Fitting irregular diameter distributions of forest stands by Weibull, modified Weibull, and mixture Weibull models

Irregular diameter frequency distributions of forest stands include multimodal structure of mixed-species stands, highly skewed and highly irregular shapes of uneven-aged stands, and rotated sigmoid form of old-growth stands. In this study, a traditional two-parameter Weibull model, a modified two-parameter Weibull model, and a finite mixture of two-parameter Weibull models were used to fit four artificial example plots. The model fitting and comparison results indicate that the mixture Weibull model is more flexible to fit various irregular diameter distributions, while the traditional Weibull model fails in every case to adequately describe these frequency distributions. The modified Weibull model is a good choice for fitting the “rotated-sigmoid” diameter distribution of an uneven-aged old-growth stand. However, it may not be sufficient when a diameter frequency distribution is multimodal or highly irregular in shape.     

Modelling stand structure in young Scots pine dominated stands

The purpose of this study was to construct models for predicting the structure of young Scots pine (Pinus sylvestris L.) stands. The two-parameter Weibull function characterized the height distribution of the stands. In young stands height was preferred to dbh as a random variable because of its continuous feature. Tree diameters were predicted using a multiplicative model, fitted as a linearized mixed-effect model. The modelling data consisted of repeatedly measured Scots pine dominated juvenile stands, carried out on a sub-sample of the 7th National Forest Inventory. The data covered a dominant height range from 0.2 up to 17 m. Two independent data sets were used to validate the models. The Weibull function was fitted using the maximum likelihood method. Four methods for predicting the distributions were compared: (1) parameter prediction models (PPM) consisting of seemingly unrelated regression equations, (2) a generalized linear model (GLM) which was a one-stage distribution and model fitting procedure, (3) a hybrid method including PPM for the shape parameter together with moment-based parameter recovery for the scale parameter, and (4) inclusion of moment-based parameter recovery for the scale parameter in the estimated GLM. Goodness-of-fit were tested in terms of Kolmogorov–Smirnov and error index statistics. Parameter recovery showed no improvement when used with PPM, but it improved GLM and gave the overall best performance for this new method. The constructed diameter–height model showed quite flexible and unbiased behaviour. Models are recommended as practical tools for Finnish forest management planning purposes.     

Stand-level diameter distribution yield model for black spruce plantations

The objectives of this study were to develop and demonstrate a stand-level diameter distribution yield model and associated algorithm for black spruce (Picea mariana (Mill.) B.S.P) plantations. Employing a parameter prediction approach within the context of a stand density management diagram (SDMD), model development consisted of four sequential steps: (1) obtaining maximum likelihood estimates for the location, scale and shape parameters of the Weibull probability density function (PDF) for 296 empirical diameter frequency distributions; (2) developing and evaluating parameter prediction equations in which the parameter estimates of the Weibull PDF were expressed as functions of stand-level variables employing stepwise regression and seemingly unrelated regression techniques; (3) explicitly incorporating the parameter prediction equations into the SDMD modelling framework; and (4) developing an associated PC-based algorithm and demonstrating its utility in density management decision-making. The results indicated that the parameter prediction equations described 74.4, 87.1 and 66.8% of the variation in location, scale and shape parameter estimates, respectively. Incorporating the parameter prediction equations into the structure of the SDMD enabled the prediction of the temporal dynamics of the diameter frequency distribution by density management regime, site quality and region. An algorithmic version of the model is provided as a decision-support aid in which forest managers are able to simultaneously contrast multiple density management regimes in terms productivity, product value and optimal site occupancy.     

Comparison of beta and weibull functions for modelling basal area diameter distribution in stands of pinus sylvestris and picea abies

The purpose of this study was to compare beta and Weibull distributions in describing basal area diameter distributions in stands dominated by Scots pine and Norway spruce. The material of the study consisted of 535 stands located in eastern Finland. Parameters for both two‐ and three‐parameter approaches of the Weibull distribution were estimated using the method of maximum likelihood. Models for these parameters were derived using regression analysis. For the beta distribution, regression models were formed for the minimum, maximum and standard deviation of diameters within individual stands. These models were used when the exponents of the beta distribution were calculated analytically. Also, some parameter models for beta and Weibull distributions from previous studies were compared with the measured diameter distributions. The distributions obtained were compared using diameter sums and an estimate of the proportion of sawtimber. The results did not reveal any major differences between the suitability of the beta and two‐parameter Weibull distributions. There are appropriate models available for both of the distributions and the more similar the original data is to the data of an application, the better are the results. The two‐parameter approach of the Weibull distribution gave better results than the three‐parameter approach. The poorest results for all the predicted distributions were obtained at the extremes of the distributions.     

