Forest Ecology and Management 292 (2013) 111–121
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Modeling volume expansion factors for temperate tree species in France Fleur Longuetaud a,c,⇑, Philippe Santenoise a,e, Frédéric Mothe a,c, Tristan Senga Kiessé a,d, Michaël Rivoire a,c, Laurent Saint-André b,g, Nina Ognouabi f, Christine Deleuze e a
INRA, UMR1092 LERFoB, 54280 Champenoux, France INRA, Unité Biogéochimie des Ecosystèmes Forestiers, 54280 Champenoux, France c AgroParisTech, UMR1092 LERFoB, 54000 Nancy, France d ONF, Département R&D, 54840 Velaine en Haye, France e ONF, Département R&D, 39100 Dole, France f Centre de Recherche sur la Durabilité et la Productivité des Plantations Industrielles, BP 1291 Pointe-Noire, Republic of Congo g CIRAD, UMR Eco & Sols – Ecologie Fonctionnelle & Biogéochimie des Sols & Agrosystèmes, Montpellier SupAgro-CIRAD-INRA-IRD, 34060 Montpellier, France b
a r t i c l e
i n f o
Article history: Received 10 August 2012 Received in revised form 12 December 2012 Accepted 15 December 2012 Available online 24 January 2013 Keywords: Volume Biomass VEF BEF Nonlinear mixed-effects model
a b s t r a c t The objective of this study was to model the volume expansion factor (VEF), deﬁned as being the ratio between the total aboveground woody volume and the stem merchantable volume of a tree, as a function of tree height and diameter at breast height. A large database (dataset #1) of detailed stem and branches volume measurements, constituted by 8192 felled trees from 19 temperate tree species, was used for calibrating the models. In addition, an independent dataset (dataset #2), constituted by 176 felled trees from 13 species, was collected for validating the models. From dataset #1, the RMSE for the prediction of the total volume varied from 0.005 to 0.476 m3, depending on the diameter class, and the corresponding relative RMSE varied from 8.0% to 13.7%, depending on the diameter class. A 10-fold cross-validation on dataset #1 gave an average RMSE of 0.136 for the prediction of the VEF and of 0.150 m3 for the prediction of the total volume, which was in the same order as average RMSE obtained from the whole dataset #1. Validation on dataset #2 gave satisfactory results, even for the application of VEF to other tree species: RMSE obtained for the different tree species were in the same order as RMSE obtained from calibration dataset #1, except for some particular cases. The largest errors were obtained when the model was clearly used in extrapolation, e.g., for ﬁve large-size Fraxinus (whereas dataset #1 included smaller trees) and several Quercus and Fagus from coppice-with-standards stands (whereas dataset #1 included mainly pure even-aged high-forest trees). The observed differences between species seemed consistent with the general knowledge about species-speciﬁc traits. For a given diameter at breast height, angiosperms were found to have a much larger volume of branches in comparison with the corresponding stem volume than gymnosperms. For a tree of 30 cm in diameter, the lowest values of VEF were obtained for Picea and Abies (VEF < 1.1) and the highest ones for Fraxinus and Carpinus (VEF > 1.3). The methodology that was developed, based on nonlinear mixed-effects modeling, is easily applicable to other deﬁnitions of VEF or biomass expansion factors. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Biomass estimation at large scales (e.g., forest stand, regional or national scales) was the objective of numerous studies in the past.
⇑ Corresponding author at: INRA, UMR1092 LERFoB, 54280 Champenoux, France. E-mail addresses: [email protected]
(F. Longuetaud), [email protected]
(P. Santenoise), [email protected]
(F. Mothe), [email protected]
(T. Senga Kiessé), [email protected]
(M. Rivoire), [email protected]
(L. Saint-André), [email protected]
(N. Ognouabi), [email protected]
(C. Deleuze). 0378-1127/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.foreco.2012.12.023
Some studies are relatively recent (e.g., Brown, 2002; FAO, 2005; Teobaldelli et al., 2009) and were performed subsequently to the United Nations Framework Convention on Climate Change (UNFCCC) and to the ratiﬁcation of the Kyoto protocol which both aimed to reduce greenhouse gas emissions at the international level. The objective of such studies was to provide estimations of the carbon storage in forests of the different countries. Some other studies were performed in the 1970s and 1980s after the ﬁrst oil crisis in order to estimate the amount of available fuel wood in forests (e.g., Auclair and Métayer, 1980; Pardé, 1980). Today, both estimations of carbon storage and fuel wood are of great ecological
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and economic interests, and this is the reason why accurate biomass estimations are needed. Depending on the available inventory data, several methods exist to estimate total biomass (above-ground only, with or without leaves, or including roots) of trees (Fig. 1). When diameter at breast height and height measurements are available, it is possible to use directly biomass allometric equations (arrow #2 in Fig. 1; e.g., Zianis et al. (2005); Genet et al. (2011)). When estimate of merchantable volume or growing stock (including stem only or stem and branches) is available, it is possible to use a biomass expansion factor (BEF) or more precisely a biomass conversion and expansion factor (BCEF) (arrow #7 in Fig. 1; e.g., Schroeder et al. (1997)). BEF is usually a ratio between total biomass and merchantable biomass, stem biomass or growing stock biomass, whereas BCEF is usually a ratio between total biomass and merchantable volume, stem volume or growing stock volume in m3 (FAO, 2005). BCEF can be converted to BEF by dividing by the basic wood density. In practice, the term BEF may be used for BCEF. Indirect methods that ﬁrst convert stem-wood volume to stem-wood biomass (arrow #3 in Fig. 1) and then stem-wood biomass to total biomass (arrow #4 in Fig. 1) were also tested (Brown and Lugo, 1992). Finally, numerous BEFs exist depending on the deﬁnition of the merchantable volume, stem volume or growing stock volume and of the total biomass compartment, making the BEFs difﬁcult to compare between the different studies (Gschwantner et al., 2009; IPCC, 2003; FAO, 2005). The general deﬁnition of BEF provided in the Good Practice Guidance for Land Use (IPCC, 2003) is: ‘‘A multiplication factor that expands growing stock, or commercial roundwood harvest volume, or growing stock volume increment data, to account for non-merchantable biomass components such as branches, foliage, and non-commercial trees’’. Some BEFs were developed at the individual tree level (e.g., Pajtik et al., 2011; Skovsgaard and Nord-Larsen, 2012), whereas some others were developed at the stand level (e.g., Brown and Lugo, 1992; Somogyi et al., 2007; Jalkanen et al., 2005; Teobaldelli et al., 2009). A constant BEF value may be used; however, it was proved that BEF varies with forest type, growing conditions, stand density and climate (IPCC, 2003). In particular, BEF was decreasing exponentially with increasing growing stock (at individual tree level in m3 or at stand level in m3/ha; e.g., Brown (2002); Teobaldelli et al. (2009)), stand age (Lehtonen et al., 2004; Jalkanen et al., 2005; Teobaldelli et al., 2009; Pajtik et al., 2011; Sanquetta et al., 2011), or tree diameter
