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Modelling bark thickness of Picea abies with taper curves
Forest Ecology and Management 206 (2005) 35–47 www.elsevier.com/locate/foreco
Modelling bark thickness of Picea abies with taper curves Jouko Laasasenaho*, Timo Melkas, Sari Alde´n Department of Forest Resource Management, University of Helsinki, Faculty of Agriculture and Forestry, Latokartanonkaari 7 (PL 27), 00014 University of Helsinki, Finland Received 29 September 2003; received in revised form 22 July 2004; accepted 12 October 2004
Abstract Models predicting bark thickness at 14 different relative stem heights were derived, and factors affecting bark thickness were analyzed. The study material was collected from the tracts of the fifth National Forest Inventory (NFI) during the years 1968– 1971. It consists of sample tree data measured from 1864 Norway spruce (Picea abies) stems. Bark thickness was measured with bark gauges at the following 14 different relative heights: 1, 2.5, 5, 7.5, 10, 15, 20, 30, 40, 50, 60, 70, 80, and 90%. Linear regression models predicting bark thickness at each relative height were derived. These models were combined into an equation group in which double bark thickness at each relative height was predicted with both endogenous variables (bark thickness at neighboring relative heights) and exogenous variables (tree age, tree height, and breast height diameter). This equation group was solved with the Thomas algorithm. Predicted bark thickness values were finally joined with spline functions to form a bark curve model. The models gave unbiased estimates for both bark thickness and bark volume. The relative error for bark thickness varied from 13.7 to 27.9% and the standard error for stem volume under bark varied from 5.27 to 7.79 dm3 depending on the amount of input data. Bark thickness was found to correlate with diameter at breast height, tree height, tree age and stem tapering. Climatic zone also clearly affected the amount of bark. By combining the models derived in this study with existing stem curve models, it is possible to accurately calculate stem volume both under and over bark for any arbitrary portion of the stem. # 2004 Elsevier B.V. All rights reserved. Keywords: Bark; Bark model; Spruce bark; Polynomial model; Taper curve; Stem form; Spline function; Equation group
1. Introduction In Nordic countries, bark-related research has concentrated on the analysis of factors influencing * Corresponding author. Tel.: +358 9 191 58176; fax: +358 9 191 58159. E-mail address: [email protected] (J. Laasasenaho).
bark thickness and the derivation of models predicting bark thickness at breast height. The stem volumes over and under bark have been traditionally presented in a tabular form although nowadays the volumes are commonly calculated with volume functions or stem curve models (e.g. Laasasenaho, 1982; Ahonen, 1996). In bark-related research, emphasis has been on the prediction of the amount of bark found in harvested timber. Research on the
0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.10.058
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amount of bark in standing trees has been very limited. In Finland, the first comprehensive set of volume tables for domestic tree species was published by Ilvessalo (1948). In these tables, bark amount was presented as bark percentage of the total stem volume as a function of tree species, tree height, diameter at breast height, and stem form. They were derived by graphical smoothing. Jonson (1912) published a respective set of tables for conifers in Sweden a few decades before this. Studies on the amount of bark in harvested timber has been carried out by Aro (1935), Heiskanen and Rikkonen (1976), and Saikku and Rikkonen (1976). In research related to bark prediction models, emphasis has been put on models predicting bark thickness at breast height, since the prediction of stem growth has been based on breast height characteristics. Such models have been presented by Jonson (1912), ¨ stlin (1963a), Pa¨ ivinen (1978), and Mielika¨ inen O (1985). Variation of bark thickness along the stem has ¨ stlin (1963a), Hakkila (1967), and been studied by O Nylinder (1973). Hakkila (1967) derived models for bark taper as a function of stem height for Norway spruce, Scots pine, and silver birch. Ojansuu (1993) derived both over and under bark stem curve models for Scots pine. Within models, stem form has been depicted as multidimensional vectors of polar coordinates. Ka¨ rki et al. (1999) derived models predicting bark proportion for Gray alder based on mixed linear models (Lappi, 1986). In Sweden, Jonsson and Nylinder (1990) derived bark functions for predicting a bark curve for Scots pine. Hakkila et al. (1995) studied the effects of bark on the value of first thinning Scots pine timber and its suitability for various purposes. In Finland, the estimation of standing tree stem volumes is currently based on stem curve models and volume functions that include bark (Laasasenaho, 1982). In order to improve estimation of stem volume and stem growth, prediction of stem form, and stem bucking, bark amount should be predicted reliably. The purpose of this study is to examine and derive models for predicting bark thickness at various stem heights of Norway spruce and to consequently derive models and methods for estimating stem volumes over and under bark. Bark thickness is calculated with a method based on the use of spline functions.
