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.     

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.     

Allometric model of the height–diameter curve for even-aged pure stands of Japanese cedar (<Emphasis Type="Italic">Cryptomeria japonica</Emphasis>)

We derived an allometric model of the height–diameter curve for even-aged pure stands, which was a modification of the earlier model proposed by Inoue (2000a). An individual-dependent allometric equation was used as the height–diameter curve. Using the discriminant analysis method, all trees composed of a stand were stratified into upper and lower trees. It was assumed that both relationships between mean tree height H _m and upper tree height H _u and between mean DBH D _m and mean DBH of upper trees D _u could be described by the time-dependent allometric power equations. The height–diameter curve showed an average relationship between tree height and DBH of a given stand at a given time, and hence it could be assumed that the height–diameter curve contained two points (D _m , H _m ) and (D _u , H _u ). With these assumptions, we derived an allometirc model of height–diameter curve, which allowed the coefficients of the curve to be estimated from mean tree height and mean DBH. The proposed model was fitted to Japanese cedar (Cryptomeria japonica D. Don) data. The error ratio of the allometric model ranged from 2.254% to 13.412% (mean = 6.785%), which was significantly smaller than that of the earlier model. When the error of mean tree height was ±1.0 m or less, the effect of the error of mean tree height on the error ratio was comparatively small. This suggested that the error of ±1.0 m in mean tree height could be accepted in the estimation of height–diameter curve using the allometric model. These features enable us to combine the allometric model with Hirata’s vertical angle-count sampling or growth models. In conclusion, the allometric model would be one of the most practical and convenient approaches for estimating the height–diameter relationship of even-aged pure stands.     

Spatial structure of deciduous forest stands with contrasting human influence in northwest Spain

Five contrasting deciduous forest stands were studied to characterize the spatial structural variability in human-influenced forests. These stands are representative of cultural forest types widely represented in western Europe: one plantation, two coppices, one wood-pasture forest and one high forest stand. All stems with DBH > 5 cm were measured and mapped, and stem DBH distributions, spatial structure of DBH, spatial point patterns and spatial associations were analysed. Spatial autocorrelation for DBH was calculated with Moran’s I correlograms and semivariograms. Complete spatial randomness hypothesis for spatial point patterns, and both independence and random labelling hypotheses for spatial associations were analysed using Ripley’s K function. The results showed that tree sizes were conditioned by particular former management systems, which determined unimodal symmetric, positively skewed or compound DBH distributions. Spatial structure was more complex when human influence became reduced. Coppice stands showed clumped spatial patterns and independence among size classes, as a consequence of sexual and vegetative establishment of new stems in open areas. The largest clumping intensity was observed in the wood-pasture with an intermediate disturbance frequency and low inter-tree competition. The high forest stand displayed spatial traits consistent with the gap-dynamics paradigm, such as clumping of smaller trees, random arrangement of larger trees, negative association between juveniles and adults, and high structural heterogeneity. It can be expected that after cessation of human interference, coppices and wood-pastures would evolve to a more heterogeneous structure, probably with a higher habitat and species diversity.     

Estimating stand structure using digitalized aerial photography

This study estimates the structure of a forest stand using high-resolution data from digitized aerial photography. The study area was conducted on the fifteen plots with the size of 20 m × 20 m or 15 m × 15 m taken as samples of sugi (Cryptomeria japonica D. Don) stands. The stand factor was investigated based on its relation with the digitalized number (DN) on the black and white aerial ortho photos (scale 1:5,000, taken in 1995). We analyzed various stand structure factors and the distribution of DN from the image data using the Weibull function, which comforms well to every shape of distributions. Using the Weibull distribution, a significant negative correlation was observed between the quantity of stand factors (Mean DBH and the stand volume) and the distribution parameterc of DN. A significant positive correlation was also observed between the distribution parameterc of the stand factor (DBH, height and volume) and the distribution parameterc of DN.     

Forest modelling: the gamma shape mixture model and simulation of tree diameter distributions

? Key message

New types of distribution functions are needed to model the dynamics of stands where important age classes are represented by few trees. In this study the gamma shape mixture model and two simulation methods were used for generating tree diameter data.

? Context

To analyse forest dynamics, it is necessary to know distribution of the characteristics (mainly tree diameters) of trees forming particular developmental phases. In many forest inventories, the measurement of large diameter at breast height (DBH) samples is practically impossible. In this case, DBH distributions can be generated using theoretical models.

? Aims

The aim of this study was to assess the precision of the approximation of empirical DBH data using the gamma shape mixture (GSM) model and kernel density estimation. The strengths and weaknesses of the two simulation methods were presented and discussed.

? Methods

The GSM model was adopted to approximate empirical DBH data collected in 20 near-natural stands. Two simulation methods were used: (a) the procedure based on a multimodal distribution and gamma random numbers (MDGR procedure) and (b) MCMC techniques with Metropolis–Hastings sampling (MH method).