Fig. 1. Modeling the total biomass of trees (inspired by Somogyi et al. (2007)). Direct models of total biomass from inventory measurements (2); indirect models starting from merchantable volume estimate (1) followed by conversion from volume to biomass and expansion from merchantable to total tree component (3– 7); indirect models starting from inventory measurements through total volume estimate and conversion to biomass (8 and 6). Volume expansion factor (VEF) as deﬁned in this paper refers to arrow 5.
and/or height (FAO, 2005; Pajtik et al., 2008; Sanquetta et al., 2011). Therefore, it appeared necessary to establish BEF equations depending on age or size of trees (Schroeder et al., 1997; Teobaldelli et al., 2009; Sanquetta et al., 2011). Surprisingly, indirect methods that deliver information about the volume distribution within trees (i.e., ratio between the total woody volume and the merchantable volume) are less common (arrows #5 and #6 in Fig. 1) although reviews about allometric equations were published both for volume and biomass estimations (Zianis et al., 2005; Henry et al., 2011). We deﬁned the volume expansion factor (VEF) as the ratio between the total volume and the merchantable volume of a tree. When the deﬁnitions of the merchantable part of the tree, on the one hand, and of the total woody compartment, on the other hand, are the same across studies (which occurs rarely in fact), and when basic wood density is assumed to be constant within trees between stem and branches, VEF is equal to BEF, and the VEF can be easily converted to BCEF by multiplying by basic wood density. When the objective is not only the estimation of carbon storage by forests, but also the potential utilization of wood for industrial purposes, it is very interesting to detail the volume distribution between the merchantable part of the tree, the price of which is most of the time related to volume, and the remaining parts of the tree (mainly branches) for estimating the fuel wood resource and for logistical aspects. In addition, if independent models are used to estimate merchantable volume on the one hand (arrow #1 in Fig. 1) and total woody volume on the other hand (arrow #8 in Fig. 1), it is possible to get at the end negative volumes of branches by subtracting the two independent predictions. Even if there exist some methods to ensure additivity between modeled components (e.g., Parresol, 1999, 2001), VEF is an insurance that the total woody volume will be bigger than the merchantable volume from which it is expanded. Somogyi et al. (2007) provided illustrations for different deﬁnitions of VEF. Pretzsch (2009) indicates that VEF values for converting from stem volume over bark to total tree volume above-ground range from 1.15 to 3, depending on tree species and tree age, diameter and height (Grundner and Schwappach, 1952; Burschel et al., 1993). Our objective was to develop a generic model of VEF suitable to be used with merchantable stem volume estimates usually provided by forest inventories for various temperate tree species, in order to get at the end reliable estimates for the total above-ground volume and the available branch volume. VEF at the individual tree level was expressed as a function of classical inventory measurements, which are the diameter at breast height and the total tree height. A large database was used for developing and validating the models (8192 trees from 19 species in dataset #1 and 176 trees from 13 species in dataset #2). For all these trees, the volumes of stem and branches, up to 7 cm in diameter, were measured in detail and the smaller branches were weighed. Therefore, in comparison with many other studies, our models were developed directly from volume measurements, i.e., not from volume predictions obtained through allometric equations, which avoided propagating modeling errors. In the literature, numerous studies have focused on the relationship between BEF and stand age, which is particularly suitable for planted or regular forests (e.g., Lehtonen et al., 2004). However, heterogeneous forests are very common in Europe (MCPFE, 2007) and there is a need for developing approaches that are suitable for various forest structures, in particular irregular and mixed forests, which present a high variability (e.g., tree size, tree age and tree species) and for which it is not possible to use the age variable in models. A generic equation of VEF was proposed that can be used for the different tree species. The equation was chosen for having meaningful parameters that could be easily interpreted. The binary variable angiosperm vs. gymnosperm was used as input in the model. We were also particularly interested in comparing the
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different tree species with respect to their VEF parameters and the results were discussed with respect to our knowledge of speciesspeciﬁc ecological behaviors. 2. Materials and methods 2.1. Deﬁnition of the volume expansion factor In this work, we focused on a volume expansion factor (VEF) – dimensionless – deﬁned as the total woody volume of a tree (Vtot), i.e., the volume of the stem and of all branches whichever their diameter, divided by the volume of the part of the stem having a diameter higher than 7 cm (Vstem7) (Eq. (1)). Volumes were measured over-bark not including the stump. Hence, with this deﬁnition VEF is >1. The stem top diameter of 7 cm corresponds to the limit used by the French National Forest Inventory and by ﬁve other European countries (Gschwantner et al., 2009).