2. Material 2.1. Characteristics of the data The research material consists of 1864 Norway spruce (Picea abies) stems measured from sample plots of the fifth National Forest Inventory (NFI), which was carried out during the years 1968–1971. The sample plots were laid out in a systematic manner representing the whole of Finland. The sample trees were selected from the plots with relascopes (BAF 2 m2/ha). Bark thickness was measured from each stem with bark gauges at 14 different relative heights. More information on the gathering and characteristics of the study material is presented in Kuusela and Salminen (1969) and in Laasasenaho (1982). The measured stem variables used in this study are: - diameter over bark at relative heights 1, 2.5, 5, 7.5, 10, 15, 20, 30, 40, 50, 60, 70, 80 and 90% (di); - bark thickness at the same relative heights (bi); - diameter over bark at breast height (d1.3); - bark thickness at breast height (b1.3); - upper diameter (d6.0); - tree height (h); - tree age (t); - date of measurements (date). Minimum and maximum values, means (x) and standard deviations (S.D.) of the central sample tree measurement characteristics (d1.3, b1.3, h, and t), and volumes (v, dm3) are presented in Table 1. In this study, bark amount is expressed either as double bark thickness or as the proportion of over bark diameter or volume. Forest type (site), climatic zone (length of the growth period), and development class were measured as stand variables. The classifications can be seen in Table 1 Characteristics of the data
d1.3 (cm) b1.3 (cm) h (m) t (years) v (dm3)
Minimum
Maximum
Mean
S.D.
1.5 0.2 1.8 4 1
61.9 3.8 32.7 299 3789
18.0 1.25 13.8 66 265
8.6 0.54 5.8 44 312
v, volume (dm3).
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Table 2 Classifications of forest type, development class, and climate zone Site class
Forest type
Class
Development class
Climatic zone
The length of growing period (days)
1 2 3 4 5 6 7 8
Very rich sites (OMaT) Rich sites (OMT) Damp sites (MT) Sub-dry sites (VT) Dry sites (CT) Barren sites (ClT) Rocky sites Hill top sites and fields
0 1 2 3 4 5 6 7 8
Forestry land other than forest land Open regeneration area and seed tree stand Seedling and sapling stands with standards Seedling and sapling stands Thinning stands Preparatory stands Mature stands Shelter wood stands Low yielding stands
1 2 3 4 5 6 7
<125 125–135 135–145 145–155 155–165 165–175 >175
(Kuusela and Salminen, 1969).
Table 2. Stem form is expressed with tapering, (d1.3 d6) and the classification of stem form is done by using 1-cm taper classes. 2.2. Correction of bark thickness values measured at relative heights and at breast height Bark thickness has been measured by using bark gauges. Laasasenaho and Sevola (1972) and Heikurainen (1984) have shown that the use of bark gauges may easily lead to biased measurement results, because the results depend on the power with which you strike the bark gauge to the surface of the tree. The authors have also found out that the results are dependent on the time of year of the measurement. Therefore, the influence of measurement date on measured bark thickness was examined in order to eliminate any time-related systematic errors in the measured bark thickness values. The material of this study was measured between May and October. Mean and standard deviation values for absolute and relative bark thickness at each relative height were calculated for each measurement month. In order to eliminate the effect of a tree’s size on the thickness of bark, the study was conducted by using relative bark thicknesses for each measurement month (Table 3). Bark thickness measurements were found to increase from May to July and in turn, decrease in October. It seems that relatively soft early growth allows the bark gauge to sink not only through the bark but also through the growing annual ring. These obvious systematic errors were corrected by using
linear regression models. First, all measurement days from the 1st of June to the end of August were numbered consecutively from 1 to 92. All other measurement days received a value zero. A regression model was then derived for each relative height. Relative bark thickness at the respective height (bi/di) was selected as the dependent variable. The independent variables included were diameter at the respective height (di), tree age (t), ratio of tree age to respective diameter (t/di), consecutive measurement date number (date), and the square of the consecutive measurement date number (date2). Next, bark thickness (best) for each day between 1.6 and 31.8 was calculated with the derived regression models. Another estimation for the same bark thickness (bzero) was calculated by setting date to zero. The measured bark thickness was finally corrected by subtracting the difference of best and bzero from the measured value. The correction was made only if the difference was positive. It was assumed that the difference of best and bzero would approach zero at both the beginning of June and the end of August. This did not, however, happen. At relative heights of 70, 80, and 90%, the difference turned negative already a week before the end of August. On the other hand, the differences were still clearly positive at the end of August at relative heights of 1–20%. Final corrections were made using time intervals beginning from June to mid-September at relative heights of 1–20% and the beginning of June to the end of August at relative heights of 30–90%. Bark thickness values measured at breast height were also corrected. Breast height was transferred to a
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Table 3 Mean and standard deviation values for relative bark thickness (bi/di) at each relative height Variables (%)
Relative height (%) 1
2.5
5
7.5
10
15
20
30
40
50
60
70
80
90
Mean
May (n = 392) Mean S.D.
