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Silva Fennica 1926-1997
Acta Forestalia Fennica

Articles containing the keyword 'Weibull function'

Category: Article

article id 5392, category Article
Pekka Kilkki, Matti Maltamo, Reijo Mykkänen, Risto Päivinen. (1989). Use of the Weibull function in estimating the basal area dbh-distribution. Silva Fennica vol. 23 no. 4 article id 5392.
Keywords: Picea abies; distribution; diameter; Weibull function; maximum likelihood; estimation
Abstract | View details | Full text in PDF | Author Info

The paper continues an earlier study by Kilkki and Päivinen concerning the use of the Weibull function in modelling the diameter distribution. The data consists of spruces (Picea abies (L.) H. Karst.) measured on angle count sample points of the National Forest Inventory of Finland. First, maximum likelihood estimation method was used to derive the Weibull parameters. Then, regression models to predict the values of these parameters with stand characteristics were calculated. Several methods to describe the Weibull function by a tree sample were tested. It is more efficient to sample the trees at equal frequency intervals than at equal diameter intervals. It also pays to take separate samples for pulpwood and saw timber.

The PDF includes an abstract in Finnish.

  • Kilkki, E-mail: pk@mm.unknown (email)
  • Maltamo, E-mail: mm@mm.unknown
  • Mykkänen, E-mail: rm@mm.unknown
  • Päivinen, E-mail: rp@mm.unknown

Category: Article

article id 7676, category Article
Jussi Saramäki. (1992). A growth and yield prediction model of Pinus kesiya (Royle ex Gordon) in Zambia. Acta Forestalia Fennica vol. 0 no. 230 article id 7676.
Keywords: diameter distribution; Weibull function; simulation models; Pinus kesiya; thinning reaction
Abstract | View details | Full text in PDF | Author Info

The study presents a growth and yield prediction model for a Pinus kesiya (Royle ex Gordon) stand by diameter classes. The material consists of temporary sample plots taken from a plantation inventory, of permanent sample plots established in commercial compartments and of an espacement trial. The mean basal area of the stand, variance and skewness of the diameter distribution are predicted. From these variables the parameters of the Weibull function are derived. Site class is assumed to be known or is calculated from measured information. Mortality is also predicted by estimating the number and mean size of dead trees. Thinnings are defined by the number of trees removed and by their relative size. If measured tree level data at the initial situation is available it can be utilized in the predictions, however, simulations can also be performed with stand level information. The minimum information needed for the prediction is planting density, site class as well as the times and removals of thinnings.

The calculations show that by decreasing the planting density of P. Kesiya from the present 1,330 stems/ha or by conducting early precommercial thinning both the relative and absolute amount of large sawlogs in the total production increase. An increase in the present planting density only slightly increases total yield. It is obvious that the presently recommended rotation of 25 years is too short for producing large sawlogs, especially on poor sites. This rotation period is suitable for small sawlog production while for pulpwood regimes shorter rotation periods can be used. If thinnings are done before the maximum current annual growth is reached stands will react well, but later on the ability to respond to thinnings decreases rapidly. Thinnings from below accelerates the production of large sawlogs more than thinning from above or systematic thinning. If all sawlog sizes are considered no great differences between thinning type exist. The study recommends different thinning regimes according to site class. Separate programs are recommended for the production of sawlogs and pulpwood.

The used thinning reaction model needs refinement and further studies with annual measured thinning trial material.

The PDF includes a summary in Finnish.

  • Saramäki, E-mail: js@mm.unknown (email)

Category: Research article

article id 10062, category Research article
Jouni Siipilehto, Miika Rajala. (2019). Model for diameter distribution from assortments volumes: theoretical formulation and a case application with a sample of timber trade data for clear-cut sections. Silva Fennica vol. 53 no. 1 article id 10062.
Keywords: bucking; optimization; simplex method; truncated Weibull function
Highlights: The Weibull distribution was solved successfully from assortment volumes using optimization; The solved distribution provided accurate assortment volume when the input variables were correct; Goodness-of-fit tests indicate the compatibility between the solved distribution and the cut trees, according to harvester data; Timber trade contracts showed overestimated average merchantable tree sizes, which resulted in an underestimation of the number of cut trees; The reason for underestimation seemed to be in the decreasing distributions.
Abstract | Full text in HTML | Full text in PDF | Author Info

This study examined a theoretical model for stand structures from the volumes of pulpwood and saw logs of clear-cut stands. The average stem size was used to estimate the number of cut trees. The distribution was solved using nonlinear derivative-free optimization. The truncated 2-parameter Weibull distribution was used to describe the stand structure of the commercial stems. This method was first tested with harvester data collected from seven clear-cut stands in southern Finland. Validation included reliability in the stand characteristics and goodness-of-fit of the species-specific distributions. The distributions provided unbiased estimates for the saw log volume, while the bias in the estimated pulpwood volume was 2%. The standard stand characteristics from the Weibull distributions corresponded notably well with the harvester data. A Kolmogorov-Smirnov (KS) test rejected two distributions out of 21 cases, when the accurate input variables were available for the theoretical model. The results of the study suggest that the presented method is a relevant option for predicting the stand structure. In practice, the reliability of the presented method was dependent on the quality of the information available from the stand prior to cutting. With a timber trade data set, the solution for the distribution for a clear-cut section was found. The goodness-of-fit was dependent on the accuracy of the visually assessed timber trade variables. Especially the average stem size proved difficult to assess due to high number of understorey pulpwood stems. Due to overestimated average stem sizes, the solved number of harvested trees was underestimated. Less than 50% of the distributions predicted for clear-cut sections passed the KS test.