Vtot ¼ VEF Vstem7
2.2. Biological material Two distinct datasets were used in this study. Dataset #1 was issued from manuscript sheets of destructive volume measurements collected between 1920 and 1985 in the network of permanent plots maintained by the French forest administration. From this material, a ﬁrst set of 4619 tree sheets was encoded and used by Vallet et al. (2006) for delivering models of total aboveground volumes for seven species. A gigantic task of data encoding and veriﬁcation was performed by the Research center on sustainability and productivity of industrial plantations (CRDPI) of the Republic of Congo within the frame of the French ANR EMERGE project which aims to evaluate the available forest biomass in France (Deleuze et al., 2010; Rivoire et al., 2010). The current database includes 44,668 trees on which tree circumference, height and Vstem7 were measured. The total wood volume Vtot was measured on about one fourth of these trees. These measurements were performed according to the protocol of Oudin (1930) (summarized in Vallet et al. (2006)). The volumes of stem and branches larger than 7 cm in diameter were calculated from circumference measurements taken every 1 m. The volume of small branches (below 7 cm diameter) was computed from weight measurements, assuming a green density of 1000 kg m3 (according to results obtained from an unpublished Master thesis performed in our lab in 2011). After removing trees with uncertain species identiﬁcation, incomplete or obviously incorrect data, and trees with diameter at breast height (DBH) below 7 cm for which Vstem7, when it exists, does not make much sense, the ﬁnal dataset #1 included 8192 trees from 10 genera and 19 species (Table 1). Dataset #2 was used for model validation. It was constituted during the above mentioned EMERGE project with the main objective to complement the available data with under-represented species, regions or diameter classes. A total of 209 trees1 was measured in 2009 and 2010. After removing missing data and trees with DBH below 7 cm, the ﬁnal dataset #2 included destructive volume measurements for 176 trees from 11 genera and 13 species (Table 2). Thirty-one Fagus sylvatica trees and 23 Acer pseudoplatanus trees were sampled within an even-aged high forest located along a soil gradient: soil #1 was alocrisol, soil #2 was oligo-satured brunisol, soil #3 was rendisol–calcisol and soil #4 was rendosol. Among 12 Quercus petraea/robur trees coming from an even-aged high forest, six were chosen for having a big fork. 1 45 stems of Eucalyptus trees were not considered here since the total volume of each coppice clump was not measured.
Only ﬁve species are common to both datasets. Compared with dataset #1, dataset #2 included less trees but more detailed measurements on each tree (such as wood density, biomass and nutrient content by tree compartments, T-LiDAR measurements, maturation strains, etc.). Moreover, basic description of the silvicultural system (high-forest, coppice, coppice-with-standards. . .) was recorded in dataset #2 whereas no stand information was available in dataset #1. Since the heights were not measured on standing trees in dataset #1, the length above stump measured on the felled tree was used as surrogate for the total standing height H for both datasets. In this study, dataset #1 was used for calibration and crossvalidation of the models. Dataset #2 was used for independent validation of the models including eight species not used for calibration.
2.3. Modeling the volume expansion factor In this section we describe step by step the modeling approach that we developed on the basis of recommendations made by Zuur et al. (2009) and Pinheiro and Bates (2000). The goodness of ﬁt of the different models was measured by the Akaike information criterion (AIC) and by computing RMSE and relative RMSE (Mayer and Butler, 1993) in terms of VEF and Vtot predictions. A 10-fold crossvalidation approach based on dataset #1 was tested as well as the use of the model to predict VEF on a completely independent dataset (dataset #2). The same methodology can be easily used for other deﬁnitions of expansion factors. On the basis of the data visualization (Fig. 2), we chose Eq. (2) to model the VEF as a function of the DBH (in cm).
VEF ¼ expðb1 DBHÞb2 þ expðb3 Þ DBH þ 1
With this equation, the estimated VEF was forced to be >1 to ensure that Vtot > Vstem7. The ﬁrst part of the equation, expðb1 DBHÞb2 , was used to model the strong decrease of VEF that was observed for small diameter trees and the second part, exp(b3) DBH + 1, was used to model the slight linear increase of VEF that was observed for bigger trees (the slope exp(b3) was >0). Looking at the model residuals (not shown) and conﬁrmed by the AIC, we observed that a reﬁnement could be done by including the tree height H (in m) in Eq. (2) and then Eq. (3) was obtained. Hence, the model was better ﬁtted to the data but one drawback was that the meaning of the slope parameter changed.
