{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T02:47:59Z","timestamp":1769222879352,"version":"3.49.0"},"reference-count":60,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T00:00:00Z","timestamp":1594598400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Central Public interest Scientific Institution Basal Research Fund","award":["CAFYBB2019QD003"],"award-info":[{"award-number":["CAFYBB2019QD003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The forest growth and yield models, which are used as important decision-support tools in forest management, are commonly based on the individual tree characteristics, such as diameter at breast height (DBH), crown ratio, and height to crown base (HCB). Taking direct measurements for DBH and HCB through the ground-based methods is cumbersome and costly. The indirect method of getting such information is possible from remote sensing databases, which can be used to build DBH and HCB prediction models. The DBH and HCB of the same trees are significantly correlated, and so their inherent correlations need to be appropriately accounted for in the DBH and HCB models. However, all the existing DBH and HCB models, including models based on light detection and ranging (LiDAR) have ignored such correlations and thus failed to account for the compatibility of DBH and HCB estimates, in addition to disregarding measurement errors. To address these problems, we developed a compatible simultaneous equation system of DBH and HCB error-in-variable (EIV) models using LiDAR-derived data and ground-measurements for 510 Picea crassifolia Kom trees in northwest China. Four versatile algorithms, such as nonlinear seemingly unrelated regression (NSUR), two-stage least square (2SLS) regression, three-stage least square (3SLS) regression, and full information maximum likelihood (FIML) were evaluated for their estimating efficiencies and precisions for a simultaneous equation system of DBH and HCB EIV models. In addition, two other model structures, namely, nonlinear least squares with HCB estimation not based on the DBH (NLS and NBD) and nonlinear least squares with HCB estimation based on the DBH (NLS and BD) were also developed, and their fitting precisions with a simultaneous equation system compared. The leave-one-out cross-validation method was applied to evaluate all estimating algorithms and their resulting models. We found that only the simultaneous equation system could illustrate the effect of errors associated with the regressors on the response variables (DBH and HCB) and guaranteed the compatibility between the DBH and HCB models at an individual level. In addition, such an established system also effectively accounted for the inherent correlations between DBH with HCB. However, both the NLS and BD model and the NLS and NBD model did not show these properties. The precision of a simultaneous equation system developed using NSUR appeared the best among all the evaluated algorithms. Our equation system does not require the stand-level information as input, but it does require the information of tree height, crown width, and crown projection area, all of which can be readily derived from LiDAR imagery using the delineation algorithms and ground-based DBH measurements. Our results indicate that NSUR is a more reliable and quicker algorithm for developing DBH and HCB models using large scale LiDAR-based datasets. The novelty of this study is that the compatibility problem of the DBH model and the HCB EIV model was properly addressed, and the potential algorithms were compared to choose the most suitable one (NSUR). The presented method and algorithm will be useful for establishing similar compatible equation systems of tree DBH and HCB EIV models for other tree species.<\/jats:p>","DOI":"10.3390\/rs12142238","type":"journal-article","created":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T09:30:49Z","timestamp":1594719049000},"page":"2238","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Prediction of Individual Tree Diameter and Height to Crown Base Using Nonlinear Simultaneous Regression and Airborne LiDAR Data"],"prefix":"10.3390","volume":"12","author":[{"given":"Zhaohui","family":"Yang","sequence":"first","affiliation":[{"name":"Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China"},{"name":"Key Laboratory of Forest Management and Growth Modeling, National Forestry and Grassland Administration, Beijing 100091, China"}]},{"given":"Qingwang","family":"Liu","sequence":"additional","affiliation":[{"name":"Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China"}]},{"given":"Peng","family":"Luo","sequence":"additional","affiliation":[{"name":"Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China"}]},{"given":"Qiaolin","family":"Ye","sequence":"additional","affiliation":[{"name":"College of Information Science and Technology, Nanjing Forestry University, Nanjing 210037, China"}]},{"given":"Guangshuang","family":"Duan","sequence":"additional","affiliation":[{"name":"Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China"},{"name":"College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8623-6211","authenticated-orcid":false,"given":"Ram P.","family":"Sharma","sequence":"additional","affiliation":[{"name":"Institute of Forestry, Tribhuwan