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Articles containing the keyword 'Weibull'.

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. https://doi.org/10.14214/sf.10062
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.

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 ORCID ID:E-mail: jouni.siipilehto@luke.fi (email)
  • Rajala, Metsä Group, Revontulenpuisto 2, P.O. Box 10, 02020 METSÄ, FI-02100 Espoo, Finland ORCID ID:E-mail: miika.rajala@metsagroup.com
article id 1568, category Research article
Jouni Siipilehto, Harri Lindeman, Mikko Vastaranta, Xiaowei Yu, Jori Uusitalo. (2016). Reliability of the predicted stand structure for clear-cut stands using optional methods: airborne laser scanning-based methods, smartphone-based forest inventory application Trestima and pre-harvest measurement tool EMO. Silva Fennica vol. 50 no. 3 article id 1568. https://doi.org/10.14214/sf.1568
Highlights: An airborne laser scanning grid-based approach for determining stand structure enabled bi- or multimodal predicted distributions that fitted well to the ground-truth harvester data; EMO and Trestima applications needed stand-specific inventory for sample measurements or sample photos, respectively, and at their best, provided superior accuracy for predicting certain stand characteristics.

Accurate timber assortment information is required before cuttings to optimize wood allocation and logging activities. Timber assortments can be derived from diameter-height distribution that is most often predicted from the stand characteristics provided by forest inventory. The aim of this study was to assess and compare the accuracy of three different pre-harvest inventory methods in predicting the structure of mainly Scots pine-dominated, clear-cut stands. The investigated methods were an area-based approach (ABA) based on airborne laser scanning data, the smartphone-based forest inventory Trestima app and the more conventional pre-harvest inventory method called EMO. The estimates of diameter-height distributions based on each method were compared to accurate tree taper data measured and registered by the harvester’s measurement systems during the final cut. According to our results, grid-level ABA and Trestima were generally the most accurate methods for predicting diameter-height distribution. ABA provides predictions for systematic 16 m × 16 m grids from which stand-wise characteristics are aggregated. In order to enable multimodal stand-wise distributions, distributions must be predicted for each grid cell and then aggregated for the stand level, instead of predicting a distribution from the aggregated stand-level characteristics. Trestima required a sufficient sample for reliable results. EMO provided accurate results for the dominating Scots pine but, it could not capture minor admixtures. ABA seemed rather trustworthy in predicting stand characteristics and diameter distribution of standing trees prior to harvesting. Therefore, if up-to-date ABA information is available, only limited benefits can be obtained from stand-specific inventory using Trestima or EMO in mature pine or spruce-dominated forests.