? Results

The GSM model precisely fitted the investigated DBH distributions. The MDGR procedure was slightly more precise than the MH method, especially in the case of the samples of 250 DBHs. The level of homogeneity within the drawn DBH sets was similar for all samples.

? Conclusion

The GSM model is very flexible. The DBH random variates, generated with the use of analysed procedures, represented all tree generations being significant from a biological point of view.

     

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。     

Modeling diameter distributions of mixed-species forest stands

This study compares three diameter distribution models to fit mixed-species forest stands using four example plots with two or three species components in Daxing'an Mountain, PR China. The methods include (1) a finite mixture model (FMM) to fit two or three species components simultaneously, (2) a single Weibull function to fit the whole plot only, and (3) a single Weibull function to fit each species component separately and the summation of the individual species produced the whole plot. Our results indicated that Method 2 is only suitable to regular and unimodal diameter distributions with a balanced reversed J-shape. Method 3 may be able to fit each species component well if its frequency distribution is known and available in the data. However, Method 3 ignores the interspecies relationships within a given plot. Thus, the summation of the species components may not produce a good fit for the whole plot. In contrast, Method 1 (FMM) fits the species component distributions simultaneously with the constraint that the individual components add up to the whole plot, without requiring the observed frequencies for each species across the diameter classes. The FMM models are more flexible to describe highly skewed and irregular diameter distributions for the whole plot, as well as provide the acceptable estimation for each species component and the mixing proportions. Thus, the FMM models can be a useful tool for effectively managing mixed-species forest stands.     

Comparison of stand structure and growth between artificial and natural forests of Pinus sylvestiris var, mongolica on sandy land

Mongolian pine (Pinus sylvestiris Linnaeus var. mongolica Litvinov) as a valuable conifer tree species has been broadly introduced to the sandy land areas in “Three North” regions (North, northwest and northeast of China), but many prob-lems occurred in the earliest Mongolian pine plantations in 7hanggutai, 7hangwu County, Liaoning Province (ZZL). In order to clarify the reason, comprehensive investigations were carried out on differences in structure characteristics, growth processes and ecological factors between artificial stands (the first plantation established in ZZL in 1950s) and natural stands (the origin forests of the tree species in Honghuaerji, Inner Mongolia) on sandy land. The results showed that variation of diameter-class distributions in artificial stands and natural stands could be described by Weibull and Normal distribution models, respectively.Chapman-Richards growth model was employed to reconstruct the growth process of Mongolian pine based on the data from field investigation and stem analysis. The ages of maximum of relative growth rate and average growth rate of DBH, height, and volume of planted trees were 11,22 years, 8, 15 years and 35, 59 years earlier than those of natural stand trees, respectively. In respect of the incremental acceleration of volume, the artificial and natural stands reached their maximum values at 14 years and 33 years respectively. The quantitative maturity ages of artificial stands and natural stands were 43 years and 102 years respectively. It was concluded that the life span of the Mongolian pine trees in natural stands was about 60 years longer than those in artificial stands. The differences mentioned above between artificial and natural Mongolian pine forests on sandy land were partially attributed to the drastic variations of ecological conditions such as latitude, temperature, precipitation, evaporation and height above sea level. Human beings‘‘ disturbances and higher density in plantation forest may be ascribed as additional reasons. Those results may be potentially useful for the management and afforestation of Mongolian pine plantations on sandy land in arid and semi-arid areas.     

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.     

Root biomass and distribution of a <Emphasis Type="Italic">Picea–Abies</Emphasis> stand and a <Emphasis Type="Italic">Larix–Betula</Emphasis> stand in pumiceous Entisols in Japan

Root biomass and root distribution were studied in Entisols derived from the thick deposition of volcanic pumice on Hokkaido Island, Japan, to examine the effect of soil conditions on tree root development. The soil had a thin (<10 cm) A horizon and thick coarse pumiceous gravel layers with low levels of available nutrients and water. Two stands were studied: a Picea glehnii–Abies sachalinensis stand (PA stand) and a Larix kaempferi–Betula platyphylla var. japonica stand (LB stand). The allometric relationships between diameter at breast height (DBH) and aboveground and belowground biomass of these species were obtained to estimate stand biomass. The belowground biomass was small: 30.6 Mg ha₋₁ for the PA stand and 24.3 Mg ha₋₁ for the LB stand. The trunk/root ratios of study stands were 4.8 for the PA stand and 4.3 for the LB stand, which were higher than those from previous studies in boreal and temperate forests. All species developed shallow root systems, and fine roots were spread densely in the shallow A horizon, suggesting that physical obstruction by the pumiceous layers and their low levels of available water and nutrients restricted downward root elongation. The high trunk/root ratios of the trees may also have resulted from the limited available rooting space in the study sites.     

