Taper Curve Research Articles

Modeling a new taper curve and form factor of tree branches using terrestrial laser scanning

Abstract Modeling branch taper curve and form factor contributes to increasing the efficiency of tree crown reconstructions: the branch taper, defined as the sequential measure of diameters along the course of the branch, is pivotal to accurately estimate key branch variables such as biomass and volume. Branch diameters or volumes have commonly been estimated from terrestrial laser scanning (TLS) based on automatized voxelization or cylinder-fitting approaches, given the whole branch length is sufficiently covered by laser reflections. The results are, however, often affected by ample variations in point cloud characteristics caused by varying point density, occlusions, and noise. As these characteristics of TLS can hardly be sufficiently controlled or eliminated in automatized techniques, we proposed a new branch taper curve model and form factor, which can be employed directly from the laser reflections and under variable point cloud characteristics. In this paper, the approach is demonstrated on primary branches using a set of TLS-derived datasets from a sample of 20 trees (six species). The results showed an R2 of 0.86 and a mean relative absolute error of 1.03 cm (29%) when validated with field-measured diameters. The approach improved the accuracy of diameter estimates for the fine branch scales (<10 cm) as compared to the quantitative structural model (QSM). Our approach also allowed branch diameter estimation for a relatively larger number of manually recognized primary branches (>85%) from point clouds when validated with panoramic images acquired simultaneously with laser scanning. Frequently used automatized crown reconstructions from QSM, on the other hand, were affected by gaps in the point clouds due to obstruction, with the crown-tops and finer branches being the most critical. Our approach reports mean form factors across the examined species of 0.35 and 0.49, with the diameters determined at 5% and 10% of the total branch length, respectively. Our approach may have the potential to produce branch volume information with reasonable accuracy from only knowing the length and respective diameter of each branch. Our model delivers a first approximation for the taper curve and form factor for primary branches but was developed on a relatively small set of samples. We believe that our approach holds the potential to improve the accuracy of the assessment of branch diameter and volume from TLS data. The approach may also be extended to other branch orders. This could expand the horizon for volumetric calculations and biomass estimates from non-destructive TLS proxies in tree crowns.

Validation of saw log and technological wood assortment recovery and reduction predictions based on cut-to-length harvester data

Abstract In this study, assortment yields were studied for Scots pine, Norway spruce, birch spp. (silver and downy birch), European aspen, black alder, and grey alder stand elements. Theoretical assortment yields were calculated using the Ozolinš’ stem taper curve, incorporating tree diameters and heights. The modelled results were compared with actual yields provided by the State Forest Management Centre. During the study, differences in timber assortments and firewood yields were modelled compared to actual data. Changes in wood assortment yields, compared with undamaged and straight trees, depended on the tree species, stand site index, and stand element age. For Scots pine, depending on the stand site index and age, the reduction in log volume ranged from 7 to 28% of the total volume of all assortments. For Norway spruce, it was 5–30%, for birch spp. 30–70%, for aspen 50–90%, for black alder 20–50% and grey alder 5–30%. The increase in firewood volume according to the volume of all assortments was 3–4% for Scots pine, 5–14% for Norway spruce and birch spp., and 5–16% for European aspen. The difference between log and firewood changes represents the change in pulpwood. Over time, the need for new studies arises to adapt to evolving industry practices, including changes in log diameters and quality criteria. The appendix outlines steps that programmers can take to utilize the developed model.

Local taper functions for natural black pine forests  in the Northwestern Rhodopes in Bulgaria Local taper functions for black pine

Unlike the heavily deforested Eastern or Low Rhodopes, the Western or High Rhodopes are one of Bulgaria’s main forestry areas, where black pine is widespread and economically important. Based on stem analyzes, an average taper curve for black pine stems in the Northwestern Rhodopes was determined. The method of relative taper curves (echte Ausbauchungsreihen) was used. Taper models based on spline functions and power functions were fitted. A comparison with the official assortment table of Nedyalkov was made. The systematic deviation of the sample stems from the long-used assortment table justifies the development of a local black pine table. 

