Models of internal knot properties for Picea abies

Forest Ecology and Management 147 (2001) 123±138

Models of internal knot properties for Picea abies Lennart Moberg* Department of Forest Management and Products, Swedish University of Agricultural Sciences, PO Box 7060, SE-750 07 Uppsala, Sweden Received 18 November 1999; received in revised form 27 March 2000; accepted 25 April 2000

Abstract Models describing the variation in internal knot properties of Norway spruce (Picea abies (L.) Karst.) in Sweden were developed. The sawlog portion of 114 mature trees was scanned using computer tomography in order to identify internal knots and measure knot dimensions (knot size, sound-knot length, loose-knot length). The variation of both maximum and mean knot diameter per whorl was described in terms of a non-linear segmented (hyperbolic±quadratic±quadratic) model, and as a function of several stand variables, tree variables, and height above ground. The number of knots per whorl was described in a linear model as a function of the distance between whorls and site index. Sound-knot length was modelled in a linear mixed model with knot diameter, height above ground, and diameter at breast height (DBH) as predictors. Loose-knot length was described as a function of relative knot diameter and DBH in a mixed log-linear model. This family of models was intended to be integrated in a decision support system for sawmill conversion simulation studies applicable to merchantable trees in mature stands. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Individual tree models; Knot diameter; Knot frequency; Knot length; Wood quality

1. Introduction Within-tree models of knot or branch properties have gained much interest in recent years. The general principle involved for many of these efforts is that stand and tree variables, generated through application of growth models or obtained through ®eld inventories, can be used to predict quality attributes (KaÈrkkaÈinen, 1986; Mitchell, 1988; Oker-Blom et al., 1988; Maguire et al., 1991; Houllier et al., 1995; MaÈkelaÈ et al., 1997). These models are often used in conjunction with sawmill conversion simulation systems (VaÈisaÈnen et al., 1989; Barbour and Kellogg, 1990; Briggs, 1992; Leban et al., 1997) in order to provide a * Tel.: ‡46-18-67-24-78; fax: ‡46-18-67-35-22. E-mail address: [email protected] (L. Moberg).

link between product recovery and different silvicultural strategies, raw material sources, or bucking and sawing patterns. Applied in this way, it is possible to study the utilisation of timber in terms of alternative decisions in the whole forest-wood chain. Many modelling efforts focus on (external) branch properties. In order to obtain data from the internal quality attributes of whole stems (i.e. below as well as above the live crown), these models need to be applied iteratively together with growth models. In other words, the attempt is to mimic the dynamic process of tree growth, crown extension, and crown recession throughout the life of individual trees. Models directly based on (internal) knot properties provide a means of simultaneously studying the effects of past and present growth conditions. These have been based on either destructive (e.g. Hùibù et al., 1997; Vestùl and Colin,

0378-1127/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 1 1 2 7 ( 0 0 ) 0 0 4 7 1 - 0

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L. Moberg / Forest Ecology and Management 147 (2001) 123±138

1998; éyen and Hùibù, 1999a; Vestùl et al., 1999) or non-destructive knot measurements (e.g. Moberg, 2000). The modelling approach adopted by Moberg (2000) is similar to that applied in many models of branch size (e.g. Colin and Houllier, 1991; Maguire et al., 1994; Meredieu et al., 1998), and knot size below the crown (Vestùl et al., 1999), but it provides an extension of these efforts in order to include the whole stem. It is also an application of the biological framework for analysing the vertical knot size variation presented by BjoÈrklund (1997). Moberg (2000) illustrates, for Scots pine (Pinus sylvestris L.), that the effect of past and present growth conditions on knot size can be depicted in a single segmented model for the whole stem. Hùibù et al. (1997) and Vestùl et al. (1997) present similar, but simpler, models for Norway spruce (Picea abies (L.) Karst.) without the use of multiple segments. However, these two latter works are based on a limited sample (a total of 12 stems from two stands), and may not be representative of a larger forest resource. The objective of the present study was to develop models describing the variation in internal knot properties of Norway spruce stems re¯ecting conditions in Sweden. The following knot properties were studied: knot diameter, knot frequency, radial sound-knot (intergrown) length, and radial loose-knot (encased) length. The intent is for the resulting system of equations to be integrated in a decision support system (SOPT) for sawmill conversion simulation (LoÈnner, 1996; BjoÈrklund and Julin, 1998). SOPT could then be used to study the product recovery from stems with simulated knot properties on the basis of Swedish forest inventory data. 2. Materials and methods This study was based on 114 Norway spruce trees from 19 stands in Sweden (Fig. 1). The stands were chosen in order to provide a variation in terms of site (site index and location) and silviculture (stand establishment and thinning). Variables (de®ned in Table 1) describing the stands and sampled trees have been summarised in Table 2. Nine stands originated from natural regeneration, and 10 stands were planted. Six trees representing

Fig. 1. Map of Sweden illustrating stand locations.