  • Siipilehto, Natural Resources Institute Finland (Luke), Latokartanonkaari 9, P.O. Box 2, FI-00790 Helsinki, Finland E-mail: (email)
  • Rajala, Metsä Group, Revontulenpuisto 2, P.O. Box 10, 02020 METSÄ, FI-02100 Espoo, Finland E-mail:
article id 332, category Research article
Marc Palahí, Timo Pukkala, Antoni Trasobares. (2006). Calibrating predicted tree diameter distributions in Catalonia, Spain. Silva Fennica vol. 40 no. 3 article id 332.
Keywords: calibration estimation; Weibull function; parameter prediction
Abstract | View details | Full text in PDF | Author Info
Several probability density functions have been used in describing the diameter distributions of forest stands. In a case where both the stand basal area and number of stems per hectare are assessed, the fitted or predicted distribution is scaled using only one of these variables, with the result that the distribution often gives incorrect values for the other variable. Using a distribution that provides incorrect values for known characteristics means wasting information. Calibrating the distribution so that it is compatible with the additional information on stand characteristics is a way to avoid such wasting. This study examined the effect of calibration on the accuracy of the predicted diameter distributions of the main tree species of Catalonia. The distributions were calibrated with and without considering the prediction errors of the frequencies of diameter classes. When prediction errors were assumed, the calibration was done with and without making allowance for estimation errors in the stand level calibration variables. Calibrated distributions were more accurate than non-calibrated in terms of sums of different powers of diameters. The set of calibration variables that gave the most accurate results included six stand variables: number of trees per hectare, stand basal area, basal-area-weighted mean diameter, non-weighted mean diameter, median diameter, and basal area median diameter. Of the tested three-variable combinations the best was: number of trees per hectare, stand basal area, and basal-area-weighted mean diameter. Means were more useful calibration variables than medians.
  • Palahí, Centre Tecnológic Forestal de Catalunya. Passeig Lluis Companys, 23, 08010, Barcelona, Spain E-mail: (email)
  • Pukkala, University of Joensuu, Faculty of Forestry, P.O. Box 111, 80101 Joensuu, Finland E-mail:
  • Trasobares, Foreco Technologies, Av. Diagonal 416, Estudio 2, Barcelona 08037, Spain E-mail:
article id 331, category Research article
Jouni Siipilehto. (2006). Height distributions of Scots pine sapling stands affected by retained tree and edge stand competition. Silva Fennica vol. 40 no. 3 article id 331.
Keywords: Pinus sylvestris; retention; height distribution; Weibull function; percentile prediction; edge effect
Abstract | View details | Full text in PDF | Author Info
The paper focused on the height structure of Scots pine saplings affected by (1) retained solitary pine trees or (2) a pine-dominated edge stand. The study material in (1) and (2) consisted of ten separate regeneration areas in southern Finland. In (1) 2-m radius study plots were located at 1, 3, 6 and 10 m distances from 10 systematically selected, solitary retained trees in each stand. In (2) the study plots were systematically located within 20 m from the edge stand. Competition of the individual trees was modelled using ecological field theory. The 24th and 93rd sample percentiles were used for estimating the height distribution using the two-parameter Weibull function. The models incorporated the effect of varying advanced tree competition on the predicted percentiles. Competition free dominant height was used as a driving variable for the developmental phase. Competition resulted in retarded height development within a radius of about 6 m from the retained tree, while it extended up to roughly half of the dominant height of the edge stand. The height distribution without external competition was relatively symmetrical, but increasing competition resulted in a more peaked and skewed distribution. Slight differences were found between northern sunny and southern shaded stand edges, while the least retarded height occurred at the north-western edge receiving morning sunlight. Kolmogorov-Smirnov goodness-of-fit tests showed acceptable and equal fit for both data sets; 2% and 8% of the distributions did not pass the test at the alpha 0.1 level when the Weibull distribution was estimated with the observed or predicted percentiles, respectively.
  • Siipilehto, Finnish Forest Research Institute, P.O. Box 18, FI-01301 Vantaa, Finland E-mail: (email)
article id 620, category Research article
Annika Kangas, Matti Maltamo. (2000). Performance of percentile based diameter distribution prediction and Weibull method in independent data sets. Silva Fennica vol. 34 no. 4 article id 620.
Keywords: stand structure; calibration estimation; Weibull function; diameter distribution prediction; distribution-free method; nearest neighbour method
Abstract | View details | Full text in PDF | Author Info
Diameter distribution is used in most forest management planning packages for predicting stand volume, timber volume and stand growth. The prediction of diameter distribution can be based on parametric distribution functions, distribution-free parametric prediction methods or purely non-parametric methods. In the first case, the distribution is obtained by predicting the parameters of some probability density function. In a distribution-free percentile method, the diameters at certain percentiles of the distribution are predicted with models. In non-parametric methods, the predicted distribution is a linear combination of similar measured stands. In this study, the percentile based diameter distribution is compared to the results obtained with the Weibull method in four independent data sets. In the case of Scots pine, the other methods are also compared to k-nearest neighbour method. The comparison was made with respect to the accuracy of predicted stand volume, saw timber volume and number of stems. The predicted percentile and Weibull distributions were calibrated using number of stems measured from the stand. The information of minimum and maximum diameters were also used, for re-scaling the percentile based distribution or for parameter recovery of Weibull parameters. The accuracy of the predicted stand characteristics were also compared for calibrated distributions. The most reliable results were obtained using the percentile method with the model set including number of stems as a predictor. Calibration improved the results in most cases. However, using the minimum and maximum diameters for parameter recovery proved to be inefficient.
  • Kangas, Finnish Forest Research Institute, Kannus Research Station, P.O. Box 44, FIN-69101 Kannus, Finland E-mail: (email)
  • Maltamo, Finnish Forest Research Institute, Joensuu Research Station, P.O. Box 68, FIN-80101 Joensuu, Finland E-mail:

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