  • Siipilehto, Natural Research Institute Finland (Luke), Management and Production of Renewable Resources, P.O. Box 18, FI-01301 Vantaa, Finland ORCID ID:E-mail: jouni.siipilehto@luke.fi (email)
  • Lindeman,  Natural Research Institute Finland, Green Technology, Kaironiementie 15, 39700 Parkano ORCID ID:E-mail: harri.lindeman@luke.fi
  • Vastaranta, University of Helsinki, Department of Forest Sciences, P.O. Box 62 (Viikinkaari 11), FI-00014 University of Helsinki ORCID ID:E-mail: mikko.vastaranta@helsinki.fi
  • Yu, Finnish Geospatial Research Institute (FGI), Department of Remote Sensing and Photogrammetry, National Land Survey of Finland, P.O. Box 15 (Geodeetinrinne 2), FI-02431, Masala, Finland ORCID ID:E-mail: xiaowei.yu@maanmittauslaitos.fi
  • Uusitalo,  Natural Research Institute Finland, Green Technology, Kaironiementie 15, 39700 Parkano ORCID ID:E-mail: jori.uusitalo@luke.fi
article id 1057, category Research article
Jouni Siipilehto, Lauri Mehtätalo. (2013). Parameter recovery vs. parameter prediction for the Weibull distribution validated for Scots pine stands in Finland. Silva Fennica vol. 47 no. 4 article id 1057. https://doi.org/10.14214/sf.1057
Highlights: A parameter recovery method (PRM) was developed for forest stand inventories and compared with previously developed parameter prediction methods (PPM) in Finland; PRM for the 2-parameter Weibull function provided compatibility for the main stand characteristics: stem number, basal area and one of the four optional mean characteristics; PRM provided comparable and at its best, superior accuracy in volume characteristics compared with PPM.
The moment-based parameter recovery method (PRM) has not been applied in Finland since the 1930s, even after a continuation of forest stand structure modelling in the 1980s. This paper presents a general overview of PRM and some useful applications. Applied PRM provided compatibility for the included stand characteristics of stem number (N) and basal area (G) with either mean (D), basal area-weighted mean (DG), median (DM) or basal area-median (DGM) diameter at breast height (dbh). A two-parameter Weibull function was used to describe the dbh-frequency distribution of Scots pine stands in Finland. In the validation, PRM was compared with existing parameter prediction models (PPMs). In addition, existing models for stand characteristics were used for the prediction of unknown characteristics. Validation consisted of examining the performance of the predicted distributions with respect to variation in stand density and accuracy of the localised distributions, as well as accuracy in terms of bias and the RMSE in stand characteristics in the independent test data set. The validation data consisted of 467 randomly selected stands from the National Forest Inventory based plots. PRM demonstrated excellent accuracy if G and N were both known. At its best, PRM provided accuracy that was superior to any existing model in Finland – especially in young stands (mean height < 9 m), where the RMSE in total and pulp wood volumes, 3.6 and 5.7%, respectively, was reduced by one-half of the values obtained using the best performing existing PPM (8.7–11.3%). The unweighted Weibull distribution solved by PRM was found to be competitive with weighted existing PPMs for advanced stands. Therefore, using PRM, the need for a basal area weighted distribution proved unnecessary, contrary to common belief. Models for G and N were shown to be unreliable and need to be improved to obtain more reliable distributions using PRM.
  • Siipilehto, Finnish Forest Research Institute, Vantaa Research Unit, P.O. Box 18, FI-01301 Vantaa, Finland ORCID ID:E-mail: jouni.siipilehto@metla.fi (email)
  • Mehtätalo, University of Eastern Finland, School of Computing, P.O. Box 111, FI-80101 Joensuu, Finland ORCID ID:E-mail: lauri.mehtatalo@uef.fi
article id 956, category Research article
Michał Zasada. (2013). Evaluation of the double normal distribution for tree diameter distribution modeling. Silva Fennica vol. 47 no. 2 article id 956. https://doi.org/10.14214/sf.956
The double normal distribution consists of two normal distributions truncated at their means and then combined in such a way, that points of truncation now become the overall distribution mode. So far, parameters of the double normal distribution have been estimated exclusively using the method of moments. This study evaluates the maximum likelihood method for the estimation of the double normal distribution parameters in Scots pine stands in Poland, and compares it to the results of the method of moments and the two-parameter Weibull distribution fitted using the maximum likelihood method and the method of moments. Presented results show that it is not recommended to use the maximum likelihood method of parameter fitting with Nelder-Mead and quasi-Newton optimization algorithms for the double normal distribution for small samples. However, it can be used for large samples, giving the fit comparable to the two-parameter Weibull distribution and providing parameters having sound practical and biological meaning. In the case of smaller samples for the double normal distribution it is recommended to apply the maximum likelihood method with the alternative simulated annealing optimization algorithm, use the method of moments or substitute the described distribution with more the flexible and robust Weibull distribution. For the smaller samples, the method of moments was superior to the maximum likelihood method.
  • Zasada, Warsaw University of Life Sciences, Faculty of Forestry, Laboratory of Dendrometry and Forest Productivity, Nowoursynowska 159, 02-776 Warsaw, Poland ORCID ID:E-mail: michal.zasada@wl.sggw.pl (email)
article id 99, category Research article
Jouni Siipilehto. (2011). Local prediction of stand structure using linear prediction theory in Scots pine-dominated stands in Finland. Silva Fennica vol. 45 no. 4 article id 99. https://doi.org/10.14214/sf.99
This study produced a family of models for eight standard stand characteristics, frequency and basal area-based diameter distributions, and a height curve for stands in Finland dominated by Scots pine (Pinus sylvestris L.). The data consisted of 752 National Forest Inventory-based sample plots, measured three times between 1976 and 2001. Of the data, 75% were randomly selected for modelling and 25% left out for model evaluation. Base prediction models were constructed as functions of stand age, location and site providing strongly average expectations. These expectations were then calibrated with the known stand variables using linear prediction theory when estimating the best linear unbiased predictor (BLUP). Three stand variables, typically assessed in Finnish forest management planning fieldwork, were quite effective for calibrating the expectation for the unknown variable. In the case of optional distributions, it was essential to choose the weighting of the diameter distribution model such that the available input variables and the model applied were based on the same scale (e.g. arithmetic stand variables for frequency distribution). Additional input variables generally improved the accuracy of the validated characteristics, but the improvements in the predicted distributions were most noteworthy when the arithmetic mean and basal area-weighted median were simultaneously included in the BLUP estimation. The BLUP method provided a flexible approach for characterising relationships among stand variables, alternative size distributions and the height–diameter curve. Models are intended for practical use in the MOTTI simulator.
  • Siipilehto, Finnish Forest Research Institute, Vantaa Research Unit, P.O. Box 18, FI-01301 Vantaa, Finland ORCID ID:E-mail: jouni.siipilehto@metla.fi (email)
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. https://doi.org/10.14214/sf.332
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 ORCID ID:E-mail: marc.palahi@ctfc.es (email)
  • Pukkala, University of Joensuu, Faculty of Forestry, P.O. Box 111, 80101 Joensuu, Finland ORCID ID:E-mail:
  • Trasobares, Foreco Technologies, Av. Diagonal 416, Estudio 2, Barcelona 08037, Spain ORCID ID: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. https://doi.org/10.14214/sf.331
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 ORCID ID:E-mail: jouni.siipilehto@metla.fi (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. https://doi.org/10.14214/sf.620
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 ORCID ID:E-mail: annika.kangas@metla.fi (email)
  • Maltamo, Finnish Forest Research Institute, Joensuu Research Station, P.O. Box 68, FIN-80101 Joensuu, Finland ORCID ID:E-mail:
article id 650, category Research article
Jouni Siipilehto. (1999). Improving the accuracy of predicted basal-area diameter distribution in advanced stands by determining stem number. Silva Fennica vol. 33 no. 4 article id 650. https://doi.org/10.14214/sf.650
The objective of this paper was to study to what extent the accuracy of predicted basal-area diameter distributions (DDG) could be improved by means of stem number observations in advanced (H > 10 m) stands. In the Finnish forest management planning (FMP) inventory practice, stem number is determined only in young stands; in older stands stand basal area is used. The study material consisted of sixty stands of Norway spruce (Picea abies Karst.) and ninety-one stands of Scots pine (Pinus sylvestris L.) with birch (Betula pendula Roth and B. pubescens Ehrh.) admixtures in southern and eastern Finland. For test data, 167–292 independent, National Forest Inventory-based, permanent sample plots were used. DDGs were estimated with the maximum likelihood method. Species-specific models for predicting the distribution parameters were derived using regression analysis. The two-parameter Weibull distribution was compared to the three-parameter Johnson’s SB distributions in predicting DDGs. The models were based on either predictors that are consistent with current FMP (model G), or assuming an additional stem number observation (model G+N). The predicted distributions were compared in terms of the derived stand variables: stem number, total and timber volumes. The results were similar in modelling and test data sets. Methods, based on the SB distribution obtained with model (G+N), proved to give the most accurate description of the stand structure. Differences were marginal in stand total volumes. However, the error variation in stem number was 20% to 80% lower than when applying model (G). SB and Weibull distributions gave very much the same results if model (G) was applied.
  • Siipilehto, Finnish Forest Research Institute, Vantaa Research Centre, P.O. Box 18, FIN-01301 Vantaa, Finland ORCID ID:E-mail: jouni.siipilehto@metla.fi (email)

