A taper function for Pseudotsuga menziesii plantations in Spain

Five stem taper models belonging to three different taper function categories were fitted to data corresponding to 282 Pseudotsuga menziesii trees. The trees were selected in the area surrounding 61 research plots installed in Galicia, Asturias and the Basque Country, northern Spain. The models were simultaneously fitted to observed values of diameter outside bark and inside bark. A third-order continuous-time autoregressive error structure was used to account for autocorrelation. Selection of the best model was based on both numerical (goodness-of-fit statistics) and graphical analysis (plots of residuals against position along the stem and against tree size). The three-segmented taper model finally selected has the advantage of being compatible with both a merchantable and a total stem volume equation.     

Modeling stem profile of Lebanon cedar,Brutian pine,and Cilicica fir in Southern Turkey using nonlinear mixed-effects models

The Max and Burkhart segmented taper model was fitted using nonlinear mixed-effects modeling techniques to account for within- and between-individual stem profile variation for Lebanon cedar (Cedrus libani A. Rich.), brutian pine (Pinus brutia Ten.), and cilicica fir (Abies cilicica Carr.) in Turkey. About 75% of the trees were randomly selected for model development, with the remainder used for model validation. Diameter measurements from various heights were evaluated for tree-specific calibrations by predicting random-effects parameters using an approximate Bayesian estimator. The procedure was tested with a validation dataset. Predictive accuracy of the model was improved by including random-effects parameters for a new tree based on upper stem diameter measurements. Prediction in stem diameter was less biased and more precise across the all sections of bole when compared to predictions based only on fixed-effects parameters. In the future, the proposed mixed models can be applied to region wide three species stands by fitting the model to a larger data set that more closely represents regional variation.     

Selection of mixed-effects parameters in a variable–exponent taper equation for birch trees in northwestern Spain

Context

Taper equations predict the variation in diameter along the stem, therefore characterizing stem form. Several recent studies have tested mixed models for developing taper equations. Mixed-effects modeling allow the interindividual variation to be explained by considering both fixed-effects parameters (common to the population) and random-effects parameters (specific to each individual).

Aims

The objective of this study is to develop a mixed-effect variable exponent taper equation for birch trees in northwestern Spain by determining which fixed-effects parameters should be expanded with random-effects parameters.

Methods

All possible combinations of linear expansions with random effects in one and in two of the fixed-effects model parameters were tested. Upper stem diameter measurements were used to estimate random-effects parameters by the use of an approximate Bayesian estimator, which calibrated stem profile curves for individual trees.

Results

Parameter estimates for more than half of the mixed models investigated were nonsignificant. A first order autoregressive error structure was used to completely remove the autocorrelation between residuals, as mixed-effects modeling were not sufficient for this purpose.

Conclusion

The mixed model with the best fitting statistics did not provide the best calibration statistics for all upper stem diameter measurements. From a practical point of view, model calibration should be considered an essential criterion in mixed model selection.     

兴安落叶松树干去皮直径预测模型

利用兴安落叶松解析木数据,比较了树干去皮直径预测的3种类型模型:Grosenbaugh的比率方程式、回归模型和削度方程。Grosenbaugh的比率方程式有很大的灵活性,没有参数不需要模型拟合。总体评价和模型分段比较表明,回归模型有较小的预测误差,尤其是Cao and Pepper提出的含有带皮直径、树高、相对树高、胸径处的带皮直径和去皮直径变量的模型。由于削度模型不含有带皮直径变量,因此产生较大的去皮直径预测误差。不同类型的模型在森林经营过程中都有一定的适应性。     

Estimation of taper rates and volume of smaller-sized logs in spotted gum saw timber plantations in Southeast Queensland,Australia