three different tree-size classes (two trees from each class) were sampled from each stand. The middle class represented the quadratic mean diameter at breast height (DBH) of the stand, and the other two classes were separated by one-half standard deviation above and below this mean, respectively. DBH was measured in two directions (nearest to 1 mm) before felling. Total height (Ht), height to the lowest live branch (Hllb), and height to the lowest dead branch (Hldb) were measured (nearest to 1 dm) after felling. Longitudinal branch diameters were measured (nearest to 1 mm) in one whorl per log. In order to avoid the branch swell near the stem, these measurements were carried out at a distance from the stem surface roughly corresponding to the diameter of the branch. The stems were delimbed and cross-cut into approximately 4.5 m long sections, and the logs were transported to a laboratory (Division of Wood Technology, LuleaÊ University of Technology Ð LTU) for computer tomography (CT-scanning using a Siemens SOMATOM AR.T.). Internal knot properties and log geometry were measured using digital image analysis techniques (Grundberg, 1994; Oja, 1999). Only the portion of the stem between the stump level (about 0.3 m above ground) and the level where stem diameter over bark was at least 12 cm was investigated. The knot

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Table 1 List of symbols and variable definitions Symbol

Description

Intra-tree variables KDmax KDmean KDrel KF KLloose KLsound Hk DHk

Knot diameter at the sound/loose knot border in the transverse plane (mm). Maximum knot size per whorl Mean knot diameter per whorl (mm) at the sound/loose knot border Relative knot diameter (KD/KDmean) Knot frequency (number per whorl) Radial loose-knot length of occluded knots. Difference between total knot length and Klsound (mm) Radial sound-knot length (mm) Height above the ground (m) for whorl k Distance between a whorl and its adjacent lower whorl (Hk Hk 1; m)

Tree-level variables CL DBH Hllb Ht a(g)ij

Crown length (Ht- Hllb; m) Diameter at breast height outside bark (cm) Height to the lowest live branch (m) Total tree height (m) Random between tree effect, nested within stands (iˆ1±19; jˆ1±114)

Stand-level variables AGE SI Tsum gi

Mean total stand age (years) Site index (m). Dominant height at 100 years Temperature sum (above a threshold of 58C), calculated as a function of altitude and latitude (MoreÂn and Perttu, 1994; 8C days) Random between stand effect (iˆ1±19)

Statistical terminology e R2 REML RMSE Scale

Residual error Coefficient of determination (1 SSresidual/SScorrected total; SSˆsum of squares) Restricted maximum likelihood (Searle et al., 1992; Anon., 1997) Root mean square error (mm) Extra-dispersion parameter (McCullagh and Nelder, 1989; Anon., 1997)

diameter values obtained from image analysis were adjusted (2 mm was added) for bias induced by the harsh threshold applied to isolate (i.e. segment) knots (Oja, 1999). Annual-ring-width data were obtained from discs (green condition), scanned with a CCD line camera (resolution 0.1 mm per pixel; performed at LTU), and measured with a semi-automatic image analysis algorithm (WinDENDROTM). Data were obtained from two opposing radial directions from pith to bark (radial directions with knots, butt ¯ares, and asymmetric radii were avoided) for each disc. Due to the poor quality image of some discs, and some missing discs, annual ring data were missing for six stems. The knots were classi®ed into whorls by subjectively identifying clusters of knots. The main criterion applied was the vertical distance between adjacent knots. The number of such identi®ed whorls in each log was checked against the difference in the total

number of annual rings between the top and bottom ends. Secondary criteria (when corrections were necessary) were knot size and even spacing between consecutive whorls. These corrections were primarily necessary for older trees from poor sites where the knots and the height increment were small. The knot dimensions studied are illustrated in the transverse plane in Fig. 2. This plane provides a much higher resolution of the image with the measurement technology used than the more conventional longitudinal-radial plane used in stem dissection techniques (e.g. Maguire and Hann, 1987). The following knot properties were studied: maximum and mean knot size of each whorl (KDmax and KDmean, respectively); the number of knots per whorl (knot frequency, KF); the distance between the pith and the sound/loose knot border (KLsound); and the distance between the sound/loose knot border and the occluded knot end (KLloose).

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Table 2 Description of the sampled trees and standsa Variable

Units

Intra-tree variables KDmax KDmean KF KLloose KLsound Hk DHk

mm mm No. per whorl mm mm m m

Tree-level variables CL DBH Hllb Ht

m cm m m

Stand-level variables AGEb DBHb Ht b SI (H100) Tsum

years cm m m 8C days

a b

Minimum

Mean

7 7 0 5 4 0.31 0.05

S.D.

Maximum

23 18 3.4 32 85 8.34 0.34

8.6 6.6 2.0 16 24 5.99 0.21

56 50 13 132 229 27.24 0.98

7.7 18.2 2.1 19.7

15.0 29.1 9.2 24.2

4.1 7.2 3.80 4.23

28.1 44.2 18.1 34.5

51 23.1 19.7 16 825

98 29.7 24.5 28 1130

37 4.6 3.7 7.1 229

152 36.7 32.3 36 1420

A total of 114 trees were sampled from 19 stands and 11 different sites. Statistics refer to the stand mean.