Fine root biomass,distribution, and production in young pine-hardwood stands

Patterns of fine root biomass, production, and distribution were estimated for pure stands and mixtures of three-year-old loblolly pine (Pinus taeda L.) with red maple (Acer rubrum L.) or black locust (Robinia pseudoacacia L.) on the Virginia Piedmont to determine the role of fine roots in interference between pine and hardwood tree species. Estimates were based on amounts of live and dead fine roots separated from monthly core samples during the third growing season after planting. Live and dead fine root biomass and production varied by species, but mixtures of loblolly pine and black locust generally had greater fine root biomass and fine root production than pure stands or loblolly pine-red maple mixtures. Hardwood species had greater live fine root biomass per tree in mixtures with pine compared to pure stands. Greater live fine root biomass in pine-locust stands may be attributed to differential utilization of the soil volume by fine roots of these species. For all stands, approximately 50% of live five root biomass was located in the upper 10 cm of soil.     

Modeling diameter distribution of Austrian black pine (<Emphasis Type="Italic">Pinus nigra</Emphasis> Arn.) plantations: a comparison of the Weibull frequency distribution function and percentile-based projection methods

The main purpose of the present investigation is to examine and compare three methods for diameter distribution modeling in terms of their fitness to predict from stand level variables the diameter distributions of even-aged Austrian black pine (Pinus nigra Arn.) plantations in Bulgaria. The percentile-based projection method involving empirical probability density function based on 12 percentiles was the first method tested. A new modified approach based on the first method was proposed as the second alternative. The third method was the 2-parameter Weibull functional model in which parameters were recovered from the first and the second raw moments and the second central moment of the empirical distributions. The Kolmogorov–Smirnov test was applied to compare the experimental distributions with the predicted ones, and estimation of the error indices was employed to evaluate the total absolute deviation of the predicted numbers from the actual ones by diameter class. The two-parameter Weibull function proved superior to the examined alternative percentile-based projection methods and the newly proposed percentile method, without a driver percentile showed improved precision compared to the classical percentile method (with a driver percentile). The parameters of the Weibull frequency distribution function can be easily recovered from the stand quadratic mean diameter. Consequently, this diameter distribution model could be incorporated as a sub-model for stand horizontal structure characterization within the Stand Density Management Diagram modeling framework.     

Diameter distributions in Picea ables described by the Weibull model

The three‐parameter Weibull function met specified statistical standards for goodness of fit as a model for the diameter distribution of moderately thinned Norway spruce stands in Denmark. Weibull distributions estimated by percentile estimators fit the majority of 522 observed diameter distributions (material made available by the Danish Forest Experiment Station) at the 10% level of significance in the Chi‐square and Kolmogoroff‐Smirnow tests. The Weibull distribution was less suited to describe the diameter distribution in unthinned stands and heavily thinned stands. Weibull parameter predictions were developed and used in connection with a Danish yield table for Norway spruce on clay‐rich coastal soils. The use of the Weibull distribution to predict stem frequencies in different diameter classes before and after a specified thinning of stands is demonstrated.     

四川江安县人工经营毛竹林分结构特征

通过对江安县毛竹人工林的典型样地调查,利用Weibull分布函数对其年龄、胸径和枝下高的密度分布进行了模拟。结果表明:(1)毛竹人工林株数分布较均匀,且结构较合理,1度竹达598株·hm-2,占林地总株数的16.5%;(2)毛竹人工林分的年龄分布符合三参数Weibull分布函数,其所有样地拟合的可决系数R2达0.99812,各年龄组误差均在3%以内;(3)毛竹人工林分胸径和枝下高分布符合二参数Weibull分布函数,其全部样地拟合的可决系数R2分别为0.99951和0.99819;由于人工干扰的原因,其胸径分布的前4个径阶组和枝下高分布的前3个高阶组的拟合误差较大,最大值分别为31.69%和-30.02%,而其余各组的拟合误差则多在3%以下;(4)毛竹人工林的胸径-年龄和枝下高-年龄具有相似的结构特征,即,成熟龄组的胸径分布呈正态分布,对应的竹高分布则呈现左截尾的非完整的正态分布,而其余龄组则无明显分布特征,这与人为经营活动及其水平是相关的。     

