","first-page":"3095153","article-title":"Upscaling in Situ Site-Based Albedo Using Machine Learning Models: Main Controlling Factors on Results","volume":"60","author":"Wang","year":"2022","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"113465","DOI":"10.1016\/j.rse.2023.113465","article-title":"Quantification of the uncertainty in multiscale validation of coarse-resolution satellite albedo products: A study based on airborne CASI data","volume":"287","author":"Wu","year":"2023","journal-title":"Remote Sens. Environ."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Chen, Y.G. (2020). Fractal Modeling and Fractal Dimension Description of Urban Morphology. Entropy, 22.","DOI":"10.3390\/e22090961"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Jiang, J., Ji, X., Yao, X., Tian, Y., Zhu, Y., Cao, W., and Cheng, T. (2018). Evaluation of Three Techniques for Correcting the Spatial Scaling Bias of Leaf Area Index. Remote Sens., 10.","DOI":"10.3390\/rs10020221"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Becker, R. (2012). The Variance Drain and Jensen\u2019s Inequality. CAEPR Work. Pap., 2004\u20132012.","DOI":"10.2139\/ssrn.2027471"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"108695","DOI":"10.1016\/j.agrformet.2021.108695","article-title":"Using fractal dimension to correct clumping effect in leaf area index measurement by digital cover photography","volume":"311","author":"Li","year":"2021","journal-title":"Agric. For. Meteorol."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Shi, Y., Wang, J., Wang, J., and Qu, Y. (2017). A Prior Knowledge-Based Method to Derivate High-Resolution Leaf Area Index Maps with Limited Field Measurements. Remote Sens., 9.","DOI":"10.3390\/rs9010013"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5359","DOI":"10.1080\/01431160600658107","article-title":"Evaluating the fraction of vegetation cover based on NDVI spatial scale correction model","volume":"27","author":"Zhang","year":"2006","journal-title":"Int. J. Remote Sens."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1016\/j.rse.2018.11.036","article-title":"LESS: LargE-Scale remote sensing data and image simulation framework over heterogeneous 3D scenes","volume":"221","author":"Qi","year":"2019","journal-title":"Remote Sens. Environ."},{"key":"ref_36","first-page":"1138","article-title":"Influence of woody elements on nadir reflectance of forest canopy based on simulations by using the LESS model","volume":"25","author":"Xu","year":"2021","journal-title":"J. Remote Sens."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/0002-1571(71)90092-6","article-title":"A theoretical analysis of frequency of gaps in plant stands","volume":"8","author":"Nilson","year":"1971","journal-title":"Agric. Meteorol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.agrformet.2006.12.003","article-title":"Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage","volume":"143","author":"Wang","year":"2007","journal-title":"Agric. For. Meteorol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.agrformet.2018.11.033","article-title":"Review of indirect optical measurements of leaf area index: Recent advances, challenges, and perspectives","volume":"265","author":"Yan","year":"2019","journal-title":"Agric. For. Meteorol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1109\/TGRS.2013.2237780","article-title":"Use of General Regression Neural Networks for Generating the GLASS Leaf Area Index Product From Time-Series MODIS Surface Reflectance","volume":"52","author":"Xiao","year":"2014","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.agrformet.2021.108374","article-title":"Canopy clumping index (CI): A review of methods, characteristics, and applications","volume":"303","author":"Fang","year":"2021","journal-title":"Agric. For. Meteorol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1016\/j.agrformet.2010.01.009","article-title":"On the correct estimation of effective leaf area index: Does it reveal information on clumping effects?","volume":"150","author":"Ryu","year":"2010","journal-title":"Agric. For. Meteorol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1016\/j.jenvman.2006.08.018","article-title":"LAI inversion algorithm based on directional reflectance kernels","volume":"85","author":"Tang","year":"2007","journal-title":"J. Environ. Manag."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/0034-4257(91)90009-U","article-title":"Potentials and limits of vegetation indices for LAI and APAR assessment","volume":"35","author":"Baret","year":"1991","journal-title":"Remote Sens. Environ."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.rse.2005.05.003","article-title":"Global mapping of foliage clumping index using multi-angular satellite data","volume":"97","author":"Chen","year":"2005","journal-title":"Remote Sens. Environ."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3767","DOI":"10.3390\/s8063767","article-title":"Impact of spatial LAI heterogeneity on estimate of directional gap fraction from SPOT-satellite data","volume":"8","author":"Ma","year":"2008","journal-title":"Sensors"}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/12\/3068\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:53:18Z","timestamp":1760125998000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/12\/3068"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,12]]},"references-count":46,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["rs15123068"],"URL":"https:\/\/doi.org\/10.3390\/rs15123068","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,12]]}}}