APPROACHES TO ESTIMATING DIAMETER DISTRIBUTIONS FROM TERRESTRIAL AND AIRBORNE LIDAR VIA COPULAS

Light Detection and Ranging (LiDAR) can create three-dimension point clouds of forest structure and maps of ground surface. These features have been shown to be useful for quantifying forest stand parameters such as density and tree height at broad scales (Means et al. , 2000). A copula is a special class of multivariate distributions where the marginal distributions are all uniform [0,1] distributions (Genest and MacKay, 1986). The uniform marginal can be stripped from the copula and replaced with any probability distribution though a statistical process known as translation (Genest and MacKay, 1986; Nelsen, 2006). By translating into any mix of distributions (Nelsen, 2006), copulas become a flexible and powerful tool for analysing dependent processes arising from a number of different underlying factors (Genest and MacKay, 1986; Wang, 1998). While copulas have been widely applied in many fields (Genest and MacKay, 1986; Frees and Valdez, 1998; Wang, 1998; Nelsen, 2006; Yan, 2007), they have only recently been applied to modelling forest structure and dynamics (Kershaw et al. , 2010) and individual tree height-diameter relationships (MacPhee et al. , 2018). LiDAR presents many opportunities for individual tree analyses and much work has focused on individual tree segmentation and attribute estimation (e.g., Li et al. , 2012). Through developing moment-based parameter recovery of Weibull distribution for predicting the parameters of the copula marginal distributions from LiDAR attributes, the copula-based diameter-height models have potential to improve individual tree attribute prediction from LiDAR data (MacPhee et al. , 2018). The Noonan Research Forest (NRF, N 45°59′12″, W 66°25′15″) located approximately 30 km northwest of Fredericton, New Brunswick, Canada, is approximately 1500 ha and is composed of a diversity of stand structures and species compositions typical of the Acadian Forest. Three 50 m by 50 m mapped plots with field DBH measurements from a black spruce stand, eastern hemlock stand,and mixed hardwood stand in NRF were used to compare with predict diameter values. The study used LiDAR-extracted heights to estimate DBH distributions for individual trees. The impacts of three LiDAR sources (airborne leaf-on, airborne leaf-off, and terrestrial) on height distributions and four approaches for predicting diameter distributions are explored. The von Bertalanffy-Richards function is widely used as height-diameter equation because of its simplicity and flexibility (Huang et al., 1992; Kershaw et al. , 2008; Russell et al., 2011). To estimate the diameter distribution via LiDAR height, the standard H-D equation form is solved for DBH and fitted to the field measured data. Four approaches were used: diameter-height (D-H) prediction using non-linear least squares approach; D-H prediction using randomForest imputation; moment–based Weibull parameter recovery based on nonlinear least squares prediction of moments; and moment–based Weibull parameter recovery based on randomForest imputation of moments. The moment-based methods used copula models to link D to H. The diameter distributions derived from copulas retained more of the original variation than did those derived from the direct prediction of DBH. Due to differences associated with the LiDAR-extracted heights, the H-D distributions did align very well. However, when field-measured heights were used with the D-H copulas the results were equivalent to the field data. Heights extracted from TLS point clouds as well as the associated point cloud metrics were much lower than those derived from airborne LiDAR and field measurements. A ration correction factor calculated as the ratio of the mean of the leaf-on airborn LiDAR heights and the mean TLS heights. Extraction of heights from LiDAR that were consistent with field measured heights was challenging despite several other researchers reporting good success in this process (Sexton et al. , 2009; Andersen et al., 2014). Although the three LiDAR height distributions are not very close to the measured height distribution, the diameter distributions estimated by the copula models performed very well. The diameter and height distributions can be used to estimate other attributes such as volume or carbon content and summed to obtain more precise area-based estimates.

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