Biomass allometric models for Larix rupprechtii based on Kosak's taper curve equations and nonlinear seemingly unrelated regression.

The diameter at breast height (DBH) is the most important independent variable in biomass allometry models based on metabolic scaling theory (MST) or geometric theory. However, the fixed position DBH can be misleading in its use of universal scaling laws and lead to some deviation for the biomass model. Therefore, it is still an urgent scientific problem to build a high-precision biomass model system. A dataset of 114 trees was destructively sampled to obtain dry biomass components, including stems, branches, and foliage, and taper measurements to explore the applicability of biomass components to allometric scaling laws and develop a new system of additive models with the diameter in relative height (DRH) for each component of a Larch (Larix principis-rupprechtii Mayr) plantation in northern China. The variable exponential taper equations were modelled using nonlinear regression. In addition, applying nonlinear regression and nonlinear seemingly unrelated regression (NSUR) enabled the development of biomass allometric models and the system of additive models with DRH for each component. The results showed that the Kozak's (II) 2004 variable exponential taper equation could accurately describe the stem shape and diameter in any height of stem. When the diameters in relative height were D0.2, D0.5, and D0.5 for branches, stems, and foliage, respectively, the allometric exponent of the stems and branches was the closest to the scaling relations predicted by the MST, and the allometric exponent of foliage was the most closely related to the scaling relations predicted by geometry theory. Compared with the nonlinear regression, the parameters of biomass components estimated by NSUR were lower, and it was close to the theoretical value and the most precise at forecasting. In the study of biomass process modelling, utilizing the DRH by a variable exponential taper equation can confirm the general biological significance more than the DBH of a fixed position.

Modelling bark thickness for Scots pine (Pinus sylvestris L.) and common oak (Quercus robur L.) with recurrent neural networks.

Variation of the bark depends on tree age, origin, geographic location, or site conditions like temperature and water availability. Most of these variables are characterized by very high variability but above of all are also affected by climate changes. This requires the construction of improved bark thickness models that take this complexity into account. We propose a new approach based on time series. We used a recurrent neural network (ANN) to build the bark thickness model and compare it with stem taper curves adjusted to predict double bark thickness. The data includes 750 felled trees from common oak and 144 Scots pine-trees representing dominant forest-forming tree species in Europe. The trees were selected across stands varied in terms of age and site conditions. Based on the data, we built recurrent ANN and calculated bark thickness along the stem. We tested different network structures with one- and two-time window delay and three learning algorithms-Bayesian Regularization, Levenberg-Marquardt, and Scaled Conjugate Gradient. The evaluation criteria of the models were: coefficient of determination, root mean square error, mean absolute error as well as graphical analysis of observed and estimated values. The results show that recurrent ANN is a universal approach that offers the most precise estimation of bark thickness at a particular stem height. The ANN recursive model had an advantage in estimating trees that were atypical for height, as well as upper and lower parts on the stem.

Mixed linear and non-linear tree volume models with regional parameters to main tree species in Finland

AbstractThe volume models that have been used in Finland for the last 40 years, while generally well thought-out, exhibit an illogical behaviour for small trees. In recent studies, tree stem form was observed to have changed in time and also involve spatial variation attributable to environmental factors. It is yet unclear how the stem taper has actually changed. To overcome these problems, we fitted a completely new set of volume and taper curve models and examined whether this change is attributable to the changes in management and environmental factors rather than to measurement errors in the previously used datasets. For the latter, we added a dataset into the analysis, which was smaller but of higher quality due to the destructive nature of the stem taper measurements. We aim at (1) developing a new non-linear variable form factor volume function that works with trees of all sizes, (2) improving the description of the variation of the stem form in time and space by including temperature sum and soil type as predictors, (3) understanding the changes in the stem form by fitting new taper curve models and (4) improving the statistical properties of the predictions by using mixed model techniques and by addressing the effect of parameter uncertainty. To assess the impact of renewing the models, we (5) predicted the mean volume and its confidence interval with each model for forest inventory data at country level. The results show that the tree stem form has a spatial trend that can be described with the temperature sum. Moreover, the changes in stem form also have a spatial trend, with largest changes in Lapland. The difference is mostly observable in the lowest part of the stem, and it is especially large in the largest pines. We conclude that environmental variables can help to improve national stem taper functions in countries with pronounced environmental gradients.