8.55 2.84
8.44 2.81
8.10 2.82
7.78 2.87
7.46 2.70
7.22 2.50
7.13 2.51
7.20 2.47
7.41 2.46
7.76 2.53
8.42 2.82
9.10 3.14
10.40 3.82
13.90 5.63
8.49 1.73
June (n = 378) Mean S.D.
9.37 3.02
9.27 3.07
8.86 3.19
8.49 3.14
8.07 2.90
7.73 2.69
7.62 2.69
7.69 2.75
7.88 2.61
8.43 2.92
8.94 3.11
9.90 3.70
11.50 4.53
14.90 6.81
9.19 1.88
July (n = 354) Mean S.D.
10.80 2.93
10.80 3.05
10.40 3.13
9.98 3.12
9.61 3.06
9.34 2.98
8.95 2.68
9.02 2.48
9.23 2.41
9.57 2.46
10.20 2.55
11.00 3.07
12.70 4.11
16.20 6.19
10.56 1.84
August (n = 292) Mean 10.90 S.D. 3.10
10.97 3.49
10.65 3.67
10.20 3.58
9.70 3.53
9.31 3.46
9.01 3.25
8.76 2.88
8.84 2.57
9.09 2.56
9.52 2.77
10.30 2.87
11.70 3.79
15.70 5.93
10.33 1.72
September (n = 275) Mean 9.45 S.D. 2.84
9.34 2.86
9.18 2.96
8.19 2.91
8.56 2.77
8.41 2.82
8.17 2.61
8.12 2.50
8.36 2.52
8.71 2.64
9.25 2.74
10.10 2.94
11.50 3.45
14.95 5.09
9.45 1.76
October (n = 173) Mean 7.60 S.D. 2.22
7.68 2.42
7.21 2.44
6.88 2.50
6.64 2.23
6.35 2.10
6.34 2.10
6.36 2.05
6.48 1.82
6.86 1.87
7.26 1.97
8.34 2.17
9.59 2.34
13.50 3.70
7.65 1.84
relative height by dividing 1.3 with tree height. The minimum relative breast height was 3.96% and the maximum was 72.2%. Each observation was assigned to the nearest fixed relative height. Bark thickness measured at breast height was corrected by using the respective regression model presented in Table 4.
2.3. Additional measurements The NFI material includes measured bark thickness data only up to the relative height of 90%. However, in order to derive complete under bark stem curves it was considered necessary to acquire some additional
Table 4 Coefficients and R2 values of the bark thickness correction models Dependent variables
b1/d1 b2.5/d2.5 b5/d5 b7.5/d7.5 b10/d10 b15/d15 b20/d20 b30/d30 b40/d40 b50/d50 b60/d60 b70/d70 b80/d80 b90/d90 a
Coefficients.
R2
Independent variables a
a
di
11.6012 11.6848 11.4768 11.4221 10.7146 10.2761 10.0127 9.5983 9.4071 9.9255 10.5445 11.5314 13.4232 18.8119
0.0163 0.0209 0.0257 0.0277 0.0278 0.0278 0.0277 0.0292 0.0319 0.0372 0.0462 0.0619 0.0950 0.2237
a
a
t
t/di
0.0175 0.0232 0.0256 0.0247 0.0255 0.0255 0.0256 0.0266 0.0281 0.0261 0.0254 0.0233 0.0192 0.0103
0.0022 0.0234 0.0203 0.0199 0.0694 0.0764 0.1016 0.2052 0.3244 0.4122 0.6180 1.0140 1.6555 4.1523
a
2a
Date
Date
0.0740 0.0744 0.0707 0.0652 0.0599 0.0559 0.0519 0.0545 0.0545 0.0537 0.0541 0.0502 0.0674 0.0546
0.0007 0.0007 0.0006 0.0006 0.0005 0.0005 0.0005 0.0006 0.0006 0.0006 0.0006 0.0006 0.0008 0.0006
0.4216 0.4539 0.4776 0.4792 0.4909 0.4879 0.4688 0.4494 0.4501 0.4209 0.3958 0.3488 0.3160 0.3215
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
measurements above this height. A total of 19 tops were measured at cutting sites in the Porvoo and Helsinki region (Southern Finland) in September 1997. Diameters under and over bark were measured at the middle of each annual height growth starting from the latest annual height growth. Measurements were taken from as many annual height growths as possible. The amount of measurements taken from a single top ranged between 4 and 10. The measurements were made by using a digital caliber gauge. Diameters under bark were measured after the bark was carefully carved away from the measurement point with a knife. Additionally, diameters over and under bark for five tops were measured with a year ring microscope having an accuracy of 0.01 mm. This was performed in order to see whether the usage of a knife caused any systematic error to the measurements. However, comparisons made between values measured manually and those measured with the microscope showed no systematic error.