In an initial screening of the data, substantial dif®culties with potential outliers were encountered when internal knot size was checked against external branch measurements. In an effort to remedy these problems, the following ®ve criteria were used to ®lter out incorrect measurements on an individual-knot basis: (1) knot diameter greater than 120% of the largest branch

measurement in each stand; (2) longitudinal angle less than 308; (3) tangential angle greater than 608 at the sound/loose knot border; (4) knots deviating greatly in diameter from surrounding knots; and (5) knots convex in shape (i.e. continuously diverging in the transverse plane). A total of 1907 measurements were removed from the data set for all properties except knot frequency (only criteria (2) and (5) were applied for KF). The resolution obtained through CT-scanning makes it dif®cult to identify and measure small knots (GroÈnlund, 1995; Oja, 1999). Therefore, only knots larger than or equal to 7 mm were analysed. This limit coincides with the lower limit of knots recognised by the Nordic grading rules for sawnwood (Anon., 1994). Furthermore, inter-whorl knots were excluded from all models, knots above the lowest live branch were excluded from the KLsound model, and knots which were not occluded were excluded from the KLloose model (see Fig. 2). 2.1. Model development

Fig. 2. Illustration of knot diameter (KD), sound-knot length (KLsound), and loose-knot length (KLloose) in the transverse plane of a stem.

The model ®tting procedure used for knot diameter is described in detail in a previous paper based on the

L. Moberg / Forest Ecology and Management 147 (2001) 123±138

same measurement methods, but concerning Scots pine (Moberg, 2000). This procedure is also similar to that developed for crown branches by Colin and Houllier (1991). In summary, a model was sought that could mimic the expected biological behaviour concerning height growth and crown expansion/recession patterns over the lifespan of individual trees (e.g. Kramer, 1962; Merkel, 1967; BjoÈrklund, 1997). Since knots below the live crown are affected by past growth conditions, and knots within the live crown are more dependent on current conditions, different predictor variables were sought for different parts of the stem. Segmented models provide this ¯exibility (Colin and Houllier, 1991; Moberg, 2000). In an explorative analysis, different number of segments, different model forms for each segment, and different restrictions at segment ends were evaluated. Predictor variables were then sought and tested using linear models for each segment and joint point from the following set of stand and tree variables (see Table 1): site index, stand density, altitude, latitude, Tsum, AGE, DBH, ring width, Ht, Hllb and Hldb as well as different combinations and transformations of these variables. The ®nal global model for the whole data set was ®tted as a nonlinear model with the SAS procedure ``NLIN'' using the Gauss±Newton method (Anon., 1989). Root mean square error (RMSE) and coef®cient of determination (R2) were used to evaluate model ®t. Knot frequency was ®tted as a log-linear model similar to Colin and Houllier (1992) and Maguire et al. (1994). A generalised linear model was applied to account for a non-normal distribution due to the discrete nature of KF values. This involved a (natural) log link function and a Poisson distribution in accordance with distribution theory (Myers, 1990). It was ®tted with the restricted maximum likelihood estimation (REML) method (see Searle et al., 1992) using the SAS procedure ``GENMOD'' (Anon., 1997). Height increment was used as the within-tree predictor variable (cf. Colin and Houllier, 1992; Maguire et al., 1994), and the same set of tree- and stand-level predictor variables as described above for knot diameter were tested. Model ®t was evaluated in terms of deviance and extra-dispertion parameter (scale). Deviance provides a statistic for evaluating goodness-of-®t, whereas the extra-dispersion parameter indicates how well the error variance follows the assumed distribution (a value of one indicates perfect

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consistency; see McCullagh and Nelder, 1989; Anon., 1997). Knot lengths were ®rst ®tted on tree-level means to identify predictor variables from the set of stand- and tree-level variables described for KD above. Using a mixed-model approach, different within-tree variables (height above ground, distance between whorls and knot diameter) were then tested on the whole data set. In addition, KLsound was tested as a predictor of KLloose. The ®nal model for the whole data set was obtained using a mixed model with the REML method through the SAS procedure ``MIXED'' (Littell et al., 1996; Anon., 1997). Model ®t was evaluated in terms of Akaike's information criterion (AIC) and residual variance (Littell et al., 1996; Anon., 1997). Ring-width variables have desirable properties for some models such as correlation with past silviculture and consistency over time (Nylinder, 1959; Heiskanen, 1965; Persson, 1977; Vestùl and Hùibù, 1998a; Moberg, 2000). A large number of ring-width variables from stump height (RWx y, where x and y are ring number from the pith, and xˆ1, 6, 11, 16, 21, 31, 41; yˆ5, 10, 15, 20, 30, 40, 50; x
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L. Moberg / Forest Ecology and Management 147 (2001) 123±138

®nalmodels. Variance homogeneity and normality of errors were evaluated through residual plots. 3. Results 3.1. Knot diameter The models for KDmean and KDmax were non-linear, segmented (hyperbolic±quadratic±quadratic) equations. In developing the models, an equation (Eqs. (3.1)±(3.3)) was ®rst ®tted for each stem separately using height above ground as the only independent variable. Stand and tree-level variables (as described in Section 2.1) were then sought in order to predict the parameters of each segment and joint point (Eqs. (3.4)±(3.8)). Finally, a global model (Eqs. (3.1)±(3.8)) was applied to the whole data set in order to obtain the parameter estimates listed in Table 3. An example of observed and predicted values for both properties are provided in Fig. 3. Residual plots (Fig. 4 and others not shown) did not reveal any misspeci®cation or dif®culties with non-constant variance. The following model was obtained for both Table 3 Parameter estimates of Eqs. (3.1) and (3.2) Parameter

Estimate

S.E.