VEF ¼ expðb1 DBHÞb2 þ expðb3 Þ
The nlsList function from the nlme package of the R statistical software (R Development Core Team, 2011) was used to ﬁt separate models for each genus. The results help to decide for which parameters random effects are needed since ﬁtting separate models requires too many parameters. We assumed that the true intergenus variability existing in the French temperate forest resource was well represented in our calibration dataset. Modeling the inter-genus variability as a random effect was useful for reducing the number of parameters to estimate and for providing reliable estimates of species-speciﬁc parameters through best linear unbiased predictions (BLUP; Robinson (1991); Pinheiro and Bates (2000)), especially when the number of trees was too low for a given genus or the complete range of DBH not represented, for instance. The mixed model was ﬁtted by using the nlme function of the R software. The model parameters were estimated by the maximum likelihood method. From the results of the nlsList ﬁtting, it appeared that parameter b3 was highly variable depending on the genus and more precisely depending on the belonging to the angiosperm or gymnosperm groups. Parameters b1 and b2 were
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Table 1 Number of trees per species in dataset #1; measured ranges of tree diameter (DBH), height and total volume (Vtot). Species common to dataset #1 and #2 are highlighted in gray.
Table 2 Number of trees per species in dataset #2; measured ranges of tree diameter (DBH), height and total volume (Vtot). Species common to dataset #1 and #2 are highlighted in gray.
much less variable. Moreover, in the case of parameter b1 the variability was probably due to a lack of small diameter trees for some species. Finally, we chose to model random effects for parameters b2 and b3 and to keep only a ﬁxed effect for parameter b1. More precisely, the parameter b3 was modeled as b03 G þ b4 þ b4i , where G is a binary variable coding for the belonging to the angiosperm (G = 1) or gymnosperm (G = 0) groups, b03 and b4 are ﬁxed parameters and b4i is used for modeling a random effect on the intercept. The correlation between random effects b2i and b4i was not statistically signiﬁcant (correlation of 0.29 with a 95% conﬁdence interval of [0.52; 0.83]) and then we deﬁned the corresponding variance-covariance matrix w as being diagonal. Last, looking at the model residuals as a function of ﬁtted values we observed that the within-group variance was increasing. High VEF ﬁtted values corresponded to small DBH values. Therefore, in order to take into account this heterogeneity we chose to model the within-group variance as a power function of the DBH, with different values of the power for angiosperms and gymnosperms. The ﬁnal model is given by Eq. (4). The AIC of the model was 11,902.
DBHij VEF ij ¼ expðb1 DBHij Þb2 þb2i þ exp b03 Gi þ b4 þ b4i H2ij þ 1 þ ij
where i represents the genus and j represents the tree within the genus. G = 1 for angiosperm, 0 for gymnosperms. b1 ; b2 , b03 and b4 are the ﬁxed parameters of the model, and b2i and b4i are parameters for the random effects. b2i and b4i are assumed to be normally distributed with mean 0 and a diagonal variance–covariance matrix w. The within-group errors ij are assumed to be normally distributed with mean 0 and variance equal to r2 jDBHij j2dGi . For practical reasons, since DBH is a variable easier to measure in the ﬁeld than tree height H, we also proposed a model based on DBH only, which had exactly the same characteristics as the model presented in Eq. (4), except that the variable H was removed from the equation. The AIC of the model was 9667. We made the choice to calibrate our models per genus rather than per species. This enabled to calibrate the models based on more observations, leading potentially to less accurate models at the species level but probably more robust. We made the
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Fig. 2. Volume expansion factor (VEF) as a function of diameter at breast height (DBH) by genus for trees of dataset #1 having a DBH P 7 cm.
assumption of certain homogeneity of the species within a given genus, even if it was not always veriﬁed in our dataset. Based on datasets #1 and #2, seven species of Pinus and six species of Quercus were represented for their relationship between VEF and DBH (Fig. 3). Pinus halepensis coming from dataset #2 showed particularly high VEF. VEF of Quercus ilex were higher than VEF of other Quercus species for comparable diameter classes. For Quercus, the biggest diameters were essentially represented by only two species Quercus robur and petraea, making comparisons difﬁcult. 3. Results 3.1. Calibration on dataset #1 3.1.1. Model based on DBH and H By ﬁtting the model corresponding to Eq. (4) on the whole dataset #1 (n = 8192), we obtained the following parameter estimates:
b1 ¼ 6:830 ð0:024Þ, b2 ¼ 1:015 ð0:121Þ, b03 ¼ 1:009 ð0:123Þ and b4 = 0.522 (0.084) for the ﬁxed effects; standard deviations of 0.310 and 0.183 for the random effect parameters b2i and b4i, respectively; r = 0.275 and dGi ¼ 0:420 for gymnosperms and 0.154 for angiosperms. The RMSE was 0.136 for the prediction of VEF and 0.151 m3 for the prediction of Vtot. However, due to the high heteroscedasticity of Vtot errors as a function of DBH (the errors in term of Vtot are logically higher for big trees than for small ones; see Fig. 4) it was meaningful to compute the RMSE and relative RMSE by diameter classes. Depending on the DBH class, the RMSE for the prediction of VEF varied from 0.095 to 0.251 and the corresponding relative RMSE varied from 8.0% to 16.2% (Table 3) to be compared with 11.1% when computed directly on the whole dataset. The error in term of VEF prediction was higher for very small trees and for big trees. Depending on the DBH class, the RMSE for the prediction of Vtot varied from 0.005 to 0.476 m3 and the corresponding relative RMSE varied
Fig. 3. Volume expansion factor (VEF) as a function of diameter at breast height (DBH) for Pinus species (on the left) and for Quercus species (on the right); datasets #1 and #2 were used.