Spotted gum (Corymbia citriodora subsp. variegata) is a popular tree species for hardwood saw timber plantations in Southeast Queensland (SEQ), Australia. In many parts of the world, logs up to 10cm top diameter are considered to be merchantable logs and acceptable at sawmills. However, due to the higher handling costs of smaller-sized logs, sawmills in SEQ are only buying logs with a top diameter of 25cm and larger. This necessitates the determination of the volume of logs with top diameters between 10 and 25cm. This paper compares the taper rates of different sizes of spotted gum logs and estimates the stem volume of spotted gum logs between 10 and 24.94cm diameter. Analysis showed that there is a statistically significant difference between the mean taper rates of bottom (30.5–64.2cm diameter) and top parts of stems (10–24.94cm diameter) at harvesting age. Using only the average taper rate of large diameter logs would underestimate the log volume of smaller-diameter logs. The average length of spotted gum logs between 10 and 24.94cm was found to be 19.27 m. Therefore, by the sawmill declining to use 10–24.94cm diameter logs, approximately 0.55m³ of log volume per tree at harvesting age would be lost.     

A simple stem taper model with mixed effects for boreal black spruce

We simplified Kozak’s taper model by setting the inflection point at 1.3 m (dbh) without losing accuracy and precision. The simplification was required to facilitate the estimation of the covariance parameters when using a mixed-effects method. This method was necessary to take into account the correlation among multiple diameter measurements on an individual stem. The simple stem taper model was fitted to an extended data set collected across the province of Quebec, Canada. Comparison of the predicted stem taper and the derived stem volume with those obtained using existing models showed a comparable predictive power for the simple model. Including a prediction of the tree random effects based on supplementary diameter measurements of the bole improves the predictive ability of the model around the extra diameter observation. This model offers welcome simplicity as a means of predicting tree taper at coarse resolution for planning tree harvesting.     

Segmented taper equations with crown ratio and stand density for Dahurian Larch (<Emphasis Type="Italic">Larix gmelinii</Emphasis>) in Northeastern China

Segmented taper equation was selected to model stem profile of Dahurian larch (Larix gmelinii Rupr.). The data were based on stem analysis of 74 trees from Dailing Forest Bureau in Heilongjiang Province, Northeastern China. Two taper equations with crown ratio and stand basal area were derived from the Max and Burkhart’s (1976) taper equation. Three taper equations were evaluated: (1) the original equation, (2) the original equation with crown ratio, and (3) the original equation with basal area. SAS NLIN a...     

Integrated individual tree biomass simultaneous equations for two larch species in northeastern and northern China

Forest biomass estimation at large scale has become an important topic in the background of facing global climate change, and it is fundamental to develop individual tree biomass equations suitable for large-scale estimation. Based on the measured data of biomass components and stem volume from 100 sample trees of two larch species (Larix gmelinii and L. principis-rupprechtii) in northeastern and northern China, an integrated equation system including individual tree biomass equations, stem volume equation and height–diameter regression model were constructed using the dummy variable model and error-in-variable simultaneous equations. In the system, all the parameters of equations were estimated simultaneously, so that the aboveground biomass equation was compatible to stem volume equation and biomass conversion factor (B_CF) function; the belowground biomass equation was compatible to root-to-shoot ratio (R_SR) function; and stem wood, stem bark, branch and foliage biomass equations were additive to aboveground biomass equation. In addition, the system also ensured the compatibility between one- and two-variable models. The results showed that: (1) whether aboveground biomass equations or belowground biomass equations and stem volume equations, the estimates for larch in northeastern China were greater than those in northern China; (2) B_CF of a larch tree decreased with the growing diameter while R_SR increased with the growing diameter; (3) the proportion of stem wood biomass to aboveground biomass increased with the growing diameter while those of stem bark, branch, and foliage biomass decreased.     

