A flexible stem taper and volume prediction method based on mixed-effects B-spline regression

Modelling stem taper and volume is crucial in many forest management and planning systems. Taper models are used for diameter prediction at any location along the stem of a sample tree. Furthermore, taper models are flexible means to provide information on the stem volume and assortment structure of a forest stand or other management units. Usually, taper functions are mean functions of multiple linear or nonlinear regression models with diameter at breast height and tree height as predictor variables. In large-scale inventories, an upper diameter is often considered as an additional predictor variable to improve the reliability of taper and volume predictions. Most studies on stem taper focus on accurately modelling the mean function; the error structure of the regression model is neglected or treated as secondary. We present a semi-parametric linear mixed model where the population mean diameter at an arbitrary stem location is a smooth function of relative height. Observed tree-individual diameter deviations from the population mean are assumed to be realizations of a smooth Gaussian process with the covariance depending on the sampled diameter locations. In addition to the smooth random deviation from the population average, we consider independent zero mean residual errors in order to describe the deviations of the observed diameter measurements from the tree-individual smooth stem taper. The smooth model components are approximated by cubic spline functions with a B-spline basis and a small number of knots. The B-spline coefficients of the population mean function are treated as fixed effects, whereas coefficients of the smooth tree-individual deviation are modelled as random effects with zero mean and a symmetric positive definite covariance matrix. The taper of a tree is predicted using an arbitrary number of diameter and corresponding height measurements at arbitrary positions along the stem to calibrate the tree-individual random deviation from the population mean estimated by the fixed effects. This allows a flexible application of the method in practice. Volume predictions are calculated as the integral over cross-sectional areas estimated from the calibrated taper curve. Approximate estimators for the mean squared errors of volume estimates are provided. If the tree height is estimated or measured with error, we use the “law of total expectation and variance” to derive approximate diameter and volume predictions with associated confidence and prediction intervals. All methods presented in this study are implemented in the R-package TapeR.     

建立相似林分样木因子回归模型推算采伐木蓄积

在安化县杉木人工林相似林分内设立样地,测量样地对角线上26株林木的地径、胸径和树高,用Excel软件建立地径与胸径、胸径与树高的回归模型,再测量采伐木地径,用回归模型预测采伐木的胸径和树高,推算采伐木蓄积量。同时,采用地径一元立木材积表法测算林木蓄积量,并与采伐作业设计的二元立木材积表法测算的林木蓄积量相比较。结果表明:采用建立相似林分样木因子回归模型推算采伐木蓄积量方法接近采伐作业设计的蓄积量。     

基于树木特征因子预测木材材性及利用价值的研究概述

系统介绍了树木特征因子与林木材性、品质、等级以及经济价值之间的关系, 以期为实现我国森林资源的高效、合理利用提供新的思路。     

Representation of tree stem taper curves and their dynamic, using a linear model and the centroaffine transformation

One main task of forestry is a reliable estimation of the stem form and its development applied in calculating total and log volume. As long as process-oriented models are not available for this practical use, empirical models must serve instead. Taper curve data of trees within stands normally show a rank maintenance,i.e., a tree which has a greater diameter than another one at a certain height, is most probably bigger at any other height, too. This property also applies to the analysed tree species, sugi (Cryptomeria japonica) and hinoki (Chamaecyparis obtusa), and will be well-represented by a linear model formulation. As model parameter sets of single stands have a very limited time validity, two approaches for future stem form prediction are discussed. The one, the centroaffine transformation of a linear taper curve function, is not suitable for representing the time-depending change of the stem form. However, this can be done by a linear unit taper curve model, the parameters of which are based on sample trees of stands of several age classes. Temporary unit parameter sets are derived for sugi and hinoki and the estimated stand volumes are compared to the real ones to evaluate the model performance, which turned out to be very good. We would like to express our gratitude to the Japan Science and Technology Corporation for the financial promotion, which enabled this research work.     

A flexible regression model for diameter prediction

We present a functional regression model for diameter prediction. Usually stem form is estimated from a regression model using dbh and height of the sample tree as predictor. With our model additional diameter observations measured at arbitrary locations within the sample tree can be incorporated in the estimation in order to calibrate a standard prediction based on dbh and height. For this purpose, the stem form of a sample tree is modelled as a smooth random function. The observed diameters are assumed as independent realizations from a sample of possible trajectories of the stem contour. The population average of the stem form within a given dbh and height class is estimated with the taper curves applied in the national forest inventory in Germany. Tree deviation from the population average is modelled with the help of a Karhunen–Loève expansion for the random part of the trajectory. Eigenfunctions and scores of the Karhunen–Loève expansion are estimated through conditional expectations within the methodological framework of functional principal component analysis (FPCA). In addition to a calibrated estimation of the stem form, FPCA provides asymptotic pointwise or simultaneous confidence intervals for the calibrated diameter predictions. For the application of functional principal component analysis modelling the covariance function of the random process is crucial. The main features of the functional regression model are discussed informally and demonstrated by means of practical examples.     