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Fig. 4. Observed versus predicted values of Vtot for dataset #1 (on the left) and for dataset #2 (on the right).
from 8.0% to 13.7% (Table 3) to be compared with 21.7% when computed directly on the whole dataset. Considering only the ﬁxed part of the model, the RMSE was 0.148 for the prediction of VEF and 0.150 m3 for the prediction of Vtot. Fig. 4 shows observed versus predicted values of Vtot. For comparison purposes, we computed the slope of the linearly increasing part of the model (i.e., the ﬁxed quantity expðb03 Gi þ b4 Þ in Eq. (4)) for both gymnosperms and angiosperms. The slopes were 1.686 and 4.624, respectively, indicating that the increase with DBH was much stronger (more than 2.5 times H2 higher) for angiosperms than for gymnosperms. Table 4 gives the parameters to use for predicting VEF for each genus individually, based on BLUP computation. RMSE of VEF and Vtot are given. In overall, the errors in term of Vtot were higher for angiosperms than for gymnosperms. For predicting VEF for species belonging to other genera, we suggest to use the ﬁxed part of the model which constituted the best mean model it was possible to obtain from dataset #1.
Among gymnosperms, Pseudotsuga species showed a relatively high increase whereas Abies and Picea species showed low increase. 3.2. Cross-validation on dataset #1 In order to have an idea of the level of ﬁt of the model when applied to an independent dataset, we performed a 10-fold cross-validation of the model based on DBH and H. By using cross-validation the RMSE of the model was 0.136 for the prediction of VEF and 0.150 m3 for the prediction of Vtot. Considering only the ﬁxed part of the model, the RMSE was 0.148 for the prediction of VEF and 0.148 m3 for the prediction of Vtot. Thus, the RMSE were in the same order as without cross-validation. From the ten model ﬁttings, the ranges obtained for parameters b1 , b2 , b03 and b4 were [6.789; 6.864], [0.983; 1.067], [0.997; 1.023] and [0.519; 0.530], respectively. The stability of the parameter estimates is an indicator of the model robustness. 3.3. Validation on the independent dataset #2
3.1.2. Model based on DBH only As mentioned in Section 2, a model corresponding to Eq. (4) was ﬁtted on the whole dataset #1 (n = 8192) with keeping only variable DBH in the equation. The following parameter estimates were obtained: b1 = 6.438 (0.051), b2 = 0.368 (0.023), b3 = 0.998 (0.249) and b4 = 5.691 (0.174) for the ﬁxed effects; standard deviations of 0.061 and 0.382 for the random effects b2i and b4i, respectively; r = 0.320 and d = 0.385 and 0.201 for gymnosperms and angiosperms, respectively. The RMSE was 0.146 for the prediction of VEF and of 0.167 m3 for the prediction of the total volume. Considering only the ﬁxed part of the model, the RMSE was 0.158 for the prediction of VEF and 0.256 m3 for the prediction of the total volume. The increase of VEF with DBH was much stronger (three times higher) for angiosperms (=0.009) than for gymnosperms (=0.003). Among angiosperms, Carpinus species showed a particularly strong increase whereas Quercus species showed the lowest increase.
Table 3 RMSE and relative RMSE (in brackets) for VEF and Vtot computed by diameter classes (model corresponding to Eq. (4) ﬁtted on the whole calibration dataset #1). DBH (cm)
n RMSE of VEF
893 0.251 (16.2%) 0.005 (12.9%)
3308 0.095 (8.0%) 0.014 (8.0%)
2336 0.102 (8.8%) 0.050 (8.7%)
939 0.136 (11.5%) 0.142 (10.4%)
716 0.180 (14.3%) 0.476 (13.7%)
RMSE of Vtot (m3)
In the following Sections 3.3.1 and 3.3.2, the model based on DBH and H was used. Similar results were obtained for the model based on DBH only. 3.3.1. Validation on tree species already used for the model calibration The results obtained from the application of the model to other trees from the same species as for calibration are given in Fig. 5 and Table 5. For Quercus petraea and robur and for Fagus sylvatica the model tended to underestimate the VEF of big trees coming from coppice-with-standards stands. The only big tree coming from coppice-with-standards that was below the predictions was declining. Hence, the silvicultural effect (i.e., even-aged high forest vs. coppice-with-standards) seemed not perfectly taken into account by the model though DBH and tree height. For Fagus sylvatica trees, it was not possible to observe any signiﬁcant difference between the four types of soils. For Quercus petraea/robur, the six forked trees showed higher VEF than not forked trees coming from the same stand and the model was not able to predict accurately VEF for these trees. These forked trees had a higher Vtot than not forked trees (3.362 m3 in average vs. 2.961 m3). For Fraxinus excelsior some observations were strongly above the predictions for DBH between about 45 and 65 cm. In overall, on this independent dataset, the results obtained from using only the ﬁxed part of the model were as good as those obtained from the species-speciﬁc coefﬁcients (Table 5). In terms of volume, the RMSE were quite high for Fraxinus and Quercus with values around 1 m3 due to some big trees.
