Height-diameter Function Research Articles

Height–diameter relationship of trees in Omo strict nature forest reserve, Nigeria

The height and diameter of trees are important variables in estimating the aboveground biomass of trees. Tree height measurements are often difficult in tropical forests hence tree height–diameter models are often used as an alternative for predicting tree heights. This study was carried out to develop height–diameter models for tree species in Omo strict nature forest reserve in Nigeria. Height and diameter data used for the study comprised of 100 tree species, which were classified into three groups using cluster analysis. Eight commonly used non-linear height–diameter functions were tested for predicting heights of the species groups using nonlinear least squares (Nls) method. The power function was selected based on its performance, and fitted using mixed-effects modeling approach to account for between-plot height variability. The Mixed-effects models were evaluated using the Akaike information criteria and Residual standard error. The models were calibrated using three subsample selection approaches. The best calibration results were obtained using subsamples of four trees from four diameter classes per plot. The best calibrated mixed-effect models outperformed the Nls models for all the tree species groups; increasing the accuracy of tree height prediction.

Development and Validation of Height-Diameter Models for Cupressus lusitanica in Gergeda Forest, Ethiopia

This study was conducted to fit and evaluate ten existing nonlinear height diameter functions for Cupressus lusitanica in Gergeda forest Ethiopia. A total of 260 trees were measured for their diameter at breast height (D) and height using destructive sampling methods. This data were randomly split in to two datasets for model development (50%) and for model validation (50%). Considering hetroscadasticity of variance, all functions were fitted using weighted nonlinear least square regression. To evaluate the performance of each model, five fit statistics-such as Coefficient of determination (R2), root mean square error (RMSE), bias (Ē), absolute mean deviation (AMD), and coefficient of variation (CV%) were used. Among all the models tested, the Weibull type function of the form H = 1.3 + a (1-exp (-bDc)) + ɛ was observed to give the best fit based on the model’s goodness of fit and predictive ability. Therefore, this model with three parameters has been conformed to provide reliable estimate of total tree height for Cupressus lusitanica in Gergeda forest, Ethiopia.

Comparison of height-diameter models based on geographically weighted regressions and linear mixed modelling applied to large scale forest inventory data

Aim of the study: The main objective of this study was to test Geographically Weighted Regression (GWR) for developing height-diameter curves for forests on a large scale and to compare it with Linear Mixed Models (LMM).Area of study: Monospecific stands of Pinus halepensis Mill. located in the region of Murcia (Southeast Spain).Materials and Methods: The dataset consisted of 230 sample plots (2582 trees) from the Third Spanish National Forest Inventory (SNFI) randomly split into training data (152 plots) and validation data (78 plots). Two different methodologies were used for modelling local (Petterson) and generalized height-diameter relationships (Cañadas I): GWR, with different bandwidths, and linear mixed models. Finally, the quality of the estimated models was compared throughout statistical analysis. Main results: In general, both LMM and GWR provide better prediction capability when applied to a generalized height-diameter function than when applied to a local one, with R2 values increasing from around 0.6 to 0.7 in the model validation. Bias and RMSE were also lower for the generalized function. However, error analysis showed that there were no large differences between these two methodologies, evidencing that GWR provides results which are as good as the more frequently used LMM methodology, at least when no additional measurements are available for calibrating.Research highlights: GWR is a type of spatial analysis for exploring spatially heterogeneous processes. GWR can model spatial variation in tree height-diameter relationship and its regression quality is comparable to LMM. The advantage of GWR over LMM is the possibility to determine the spatial location of every parameter without additional measurements. Keywords: Spatial analysis; Pinus halepensis Mill; forest modelling.Abbreviations: GWR (Geographically Weighted Regression); LMM (Linear Mixed Model); SNFI (Spanish National Forest Inventory).

Nonlinear height–diameter models for three woody, understory species in a temperate oak forest in Hungary

Information about the diameter and the height of woody species is fundamental to developing growth and yield models in forest stands. Ten nonlinear height–diameter functions were fitted and evaluat...