Eq. (3.1) Dependent variable: KDmean, No. of observations: 4813, statistics of fit: R2ˆ0.558; RMSEˆ4.4 mm 0.0161 0.000293 a1 a2 3.15 0.807 b0 0.334 0.0198 c1 0.206 0.0056 0.00469 0.00117 c2 c3 0.0838 0.0167 c4 0.00298 0.000361 h11 0.195 0.02 0.877 0.0173 h21 Eq. (3.2) Dependent variable: KDmax, No. of observations: 4813, statistics of fit: R2ˆ0.618; RMSEˆ5.5 mm 0.0196 0.000363 a1 a2 5.45 1.02 b0 0.349 0.0204 0.25 0.00598 c1 c2 0.00397 0.00131 c3 0.112 0.0186 c4 0.00282 0.000404 0.2 0.0167 h11 h21 0.952 0.0141

Fig. 3. Observed and predicted values of mean and maximum knot diameter (obtained with Eqs. (3.1) and (3.2)) plotted against height above ground for a sample stem.

maximum and mean knot diameter (variable de®nitions are provided in Table 1): if Hkh1, KD ˆ a

Hk ‡ eijk b0 ‡ Hk

if h1
Hk h2

h1 h1

2

(3.1)

Ht †

2

h1 a b0 ‡ h1

‡ eijk



(3.2)

if Hkh2, KD ˆ c…Hk

Ht † ‡ d…Hk

Ht †2 ‡ eijk

(3.3)

where a ˆ a1 Tsum ‡ a2

DBH AGE

(3.4)

c ˆ c1 DBH ‡ c2 AGE ‡ c3 CL ‡ c4 DBH CL (3.5) dˆ

 c…Ht 

…Ht

 h1 …h2 h1 † b ‡ h1 ! 1 …Ht h2 †2 h2 † ‡ h2 h1 h2 † ‡ a

1

‡

c 2



(3.6)

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Fig. 4. Residual values of mean and maximum knot diameter (observed Ð predicted) for Eq. (3.1) (a and c) and Eq. (3.2) (b and d).

h1 ˆ MINfh11 SI; h2 h2 ˆ Ht

h21 CL

1g

(3.7) (3.8)

In Fig. 3, knot diameter increased from the ground towards a limit de®ned by a (Eq. (3.4)). This ®rst segment (Eq. (3.1)) ended at h1 (Eq. (3.7)). A quadratic segment (Eq. (3.2)) was used to join the lower hyperbolic segment with an upper quadratic segment (Eqs. (3.3) and (3.5)). This uppermost segment was restricted to pass through the horizontal axis (i.e. give a knot diameter of zero) at the top of the stem, and to have the same slope as the middle segment (through Eq. (3.6)) at their joint point h2 (Eq. (3.8)). 3.2. Knot frequency Using a generalised linear modelling approach with a Poisson distribution, a (natural) log-linear model for knot frequency was obtained. It had the following form (parameter values are listed in Table 4; model behaviour is illustrated in Fig. 5; residuals are plotted in Fig. 6): KF ˆ exp…a0 † DHka1 SIa2

(3.9)

3.3. Sound-knot length Sound-knot length was described with a linear mixed model. The model obtained the following form (variable de®nitions are provided in Table 1; parameter values are listed in Table 4; model behaviour is illustrated in Fig. 7; residuals are plotted in Fig. 8) 1=2 ‡ a3 H k KLsound ˆ a1 KDrel ‡ a2 KDrel KDmean

‡ a4 Hk KD2rel ‡ a5 Hk KDrel KD1=2 mean ‡ a6 DBH ‡ gi ‡ a…g†ij ‡ eijk

(3.10)

Almost all of the random stand-level (g) and treelevel (a) variation, and a smaller part of the withintree variation (e), was accounted for by the model (Table 4). 3.4. Loose-knot length It was dif®cult to ®nd signi®cant explanatory variables for the within-tree variation of loose-knot length. However, the following log-linear mixed model was chosen (parameter values are listed in Table 4; model

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Table 4 Parameter estimates of Eqs. (3.9)±(3.11) Parameter

Estimate

S.E.