Stem-Level Bucking Pattern Optimization in Chainsaw Bucking Based on Terrestrial Laser Scanning Data

Cross-cutting of a tree into a set of assortments (»bucking pattern«) presents a large potential for optimizing the volume and value recovery; therefore, bucking pattern optimization has been studied extensively in the past. However, it has not seen widespread adoption in chainsaw bucking, where time consuming and costly manual measurement of input parameters is required for taper curve estimation. The present study investigated an alternative approach, where taper curves are fit based on terrestrial laser scanning data (TLS), and how deviations from observed taper curves (REF) affect the result of bucking pattern optimization. In addition, performance of TLS was compared to a traditional, segmental taper curve estimation approach (APP) and an experienced chainsaw operator’s solution (CHA).A mature Norway Spruce stand was surveyed by stationary terrestrial laser scanning. In TLS, taper curves were fit by a mixed-effects B-spline regression approach to stem diameters extracted from 3D point cloud data. A network analysis technique algorithm was used for bucking pattern optimization during harvesting. Stem diameter profiles and the chainsaw operator’s bucking pattern were obtained by manual measurement. The former was used for post-operation fit of REF taper curves by the same approach as in TLS. APP taper curves were fit based on part of the data. For 35 trees, TLS and APP taper curves were compared to REF on tree, trunk and crown section level. REF and APP bucking patterns were optimized with the same algorithm as in TLS. For 30 trees, TLS, APP and CHA bucking patterns were compared to REF on operation and tree level.Taper curves were estimated with high accuracy and precision (underestimated by 0.2 cm on average (SD=1.5 cm); RMSE=1.5 cm) in TLS and the fit outperformed APP. Volume and value recovery were marginally higher in TLS (0.6%; 0.9%) than in REF on operation level, while substantial differences were observed for APP (–6.1%; –4.1%). Except for cumulated nominal length, no significant differences were observed between TLS and REF on tree level, while APP result was inferior throughout. Volume and value recovery in CHA was significantly higher (2.1%; 2.4%), but mainly due to a small disadvantage of the optimization algorithm.The investigated approach based on terrestrial laser scanning data proved to provide highly accurate and precise estimations of the taper curves. Therefore, it can be considered a further step towards increased accuracy, precision and efficiency of bucking pattern optimization in chainsaw bucking.

Empirical observations of the yield of logs from trees of the boreal region

The yield of sawlogs and plywood logs from boreal trees is discussed. First, taper curves are used to compute geometric (maximal) sawlog and plywood log contents. Then, a quality reduction is implemented, according to previously published models. Finally, the outcome is verified using empirical observations. Regarding spruce trees, the different estimates of sawlog content agree. In the case of birch and pine trees, the estimates differ. The sawlog and plywood log yield from large trees is greater than prognosticated one, probably due to missing over-aged trees in the empirical data. The yield of sawlogs and plywood logs from small trees is smaller than prognosticated one, most of the empirical data originating from thinnings with a focus on the quality of the remaining trees. The sawlog content of individual trees from clearcutting displays a skew distribution around the expected value. The distribution in thinning trunks shows a binary component, with a large proportion of trees with zero sawlog content. Keywords: sawlog; plywoodlog; Picea abies; Betula species; Pinus sylvestris