Height-Diameter Models for Mixed-Species Forests Consisting of Spruce, Fir, and Beech

Abstract Height-diameter models define the general relationship between the tree height and diameter at each growth stage of the forest stand. This paper presents generalized height-diameter models for mixed-species forest stands consisting of Norway spruce (Picea abies Karst.), Silver fir (Abies alba L.), and European beech (Fagus sylvatica L.) from Slovakia. The models were derived using two growth functions from the exponential family: the two-parameter Michailoff and three-parameter Korf functions. Generalized height-diameter functions must normally be constrained to pass through the mean stand diameter and height, and then the final growth model has only one or two parameters to be estimated. These “free” parameters are then expressed over the quadratic mean diameter, height and stand age and the final mathematical form of the model is obtained. The study material included 50 long-term experimental plots located in the Western Carpathians. The plots were established 40-50 years ago and have been repeatedly measured at 5 to 10-year intervals. The dataset includes 7,950 height measurements of spruce, 21,661 of fir and 5,794 of beech. As many as 9 regression models were derived for each species. Although the “goodness of fit” of all models showed that they were generally well suited for the data, the best results were obtained for silver fir. The coefficient of determination ranged from 0.946 to 0.948, RMSE (m) was in the interval 1.94-1.97 and the bias (m) was -0.031 to 0.063. Although slightly imprecise parameter estimation was established for spruce, the estimations of the regression parameters obtained for beech were quite less precise. The coefficient of determination for beech was 0.854-0.860, RMSE (m) 2.67-2.72, and the bias (m) ranged from -0.144 to -0.056. The majority of models using Korf’s formula produced slightly better estimations than Michailoff’s, and it proved immaterial which estimated parameter was fixed and which parameters were free

Modelling tree height–diameter relationships in multi-species and multi-layered forests: A large observational study from Northeast China

Projection of stand development over time relies on accurate height–diameter functions. In complex forests where trees of various ages, species, sizes, vigor and crown classes, and shade tolerance levels co-exist, tree height predictions are more difficult than in uniform conditions. In such complex situations, individual tree height–diameter relationships and predictions are often further complicated by structure (e.g., relative position of trees) and stand density which sometime are exacerbated by selective harvesting and other disturbances, and changing stand dynamics. Height–diameter (H–D) relationships are significantly modulated by microsite (elevation, aspect and slope), climate, and competition. Traditionally, tree height–diameter relationships are modeled or examined using linear or nonlinear regression models, in which the microsite influences or species–habitat relationships are largely ignored. Studies that have examined H–D relationships for multi-species and multi-layered forests using species–habitat relationships or microsite influences are limited or non-existent.Using an extensive dataset, we evaluated two sets of height–diameter equations for estimating tree heights in multi-species and multi-layered forests in north-eastern China. The first set included three equations for estimating height as a function of individual tree diameter, the second set included nine equations for estimating height as a function of individual tree diameter, relative tree position, and stand density variables. The inclusion of relative position of a tree and stand density into the base height–diameter equations increased the accuracy of prediction for all species. On the average, root mean square values were reduced by 12.2% by including these variables. After comparing and selecting a base H–D equation, we evaluated its statistical attributes among habitat types constructed for forest a large observational study site in north-eastern China.Nonlinear mixed effect (NLMEM) was used to account for the hierarchical data structure and to provide predictive equations. Dummy variables were used to indicate species effect and allow simultaneous estimation of all species. The NLMEM framework along with species effect made tree height predictions compatible for 23 tree species found in multi-layered forests in NE China. Model coefficients for selected equations are documented for future use.