Eq. (3.9) Dependent variable: statistics of fit: devianceˆ4928, a0 0.693 a1 0.321 a2 0.707

Reduced (%)a

KF, No. of observations: 5434, scaleˆ0.95 0.112 0.0129 0.0312

Eq. (3.10) Dependent variable: KLsound, No. of observations: 11163 a1 25.9 1.28 15.1 0.307 a2 a3 2.73 0.103 a4 0.813 0.0892 a5 0.562 0.038 1.83 0.0239 a6 g 1.15 1.03 99 a(g) 9.64 1.61 95 e 102.9 57 Eq. (3.11) Dependent variable: KLloose, No. of observations: 5639 a1 0.222 0.0183 a2 0.986 0.00594 0.00454 0.00257 73 gb a(g)b 0.012 0.00255 79 eb 0.202 3 a b

Relative reduction in variance by the parameters as listed. Values are given in (natural) log scale.

behaviour is illustrated in Fig. 9; residuals are plotted in Fig. 10) KLloose ˆ

KDarel1

a2

DBH expfgi ‡ a…g†ij ‡ eijkl g (3.11)

Fig. 5. Simulated values of knot frequency (obtained with Eq. (3.9)) plotted against height increment and site index.

The (natural) log transformation was applied on KLloose when fitting the model in order to stabilise the variance (i.e. to alleviate difficulties with nonconstant variance) and to avoid the possibility of negative estimates of the dependent variable. The model accounted for much of the random stand-level (g) and tree-level (a) variation, but hardly explained any of the within-tree variation (e; Table 4). 4. Discussion The non-destructive measurements used to study internal knot properties (i.e. CT-scanning and digital

Fig. 6. Residual values of knot frequency (observed Ð predicted) for Eq. (3.9) plotted against predicted values (a) and distance between whorls (b).

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Fig. 7. Simulated values of sound-knot length (obtained with Eq. (3.10)) as a function of mean knot diameter, relative knot diameter, height above ground (a), and DBH (b).

Fig. 8. Residual values of sound-knot length (observed Ð predicted) for Eq. (3.10) plotted against predicted values (a) and height above ground (b).

Fig. 9. Simulated values of loose-knot length (obtained with Eq. (3.11)) plotted against DBH and relative knot diameter.

Fig. 10. Residual values of loose-knot length (observed Ð predicted) for Eq. (3.11).

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image analysis) have been compared with destructive measurements for Scots pine by GroÈnlund (1995) and for Norway spruce by Oja (1999). The non-destructive method is said be competitive in terms of time ef®ciency and accuracy of results. However, since the method relies on (green) density variations to identify different objects (such as knots), and involves a low longitudinal resolution (i.e. one scan per centimeter), the reliability of the method differs for different knot properties. For the measurement method used in the present study, Oja (1999) reports that 94% of knots larger than 7 mm can be detected, and very few false knots (i.e. artefacts) are identi®ed (nine knots in 12 logs). Knot diameter measurements are judged to be acceptable (Oja, 1999) with a mean error of 2 mm, and a standard deviation of 3 mm. KLsound is slightly overestimated, especially for small knots in the butt log, and the standard deviation is about 10 mm. Total knot length of occluded knots is dif®cult to measure: mean error is ‡6.6 mm and standard deviation is 15 mm. The parameter description of knots provides the horizontal diameter (Oja, 1999). Lehtonen (1978) found that the vertical diameter is 7.1% larger than its horizontal counterpart for Norway spruce. Although this could perhaps account for the negative difference reported above, Oja (1999) attributes the difference to the harsh threshold value applied through image analysis in order to segment the knots. At any rate, it is dif®cult to obtain the true value of knot properties, and thus to estimate the true bias due to measurement errors associated with each individual method. The models presented in this study were intended to be used together in order to estimate the knot structure within the framework of sawmill conversion simulation. They would then be linked together in a system of equations, and the error structure would be correlated across models. Borders (1989) suggests the application of three-stage least squares (3SLS) in order to obtain unbiased and consistent results. This is successfully applied for branch characteristics by Roeh and Maguire (1997) and MaÈkinen and Colin (1998). However, the complexity of the models involved precluded this. The data were sampled from stands representing a broad range of sites and silvicultural practice (Fig. 1 and Table 1). However, only 114 trees from 19 stands and 11 locations were included in the study; this

material cannot be said to be representative of the whole Swedish Norway spruce resource in any statistical sense. The study was only concerned with stands, trees, and logs which were of merchantable maturity for the sawmilling industry, but the CTscanning method could not be used for very large trees. Therefore, the upper live crown of individual trees, young trees (below 50 years), truly suppressed trees, trees with DBH greater than 45 cm, and stands from the high-north and northern-coastal areas (Fig. 1) were not represented. Although lacking extremes, perhaps the sample can be regarded as a cross-section of typical, merchantable stand and tree types re¯ecting various Swedish conditions. The results of this study has not been validated against an independent data set. Given the few number of trees and stands, caution should be exercised when applying these models. Validation of these results will be a priority in future work. 4.1. Knot diameter The model describing the variation in mean and maximum knot diameter obtained essentially the same form and explanatory variables as the model presented for Scots pine by Moberg (2000). This modelling approach is an extension of models of the vertical variation of branch size within the crown for different coniferous species (e.g. Colin and Houllier, 1991; Colin and Houllier, 1992; Maguire et al., 1994; Roeh and Maguire, 1997; Meredieu et al., 1998), and knot size below the crown (Vestùl et al., 1999). Within the live crown, variables were sought which re¯ect current growth conditions (such as DBH, AGE, Ht and CL). Below the live crown, the goal was to mimic the historical pattern of height growth and crown recession by using variables which are relatively stable over time (such as mean ring width and site factors). Above an initial establishment phase near the ground, the nearly constant knot size depicts a period of similar rate of height growth and crown recession (Kramer, 1962; BjoÈrklund, 1997). The upper limit of the lower segment was a function of site index, roughly re¯ecting a height of the crown base when the residual trees in a stand experience reduced inter-tree competition, and increased branch longevity, due to thinnings or competition-induced mortality (Kramer, 1962; Merkel, 1967; BjoÈrklund, 1997; Moberg, 2000). By