Evaluation of a novel mobile device app for value-maximized bucking by chainsaw

ABSTRACT Cross-cutting of felled trees into logs (“bucking”) is a critical step in converting standing timber into end products, as it predetermines and limits number, type and value of products that can be produced from the logs. Bucking by chainsaw is currently not aided by information technology and volume and value recovery depend on the chainsaw operator’s proficiency. A recently presented novel mobile device-based application (“T4E Bucking App”) for assisting value-maximized bucking by chainsaw was evaluated during harvesting in a mature beech stand regarding taper curve estimation accuracy. Further, its bucking patterns were benchmarked against an experienced chainsaw operator’s choice and time consumption for app operation was determined. Taper curves were inadequately approximated and log mid diameters (1.0 cm, SD=2.4) and log volume (0.02 m3 under bark [UB] m3, SD=0.04) were significantly overestimated. Consequently, total observed volume (–8.7%) and value (–8.1%) were smaller than estimated. Inappropriate pricing and insufficient top diameters caused further reductions. Thus, total recovered volume (59.61 m3 UB; –8.8%) and value (€2,099.66; –8.6%) were smaller than estimated. Chainsaw operator’s totals differed marginally (58.59 m3 UB; €2,117.99). About 85% of the variation of time consumption for working with the app was explained by tree volume and number of grade sections. After accounting for time consumption, net value recovery was 5.1% lower than in the chainsaw operator’s case. The T4E Bucking App nevertheless represents a first step towards digitalization of motor-manual bucking and may soon be enhanced by more accurate taper curves derived from terrestrial laser scanning data.

Revealing Changes in the Stem Form and Volume Allocation in Diverse Boreal Forests Using Two-Date Terrestrial Laser Scanning

Tree growth is a multidimensional process that is affected by several factors. There is a continuous demand for improved information on tree growth and the ecological traits controlling it. This study aims at providing new approaches to improve ecological understanding of tree growth by the means of terrestrial laser scanning (TLS). Changes in tree stem form and stem volume allocation were investigated during a five-year monitoring period. In total, a selection of attributes from 736 trees from 37 sample plots representing different forest structures were extracted from taper curves derived from two-date TLS point clouds. The results of this study showed the capability of point cloud-based methods in detecting changes in the stem form and volume allocation. In addition, the results showed a significant difference between different forest structures in how relative stem volume and logwood volume increased during the monitoring period. Along with contributing to providing more accurate information for monitoring purposes in general, the findings of this study showed the ability and many possibilities of point cloud-based method to characterize changes in living organisms in particular, which further promote the feasibility of using point clouds as an observation method also in ecological studies.

Ozolinši tüvemoodustaja matemaatiline analüüs ja modifitseerimise võimalused Hiiumaa männikute näitel

Abstract This paper discloses algorithms for calculating the volume and lateral area of a stem or part of the stem (assortment) using the integral of the Ozolinš’ stem taper curve. An algorithm was also developed to change the taper curve constants in case another diameter was measured higher than the breast height. The experiment was performed with 580 Scots pines (Pinus Sylvestris L.) measured in Hiiumaa, which included height, diameter at breast height and diameter from the root collar at a height of 5 m. The volumes of tree stem and assortments were estimated with both – the original and corrected (with the measured additional diameter) Ozolinš’ taper curve. Using the original Ozolinš’ taper curve, the stem volume of pines in Hiiumaa is estimated at 4% higher according to the analysis results of 580 measured trees. In the case of Hiiumaa Scots pines (Pinus Sylvestris L.), it is recommended to use 119.717, −289.805 and 1152.016 as the values of the coefficients a0, a1 and a2, respectively, of the Ozolinš’ taper curve equation. In conclusion, for a more accurate assessment of the forest reserve, it is recommended to measure one additional diameter at 5 to 8 m above the root collar on some or all trees.