Biomass, basic density and biomass expansion factor functions for European beech (Fagus sylvatica L.) in Denmark

The objective of this study was to analyse the within-tree allocation of biomass and to develop biomass functions for above- and below-ground components of European beech in Denmark. Separate functions were developed for stem, branches, below-ground stump and root system, total above-ground biomass and total tree biomass. For each of these components or aggregate components, models were also developed for the average basic density of wood and bark. To enhance the versatility of the models, a function for estimating the biomass expansion factor (BEF) was also developed. The functions were based on 66 trees measured for total biomass. Model performance was evaluated based on 74 trees measured only for above-ground biomass. The trees were sampled in 18 different forest stands covering a wide range of tree sizes and stand treatments. Models were estimated using a linear mixed-effects procedure to account for within-stand correlations. The functions for biomass and BEFs included only diameter at breast height and total tree height for individual trees as predictor variables. Inclusion of additional variables reflecting site quality or stand density did not improve model performance. The functions for basic density included individual tree diameter, tree height and quadratic mean diameter as predictor variables, indicating an effect of stand density on the basic density of wood and bark.     

Estimating stand water use of large mountain ash trees and validation of the sap flow measurement technique

Mountain ash (Eucalyptus regnans F.J. Muell.) forest catchments exhibit a strong relationship between stand age and runoff, attributed inter alia to differences in tree water use. However, the tree water use component of the mountain ash forest water balance is poorly quantified. We have used the sap flow technique to obtain estimates of daily water use in large mountain ash trees. First, the sap flow technique was validated by means of an in situ cut tree experiment. Close agreement was obtained between the sap flow estimate of water use and the actual uptake of water by the tree from a reservoir. Second, we compared the variability in sap velocity between a symmetric and an asymmetric tree by using multiple sap flow loggers. In the symmetric tree, velocity was fairly uniform throughout the xylem during the day, indicating that accurate sap flow estimates can be obtained with a minimal number of sampling points. However, large variations in sap velocity were observed in the asymmetric tree, indicating that much larger sampling sizes are required in asymmetric stems for an accurate determination of mean sap velocity. Finally, we compared two procedures for scaling individual tree sap flow estimates to the stand level based on stem diameter and leaf area index measurements. The first procedure was based on a regression between stem diameter and tree water use, developed on a small sample of trees and applied to a stand-level census of stem diameter values. Inputs to the second procedure were tree water use and leaf area of a single tree and the leaf area index of the stand. The two procedures yielded similar results; however, the first procedure was more robust but it required more sampling effort than the second procedure.     

Structure, allometry, and biomass of plantation Metasequoia glyptostroboides in Japan

We quantified structural features and the aboveground biomass of the deciduous conifer, Metasequoia glyptostroboides (Hu and Cheng) in six plantations in central Japan. In order to derive biomass estimates we dissected 14 M. glyptostroboides trees into three structural components (stem wood, branch wood and foliage) to develop allometric equations relating the mass of these components and of the whole tree to diameter at breast height (DBH). We found robust relationships at the branch and whole tree level that allow accurate prediction of component and whole tree biomass. Dominant tree height was similar within five older (>40 years) plantations (27–33 m) and shorter in a 20-year-old plantation (18 m). Average stem diameter varied from 12.8 cm in the youngest stand to greater than 35 cm in the oldest stand.

Metasequoia have relatively compact crowns distributed over the top 30% of the tree although the youngest stand had the deepest crown relative to tree height (up to 38%). At the individual tree level in older stands, 87% of the aboveground biomass was allocated to the stem, 9% to branch wood and 4% to foliage. We found little difference in the relative distribution of above ground biomass among the stands with the exception of lower foliage biomass in larger diameter trees. Total aboveground biomass of the older stands varied twofold, ranging from a maximum of 450 Mg ha⁻¹ in a 42-year-old stand to a minimum of 196 Mg ha⁻¹ in a 48-year-old stand. Total above ground biomass of the 20-year-old stand was 176 Mg ha⁻¹.     

