Key message Using the three characteristic points of a forest stand, dg (mean quadratic diameter), d min (diameter of the smallest tree) and d max (diameter of the largest tree), appears informative enough to determine the parameters of the whole diameter distribution and, hence, the standing volume, with an accuracy of 2–3%. This is related principally with a particular feature of the Weibull distribution function, and the empirical dependency of the main scale parameter α + β from the mean quadratic diameter (dg): This allows the prediction of the parameter β with an unexpectedly high likelihood. This feature could be used for growth modelling as well as inventory purposes, at least for monospecific and even-aged stands and, maybe more, because this feature is proper to the function itself. Context One of the most appealing applications of diameter distribution functions is to predict compliant stand diameters without needing to tally all stems, but in determining the function parameters only on the base of simple stand characteristics. This can be applied for yield model construction or inventory purposes. Aims The aim of this paper is to compare different methods of estimating the Weibull distribution parameters, partly based on parameter recovery method (PRM). They use a remarkable, empiric property of the Weibull function. Their performances are assessed in comparison to real distributions from a wide database of permanent Swiss yield plots repeatedly measured (time series) for Norway spruce ( Picea abies (L.) Karst.) and European beech ( Fagus sylvatica L.). Methods The Weibull distribution offers the advantage of simple but reliable estimation procedures. One of these is the main (scale) parameter β being given at a remarkable point of the function free, i.e. independent from the shape parameter γ . Because dg lies very close to this point, it correlates empirically very tightly with this parameter and thus allows for a trustful simple estimation. We compare, with appropriate statistic tests, real distribution with such obtained with the usual maximum likelihood estimation (MLE) of the Weibull parameters and those obtained with these new procedures. Results The results obtained from a set of 800 yield plots of regular spruce stands and 596 of beech in Switzerland illustrate the good performances of the two much simpler procedures. The accuracy of estimating the standing volume is about 1.4% for beech and 2.8% for spruce when the site index (SI) is known. Conclusion The three considered characteristic stand attributes (dg, d min and d max ) appear robust enough for determining the diameter distribution with a respectable accuracy. This is enough reason for a revival of the old, but very ingenious, method of angle count sampling of Walter Bitterlich ( 1947 ).

中文翻译：

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基于单种正温带欧洲森林简单林分结构变量估计直径分布的不同方法的性能

关键信息 使用林分的三个特征点 dg（平均二次直径）、d min（最小树的直径）和 d max（最大树的直径），似乎提供了足够的信息来确定整个直径的参数分布，因此，站立体积，准确度为 2-3%。这主要与 Weibull 分布函数的特定特征以及主要尺度参数 α + β 与平均二次直径 (dg) 的经验相关性有关：这允许以意外高的可能性预测参数 β。此功能可用于生长建模和库存目的，至少对于单一品种和偶龄林分，也许更多，因为此功能适用于功能本身。上下文 直径分布函数最吸引人的应用之一是预测符合性林分直径，而无需统计所有茎，而仅根据简单林分特征确定函数参数。这可以应用于产量模型构建或库存目的。目的 本文的目的是比较估计威布尔分布参数的不同方法，部分基于参数恢复方法 (PRM)。他们使用了威布尔函数的一个显着的经验属性。它们的性能是通过与挪威云杉（Picea abies (L.) Karst.）和欧洲山毛榉（Fagus sylvatica L.）反复测量（时间序列）的瑞士永久产量图的广泛数据库的实际分布进行比较来评估的。方法 Weibull 分布提供了简单但可靠的估计程序的优点。其中之一是主要（尺度）参数 β 在函数的显着点自由给出，即独立于形状参数 γ 。由于 dg 非常接近这一点，因此它与此参数在经验上非常紧密地相关联，因此允许进行可靠的简单估计。我们通过适当的统计检验将真实分布与通过 Weibull 参数的通常最大似然估计 (MLE) 获得的分布和通过这些新程序获得的分布进行比较。结果 从一组 800 个普通云杉林分和 596 个瑞士山毛榉产量图获得的结果说明了两种简单得多的程序的良好性能。估计站立体积的准确度约为 1。当场地指数 (SI) 已知时，山毛榉为 4%，云杉为 2.8%。结论 所考虑的三个特征林分属性（dg、d min 和 d max ）似乎足够稳健，可以以相当高的精度确定直径分布。这足以让 Walter Bitterlich (1947) 重新使用旧的但非常巧妙的角度计数采样方法。