Univeristy, Kritipur, Kathmandu-44600, Nepal"}]},{"given":"Huiru","family":"Zhang","sequence":"additional","affiliation":[{"name":"Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China"},{"name":"Key Laboratory of Forest Management and Growth Modeling, National Forestry and Grassland Administration, Beijing 100091, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5419-4547","authenticated-orcid":false,"given":"Guangxing","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Geography and Environmental Resources, Southern Illinois University at Carbondale, Carbondale, IL 62901, USA"}]},{"given":"Liyong","family":"Fu","sequence":"additional","affiliation":[{"name":"Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China"},{"name":"Key Laboratory of Forest Management and Growth Modeling, National Forestry and Grassland Administration, Beijing 100091, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,13]]},"reference":[{"key":"ref_1","unstructured":"Daniels, R.F., and Burkhart, H.E. (1975). Simulation of Individual Tree Growth and Stand Development in Managed Loblolly Pine Plantations, Division of Forestry and Wildlife Resources, Virginia Polytechnic and State University."},{"key":"ref_2","unstructured":"Navratil, S. (1997, January 17\u201318). Wind damage in thinned stands. Proceedings of the A Commercial Thinning Workshop, Whitecourt, AB, Canada."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.foreco.2004.07.067","article-title":"Development of an individual tree-based mechanical model to predict wind damage within forest stands","volume":"203","author":"Ancelin","year":"2004","journal-title":"For. Ecol. Manag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.foreco.2008.08.024","article-title":"MeaNSURing heights to crown base and crown median with LiDAR in a mature, even-aged loblolly pine stand","volume":"257","author":"Dean","year":"2009","journal-title":"For. Ecol. Manag."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Scott, J.H., and Reinhardt, D. (2001). Assessing Crown Fire Potential by Linking Models of Surface and Crown Fire Potential.","DOI":"10.2737\/RMRS-RP-29"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1139\/x95-007","article-title":"Predicting tree crown ratio model for Austrian forests","volume":"25","author":"Hynynen","year":"1995","journal-title":"Can. J. For. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1080\/01431160903380615","article-title":"Estimating crown base height for Scots pine by means of the 3D geometry of airborne laser scanning data","volume":"31","author":"Vauhkonen","year":"2010","journal-title":"Int. J. Remote Sens."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1139\/x87-096","article-title":"Compatible crown ratio and crown height models","volume":"17","author":"Michael","year":"1987","journal-title":"Can. J. For. Res."},{"key":"ref_9","unstructured":"Ritchie, M.W., and Hann, D.W. (1987). Equations for Predicting Height to Crown Base for Fourteen Tree Species in Southwest Oregon, Oregon State University, Forestry Research Laboratory."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"60","DOI":"10.5558\/tfc2012-011","article-title":"Development of height to crown base models for thirteen tree species of the North American Acadian Region","volume":"88","author":"Baburam","year":"2012","journal-title":"For. Chron."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Sharma, R.P., Vacek, Z., Vacek, S., Podr\u00e1zsk\u00fd, V., and Jansa, V. (2017). Modelling individual tree height to crown base of Norway spruce (Picea abies (L.) Karst.) and European beech (Fagus sylvatica L.). PLoS ONE, 12.","DOI":"10.1371\/journal.pone.0186394"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.foreco.2016.12.034","article-title":"A generalized interregional nonlinear mixed-effects crown width model for Prince Rupprecht larch in northern China","volume":"389","author":"Fu","year":"2017","journal-title":"For. Ecol. Manag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1016\/j.biombioe.2007.06.022","article-title":"Estimating biomass of individual pine trees using airborne LiDAR","volume":"31","author":"Popescu","year":"2007","journal-title":"Biomass Bioenergy"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2602","DOI":"10.1016\/j.foreco.2008.01.044","article-title":"Spatial partitioning of biomass and diversity in a lowland Bolivian forest: Linking field and remote sensing measurements","volume":"255","author":"Broadbent","year":"2008","journal-title":"For. Ecol. Manag."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2416","DOI":"10.1016\/j.foreco.2008.01.022","article-title":"Automatic recognition and measurement of single trees based on data from airborne laser scanning over the richly structured natural forests of the Bavarian Forest National Park","volume":"255","author":"Heurich","year":"2008","journal-title":"For. Ecol. Manag."},{"key":"ref_16","first-page":"789","article-title":"Evaluation of nonlinear equations for predicting diameter from tree height","volume":"42","author":"Bi","year":"2012","journal-title":"Can. J. Remote Sens."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/S0034-4257(99)00052-8","article-title":"LiDAR Remote Sensing of the Canopy Structure and Biophysical Properties of Douglas-Fir Western Hemlock Forests","volume":"70","author":"Lefsky","year":"1999","journal-title":"Remote Sens. Environ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/S0034-4257(01)00290-5","article-title":"Predicting forest stand characteristics with airborne scanning laser using a practical two-stage procedure and field data","volume":"80","year":"2002","journal-title":"Remote Sens. Environ."