Crown radius of pedunculate oak (Quercus robur L.) depending on stem size,stand density and site productivity

Crown size is a good indicator of the growth potential of trees and is often used in forest management for outlining thinning guidelines or constructing forest growth models. The aim of this study was to analyse mean crown radius as a function of stem size, stand density and site productivity in even-aged stands of pedunculate oak (Quercus robur L.). Data included measurements of 620 trees from 53 plots in nine thinning experiments and one operational stand in Sweden, Denmark and Great Britain, representing a wide spectrum of thinning practices ranging from the strictly unthinned control to extremely heavy thinning with essentially solitary trees. Three sets of models were constructed based on different predictor variables, including indicators of individual stem size (diameter at breast height, DBH), stand density/thinning grade (quadratic mean diameter and stand basal area) and site productivity (stand top height). Preliminary results indicated a significant effect of DBH and (nominal) thinning grade on crown radius. The response pattern of the final models indicated an increasing crown radius with increasing DBH, with increasing thinning grade (decreasing stand density) and with decreasing site productivity. The models are valid for predicting the crown radius of pedunculate oak in even-aged forest stands.     

Combining ALS and NFI training data for forest management planning: a case study in Kuortane,Western Finland

Forest inventories based on airborne laser scanning (ALS) have already become common practice in the Nordic countries. One possibility for improving their cost effectiveness is to use existing field data sets as training data. One alternative in Finland would be the use of National Forest Inventory (NFI) sample plots, which are truncated angle count (relascope) plots. This possibility is tested here by using a training data set based on measurements similar to the Finnish NFI. Tree species-specific stand attributes were predicted by the non-parametric k most similar neighbour (k-MSN) approach, utilising both ALS and aerial photograph data. The stand attributes considered were volume, basal area, stem number, mean age of the tree stock, diameter and height of the basal area median tree, determined separately for Scots pine, Norway spruce and deciduous trees. The results obtained were compared with those obtained when using training data based on observations from fixed area plots with the same centre point location as the NFI plots. The results indicated that the accuracy of the estimates of stand attributes derived by using NFI training data was close to that of the fixed area plot training data but that the NFI sampling scheme and the georeferencing of the plots can cause problems in practical applications.     

Effect of wide spacing on increment and branch properties of young Norway spruce

The effects of stand density on increment and branch properties were studied in three spacing experiments of Norway spruce [Picea abies (L.) Karst.]. The stand densities ranged from 350 stems ha⁻¹, regarded as open-grown trees, up to 1,600 stems ha⁻¹, corresponding to the density recommended for forestry practice. Properties of all the branches were measured from the stem apex downwards. The study material included a total of 5,661 branches from 45 trees. Increasing stand density resulted in a decrease in radial increment as well as shorter and narrower crowns, but it had no effect on height increment. The average number of spike knots per tree was 0.87, 0.27, and 0.33 in densities of 350, 700 and 1,600 ha⁻¹, respectively. Additionally, in the widely spaced stands of 350 stems ha⁻¹, the fraction of trees having spike knots was high (over 50%). At a density of 1,600 ha⁻¹, the sample trees had somewhat less branches in a whorl compared with the more widely spaced plots. The most pronounced effect of stand density was the increase in branch diameter with decreasing stand density. At a density of 350 ha⁻¹, the maximum branch diameter of all the sample trees exceeded the diameter limit of quality class B in the European quality requirements for round wood. The results give some indication that trees subjected to severe competition would produce smaller branches per unit of crown projection area. However, the possibilities for reducing branch dimensions relative to stem and crown size through competition appear quite restricted.     

基于非线性混合模型的落叶松树干削度模型

以黑龙江省带岭林业局大青川林场84株人工落叶松解析木数据为例,采用Max和Burkhart分段削度模型作为基础模型,利用SAS软件中的似乎不相关回归过程得到该分段削度模型的4个参数和2个拐点参数同时估计。参数估计显著性检验(P<0.000 1)以及模型检验(F=31 392.30,P<0.000 1)都证明该分段模型能较好地描述落叶松树干干形变化。然后以该分段模型为基础模型,采用非线性混合模型的方法,建立落叶松人工林树干削度混合效应模型。结果表明:当考虑样地效应影响时,b1,b2同时作为混合参数时模型拟合最好;当考虑树木效应影响时,b2,b4同时作为混合参数时模型拟合最好。无论考虑样地效应影响还是考虑树木效应影响,混合模型的拟合精度都比基本模型的拟合精度高,并且考虑树木效应影响要比考虑样地效应影响的精度更高。模型检验结果表明:混合模型通过校正随机参数值能提高模型的预测精度。     