F. Longuetaud et al. / Forest Ecology and Management 292 (2013) 111–121 Table 4 Parameter estimates by genus based on BLUP computation in the model corresponding to Eq. (4) ﬁtted on the whole dataset #1. Parameter b03 for gymnosperms is grayed since it does not play any role in the model due to the multiplication by G = 0.
were quite similar for the species pubescens whatever the model parameters, ﬁxed part only or species-speciﬁc coefﬁcients. After veriﬁcation, the outlier Quercus pubescens tree visible in Fig. 6 was a very particular tree with a big fork at about 4 m height. It was not possible to observe an effect of the type of soil on the relationship between VEF and DBH for Acer pseudoplatanus trees (visually and statistically assessed). In terms of volume, the RMSE were satisfactory (Table 5). The results for the prediction of Vtot were comparable for datasets #1 and #2 (Fig. 4). 3.4. Extended model ﬁtted on datasets #1 and #2
3.3.2. Validation on other tree species The results obtained from the application of the model to trees of species not used for calibration are given in Fig. 6 and Table 5. From the model calibration on dataset #1, two generic models were obtained corresponding to the ﬁxed parts of the mixed-effect models, one for gymnosperms and one for angiosperms. Applying these generic models to other species is a way to assess their validity and their genericity. The genera that were not present in the calibration dataset #1 are Acer, Alnus, Populus, Robinia and Tilia. In overall, the results were satisfactory with sometimes slight underestimations or overestimations. In particular, for the genus Quercus it was interesting to apply our model with species-speciﬁc coefﬁcients to other Quercus species as those used for the calibration. Even if the Quercus species were not the same as those used for the calibration (ilex and pubescens here vs. robur and petraea for the calibration), the results were much better for the species ilex than by using only the ﬁxed part of the model and the results
In order to summarize the behavior of the different tree genera, and in particular the trend of VEF with DBH, we come back in this Section to the model based on DBH only, easier to represent graphically. Fig. 7 shows the predictions obtained for the 10 genera of dataset #1, completed with trees from dataset #2 to get a better representation of the reality by extending the range of DBH when it was possible. Differences of VEF between species increased with tree size. The slope of the relationship between VEF and DBH was almost null for Picea and Abies species, and was in overall relatively low for all gymnosperms, except Pseudotsuga. The slopes were stronger for angiosperms with the lowest slope for Quercus and the highest slope for Carpinus. For a medium tree of 30 cm in diameter, the VEF computed with the model represented in Fig. 7 were 1.062 for Picea, 1.072 for Abies, 1.100 for Larix, 1.129 for Pinus, 1.178 for Quercus, 1.196 for Pseudotsuga, 1.238 for Betula, 1.292 for Fagus, 1.322 for Fraxinus and 1.463 for Carpinus. 4. Discussion Our calibration dataset (dataset #1) contained detailed volume measurements for 8192 trees from 19 species. Many references about expansion factors found in the literature were based on
Fig. 5. Validation of the model to tree species used for the calibration. Field measurements are represented in black, green, cyan and magenta ﬁlled circles or triangles. Predictions of VEF on the basis of the ﬁxed part of the model are represented in blue circles or triangles. Predictions of VEF on the basis of the species-speciﬁc coefﬁcients estimated from the mixed effects model are represented in red circles or triangles. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
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Table 5 Validation of the model on the independent dataset #2. Genus
Number of trees
Fixed part of the model
With species speciﬁc coefﬁcients
RMSE of the VEF
RMSE of Vtot (m )
RMSE of VEF
RMSE of Vtot (m3)
13 2 34 8 24
0.137 0.162 0.168 0.367 0.214
0.267 0.107 0.554 1.091 0.749
0.143 0.152 0.150 0.379 0.249
0.282 0.068 0.486 1.127 0.950
Tree species not used for the model calibration Acer pseudoplatanus 23 Alnus sp. 16 Pinus halepensis 8 Populus tremula 2 Quercus ilex 23 Quercus pubescens 8 Robinia pseudoacacia 8 Tilia cordata 7
0.130 0.159 0.157 0.103 1.129 0.243 0.083 0.200
0.099 0.215 0.178 0.062 0.047 0.193 0.030 0.278
– – 0.203 – 0.732 0.283 – –
– – 0.235 – 0.031 0.227 – –
Tree species used for the model calibration Betula pendula Carpinus betulus Fagus sylvatica Fraxinus excelsior Quercus petraea/robur
Fig. 6. Validation of the model to tree species not used for the calibration. Field measurements are represented in black, green and cyan ﬁlled circles or triangles. Predictions of VEF on the basis of the ﬁxed part of the model are represented in blue circles or triangles. For the genera Quercus and Pinus it was possible to apply to species-speciﬁc coefﬁcients estimated on the basis of both ﬁxed and random effects and the corresponding predictions are represented in red circles or triangles. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)
volume and/or biomass estimates obtained through allometric equations and thus were not directly based on ﬁeld measurements (e.g., Schroeder et al., 1997; Lehtonen et al., 2004; Jalkanen et al., 2005), especially when the studies were oriented toward large scale estimates. When BEFs were estimated directly from ﬁeld measurements, relatively few species were generally considered (e.g., Pajtik et al., 2008, 2011; Skovsgaard and Nord-Larsen, 2012) or the range of tree size was limited (see some examples of studies below) and consequently the equations were probably not suitable for other species and for bigger trees. Modeling VEF directly from ﬁeld measurements enabled to avoid propagation of modeling errors. Working with a large range of tree species was important for the generic aspect of our model.