Application of the 3PG forest growth model to a <Emphasis Type="Italic">Eucalyptus globulus</Emphasis> plantation in Northwest Spain

This paper presents an assessment of the ability of the 3PG process-based model to simulate height (h) and diameter at breast height (DBH) growth in three plots located in a Eucalyptus globulus plantation on the Atlantic coast of Galicia (Northwest Spain). The obtained results show how this model estimates adequately height (h) and DBH growth, and determination coefficients (R ²) between observed data and data predicted by the model are always higher than 0.97, root mean square error (RMSE) of 0.45–1.05 cm in DBH and 0.46–0.85 in h, and a model efficiency (EF) close to 1 (0.98–0.99). In order to improve the results predicted by the model, we propose modifying the nS and fullCanAge parameter values given by default so as to adjust growth rates in the plantation to the weather conditions in the site. The ability of the model to discriminate growth rates in the three plots must be noted, even when plots are located in the same plantation and they only show differences in aspects related to site factors. Given the obtained results in the three plots used in this study, its simplicity and the small number of parameters needed as input data, the 3PG model stands out as a very useful tool for forest plantation management.     

Appropriate density of water and soil conservation of Pinus tabulaeformis and Robinia pseudoacacia forests in loess area, North China

In this paper, based on a long-term monitoring of water cycle in the water and soil conservation forest stands of Pinus tabulaeformis and Robinia pseudoacacia, the soil moisture deficit is calculated. Following the principles of runoff-collecting forestry and applying the forest structure investigation results, the authors developed a formula to calculate appropriate density for forests on the basis of different diameters at breast height (DBH). Using this method to manage forests, the natural water requirement of forests can be met and soil drought can be avoided. In addition, with long-term monitoring of soil moisture in stands, the authors also give an appropriate managing density specifically for the water and soil conservation forests of P. tabulaeformis and R. pseudoacacia in the loess area which is according to soil moisture content, or with the lowest soil moisture content and invalid moisture frequency as the indexes. __________ Translated from Science of Water and Soil Conservation, 2007, 5(2): 55–59 [译自: 中国水土保持科学]     

Oak (<Emphasis Type="Italic">Quercus robur</Emphasis> L.) regeneration as a response to natural dynamics of stands in European hemiboreal zone

The oak (Quercus robur L.) regeneration intensity was assessed in the core area of the Białowieża National Park (BNP) in Poland with respect to the selected ecological factors. The emphasis was placed on the response of oak regeneration to disturbances, including the large-scale dieback of spruce stands. Defining their effect could help predicting the role of oak in naturally developing lowland forest ecosystems in the European hemiboreal zone. The results of the study challenge the opinion that the ‘lime-oak-hornbeam forest’ is a ‘climax’ community, confirming a very poor regeneration represented by only two saplings taller than 0.5 m per hectare. By contrast, in spruce-dominated communities, from 49 to 848 taller saplings per hectare were found. The occurrence of saplings was associated with discontinuous canopy of late seral stage of stands, as well as with large gaps. Most of the best quality grown-up oak saplings developed in the immediate neighbourhood of spruce logs. The results of the research indicate that ‘lime-oak-hornbeam forest’ (Tilio-Carpinetum) should be rather perceived as a transient community, evolved from relict, culturally modified, oak woodlands. Because spruce had become a dominating species only after abandoning in mid-1800s the historical regime of anthropogenic disturbances (involving frequent forest fires), the observed phenomena related to the disintegration of spruce stands had probably no precedent over the last 500 years. To confirm whether the massive decline of spruce stands will finally result in the successful establishment of the new canopy oak generation, both in Białowieża and other forests of hemiboreal zone, further research is needed.     

Juvenile diameter distributions of loblolly pine characterized by the two-parameter Weibull function

Diameter distributions of juvenile loblolly pine (Pinus taeda L.) were characterized utilizing a two-parameter Weibull distribution to aid in forecasting and simulation of young stands. Juvenile diameter distributions were studied to gain insight into the effects of various stand-level factors. Results show that diameter distributions in juvenile loblolly pine stands can be successfully characterized with the two-parameter Weibull function. Repeated measures analysis detected significant planting density, age, and age by planting density interaction effects for the scale and shape parameter estimates from the two-parameter Weibull distribution. Using parameter recovery techniques, estimated diameter distributions were derived from easily attainable stand-level characteristics (i.e. basal area per hectare, planting density, age, and quadratic mean diameter). A thorough understanding of juvenile diameter distributions should prove especially useful for operational planning of stands on short rotations that require estimates of productivity at early ages.     