},{"key":"ref_19","first-page":"218","article-title":"Reconstructing tree crowns from laser scanner data for feature extraction. Int. Archives of Photogram","volume":"34","author":"Pyysalo","year":"2002","journal-title":"Remote Sens. Spat. Inf. Sci. XXXIV Part 3B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1080\/01431160701736471","article-title":"Species identification of individual trees by combining high resolution LIDAR data with multi-spectral images","volume":"29","author":"Holmgren","year":"2008","journal-title":"Int. J. Remote Sens."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1016\/j.rse.2007.06.011","article-title":"A voxel-based LiDAR method for estimating crown base height for deciduous and pine trees","volume":"112","author":"Popescu","year":"2008","journal-title":"Remote Sens. Environ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1093\/forestry\/cpq008","article-title":"Comparing different methods for prediction of mean crown height in Norway spruce stands using airborne laser scanner data","volume":"83","author":"Maltamo","year":"2010","journal-title":"Forestry"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"A562","DOI":"10.1364\/OE.26.00A562","article-title":"Simple method for direct crown base height estimation of individual conifer trees using airborne LiDAR data","volume":"26","author":"Luo","year":"2018","journal-title":"Opt. Express."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.14358\/PERS.72.12.1369","article-title":"Single tree segmentation using airborne laser scanning data in a structurally heterogeneous spruce forest","volume":"72","author":"Solberg","year":"2006","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/S0034-4257(01)00243-7","article-title":"Estimating tree height and tree crown properties using airborne scanning laser in a boreal nature reserve","volume":"79","year":"2002","journal-title":"Remote Sens. Environ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.rse.2004.10.013","article-title":"Estimating forest canopy fuel parameters using LiDAR data","volume":"94","author":"Andersen","year":"2005","journal-title":"Remote Sens. Environ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1080\/02827580600700353","article-title":"A comparative study of the use of laser scanner data and field measurements in the prediction of crown height in boreal forests","volume":"21","author":"Maltamo","year":"2006","journal-title":"Scand. J. Forest Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14214\/sf.10006","article-title":"Incorporating tree- and stand-level information on crown base height into multivariate forest management inventories based on airborne laser scanning","volume":"52","author":"Maltamo","year":"2018","journal-title":"Silva Fennica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1139\/X10-073","article-title":"Using error-in-variable (EIV) regression to predict tree diameter and crown width from remotely sensed imagery","volume":"40","author":"Zhang","year":"2010","journal-title":"Can. J. For. Res."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Rechenr, A.C., and Schaalje, G.B. (2008). Linear Models in Statistics, Woley. [2nd ed.].","DOI":"10.1002\/9780470192610"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/S0378-1127(97)00161-8","article-title":"Effect of errors-in-variables on coefficients of a growth model and on prediction of growth","volume":"102","author":"Kangas","year":"1998","journal-title":"For. Ecol. Manag."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Carroll, R.J., Ruppert, D., Stefanski, L.A., and Crainiceanu, C.M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective, Taylor & Francis Group LLC.","DOI":"10.1201\/9781420010138"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Fu, L.Y., Liu, Q.W., Sun, H., Wang, Q.Y., Li, Z.Y., Chen, E.X., Pang, Y., Song, X.Y., and Wang, G.X. (2018). Development of a system of compatible individual tree diameter and aboveground biomass prediction models using error-in-variable regression and airborne LiDAR data. Remote Sens., 10.","DOI":"10.3390\/rs10020325"},{"key":"ref_34","unstructured":"Tang, S., Li, Y., and Fu, L.Y. (2015). Statistical Foundation for Biomathematical Models, Higher Education Press. [2nd ed.]."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Fuller, W.A. (1987). Meansureement Error Models, John Wiley and Sons.","DOI":"10.1002\/9780470316665"},{"key":"ref_36","first-page":"161","article-title":"Measurement error models and their applications","volume":"13","author":"Tang","year":"1998","journal-title":"J. Biomath."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/S0304-3800(01)00326-X","article-title":"Simultaneous equations, errors-invariable models, and model integration in systems ecology","volume":"142","author":"Tang","year":"2001","journal-title":"Ecol. Model."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"218","DOI":"10.2307\/2984115","article-title":"Regression lines and the linear functional relationship","volume":"9","author":"Lindley","year":"1947","journal-title":"J. R. Stat. Soc. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/S0304-3800(02)00173-4","article-title":"A parameter estimation program for the errors-in-variable model","volume":"156","author":"Tang","year":"2002","journal-title":"Ecol. Model."},{"key":"ref_40","unstructured":"Walters, D.K., and Hann, D.W. (1986). Taper Equations for Six Conifer Species in Southwest Oregon, Forest Research Laboratory, Oregon State University."},{"key":"ref_41","unstructured":"Pang, Y., Chen, E., Liu, Q., Xiao, Q., Zhong, K., Li, X., and Ma, M. (2008). WATER: Dataset of airborne LiDAR mission at the super site in the Dayekou watershed flight zone on Jun. 23, 2008. Chinese Academy of Forestry; Institute of Remote Sensing Applications, Chinese Academy of Sciences, Heihe Plan Science Data Center."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"357","DOI":"10.14358\/PERS.72.4.357","article-title":"Detection of Individual Tree Crowns in Airborne Lidar Data","volume":"72","author":"Koch","year":"2006","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.isprsjprs.2014.06.009","article-title":"Assessing the potential for leaf\u2013off LiDAR data to model canopy closure in temperate deciduous forests","volume":"95","author":"Parent","year":"2014","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1109\/TGRS.2019.2938017","article-title":"Improving estimation of forest canopy cover by introducing loss ratio of laser pulses using airborne LiDAR","volume":"58","author":"Liu","year":"2020","journal-title":"IEEE Trans. Geosci. Remote."},{"key":"ref_45","unstructured":"Liu, Q. (2009). Study on the Estimation Method of Forest Parameters Using Airborne Lidar. [Ph.D. Thesis, Chinese Academy of Forestry]. (In Chinese)."},{"key":"ref_46","first-page":"83","article-title":"Extracting individual tree heights and crowns using airborne LIDAR data","volume":"30","author":"Liu","year":"2008","journal-title":"J. Beijing For. Univ."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1007\/s00468-015-1325-x","article-title":"Comparison of seemingly unrelated regressions with multivariate errors-in-variables models for developing a system of nonlinear additive biomass equations","volume":"30","author":"Fu","year":"2016","journal-title":"Trees"},{"key":"ref_48","first-page":"573","article-title":"Assessing tree and stand biomass: A review with examples and critical comparisons","volume":"45","author":"Parresol","year":"1999","journal-title":"For. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1139\/x00-202","article-title":"Additivity of nonlinear biomass equations","volume":"31","author":"Parresol","year":"2011","journal-title":"Can. J. For. Res."},{"key":"ref_50","unstructured":"Judge, G.G., Hill, R.C., Griffifiths, W.E., Lutkepohl, H., and Lee, T.C. (1988). Introduction to the Theory and Ractice of Econometrics, Wiley. [2nd ed.]."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"54","DOI":"10.2307\/1911287","article-title":"Three Stage Least Squares: Simultaneous Estimation of Simultaneous Equations","volume":"30","author":"Zellner","year":"1962","journal-title":"Econometrica"},{"key":"ref_52","unstructured":"Fumio, H. (2005). Econometrics, Shanghai University of Finance and Economics Press."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1080\/02827580902795036","article-title":"Site-specific height growth models for six common tree species in Denmark","volume":"24","author":"Meilby","year":"2009","journal-title":"Scand. J. For. Res."},{"key":"ref_54","first-page":"27","article-title":"Individual tree-based diameter growth model of slash pine in Florida using nonlinear mixed modeling","volume":"59","author":"Timilsina","year":"2013","journal-title":"For. Sci."},{"key":"ref_55","first-page":"1","article-title":"Compatibility, development and estimation of taper and volume equation systems","volume":"65","author":"Zhao","year":"2019","journal-title":"For. Sci."},{"key":"ref_56","unstructured":"R Core Team (2018). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: https:\/\/www.R-project.org\/."},{"key":"ref_57","unstructured":"Greene, W.J. (2001). Econometric Analysis, Pearson Education, Inc.. [7th ed.]."},{"key":"ref_58","first-page":"229","article-title":"The importance of mesauement error for certain procedures in remote sensing at optical wavelengths","volume":"52","author":"Curran","year":"1986","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_59","first-page":"1077","article-title":"A basal area increment model for individual conifers in the northern Rocky Mountains","volume":"136","author":"Wykoff","year":"1990","journal-title":"For. Sci."},{"key":"ref_60","unstructured":"Van Deusen, P.C., and Biging, G.S. (1985). STAG, A Stand Generator for Mixed Species Stands, Northern 55 California Forest Yield Cooperative, Department of Forestry and Resource Management, University of California. Research Note."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/14\/2238\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:50:42Z","timestamp":1760176242000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/14\/2238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,13]]},"references-count":60,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["rs12142238"],"URL":"https:\/\/doi.org\/10.3390\/rs12142238","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,13]]}}}