Evaluation of nonlinear equations for predicting diameter from tree height

With the emergence and advancement of airborne laser scanning technology over the past decade, individual tree height can be easily measured over a large area of forests with a comparable degree of accuracy to conventional ground-based methods. In laser scanning based large-scale forest inventories, the need to predict diameter from remotely sensed tree height calls for a systematic evaluation of equation forms as the first step towards a well-developed approach to developing diameter–height equations. This study evaluated more than 30 height–diameter equations in the forest biometrics literature to select candidates for deriving equation forms for diameter–height equations. The evaluation was based on four criteria: (i) the height–diameter function is inversable; (ii) the inverse function is continuous and monotonically increasing over a specified working range of total tree height; (iii) diameter at breast height is equal to zero when tree height equals breast height in the inverse function; and preferably, (iv) the inverse function has an inflection point that is consistent with biological expectations. A total of 12 candidate equation forms were derived, which included five two-parameter and seven three-parameter equations. The estimation properties and predictive performance of these 12 equation forms were further evaluated and compared through repeated sampling and fitting using data from 3581 trees destructively sampled for taper measurements from Pinus radiata D. Don plantations across New South Wales, Australia. Three equation forms, including the constrained Richards, Weibull, and the combined power and exponential function, displayed superior prediction accuracy and estimation properties and so were recommended as the primary equation forms for developing diameter–height equations. The remaining equation forms were marred by either lower prediction accuracy or poorer estimation properties or both. The three recommended equation forms should only serve as basic deterministic specifications upon which other tree and stand variables should be incorporated as predictors to further improve their predictive performance.

Assessing potential impacts of climatic change on subalpine forests on the eastern Tibetan Plateau

Forest gap models have been used widely in the study of forest dynamics, including predicting long-term succession patterns and assessing the potential impacts of climate change on forest structure and composition. However, little effort is devoted to predict forest dynamics in the high elevation areas, although they have the sensitive response to global climate change. In the present study, based on a modified height-diameter function, we developed a new version (FAREAST-GFSM) of the forest patch model, FAREAST for simulating the changes of subalpine forests. The observed data from the Gongga Mt. Alpine Station were also used to test model precision. With the improved performance of FAREAST-GFSM, we explored the impact of three warming scenarios on subalpine forest on the eastern Tibetan plateau within a 100-year period. The study result indicates that the effects of climate change were evident on subalpine forests in the high elevation areas. The response of different species to the warming climate might eventually transform the subalpine Abies fabric forest into Betula utilis forest similar to that which is now widely distributed in the eastern Tibetan Plateau mountainous areas with the relatively lower elevation. Subalpine forests could move to higher and colder areas, which are currently tundra.

Development and applications of a growth model for Pinus radiata D. Don plantations in El Bierzo (Spain)

A dynamic growth model for Pinus radiata D. Don plantations in El Bierzo (Spain) was developed with data from two inventories of permanent plots, of between 7 and 36 years old, established by the University of León. In this model, stand conditions at any point in time are defined by three state variables (stand basal area, number of trees per hectare and dominant height).Themodel includes three transition functions derived by the generalized algebraic difference approach to enable projection of the state variables at any particular time. Once they are known, the number of trees in each diameter class is estimated with a distribution function, by recovery of the parameters of theWeibull function by use of the moments method. Finally, a generalized height-diameter function and a taper function allow estimation of total or merchantable stand volume. The model provides satisfactory predictions for a time interval of three years. Simulation of the growth of four stands under two silvicultural regimes and two different sites confirm that the estimates provided by the overall model adequately represent the effects of both stand density and site quality. Other applications for the model are analysed and discussed.

A growth model for Pinus radiata D. Don stands in north-western Spain

A dynamic whole-stand growth model for radiata pine (Pinus radiata D. Don) stands in north-western Spain is presented. In this model, the initial stand conditions at any point in time are defined by three state variables (number of trees per hectare, stand basal area and dominant height), and are used to estimate total or merchantable stand volume for a given projection age. The model uses three transition functions derived with the generalized algebraic difference approach (GADA) to project the corresponding stand state variables at any particular time. These equations were fitted using the base-age-invariant dummy variables method. In addition, the model incorporates a function for predicting initial stand basal area, which can be used to establish the starting point for the simulation. Once the state variables are known for a specific moment, a distribution function is used to estimate the number of trees in each diameter class by recovering the parameters of the Weibull function, using the moments of first and second order of the distribution. By using a generalized height-diameter function to estimate the height of the average tree in each diameter class, combined with a taper function that uses the above predicted diameter and height, it is then possible to estimate total or merchantable stand volume.