Stem taper equations for poplars growing on farmland in Sweden

We developed a simple polynomial taper equation for poplars growing on former farmland in Sweden and also evaluated the performance of some well-known taper equations. In Sweden there is an increasing interest in the use of poplar. Effective management of poplar plantations for high yield production would be facilitated by taper equations providing better predictions of stem volume than currently available equations. In the study a polynomial stem taper equation with five parameters was established for individual poplar trees growing on former farmland. The outputs of the polynomial taper equation were compared with five published equations. Data for fitting the equations were collected from 69 poplar trees growing at 37 stands in central and southern Sweden (lat. 55-60° N). The mean age of the stands was 21 years (range 14-43), the mean density 984 stems ha -1 (198 3,493), and the mean diameter at breast height (outside bark) 25 cm (range 12-40). To verify the tested equations, performance of accuracy and precision diameter predictions at seven points along the stem was closely analyzed. Statistics used for evaluation of the equations indicated that the variable exponent taper equation presented by Kozak (1988) performed best and can be recommended. The stem taper equation by Kozak (1988) recommended in the study is likely to be beneficial for optimising the efficiency and profitability of poplar plantation management. The constructed polynomial equation and the segmented equation presented by Max & Burkhart (1976) were second and third ranked. Due to the statistical complexity of Kozak’s equation, the constructed polynomial equation is alternatively recommended when a simple model is requested and larger bias is accepted.     

A set of models for individual tree merchantable volume prediction for <Emphasis Type="Italic">Pinus pinaster</Emphasis> Aiton in central inland of Portugal

The study purpose selected among several candidate models for best individual tree, over bark, total volume model, volume ratio model to any top height limit and taper model for maritime pine (Pinus pinaster Aiton) in the regions of Pinhal Interior Sul and Beira Interior Sul, Portugal. The data used in the study were collected from 144 felled trees, corresponding to 995 diameter/height measurements. To select among the best models, several statistics were computed during model fitting, and the independent validation procedure was used to evaluate model fitting, collinearity and prediction performance. A ranking index was used to support the final decision. The analysis of models studentized residuals distribution showed that some regression model assumptions, such as normality and homogeneity, were not met. To overcome this unideal situation, the models selected were then fitted again using robust regression and weighted regression techniques. The set of adjusted models will allow the prediction of individual tree, over bark, total volume and merchantable volume to any merchantable limit, for both species and region to support management decisions.     

Comparison of model forms for estimating stem taper and volume in the primary conifer species of the North American Acadian Region

? The performance of ten commonly used taper equations for predicting both stem form and volume in balsam fir [Abies balsamea (L.) Mill], red spruce[Picea rubens (Sarg.)], and white pine[Pinus strobus (L.)] in the Acadian Region of North America was investigated.

? Results show that the Kozak (2004) and Bi (2000) equations were superior to the other equations in predicting diameter inside bark for red spruce and white pine, while the Valentine and Gregoire (2001) equation performed slightly better for balsam fir.

? For stem volume, the Clark et al. (1991) equation provided the best predictions across all species when upper stem diameter measurements were available, while the Kozak (2004) and compatible taper equation of Fang et al. (2000) performed well when those measurements were unavailable.

? The incorporation of crown variables substantially improved stem volume predictions (mean absolute bias reduction of 7–15%; root mean square error reduction of 10–15%) for all three species, but had little impact on stem form predictions.

? The best taper equation reduced the predicted root mean square error by 16, 39, and 45% compared to estimates from the widely used Honer (1965) regional stem volume equations for balsam fir, red spruce, and white pine, respectively.

? When multiple taper equations exist for a certain species, the use of the geometric mean of all predictions is an attractive alternative to selecting the “best” equation.

     

An evaluation of the conical approximation as a generic model for estimating stem volume, biomass and nutrient content in young Eucalyptus plantations