Comprehensive yield model for plantation teak in Panama

The purpose of this study was to prepare a comprehensive, computerized teak ( L.f) plantation yield model system that can be used to describe the forest dynamics, predict growth and yield and support forest planning and decision-making. Extensive individual tree and permanent sample plot data were used to develop tree-level volume models, taper curve models and stand-level yield models for teak plantations in Panama. Tree volume models were satisfactorily validated against independent measurement data and other published models. Tree height as input parameter improved the stem volume model marginally. Stand level yield models produced comparable harvest volumes with models published in the literature. Stand level volume product outputs were found like actual harvests with an exception that the models marginally underestimate the share of logs in very large diameter classes. The kind of comprehensive model developed in this study and implemented in an easy to use software package provides a very powerful decision support tool. Optimal forest management regimes can be found by simulating different planting densities, thinning regimes and final harvest ages. Forest practitioners can apply growth and yield models in the appropriate stand level inventory data and perform long term harvest scheduling at property level or even at an entire timberland portfolio level. Harvest schedules can be optimized using the applicable financial parameters (silviculture costs, harvesting costs, wood prices and discount rates) and constraints (market size and operational capacity).Tectona grandis

Evaluation of Different Calibration Approaches for Merchantable Volume Predictions of Norway Spruce Using Nonlinear Mixed Effects Model

Research Highlights: Determination of merchantable wood volume is one of the key preconditions for sustainable forest management. This study explores accuracy of calibrated predictions of merchantable wood volume of Norway spruce (Picea abies (L.) H. Karst.) using stem taper curves (STC) in a form of a mixed model. Background and Objectives: The study is devoted to the determination of merchantable wood volume (over bark) of individual standing stems based on the integration of an STC model calibrated using upper diameter measurements. Various options of upper diameter measurement were tested and their impact on the accuracy of merchantable wood volume prediction was evaluated. Materials and Methods: To model stem taper curves, a Kozak 02 function was applied in a form of a nonlinear, mixed effects model. Accuracies of calibrated merchantable wood volume predictions obtained through remote (optical) upper diameter measurements were compared to accuracies corresponding to contact measurements by a caliper. The performance of two alternative methods used in the Czech National Forest Inventory (NFI) and forestry practice, involving diameter at breast height and total tree height as the only predictors, were also tested. The contact measurements were performed at identical stem positions after felling the respective sample tree. The calibration was done in order to account for factors inherent in particular location, and, optionally, also in a particular sample stem (within the respective location). Input data was sourced as part of a dedicated survey involving the entire territory of the Czech Republic. In total, 716 individual spruce trees were measured, felled and analysed at 169 locations. Results: In general, the best merchantable volume predictions were obtained by integrating the STC fitted (and calibrated) by minimising errors of stem cross-sectional areas instead of diameters. In terms of calibrated predictions, using single-directional, caliper measurement of upper diameter at 7 m (after felling) led to the best accuracy. In this case, the observed mean bias of merchantable volume prediction was only 0.63%, indicating underestimation. The best optical calibration strategy involved upper diameter measurements at two heights (5 and 7 m) simultaneously. Bias of this volume prediction approach was estimated at 2.1%, indicating underestimation. Conclusions: Concerning the prediction of merchantable stem volume of standing Norway spruce trees, STC calibration using two optical upper diameter measurements (at 5 and 7 m) was found to be practically applicable, provided a bias up to 3.7% can be accepted. This method was found to be more accurate than the existing national alternatives using diameter at breast height and the total tree height as the only predictors.