Dynamic growth model for Scots pine ( Pinus sylvestris L.) plantations in Galicia (north-western Spain)

In this study we developed a dynamic growth model for Scots pine ( Pinus sylvestris L.) plantations in Galicia (north-western Spain). The data used to develop the model were obtained from a network of permanent plots, of between 10 and 55-year-old, which the Unidade de Xestión Forestal Sostible (Sustainable Forest Management Unit) of the University of Santiago de Compostela has set up in pure plantations of this species of pine in its area of distribution in Galicia. In this model, the initial stand conditions at any point in time are defined by three state variables (number of trees per hectare, stand basal area and dominant height), and are used to estimate stand volume, classified by commercial classes, for a given projection age. The model uses three transition functions expressed as algebraic difference equations of the three corresponding state variables used to project the stand state at any point in time. In addition, the model incorporates a function for predicting initial stand basal area, which can be used to establish the starting point for the simulation. This alternative should only be used when the stand is not yet established or when no inventory data are available. Once the state variables are known for a specific moment, a distribution function is used to estimate the number of trees in each diameter class, by recovering the parameters of the Weibull function, using the moments of first and second order of the distribution (arithmetic mean diameter and variance, respectively). By using a generalized height–diameter function to estimate the height of the average tree in each diameter class, combined with a taper function that uses the above predicted diameter and height, it is then possible to estimate total or merchantable stand volume.

Simulating structural forest patterns with a forest gap model: a model evaluation

Ecological models can be characterized by their degree of generality, reality and precision. Any model, being a deliberate simplification of reality, cannot excel in all three aspects. Forest gap models have been widely used for studying tree population dynamics, but their predictions have not often been tested for their local precision, but rather for their broad agreement with descriptions of near-natural vegetation. The objectives of our study were (1) to evaluate the performance of the forest gap model ForClim, which had been developed striving for generality and realism, in simulating the long-term development of structural features in Swiss mountain forests; and (2) to examine whether and how the model needs to be changed to improve its precision for a specific site. We used long-term forest data (45 years) from three different forest types in the Swiss National Park. Initial simulation runs for the most dominant forest type in the study area failed to reproduce the observed structural patterns. A detailed analysis of the growth performance of individual trees led to the conclusion that a modified height–diameter function was required, which presumably increases the generality of the model. The new model structure led to simulated stand features that were broadly consistent with observations. After, in addition, taking local variations of model parameters (on mortality, browsing, and seedling establishment rates) into account, we were able to considerably improve the performance of ForClim in simulating the structural features of the different mountain forest stands. We suggest that from the point of view of its revised structure, the ForClim model is principally suitable for site-specific applications, but local precision can only be achieved by site-specific parameter estimation procedures. We conclude that model evaluation and validation as conducted in this study could be quite useful for increasing the reliability of simulations performed with this class of models.

Modelling the diameter-height relationship for Pinus pinaster Ait. in Galicia using SBB distribution

The bivariate density function SBB has been fitted to height and diameter data of maritime pine (Pinus pinaster Ait.) from 24 sample plots established in even-aged stands of this specie in Galicia (Northwestern Spain). The height-diameter model in maritime pine was estimated using a property of the SBB bivariate distribution. In order to get this, a median regression of SBB function was established between heights and diameters data of each one of the 24 sample plots. The estimated values obtained in the fit has also been used to obtain the curves corresponding to the percentiles of the 5 and 95 % of the height distribution. Finally, a generalized height-diameter function has been developed for Pinus pinaster in Galicia from the equations fitted to each plot, relating the SBB function with the principal stand variables.