Accurately and non-destructively quantifying the volume, mass or nutrient content of tree components is fundamental for assessing the impact of site, treatment, and climate on biomass, carbon sequestration, and nutrient uptake of a growing plantation. Typically, this has involved the application of allometric equations utilising diameter and height, but for accurate results, these equations are often specific to species, site, and silvicultural treatment. In this study, we assessed the value of incorporating a third piece of information: the height of diameter measurement. We derived a more general volume equation, based on the conical approximation, using a diameter projected to the base of the tree. Common equations were developed which allowed an accurate estimate of stem volume, dry weight and nutrient content across two key plantation grown eucalypt species, Eucalyptus grandis W. Hill ex Maiden and Eucalyptus globulus (Labill.). The conical model was developed with plantation-grown E. grandis trees ranging from 0.28 to 15.85 m in height (1.05 g to 80.3 kg stem wood dry weight), and E. globulus trees ranging from 0.10 to 34.4 m in height (stem wood dry weight from 0.48 g to 652 kg), grown under a range of contrasting cultural treatments, including spacing (E. grandis), site (E. globulus) and fertilization (nitrogen and phosphorus) for both species. With log transformed data the conical function (V_con) was closely related to stem sectional volume over bark and stem weight (R² = 0.996 and 0.990, respectively) for both E. grandis and E. globulus, and the same regressions can be applied to both species. Back transformed data compared with the original data yielded modelling efficiencies of 0.99 and 0.97, respectively. Relationships between V_con and bark dry weight differed for the two species, reflecting differing bark characteristics. Young trees with juvenile foliage had a different form of relationship to older trees with intermediate or adult foliage, the change of slope corresponding to heights about 1.5 m for E. grandis and age 1 year for E. globulus. The V_con model proved to be robust, and unlike conventional models, does not need additional parameters for estimating biomass under different cultural treatments. More than 99% of the statistical variance of the logarithm of biomass was accounted for in the model. V_con captures most of the change in stem taper associated with cultural treatments and some of the change in stem form that occurs after the crown base has lifted appreciably. Fertilization increased N and P concentrations in stem wood and bark, and regressions to estimate N and P contents (the products of biomass and concentration) were dependent on treatment. For instance, there was a large growth response to N fertilization in E. globulus corresponding with a change (P < 0.05) in the intercept of the regression to estimate N content.     

Mixed-effect non-linear modelling for diameter estimation along the stem of Tectona grandis in mid-western Brazil

This study evaluated the efficiency of taper functions and the application of mixed-effect modelling for diameter estimation along the stems of Tectona grandis. We sampled 266 trees of Tectona grandis, measuring the diameter at relative heights for volume determination, grouping the data according to three form-factor classes. Six taper functions were fitted, selecting the function with better fit performance. Six taper functions were fitted, selecting the function with better fit performance. The selected function was fitted in its basic formulation, and with the mixed non-linear modelling technique in different scenarios, and for the stem stratified in three portions of the total height. The precision and selection of the adjusted models were evaluated regarding the coefficient of determination, standard error of estimate, the Akaike information criterion, bias, quadratic error and absolute bias. According to the statistical criteria used, the model of Kozak was selected for the adjustments. For diameter estimation, the scenario with two coefficients as random effects provided an accuracy increase of 11.91%, and the mixed non-linear modelling better estimated the stem diameter for the stratified stems. In conclusion, the model of Kozak can be used to describe the stem shape of Tectona grandis, and the mixed-effect non-linear model approach was the best technique to estimate diameter along the stem of Tectona grandis.     

Tree taper models for Cupressus lusitanica plantations in Ethiopia

To provide a taper model for planning and management of Cupressus lusitanica plantations in Ethiopia, seven taper models were compared. Four performance indicator statistics were used for comparing the models in their ability to estimate tree diameter, and total and merchantable volumes. For the selected species, Kozak (1988) was found to be the best, followed by modified Lee et al. (2003) and Kozak (2004) as second and third best taper models, respectively. Both the Kozak (Kozak, 1988, 2004) and the modified Lee et al. (2003) models were very flexible in capturing the different shapes of trees. In particular, Kozak (2004) proved to be best of all models in diameter estimation even though it was found to be inferior to the Kozak (1988) and modified Lee et al. (2003) models for total and merchantable volume estimation. To understand the influence of the inflection point p in the Kozak (1988) model, Monte Carlo simulation was used to examine sensitivity of the model performance against inflection point p. In this effort, slightly lower standard errors were observed for p between 0.1 and 0.3. Mixed effects modeling was also used to further study the inflection point p as a random effect. The empirical Bayes estimates of the random effects were found to vary from tree to tree and also appear to depend on tree size. The results from both the Monte Carlo and mixed effects modeling study seem to indicate the need to estimate p from the data.     