Derivation of stem taper from the pipe theory in a carbon balance framework

A dynamic tree growth model is described. The model derives the development of stem taper and vertical distribution of branch basal area from the pipe model, assuming that reuse of active pipes is regulated by foliage dynamics in a vertically explicit crown with a foliage distribution of constant shape. Based on empirical findings, the pipe model was modified slightly to allow the foliage/sapwood ratio to vary as a function of distance from the treetop. Growth was derived from carbon balance in a stand of different size trees that may shade each other. The model was applied to old and middle-aged trees growing in dense and sparse stands of Scots pine for which stand-level measurements are available as a chronosequence, but individual trees have been measured only once. Measured trees were compared with corresponding simulated trees for stem taper and vertical distribution of branch basal area. The results indicated that the pipe model assumptions, combined with a model of tree growth, are capable of producing realistic predictions of the vertical distribution of stem and branch diameter in trees of different sizes in the stand. A comparison of the results with a simple form of the uniform stress theory showed good agreement between the two models. However, a significant difference was found between the measured relative contribution of heartwood to total stem diameter and the predicted share of disused pipes in the stem. A possible explanation for this discrepancy is that the transition from sapwood to heartwood is gradual rather than abrupt as assumed in the model. A modification of the pipe model to incorporate a gradual transition is outlined.     

黑龙江东部地区落叶松人工林节子大小的研究

以不同年龄、不同密度、不同立地条件的落叶松人工林作为研究对象,基于10块标准地中50株样木所获得的死枝和节子大小数据,利用数据统计分析模型和模拟技术,分析并研究了节子大小在树干上的分布规律,并构建了节子大小的预测模型。研究结果表明:1)节子直径随着节子着生高度的增加而持续增加,在树冠基部达到一渐进值,可用Mitscherlich方程对节子直径进行预测;2)在树高约40%处以下的健全节长度随着节子着生位置的增加而逐渐增加,然后再向上有减小的趋势;3)疏松节随着胸径的增大而增大,随着节子着生角度的增大而减小。     

可变参数相对树高曲线模型及其应用研究

用我国现行一元立木材积表估计林分或样地的立木材积偏差很大，而直接利用二元立木材积表，则野外实测树高的工作量又太大。本研究针对这一实际问题，提出了通过建立可变参数相对树高曲线模型，将二元立木材积表直接用于林分或样地立木材积估计的方法，其估计精度可达到接近实测树高曲线法的水平。研究提出的可变参数相对树高曲线模型结构通式为：ＲＨｉ＝。     

西南桦优树选择技术研究

以胸径、树高、材积为数量评价指标,结合干形、冠高比和分枝粗细形质指标,采用5株优势木对比法开展了西南桦人工林的优树选择技术研究。通过对西南桦人工林10 21年生不同林龄阶段的优树选择标准的总结,提出了优树选择标准,即优树胸径、树高、单株材积分别大于5株对比优势木平均值的9%11%、5%7%、26%31%以上,形质指标综合得分大于7.5,共选择出西南桦优树37株,入选率为33.9%。西南桦优树的选择标准在实际应用中可根据林分状况适当调整,选优即要考虑其生长性状,同时注意材性、抗性的选择以保证西南桦育种群体的遗传多样性。     

落叶松人工林林分因子模型的建立

利用落叶松人工林所设标准地的观测数据, 求算各标准地的威布尔分布参数a、b、c, 并将其与林分因子: 平均直径、上层平均树高、株数密度结合, 建立了Richards3种林分因子模型。经检验其精度达到96 7 % ~97 7%, 通过进一步检验表明, 密度、上层高、直径模型优于其他两种模型, 可在森林经营上应用。     

邓恩桉优树的选择标准

以1988、1991年种植在广西柳州、桂林两地的邓恩桉为选优林分对象,用5株优势木对比法进行选优,同时研究了优树的入选标准.通过对50株候选优树和250株5株优势木生长量数据、分枝与干形得分值的分析,确定了本次选优的标准为:胸径≥优势木平均胸径1.2倍(或树高≥优势木平均树高1.06倍)、单株材积≥优势木平均单株材积1.44倍、分枝与干形二者综合得分5分以上(含),符合该选优标准的候选优树有26株,入选率为52%.该选优标准适合所研究的林分及与研究林分情况相近的邓恩桉林分的选优.     

高阿丁枫人工林幼中林期的生长规律

对西双版纳普文试验林场的21年生高阿丁枫人工林进行了样地调查, 选取平均木作树干解析, 据此全面分析了高阿丁枫人工林幼中林期的直径、树高、材积生长过程以及林分状态、林木的年生长节律和结实状况,结果表明在此期间高阿丁枫人工林林木的树高生长旺盛期在3~5年, 胸径在3~7年, 材积在15年以后增长较快; 林分中林木分化不明显; 年生长期长。林分尚未进入数量成熟期。