Category: Article

article id 5609, category Article
Matti Maltamo. (1997). Comparing basal area diameter distributions estimated by tree species and for the entire growing stock in a mixed stand. Silva Fennica vol. 31 no. 1 article id 5609. https://doi.org/10.14214/sf.a8510

The purpose of this study was to compare the Weibull distributions estimated for the entire growing stock of a stand and separately for Scots pine (Pinus sylvestris L.) and Norway spruce (Picea abies (L.) H. Karst.) in describing the basal area diameter distributions in mixed stands. The material for this study was obtained by measuring 553 stands located in eastern Finland. The parameters of the Weibull distribution were estimated using the method of maximum likelihood. The models for these parameters were derived using regression analysis. Also, some parameter models from previous studies were compared with the measured distribution. The obtained distributions were compared using the diameter sums of the entire growing stock, diameter sums by tree species and of the sawtimber part of the growing stock. The results showed that far more accurate results were obtained when the distributions were formed using parameter models separately for the different tree species than when using parameter models for the entire growing stock. This was already true when considering the entire growing stock of the stand and especially when the results were examined by tree species. When the models for the entire growing stock were applied by tree species in relation to basal areas, the results obtained were overestimates for Norway spruce and underestimates for Scots pine. The models from earlier studies, where parameter models were estimated separately for tree species from the National Forest Inventory data, showed good fits also in regard to the data of this study.

  • Maltamo, ORCID ID:E-mail:
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. https://doi.org/10.14214/sf.a15550

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, ORCID ID:E-mail:
  • Maltamo, ORCID ID:E-mail:
  • Mykkänen, ORCID ID:E-mail:
  • Päivinen, ORCID ID:E-mail:
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 no. 230 article id 7676. https://doi.org/10.14214/aff.7676

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, ORCID ID:E-mail:

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