Comparison of selected statistical distributions for modelling the diameter distributions in near-natural <Emphasis Type="Italic">Abies</Emphasis>–<Emphasis Type="Italic">Fagus</Emphasis> forests in the Świętokrzyski National Park (Poland)

The aim of the study is to compare selected theoretical distributions (normal, lognormal, Weibull, gamma, logistic, and exponential) in describing the tree diameter (DBH) distributions of mixed near-natural forests consisting of fir Abies alba Mill. and beech Fagus sylvatica L. growing in various vertical structures. Tree DBH data were collected between 1997 and 2008 from 51 sample plots established in the Świętokrzyski National Park in Poland. The empirical data represent differentiated DBH distributions, ranging from almost symmetric to extremely asymmetric ones. The chi-square test and the modified Kolmogorov–Smirnov test were chosen for the goodness-of-fit testing. In addition to the test statistics, the bias (B), the root mean square error (RMSE) and the graphical method (quantile–quantile plots) were used. In one-storied stands, the most suitable distributions were the normal and logistic distributions; in two-storied and multilayered stands, the Weibull and gamma distribution were the most suitable; and in selection stands, the exponential distribution was the most appropriate to describe the DBH distribution. The order of precision of the tested distributions (from the highest to the lowest) was Weibull, gamma, logistic, normal, exponential, and lognormal. The normal and exponential distribution should be applied only to one-storied and selection forests, respectively. The least suitable distribution for DBH distribution modelling was the lognormal one.     

Phosphorus fertilizer,soils, and site preparation influence loblolly pine productivity

Growth and yield of 12-year-old loblolly pine (Pinus taeda L.) plantations were compared among five subsoil textures, seven site preparation methods, with and without phosphorus fertilizer. Phosphorus broadcast before planting increased the mean size of the loblolly pine trees. However, soil differences significantly influenced stand survival, and the method of site preparation significantly affected total stand yield. Therefore, the influence of phosphorus on total stand yield depended on soil type and site preparation method. The site preparation-fertilizer combinations yielding the most volume per soil type were: on the Lucy, Ruston, and Wagram soils, shear-windrow site preparation with phosphorus; on the Kirvin and Shubuta soils, chop-burn-harrow site preparation without phosphorus; on the Calloway and Henry soils, double chopping without phosphorus; on the Sawyer soil, chopburn-harrow site preparation with phosphorus; and on the Boswell, Cadeville, Gore, and Susquehanna soils, shear-windrow-harrow site preparation without phosphorus. Soil differences and phosphorus fertilization influenced the incidence of fusiform rust in 12-year-old stands of loblolly pine.     

Loblolly and slash pine height and diameter are related to soil drainage in winter on poorly drained silt loams

Soil drainage characteristics in winter and the heights and diameters of 10-year-old loblolly and slash pines (Pinus taeda L. and P. elliottii Engelm. var. elliottii) were measured on three poorly-drained silt loam sites that had been bedded, furrowed, or disked before planting. Quadratic response functions were used to determine the relationship between depth to water table or volume of drained soil in winter and mean tree height or diameter. These quadratic response functions (species × height or diameter: drainage characteristics) showed that diameter and height were significantly related to soil drainage in winter on these silt loams soils. Slash pines grew better than loblolly pines under more poorly drained conditions while loblolly pines grew better under the better drained conditions. On unbedded silt loams, the soil drainage in winter needed to maximize tree height was 42 cm for loblolly and 49 cm for slash pine. Pines planted on beds did not generally grow better than those on flat-disked plots because where bedding would be most benefical the practice did not form enough additional drainage to significantly influence pine development.     