The data used for calibrating the models were collected between 1920 and 1985. It is possible that the tree shape has changed since these measurements, mainly due to changes in silvicultural practices and less likely due to the climate change effect. It was not possible to investigate that point with our dataset. The major weakness of the calibration dataset is that it represents at 90% pure, even-aged high forests (Vallet et al., 2006) and rather small and medium trees in diameter compared with the French National Forest Inventory data (not shown). It is important to remind that the dataset #2 used for validation was specially chosen (see Section 2) to complement the available data with under-represented species, regions or DBH values, which often led to use the model in extrapolation:
F. Longuetaud et al. / Forest Ecology and Management 292 (2013) 111–121
Fig. 7. Predicted volume expansion factor (VEF) as a function of diameter at breast height (DBH) by genus of dataset #1 (angiosperms on the left, gymnosperms on the right). The predictions were obtained by using the model based on DBH only ﬁtted on dataset #1 with additional trees from dataset #2 for the common genera. Dashed lines show the DBH values for which the model is used in extrapolation.
For Fraxinus species, the model was calibrated with trees ranging between 8.3 and 43.4 cm in diameter (dataset #1) whereas ﬁve trees from dataset #2 had DBH ranging between 44.7 and 64.9 cm. The model was therefore used in extrapolation for these trees leading to big errors both in terms on VEF and Vtot. For Pseudotsuga, the trees from the calibration dataset were young with a maximum DBH of 34.1 cm since the species was recently introduced in France at the date of measurements. It would therefore be preferable to use the generic gymnosperm model (i.e., ﬁxed-effects model and G = 0) for this species. For Fagus and Quercus species, the model was calibrated mostly on high forest trees. The big trees of dataset #2 coming from coppice-with-standards stands tended to have underestimated values of VEF, except for one declining tree. The number of trees was particularly low for some genera from dataset #2 (e.g., only two trees for Carpinus and Populus species). However, it was interesting to visualize the behavior of the model for these few trees, even if it did not enable to validate the model for an application on these genera. In addition, it is important to remind that one advantage of using mixed-effects models and BLUP estimations was the possibility to estimate parameters even for genera with small sample size by using the information available from the general population. The predictions obtained for trees consistent with the calibration dataset #1 (i.e., issued from even-aged high forests, with a comparable range of diameters, not forked trees) were satisfactory in term of RMSE (values of the same order). To go further, we have tested a calibration of the model based on dataset #1 and #2 pooled together since the two datasets were complementary. However, dataset #2 was too small in comparison with dataset #1 to impact signiﬁcantly the parameter estimates. In the model with DBH only it was observed that the slopes were slightly increasing for Fraxinus and for Quercus, which tended to improve the model for big trees. For genera represented by a large number of trees like Pinus, Fagus or Quercus, a quite high variability of VEF was observed. For Pinus genus, this variability could result mainly from differences in origin of the trees (e.g., region, site) and differences between species. For angiosperm genera, it could be explained in addition by a larger range of silviculture practices (e.g., coppicewith-standards, pure even-aged high forests, mixed-species forests). Van Camp et al. (2004) who have compared several BEFs published in Europe (Löwe et al., 2000) and their accuracy for predicting total above-ground biomass for oak, beech and ash
concluded that there was a need to reﬁne BEF for oak due to a high variability observed in the oak population. It was not possible to get easily a reliable information about the silviculture for each tree of the dataset #1 and therefore to take this aspect into account in our models. The silvicultural effect was at least partially, but not perfectly, taken into account in our model through the variable DBH . This variable is also related to H2 the hardiness used by Vallet et al. (2006) to model Vtot directly from pcircumference at breast height (C130) and height. The variﬃﬃﬃﬃﬃﬃﬃﬃ able C130 was called hardiness by Vallet et al. (2006) because the H higher it is, the more cone-shaped is the tree, and the smaller it is, the slenderer is the tree. And it is known that the slenderness H of a tree, deﬁned as DBH , is related to the silviculture and in particular to stand density (Wang et al., 1998; Jagodzinski and Oleksyn, 2009). In addition, the hardiness was relatively independent of the DBH and therefore added new information in comparison with DBH (Vallet et al., 2006). The presence of a fork also had a big impact on VEF. For a given DBH, a forked tree had a greater VEF and a greater total woody volume than a not forked tree. This was observed in our study for Quercus robur/petraea and Quercus pubescens. Adu-Bredu et al. (2008) have shown for teak trees that the stem volume was decreasing with the number of forks and this suggested that forked trees, if they have in addition a greater total volume, have consequently much more volume in their branches and greater VEFs. Regarding the parameter estimates, the model seemed to be consistent at several levels. The value of 6.830 for b1 (6.438 in the model with DBH only), which controlled the curve location along the X-axis, was in the order of the lower diameter limit of 7 cm that was ﬁxed in the study. The error variance power values dGi obtained for angiosperms and gymnosperms were negative, which was consistent with an increasing of the VEF variability for small diameter trees. It was shown that the variability was higher for angiosperms than for gymnosperms, which reﬂected a greater variability in tree architecture and volume distribution for angiosperms. This could be related to a greater control of the volume allocation in gymnosperms, for which the growth is largely dominated by the stem (Barthélémy and Caraglio, 2007), and maybe also to more complex silvicultural practices for angiosperms than for gymnosperms (e.g., coppice-with-standards vs. high forests, mixed-species stands vs. pure stands). The slopes of the relationship between VEF and DBH were higher for angiosperms than for gymnosperms indicating that, for a given diameter at breast height, angiosperms had much more volume in their branches in comparison with the corresponding stem volume.