L. Moberg / Forest Ecology and Management 147 (2001) 123±138

connecting these segments, it was possible to describe the vertical variation of knot size within the whole stem in a single model for Norway spruce. Using a similar approach, but destructive methods, Hùibù et al. (1997) and Vestùl et al. (1997) present models describing the vertical knot-size pro®le in terms of single second-order polynomials for the whole stem. They report signi®cant effects on maximum knot diameter per whorl of mean ring width (®rst 40 years) and mean height increment (or site index), and of within-stand social status, respectively. Moberg (2000) also identi®es separate effects of within-stand differentiation, and ring-width near the pith (11±20 rings from the pith) for Scots pine. This ring-width variable is compared with DBH divided by AGE, and it is found that ring-width gave more accurate prediction of knot size below the live crown (Moberg, 2000). A positive correlation between branch size near the stem base and ring-width variables is also identi®ed by Nylinder (1959), Heiskanen (1965) and Persson (1977). However, in the present study it was felt that the very slight additional reduction of residual variation obtained through the inclusion of ring-width variables (not shown) did not warrant the inclusion of a variable which is dif®cult to measure in a practical application (i.e. in ®eld inventories). The dif®culty in using ring-width variables to predict knot size at different heights below the live crown for Norway spruce is illustrated by Vestùl and Hùibù (1998a). A signi®cant relationship is identi®ed at individual height levels, but it is dif®cult to identify a single, universal ring-width interval for all levels. DBH often performs as well as, or better than, a single ring-width variable. In an even-aged, combined spacing-progeny trial, Vestùl et al. (1999) ®nd that most of the variation in knot size below the live crown among treatments is accounted for by DBH. These results support the use of DBH below the live crown in Eq. (3.4). However, although the data in this study do not re¯ect a true time series, knot diameter at the sound/loose knot border would not be expected to change over time below Hllb, whereas DBH will continue to increase over the life of a tree. DBH is therefore not consistent over time when used to predict knot length below the live crown; it cannot provide a robust estimate of knot diameter in a dynamic simulation (i.e. recursive runs for the same tree over time).

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One reason why ring-width variables in the present study did not consistently outperform DBH divided by AGE could be due to the species. Norway spruce is very shade tolerant (Lewerenz and Hinckley, 1990), and small trees can survive for a substantial amount of time in the seedling/sapling stage within the understorey before release. The most relevant ring-width interval to study might then be after release, not necessarily any at universal interval near the pith. This would be especially true for uneven-aged, naturally regenerated stands. Perhaps the problem was exaggerated in this study since only ring-widths from the butt end of the stem were used. Much of the below crown variation among stands was accounted for by temperature sum, which can be described as a function of altitude and latitude in Sweden (MoreÂn and Perttu, 1994). A similar effect on knot size is also found by Moberg (2000), and on tree or branch growth by Linder and Flower-Ellis (1992) and Bergh et al. (1998). In contrast, within the live crown (Eq. (3.6)), there was no additional effect of stand-level variables given the tree-level variables DBH, AGE and CL. This is in agreement with the results of Colin and Houllier (1991), Colin and Houllier (1992) and Maguire et al. (1994) who also do not ®nd any signi®cant stand-level effects beyond tree-level variables. It suggests that tree-level variables re¯ect current growth conditions (i.e. within the live crown) with more precision than past conditions (i.e. below the live crown). This is perhaps not surprising when considering the age of some trees (>150 years), and the dif®culties in identifying a relevant ring-width interval near the pith as was discussed above. 4.2. Knot frequency Height increment was used as the within-tree, explanatory variable for predicting knot frequency (Eq. (3.9) and Fig. 5). The model form was very similar to those presented for Norway spruce (Colin and Houllier, 1992), Douglas-®r (Pseudotsuga menziesii (Mirb.) Franco; Maguire et al., 1994) and balsam ®r (Abies balsamea (L.) Miller; Gilmore and Seymour, 1997). In fact, the positive correlation between the number of knots (or branches) per whorl and height increment (or shoot length) is found for many other