A model for predicting log yield from stand characteristics

A method for forecasting the log‐class distribution resulting from harvesting forest stands is investigated. The diameter distribution of a stand is approximated by a two‐parameter Weibull density function, the parameters of which are recovered from stand level characteristics. Theoretical bucking is performed for actual and simulated distributions and the resulting log yields are compared. An application to 40 stands indicates that an accurate prognosis can be expected, given a good approximation of the diameter distribution. For cases with multimodal diameter distributions or many small trees, the prognosis is unreliable.     

Modeling production and decay of coarse woody debris in loblolly pine plantations

Forest management can have large impacts on the production and yield of coarse woody debris (CWD) in terrestrial ecosystems, yet few modeling tools exist to inform such efforts. The goal here was to develop a set of prediction equations for use in conjunction with loblolly pine (Pinus taeda L.) modeling and inventory systems to estimate CWD yields at scales ranging from individual trees to whole plantations. Permanent field plots from a 21-year study of thinning effects on plantation growth and yield across the commercial range of the species in the southern United States were surveyed to obtain sample data on CWD volume, density, and mass. Measured CWD properties were combined with inventory records of tree mortality over the study duration to characterize CWD production, decay and yield in a series of prediction equations. The resulting equations predict CWD attributes of dead trees including dry weight (kg) and fraction of standing versus downed woody material based on the time since death (years), tree diameter at breast height (cm) and height (m) at time of death and geographic coordinates of latitude and longitude. A stand-level equation predicts total CWD yield (Mg ha⁻¹) for thinned or unthinned stands based on plantation age, stem density (trees ha⁻¹), and the average height of dominant and codominant trees (m). Piece-level equations predict dry density (kg m⁻³) or nitrogen concentration (%) of CWD pieces based on their position (standing or down), ordinal decay classes, and latitude. The tree and stand-level prediction equations are designed for use in GIS or growth and yield modeling systems. The piece-level equations are designed to be used in inventory applications that survey CWD. The equations should facilitate the accurate and facile determination of mass, carbon, and nitrogen contents of CWD in planted loblolly pine forests of the southern United States.     

Effect of Betula pendula Admixture on Tree Growth and Branch Diameter in young Pinus sylvestris Stands in Southern Finland

The effects of a birch admixture on the height and diameter growth and maximum branch diameter in planted Scots pine stands was studied using models constructed with a data set from 13 stands of 9–16 yrs of age and 2–8 m dominant height on average sites on mineral soils in southern Finland. The density and height of the birch varied highly between and within stands. Simulated results indicated that the pines were capable of keeping up in height growth with birches that had originated from seed. Even a very high number of birches (10?000 stems ha^?1) had virtually no effect on the height growth of the pines. The number of birches had a pronounced effect on the diameter growth and the maximum branch diameter in pine. Retention of a temporary birch component in young pine stands seems a feasible way of mitigating the adverse effects of low planting densities on the external quality of pine.     

Modelling the diameter distribution of Pinus sylvestris, Pinus nigra and Pinus halepensis forest stands in Catalonia using the truncated Weibull function

Parameter prediction models for the diameter distribution ofPinus sylvestris L., Pinus nigra Arn. and Pinus halepensis Mill.in Catalonia were developed using the truncated Weibull functionas the theoretical distribution. The parameter models allowone to use individual-tree models in the simulation of standdevelopment when only stand-level data are collected in forestinventories. Parameter models for the diameter distributionof stand basal area were developed. The data consisted of permanentsample plots from the Spanish National Forest Inventory in Catalonia.A total of 1780 empirical distributions of P. sylvestris, 1204distributions of P. nigra and 1535 distributions of P. halepensiswere used as modelling data. The empirical data represent left-truncateddistributions, as the smallest diameter measured in the fieldwas 7.5 cm. Two different approaches, namely, regression (two-stepmethod) and optimization approach (one-step method), were usedto find the coefficients of the parameter models. In the two-stepmodelling method, the Weibull parameters were first estimatedseparately for every empirical distribution by maximizing thelog-likelihood function of the Weibull density function. Inthe second-step, regression analysis was used to find the relationshipbetween Weibull parameters and stand basal area, number of treesper hectare and elevation of the site. The one-step method usedoptimization to find such coefficients for the parameter models,which minimized the mean of the squared differences betweenempirical and predicted cumulative tree frequencies in the wholemodelling data. The one-step optimization method performed betterthan the two-step regression method for all tree species. Theparameter prediction models developed in this study enable theprediction of the diameter distribution of P. sylvestris, P.nigra and P. halepensis in Catalonia from limited stand information.     