Assessing the Effects of Sample Size on Parametrizing a Taper Curve Equation and the Resultant Stem-Volume Estimates

Large and comprehensive datasets, traditionally based on destructive stem analysis or other labor-intensive approaches, are commonly considered as a necessity in developing stem-volume equations. The aim here was to investigate how a decreasing number of sample trees affects parametrizing an existing taper curve equation and resultant stem-volume estimates. Furthermore, the potential of terrestrial laser scanning (TLS) in producing taper curves was examined. A TLS-based taper curve was derived for 246 Scots pines (Pinus sylvestris L.) from southern Finland to parametrize an existing taper curve equation. To assess sensitivity of the parametrization regarding sample size, the number of Scots pines included in the parametrization varied between full census and 1 Scots pine at a time. Root mean square error of stem-volume estimates remained ≤20.9% and the mean absolute difference was relatively constant (≤9.0%) between stem-volume estimates when the sample size included ≥46 Scots pines. Thus, it can be concluded that, with a rather small sample size, a taper curve equation can be re-parametrized for local conditions using point clouds from TLS to produce consistent stem-volume estimates.

The utility of terrestrial photogrammetry for assessment of tree volume and taper in boreal mixedwood forests

Key MessageThis study showed that digital terrestrial photogrammetry is able to produce accurate estimates of stem volume and diameter across a range of species and tree sizes that showed strong correspondence when compared with traditional inventory techniques. This paper demonstrates the utility of the technology for characterizing trees in complex habitats such as boreal mixedwood forests.ContextAccurate knowledge of tree stem taper and volume are key components of forest inventories to manage and study forest resources. Recent developments have seen the increasing use of ground-based point clouds, including from digital terrestrial photogrammetry (DTP), to provide accurate estimates of these key forest attributes.AimsIn this study, we evaluated the utility of DTP based on a small set of photos (12 per tree) for estimating stem volume and taper on a set of 15 trees from 6 different species (Populus tremuloides, Picea glauca, Pinus contorta latifolia, Betula papyrifera, Picea mariana, Abies balsamea) in a boreal mixedwood forest in Alberta, Canada.MethodsWe constructed accurate photogrammetric point clouds and derived taper and volume from three point cloud–based methods, which were then compared with estimates from conventional, field-based measurements. All methods were evaluated for their accuracy based on field-measured taper and volume of felled trees.ResultsOf the methods tested, we found that the point cloud–derived diameters in a taper curve matching approach performed the best at estimating diameters at the lowest parts of the stem (< 30% of total tree height), while using known DBH and height provided more accurate estimates for the upper parts of the stem (> 50% of total height). Using the field-measured DBH and height as inputs to calculate stem volume yielded the most accurate predictions; however, these were not significantly different from the best point cloud-based estimates.ConclusionThe methodology confirmed that using a small set of photographs provided accurate estimates of individual tree DBH, taper, and volume across a range of species and size gradients (10.8–40.4 cm DBH).

Measuring stem diameters with TLS in boreal forests by complementary fitting procedure

Point clouds generated by terrestrial laser scanners (TLS) have enabled new ways to measure stem diameters. A common method for diameter calculation is to fit cylindrical or circular shapes into the TLS point cloud, which can be based either on a single scan or a co-registered combination of several scans. However, as various defects in the point cloud may affect the final diameter results, we propose an automatized processing chain which takes advantage of complementing steps. Processing consists of two fitting phases and an additional taper curve calculation to define the final diameter measurements. First, stems are detected from co-registered data of several scans using surface normals and cylinder fitting. This provides a robust framework for localizing the stems and estimating diameters at various heights. Then, guided by the cylinders and their indicative diameters, another fitting round is performed by cutting the stems into thin horizontal slices and reassessing their diameters by circular shape. For each slice, the quality of the cylinder-modelled diameter is evaluated first with co-registered data and if it is found to be deficient, potentially due to modelling defects or co-registration errors, diameter is detected through single scans. Finally, slice diameters are applied to construct a spline-based taper curve model for each tree, which is used to calculate the final stem dimensions. This methodology was tested in southern Finland using a set of 505 trees. At the breast height level (1.3 m), the results indicate 5.2 mm mean difference (3.2%), −0.4 mm bias (-0.3%) and 7.3 mm root mean squared error (4.4%) to reference measurements, and at the height of 6.0 m, respective values are 6.5 mm (3.6%), +1.6 mm (0.9%) and 8.4 mm (4.8%). These values are smaller compared to most of the corresponding contemporary studies, and outperform the initial cylinder models. This indicates that the applied processing chain is capable of producing relatively accurate diameter measurements, which can, at the cost of computational heaviness, remove various defects and improve the modelling results.