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coniferous species as well (e.g. Cannell, 1974; Remphrey and Powell, 1984; PietilaÈ, 1989). In the present study, there was an additional standlevel effect of site quality. Maguire et al. (1994) suggest this possibility, but could not test it due to the few number of stands sampled. Madsen et al. (1978) and LaÈmsaÈ et al. (1990) also report a positive effect of site quality on branch frequency. But, since better sites are associated with larger knots (PietilaÈ, 1989; Moberg, 1999), and a disproportionate amount of small knots are missed by the CT-scanning method (Oja, 1999), it is possible that some of the site quality effect indicated in Eq. (3.9) (parameter a2) and Fig. 5 was actually due to measurement bias. An additional effect of within-stand differentiation is reported by Maguire et al. (1994), but no such effect is found in several other studies (Colin and Houllier, 1992; Gilmore and Seymour, 1997). The present study also failed to ®nd an additional effect due to within-stand differentiation. It is likely that much of the effect of within-stand social status is accounted for by height increment, making additional treelevel variables (such as relative DBH) sometimes redundant. 4.3. Sound-knot length A sound-knot is composed of wood within a stem which is (or was) attached to a living branch (Fig. 2). It therefore follows that KLsound, like knot diameter is related to branch longevity (Dietrich, 1973). It is logical that, as in the present study (Eq. (3.10) and Fig. 7), several studies have identi®ed a strong correlation between internal knot length and knot diameter (Dietrich, 1973; Hùibù et al., 1997). The vertical variation in mean knot length below the live crown for Norway spruce is found to be quite small above an initial increase near the ground (Vestùl and Hùibù, 1998b; Vestùl et al., 1999; éyen and Hùibù, 1999a). This seems to agree with the general trend of KDmean found in the present study (Fig. 3). There was a slight (but signi®cant) separate negative effect of height above ground (Eq. (3.9) and Fig. 7). These two effects counteracted each other in such a way that it was possible to obtain a slightly decreasing mean KLsound pattern with increasing height below the live crown. This would be more likely to occur at better sites where trees are taller, and the ¯at part

of the KDmean±Hk curve extends higher up in the tree. This vertical pattern is also described for Norway spruce by Vestùl and Hùibù (1998b). SchoÈpf (1954) describes the vertical variation in the sound-knot core in terms of the development of crown morphology. Broad-crown trees show a substantial increase in KLsound near the ground followed by a constant or slightly decreasing pattern with increasing height up to the live crown. Narrow-crown trees have a much smaller sound-knot core near the ground, followed by a slowly increasing pattern, and then a substantial increase just below the live crown of older trees. This seems to agree with the behaviour of Eq. (3.10). In contrast, Dietrich (1973) indicates a continuously increasing pattern of KLsound with height for Abies alba, and attributes this to a similarly increasing branch longevity. On the basis of joint tree growth, branch-growth and crown-recession simulations, KaÈrkkaÈinen (1986) and Maguire et al. (1991) also show continuously increasing patterns. The slight negative effect of height found in the present study might also re¯ect an overestimation of sound-knot length near the stem base identi®ed for CT-scanning by Oja (1999). Whereas KD represents twice the accumulated radial branch growth, KLsound represents the accumulated radial stem growth until branch mortality. Branches can survive for a substantial period of time after branch radial growth ceases (Andrews and Gill, 1939; Dietrich, 1973; Kershaw et al., 1990; Weslien, 1995). Although no studies of Norway spruce have been found in the literature, this should be especially true of this species since it is very shade tolerant (Lewerenz and Hinckley, 1990). This means that there will be a discrepancy in the relationship between KLsound and KD according to: (1) the difference between duration of branch radial growth and branch longevity, and (2) the stem radial growth during this time. The inclusion of DBH in Eq. (3.10) should be viewed in this context. A substantial part of the between-stand and -tree variation was accounted for by knot size. Much of the expected effects of site and mean ring width on KLsound was likely compounded in knot size. DBH, being the same scale as of KLsound (i.e. absolute distance from the pith), had the effect of adjusting the level of the relationship between KLsound and KD. When DBH was included, almost all of the remaining tree- and stand-level variation was

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accounted for. No other combination of stand- and tree-level variables provided similar ®t statistics. The dif®culty of identifying a single, universal ringwidth variable is indicated by Vestùl and Hùibù (1998b) and éyen and Hùibù (1999b). DBH often performs better than, or equal to, ring-width variables in these studies. The most relevant ring-width interval seems to be stand speci®c, perhaps related to the timing of crown closure and start of crown recession (Vestùl and Hùibù, 1998b). It should be recognised, however, that the use of DBH to estimate knot length in a dynamic simulation may lead to inconsistent conclusions as was described above for knot diameter. Relative knot diameter (knot diameter divided by KDmean) was used in order to account for the withinwhorl variation. The implied assumption was that, within a whorl, small-diameter branches will die before larger branches. The intra-whorl variation was found to be larger, in absolute terms, for larger mean knot sizes. As is also reported by Vestùl and Hùibù (1998b) and Vestùl and Colin (1998), the effect of KDrel on sound-knot length was found to be larger at greater heights from the ground and for larger trees (as measured by DBH).