Simultaneous prediction of the development of loblolly pine and woody competitors in young plantations

A system for examining the effects of hardwood density and cover of herbaceous components on mean size in newly established loblolly pine plantations was developed using a site preparation study located in the piedmont province of Georgia (USA). Multispecies density models were derived to predict the sum of crown heights for eight important hardwood species using both the current or age 1 number of rootstocks and herbaceous vegetation to account for intra- and interspecific effects. The predicted sums of crown heights for the woody species were then used as predictors of mean height, diameter, and volume for loblolly pine. A seemingly unrelated regression procedure was used to compensate for correlations in error components within each system of equations that result from using predicted crown heights as regressor variables. The effects of varying sweetgum density and andropogon cover on the inter-related components of the systems based on current and age 1 competition were examined for mean height and mean diameter of the planted loblolly pines. The cumulative impacts of associated vegetation on loblolly pine growth and the relative sensitivity of height and diameter to competing vegetation were demonstrated.     

Ice Damage in a Chronosequence of Agroforestry Pine Plantations in Arkansas,USA

Abstract

Acute (broken and leaning) and transient (bending) damage to loblolly pine (Pinus taedaL.) were assessed in a case study of experimental agroforestry plantations following a December 2000 ice storm. Stand ages were 7-, 9-, and 17-years-old and tree density ranged from 150 to 3,360 trees ha^?1 in rectangular and multi-row configurations. Wider tree spacing or lower stand density of 7-year-old trees increased stem breakage, while closer spacing increased bending. There was substantial straightening of bent 7-year-old trees 8 months after the storm, and this recovery was determined more by degree of initial bend rather than height or diameter. Nine-year-old loblolly pine had 19% more top breakage and 59% more stem breakage than shortleaf pine (P. echinataMill.) (P < 0.001). Agroforestry design influenced ice damage in 7-year-old stands, but no design had catastrophic loss. Thinning from above caused an increased susceptibility of ice damage to a 17-year-old stand compared to a nonthinned stand. The case study supports the cultivation of loblolly pine in areas prone to ice damage.     

Predicted leaf area growth and foliage efficiency of loblolly pine plantations

Foliage dynamics research is helpful for better understanding the process of forest production and improving silvicultural practice. However, the difficulty of measuring foliage amount has slowed down the research progress. Since leaf area of an individual tree can be reliably predicted from its diameter, growth and yield models that provide detailed information for each diameter class can be used to benefit foliage dynamics research. Simulation results from a growth and yield system for unthinned loblolly pine plantations indicated that foliage area increased with stand age, peaked between ages 36 and 51, and decreased after that. Volume growth increased with leaf area for young stands and decreased for older stands, whereas foliage efficiency consistently decreased with age. Better sites supported higher levels of leaf area index, volume growth, and foliage efficiency. Higher planting densities led to higher maximum leaf area indices and shorter time to reach that level. Initial density had no effect on foliage efficiency through time.     

Predicting and calibrating diameter distributions of Eucalyptus grandis (Hill) Maiden plantations in Zimbabwe

Diameter distribution models for even-aged Eucalyptus grandis plantations in Zimbabwe were developed using the two-parameter Weibull function. The analysis was based on data from Correlated Curve Trend (CCT) experiments replicated on four different sites. Parameters of the Weibull distribution were predicted using stand characteristics as regressors. Two sets of parameter models were estimated: a set with and one without stand basal area as a predictor. Stand variables such as dominant height, age, site index and number of stems were used in both sets. The models were further calibrated to result in a given number of stems and stand basal area simultaneously. The usability of constructed models was tested both in prediction of yield in a stand inventory situation and in simulation of growth in connection with different growth models. The results indicated that models not including stand basal area produce considerably less precise stand volume estimates compared to models including also stand basal area. Calibration improved the accuracy of diameter distribution models. In growth simulation diameter distribution models can be connected both to single tree growth models and to stand projection models. The usability of calibration in growth simulation depends on the accuracy of the prediction of stand characteristics.