Coupling Mechanism of Control on Strip Profile and Flatness in Single Stand Universal Crown Reversible Rolling Mill

Taper roll contours are widely applied in the single stand universal crown mill (UCM) to control the edge‐drop of silicon steel, but bilateral symmetrical quarter buckles always happen in the last pass. It shows that the control on the strip profile and flatness has strong coupling nature. In this paper, the strip profile and the rolling parameters are obtained by simulations, and influences of the rolling parameters on the strip profile and flatness are revealed. The authors put forward a cantilever end model of the work roll influenced by the bending force, and analyze the rule of the influence the work roll taper curve on the additional pressure on the strip. Moreover, the rolling process characteristics of the UCM reversible rolling are expounded. One of the reasons for the coupling nature of the control on the strip profile and flatness is that, the roll taper curve increases the concentration and peak value of the additional pressure on the strip, and another is that the roll taper curve limits the lateral displacement of metal, increasing the longitudinal prolongation of metal. The optimization principle of the roll contour is determined based on the above influence mechanism.

Evolution of tree stem taper in Pinus taeda stands

ABSTRACT: This study aimed to evaluate the evolution of stem taper of trees in Pinus taedastands using stem analysis data from 126 and 120 trees sampled from established stands in Paraná and Santa Catarina States, respectively. The integer and fractional power polynomial was fitted to estimate diameters inside bark along the stem and was used to construct taper curves by age class. The model identity test was applied to verify the change in stem form of trees. The evolution of natural and artificial form factors in the trees was also analyzed. The curves constructed from the fitted taper functions and form factors showed that the stems become more cylindrical with increasing age; the identity test indicated that, in general, the taper curves statistically differ between themselves. This showed that stratifying the data into age classes for fitting in taper functions can produce more accurate assortment estimates.

Hacia una descripción de fuste para el Laurel Cordia alliodora en sistemas agroforestales en el Bosque Protector Sumaco, Ecuador

El Bosque Protector Sumaco esta caracterizado por extensas áreas de bosque secundario en la zona de amortiguamiento del Parque Nacional Sumaco-Napo-Galeras a una altitud de 1160 m y con una precipitación promedio de 4000 mm año-1. Un adecuado manejo de estas áreas requiere información básica sobre el crecimiento y producción dependiendo de la regeneración del Laurel Cordia alliodora en sistemas agroforestales. El presente estudio muestra los resultados sobre diferentes modelos para determinar el coeficiente de forma, volumen, perfil de fuste y desarrolla un sistema de ecuaciones matemáticamente y numéricamente consistente para Cordia alliodora bajo un sistema agroforestal en la comunidad indígena de Wamaní. Para la búsqueda de los parámetros de las funciones de volumen total, el cociente de forma, la descripción del perfil de fuste y la función de conicidad, un total de 308 árboles fueron medidos el diámetro a 1, 3, 2, 4, 6, 8, metros, a la mitad del árbol y la altura total en metros. Se presenta un modelo para determinar el volumen del Laurel en base al cociente de forma, diámetro y altura total. El modelo de perfil de fuste a través de la función Pain demuestra ser un modelo muy apropiado para describir este parámetro, así como también para calcular el volumen con mayor exactitud. En este estudio la consistencia algebraica y numérica para determinar el volumen y el diámetro a una establecida altura del fuste a través de la función de conicidad fue verificada. Todos estos modelos pueden ser utilizados en la práctica, constituyéndose estos en la esencia del negocio de la madera.

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.