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The differences observed between species for the relationship between VEF and DBH probably reﬂected differences in tree architecture and possibly in the silviculture traditionally associated with the considered species. Regarding our results and in particular Fig. 7, the approximate value of 1.5 given by Pretzsch (2009) seems appropriate for angiosperms but not for gymnosperms for which the VEF and Vtot would be highly overestimated. IPCC (2003) gives an average BEF of approximately 2 and FAO (2005) estimates an average BEF of 2.2. However, these BEFs generally developed at the stand scale integrate data from numerous species of different countries, and therefore are not directly comparable with our results. Some groups of species appeared like Picea and Abies or like Fraxinus and Fagus and in a further study it would be interesting to link these groups to architectural and ecological traits known for these species. The variability that was observed within the genera Pinus and Quercus raises the question of developing models at the species level rather than at the genus level, at least for atypical species like Quercus ilex or for species sufﬁciently well represented in our dataset. Direct comparison of our results to the literature is not easy since various deﬁnitions of VEF, BEF or BCEF were used. However, VEF and BEF are quite equal and databases and publications exist that provide wood basic density for numerous temperate tree species (e.g., Chave et al., 2009; Zanne et al., 2009), which could be used to convert easily VEF to BCEF. Regarding the relationship with DBH, the trends should be comparable for VEF and BCEF since multiplying by a mean wood density would not change the sign of the slopes. In our model the slope of the relationship was constrained to be positive for trees above a given diameter depending on the model parameters. In the literature, the slopes obtained for BEF were often negative (Somogyi et al., 2007; Pajtik et al., 2008, 2011; Sanquetta et al., 2011) because the deﬁnitions used for expansion factors were different or because the trees were still young or small and only the ﬁrst part of the relationship (i.e., the decreasing part) was obtained. Sanquetta et al. (2011) showed aboveground biomass over stem biomass ratio decreasing with DBH up to 20 cm in diameter and then staying approximately constant between 1.0 and 1.5 up to 40 cm in diameter for Pinus in Brazil. Due to the variability that we observed for the genus Pinus, our results could be considered as rather consistent with this study. Somogyi et al. (2007) showed in their Fig. 2 a stem volume over merchantable volume (stem and branches greater than 7 cm in diameter) ratio as a function of DBH for beech, oak, pine and spruce. This volume ratio corresponded approximately to the inverse of our VEF. Considering this, our results were consistent with the plots of Somogyi et al. (2007) with a VEF increasing with DBH and with lower slopes for spruce and pine than for oak and beech. In their Fig. 3, Somogyi et al. (2007) showed total volume over stem volume (stem part greater than 5 cm in diameter) ratio decreasing with DBH for Robinia pseudoacacia, with DBH ranging from 5 to 15 cm, whereas in our study and for the same species DBH varied between 15 and 24 cm. Pajtik et al. (2008) showed a total aboveground biomass over stem biomass ratio decreasing with stem base diameter for Picea abies and diameters up to approximately 7 cm, whereas in our study DBH for the same species ranged between 8 and 67 cm. Pajtik et al. (2011) showed total aboveground biomass over stem volume ratio decreasing with stem base diameter for beech, pine, oak and spruce and diameters up to approximately 8 cm, whereas in our study we considered bigger diameters for all these species. Previous cited studies emphasize the lack of studies based on medium and big trees and the fact that our dataset including bigger trees was particularly valuable. Plots of BEF as a function of growing stock at the stand level (in m3/ha) show negative or almost null slopes as well (Brown, 2002; Lehtonen et al., 2004; FAO, 2005; Teobaldelli et al., 2009; Guo et al., 2010). However, these studies are difﬁcult to compare directly
with others because relationships are studied at the stand level and not at the individual tree level and the deﬁnitions of compartments ‘‘from what’’ and ‘‘to what’’ conversions are done may differ between studies (Teobaldelli et al., 2009). Acknowledgements We would like to thank the French National Research Agency which founded the research project EMERGE through its bioenergy program. Thanks also to all the people who were involved in this project, especially all the people who contributed to ﬁeld measurements (teams from LERFoB, BEF, UEFL at INRA Nancy and teams from ONF) and all the people at CRDPI (Republic of Congo), managed by Nina Ognouabi, who created the electronic ﬁles from handwritten data. References Adu-Bredu, S., Foua Tape Bi, A., Bouillet, J.-P., Kouame Me, M., Yamoah Kyei, S., Saint-André, L., 2008. An explicit stem proﬁle model for forked and un-forked teak (Tectona grandis) trees in West Africa. Forest Ecology and Management 255, 2189–2203. 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