Modelling the economically viable wood in the crown of European beech trees

Long-term forest development programs in Germany aim on an increase of close-to-nature broadleaf forest stands. This means that the economic importance of European beech is expected to increase. The economic potential of a tree basically consists of the stem as well as the economically viable wood volume in the crown. Due to the high morphological variability of European beech crowns, taper models are often not satisfactory for predicting the economically viable wood volume arising from crowns. Prediction models with a higher precision are recently still lacking. Aim of this study is thus the development of prediction model for the economically viable crown wood volume of European beech trees.We determined the distribution of the wood volume in the crown over the branch diameters using the multistage ‘randomized branch sampling’ method (RBS). The tree-specific wood volume distribution on the branch diameters were used to cluster all sampled trees into 3 groups. Additionally, we developed a method able to distinguish between economically viable and unviable crown branches. Basing on the RBS measurements as well as revenues and processing costs, we modeled the economically viable wood volume from the crown for each tree. To calculate the wood volume under bark, we parameterized a bark thickness function from disk samples of the trees.We showed that the European beech crowns could be clustered into 3 groups differing in their wood volume distribution. The economically viable wood volume in the crown significantly depended on this grouping parameter as well as diameter at breast height (DBH). By contrast, the total amount of wood in the crown only depended on DBH. The differing viable wood volumes in the crowns were thus explained by different wood distributions and not by differing total crown wood volume. To make the results applicable in practice forestry, the modelling results were used to develop a regression formula able to predict the economically viable wood volume in the crown depending on the DBH and the crown type. As the crown type can also be predicted via measurable tree covariates, the regression model of the viable wood volume in the crown can be used as a support tool for the management of European beech stands. Sensitivity analysis quantifies how harvest revenues and costs translate into different viable tree volume.     

理论造材:削度方程和出材率表的编制

本文提出了计算机理论造材仅根据削度方程和单株带,去皮胸径的转换关系式两个模型,可不必由削度方程推导材长方程和材积比方程。比相关指数,均方程为指标,结合残差图分析,选择了杉木、落叶松的削度方程;结合标准树高曲线,是可仅根据优势高为任一具体林分编制（一元）单株出材率表（特称之为自动调整出材率表）。     

A merchantable volume system for major pine species in El Salto,Durango (Mexico)

A compatible volume system for the major pine species in El Salto, Durango (Mexico) was developed from data corresponding to 1930 destructively sampled trees. Several well-known taper functions were evaluated and compared against the model selected in a previous study of these pine species in the same area. Appropriate statistical procedures were used in model fitting to account for the problems of autocorrelation and multicollinearity that are associated with the construction of taper functions. A compatible segmented model best described the experimental data and was found to be better than the previously selected model. It is therefore recommended for estimating diameter at a specific height, height to a specific diameter, merchantable volume, and total volume for the five species analyzed. The non-linear extra sum of squares method indicated differences in species-specific taper functions. A different taper function should therefore be used for each pine species.     

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 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.     

兴安落叶松立木材积和树皮材积可加性模型研建

【目的】确保立木材积和树皮材积预测的一致性并提高预测精度。【方法】以大兴安岭兴安落叶松为研究对象,分别采用控制法和分解法研建了可加性模型系统。利用SAS统计软件模型模块proc model中的NSUR法进行拟合及参数估计。拟合结果采用确定系数(R2)和均方根误差(RMSE)进行评价;检验结果则通过确定系数(R2)、均方根误差(RMSE)、平均相对误差(MRE)、平均误差绝对值(MAB)和相对误差绝对值(MPB)进行评价。【结果】从模型的整体评价结果来看,两种方法的拟合和检验效果均很好,基于分解法构建的模型略优于基于控制法构建的模型;不同径阶的检验表明,对于中等径阶的树木(20≤D<36 cm),基于控制法的模型相对较好,而对于小径阶(5≤D<20 cm)和大径阶的树木(D≤36 cm),基于分解法的带皮、去皮、树皮材积模型的预测精度要比基于控制法的各立木材积模型要稍好。【结论】总的来说,两种可加性模型系统均能很好地预测单木带皮材积、去皮材积和树皮材积,并确保得到满足一致性的预测结果,在具体应用时可根据实际情况选择适合的可加性材积模型系统。