Development of ecoregion-based height–diameter models for white spruce in boreal forests

A large number of height–diameter functions were evaluated using felled tree data for white spruce ( Picea glauca (Moench) Voss) grown in Alberta's boreal forests. The logistic-type function was found to produce the most satisfactory fits. Residual analyses were conducted to identify the error structure of the model. A weighting factor of w i=1/ H ̂ 3/2 i was found appropriate for satisfying the equal error variance assumption. Regional differences in the height–diameter models were examined and tested using the nonlinear extra sum of squares method. It was found that height–diameter models were different among different ecoregions. Incorrectly applying a height–diameter model fitted from one ecoregion to different ecoregions resulted in overestimations between 1.10% and 29.05%, or underestimations between 1.92% and 21.92%.

Ecoregion-Based Individual Tree Height-Diameter Models for Lodgepole Pine in Alberta

Abstract Using the felled tree data, ecoregion-based height-diameter models were developed for lodgepole pine (Pinus contorta var. latifolia) in Alberta. A large number of height-diameter functions were evaluated, and the Chapman-Richards function was found to produce some of the most satisfactory fits. Residual analysis was conducted to identify the error structure of the models. A weighting factor of wi = 1/Di was found appropriate for achieving the equal error variance assumption. Differences of the height-diameter models among different ecoregions were examined and tested using the nonlinear extra sum of squares method. Most height-diameter relationships were found to be different among different ecoregions. Ecoregions of similar height-diameter relationships were combined to provide a composite model to facilitate the practical use of such relationships. West. J. Appl. For. 14(4):186-193.

Effect of competition control treatments on height-age and height-diameter relationships in young Douglas-fir plantations

Height-age and height-diameter models for plantations of young Douglas-fir ( Pseudotsuga menziesii [Mirb.] Franco var. menziesii) were examined in relation to vegetation management treatments. The models were developed from 10 years of measurements in a competition release study installed on six sites in the Coast Ranges of Oregon and Washington. Analysis of height growth patterns for dominant trees indicated significant differences between the total vegetation control treatment and operational release treatments or no treatment. The resulting height-age function depicted exponential growth patterns for the total vegetation control treatment and nearly linear patterns for the operational release treatment and no treatment. The height-diameter function was compatible with dominant height growth and quadratic mean diameter prediction functions. Different height-diameter curve shapes were associated with total vegetation control and the operational release and no treatments. The resulting function implied that Douglas-fir trees of a given diameter and age were slightly taller when under interspecific competition, especially for trees with smaller diameters. The height-age and height-diameter functions may be used in conjunction with diameter distribution or stand table projection models developed for those data to predict dynamics and stand structure in young Douglas-fir plantations.

Comparison of nonlinear height–diameter functions for major Alberta tree species

Twenty nonlinear height–diameter functions were fitted and evaluated for major Alberta species based on a data set consisting of 13 489 felled trees for 16 different species. All functions were fitted using weighted nonlinear least squares regression (wi = 1/DBHi) because of the problem of unequal error variance. The examination and comparison of the weighted mean squared errors, the asymptotic t-statistics for the parameters, and the plots of studentized residuals against the predicted height show that many concave and sigmoidal functions can be used to describe the height–diameter relationships. The sigmoidal functions such as the Weibull-type function, the modified logistic function, the Chapman–Richards function, and the Schnute function generally gave the most satisfactory results.

Identifying and Evaluating the Components of Non-Measurement Error in the Application of Standard Volume Equations

We consider the prediction of stem volume of a tree by means of simple linear regression wherein the explanatory variable is D2H; D is the bole diameter measured outside bark at breast height and H is the total height of the tree. Given a fitted equation, a tree whose volume is to be predicted must be measured for D and H. Two sources of error account for the deviation of predicted tree volume from V' the actual tree volume, i.e. (1) error in the estimated coefficients of the volume function and (2) variation of V around its expected value conditional upon D2H. Owing to the greater difficulty and expense of measuring H in forest conditions, a standard practice is to predict its value from a regression on diameter. Consequently the prediction of V, V say, is subject to additional sources of error due to (1) the error in the estimated coefficients of the height-diameter function and (2) the variation of H around its expected value conditional on D. An empirical study of a population of more than 14 000 loblolly pine trees was undertaken with the objective of quantifying the average proportion of overall prediction error which was due to these components. The exact mean square error of the vector of predicted volume, V, is also derived.