As was found for KLsound, larger, within-whorl relative knot size was associated with larger looseknot lengths. But there was no such universal effect of knot size, as is reported by Dietrich (1973). In fact, Eq. (3.11) only marginally reduced the within-tree variation in KLloose (Table 4). This could be due to large measurement errors. Oja (1999) concludes that CT-scanning provides neither accurate nor precise measurements of the total knot length of occluded knots. The location of this border is often in the sapwood, and there may not be suf®cient density contrast between the knot and its surrounding stemwood to segment the knot through image analysis (Grundberg, 1994). Moreover, since KLloosewas derived from the difference in sound-knot length and total knot length, there are two sources of measurement error in the value obtained. It was therefore dif®cult to evaluate the true nature of KLloose in the present study. Validation of Eq. (3.11) is especially important given these dif®culties. It would probably be bene®cial to carry out a separate study on this property using stem dissection (e.g. Maguire and Hann, 1987) techniques.

4.4. Loose-knot length

5. Conclusions

There are very few published results on the variation in loose-knot length. KLloose represents the radial stem growth during the time between branch mortality and natural pruning (Wegelius, 1940; Dietrich, 1973). This process is dependent on biological degradation and stochastic events such as extreme snow or wind loads. This time is found to be shorter for better sites, lower latitude and, within stands, for lower crown classes (Romell, 1937; LaÈmsaÈ et al., 1990; HaÈgg and Weslien, 1995). When DBH was included in Eq. (3.11), no additional tree- or stand-level variables were signi®cant. The effect of within-stand variation was likely compounded with DBH, but the model could not accommodate any site-speci®c effects related to biodegradation. This might be due to the few number of sites in the study (11) and the stochastic nature of natural pruning. In a dynamic application of Eq. (3.11), the use of DBH may provide inconsistent results (as was described for knot diameter and sound-knot length above).

Using non-destructive internal knot measurements, it was possible to develop models describing mean and maximum knot diameter, knot frequency, sound-knot length and loose-knot length for Norway spruce. A nonlinear segmented (hyperbolic±quadratic±quadratic) model was developed for knot diameter. This enabled the use of different predictor variables above (AGE, DBH and crown length) and below the live crown (temperature sum and DBH divided by AGE). Knot frequency could be described as a function of the distance between whorls and site index. Sound-knot length was modelled in a linear mixed model using knot diameter, height above ground, and DBH as independent variables. Similarly, loose-knot length was described as a function of knot diameter and DBH in a mixed log-linear model. However, this latter model had a substantial residual error, which might be due to large measurement errors. Application of these models as a system of equations in conversion simulation software would enable product recovery studies for a standing timber

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resource (cf. KaÈrkkaÈinen, 1986; VaÈisaÈnen et al., 1989; Briggs, 1992; Houllier et al., 1995). Given the possibility of such applications, it is important to explore the following considerations in future research work: (1) validation of the models against an independent data set; (2) statistical methods which account for error correlation across equations (Borders, 1989); (3) more precise measurement methods for loose-knot length; (4) knot measurements within the upper live crown; and (5) properties of inter-whorl knots. Acknowledgements This study was supported by the European Commission (FAIR CT 96-1188, FAIR CT 96-1915). The contents of the present publication is the sole responsibility of its author and in no way represents the view of the Commission or its services. The work was also carried out within the framework of Wood and Wood Fibre Science, a post-graduate school sponsored by the Swedish Council for Forestry and Agricultural Research and the Swedish University of Agricultural Sciences (SLU). The author appreciates the valuable collaboration with other researchers at the Department of Forest Management and Products, SLU and at the Division of Wood Technology, LTU. References Andrews, S.R., Gill, L.S., 1939. Determining the time branches on living trees have been dead. J. For. 37, 930±935. Anon., 1989. SAS/STAT User's Guide, Version 6, Vol. 2. SAS Institute Inc., Cary, NC, 846 pp. Anon., 1994. Nordiskt traÈ Ð Sorteringsregler, Upplaga 1. FoÈreningen Svenska SaÊgverksmaÈn, Markaryd, 64 pp. Anon., 1997. SAS/STAT Software: Changes and Enhancements, Release 6.12. SAS Institute Inc., Cary, NC, 1167 pp. Barbour, R.J., Kellogg, R.M., 1990. Forest management and endproduct quality: a Canadian perspective. Can. J. For. Res. 20, 405±414. Bergh, J., McMurtrie, R.E., Linder, S., 1998. Climatic factors controlling the productivity of Norway spruce: a model-based analysis. For. Ecol. Mgmt. 110, 127±139. BjoÈrklund, L., 1997. The interior knot structure of Pinus sylvestris stems. Scand. J. For. Res. 12, 403±412. BjoÈrklund, L., Julin, B., 1998. VaÈrdeoptimerad soÈnderdelning av datortomograferade tallstammar. Department of Forest Industry and Market Studies, Swedish University of Agricultural Science, Uppsala, Report No. 48, 1998, 37 pp.

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