Modelling the edge effect in even-aged Monterey pine (Pinus radiata D. Don) stands

Six edge effect models are presented. Modelling results for the edge effect on the DBH, total height, crown height, and basal area in even-aged stands of Monterey pine (Pinus radiata D. Don) are provided. Free edges and edges under external competition pressure were analysed. Model fitting was carried out in two phases. In the first phase, independent fittings for each border type by sampled cardinal point in each stand were carried out to detect differences in edge effect intensity and depth according to edge aspect. In the second fitting phase, the best models selected by edge aspect in the first phase were restricted to estimate the same value for each variable in the stand interior, independently of the border where it was analysed.

The edge effect intensity and depth varied depending on the analysed variable and edge type. Modelling of the edge effect on DBH and total height on free edges of old stands requires a model with independent parameters for each border aspect; the models can be restricted to estimate the average value of each variable in stand interior. A model with independent parameters for each border aspect should be also used for modelling the effect on total height, crown height and basal area for the edges of young stands under external competition pressure. The modelling of basal area and crown height on free edges of old stands, DBH on edges of young stands under external competition pressure, and all variables mentioned on free edges of young stands can be carried out through an average fitting. DBH and basal area are directly affected by stand density; thus, modelling of edge effect on these and others variables which are affected by stand density can improve with the inclusion of a competition index or a point density index.     

Use of Gumbel and Weibull functions to model extreme values of diameter distributions in forest stands

Context

Families of the Gumbel (type I), Fréchet (type II) and Weibull (type III) distributions can be combined in the generalized extreme value (GEV) family of distributions. Maximum and minimum values of diameters in forest stands can be used in forest modelling, mainly to define parameters of the functions used in diameter class models as well as in some practical cases, such as modelling maximum diameters for sawing and processing purposes.

Aims

The purpose of this study was to examine and compare two extreme value distribution functions (the Gumbel and the Weibull functions) in modelling the distribution of the minimum and the maximum values of representative sets of tree diameter samples. Both of these functions were applied to the lower and upper values of the diameter distributions of the main forest species in northwest Spain: Quercus robur L., Betula pubescens Ehrh., Pinus radiata D. Don, Pinus pinaster Ait. and Pinus sylvestris L.

Methods

Parameters of the Gumbel function were estimated using the mode and the moments of the distributions, and parameters of the Weibull function were estimated using the moments method.

Results

In general, the Weibull distribution was the most suitable model for describing the maximum diameters. The mode method of the Gumbel yielded the best results for minimum diameters of birch and Monterrey pine. The Gumbel distribution, fitted by either the mode- or moments-based methods, proved more suitable than the Weibull distribution for describing the minimum diameters in maritime pine and Scots pine stands.

Conclusion

In some cases, better results were obtained with the Gumbel than the Weibull distribution for describing the distribution of extreme diameter values in forest stands in northwest Spain. This is the first example of the application of the Gumbel distribution in forest modelling.     

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.     

Forest biomass at Xiaolong Mountain in Gansu Province, China

In order to accurately estimate the size of the carbon pool and the capacity of the carbon sink in the forested areas of Xiaolong Mountain in Gansu Province, we have established regression equations of organ biomass of eight tree species. We measured and investigated the biomass of different forest stand types based on data from 1259 standard sample plots and 836 standard sample trees. The results show that stand biomass, expressed in t·hm⁻² for eight types of forest stands on Xiaolong Mountain, are as follows: Quercus aliena var. acuteserrata 84.05, Pinus tabulaeformis 62.44, Quercus variabilis 81.77, Populus sp. and Betula sp. combined 77.44, Larix sp. 69.00, Pinus armandii 70.07, Picea sp. 96.49 and Abies sp. 98.72. We also looked at other broad-leaved mixed forests. Our study shows that the biomass of a single tree of each tree species is closely related to the diameter at breast height (DBH) and to tree height. The biomass of single trees as well as stand volumes is closely related to average DBH, average tree height and to stand density. __________ Translated from Journal of Beijing Forestry University, 2007, 29(1): 31–36 [译自: 北京林业大学学报]     

Modeling frequency distributions of tree height,diameter and crown area by six probability functions for open forests of <Emphasis Type="Italic">Quercus persica</Emphasis> in Iran

The prediction of the distribution of quantitative variables in a forest stand is of great interest to forest managers, for the evaluation of forest resources and scheduling of future silvicultural treatments. The aim of this research was to model the distribution of quantitative variables for Quercus persica in open forests in Iran. To investigate the probability distribution of trees in natural stands, 642 trees were selected for measurement using a systematic random sampling method. Selected trees were measured and data were analyzed. Gamma, beta, normal,lognormal, exponential and Weibull probability distributions were fitted to the height distribution of trees. Variables of distribution functions were estimated using the maximum likelihood estimation method. Actual probability and probability which derived from functions was compared using Kolmogorov–Smirnov and Anderson–Darling tests. Beta, Weibull and Weibull probability distributions explained the distributions of tree height, DBH and crown area.