3. Methods 3.1. Modelling of bark thickness Factors influencing bark thickness were studied by analyzing correlation matrices and scatter plots of various variables. Covariance analysis was used in the case of continuous and non-continuous variables such as site class and climatic zone. All statistical analyses were carried out with the SAS statistical software package. The bark thickness functions were created by deriving regression models depicting bark thickness at each relative height. The independent variables included in these models were bark thickness at the preceding and following relative heights, diameter over bark at breast height, tree height, and tree age. The independent variables included were selected by studying the correlation between the dependent and various potential independent variables and by comparing the R2 values of models including various combinations of the independent variables. The inclusion of each independent variable was verified with t-tests. Regression coefficients were solved by using the least sum of squares method. The general
39
form of the derived model is presented in Eq. (1). The main characteristics of the derived models are presented in Table 9. bi ¼ a þ b1 bi1 þ b2 biþ1 þ b3 d1:3 þ b4 h þ b5 t (1) where bi is the bark thickness at relative height i, bi 1 the bark thickness at relative height i 1, bi + 1 the bark thickness at relative height i + 1, d1.3 the diameter over bark at breast height, h the tree height, t the tree age, a the constant, and b1, . . ., b5 are the coefficients. The individual bark thickness functions were then included in an equation group (Laasasenaho, 1982, p. 26 and 33) in which bark thickness at each relative height is depicted with both endogenous and exogenous variables. The endogenous variables included in the equation group were bark thickness at the preceding and following relative heights, and the respective exogenous variables were diameter over bark at breast height, tree height, and tree age. The equation group model used in this study requires far less calculations than the model proposed by Kilkki and Varmola (1981), which includes bark thickness at all other relative heights as endogenous variables. The resulting equation group includes equations for estimating bark thickness at each fixed relative height. The equation group can be solved if the exogenous variables (d1.3, h, and t) in the equation group are known or estimated. In addition, we can get more accurate results if bark thickness for one or more of the fixed relative heights is known. However, in practice, bark thickness at any specific relative height is not known. Therefore, we used measured or estimated bark thickness at the breast height to estimate bark thickness at the nearest relative height. Estimation of bark thickness at the nearest relative height was done by multiplying bark thickness at breast height with the ratio di/d1.3. Diameter for the fixed relative height (di) that is closest to breast height was calculated by using Laasasenaho’s polynomial stem curve function. The equation group was solved using Thomas’s algorithm. A detailed description of this algorithm is presented by Lahtinen and Laasasenaho (1979, p. 12) and Laasasenaho (1982). Bark thickness curves were derived by using a cubic spline function of the bark thickness values estimated by the equation group. We tested this calculation system using either measured or estimated bark thickness at breast height in the following three different ways:
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Model 1 Model 2 Model 3
b1.3 bˆ 1:3 = f(d1.3) bˆ 1:3 = f(d1.3, h, t)
3.2. Model reliability The reliability of the bark thickness models was assessed by comparing the predicted bark thickness at relative height with the respective measured values. Bias (b, cm) and standard deviation (S.D., cm) were calculated for the observed differences at each fixed relative height. Standard errors for bark thickness and absolute and relative bark volumes were calculated by using different variable classes. Under bark stem volumes estimated with cubic splines (Lahtinen and Laasasenaho, 1979; Lahtinen, 1988) were regarded as measured under bark stem volumes. Bias and standard error values were calculated using the following formulas. Pn ðˆy yi Þ b ðcmÞ ¼ i¼n i (2) n1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn yi yi Þ2 i¼n ðˆ S:D:yˆ ðcmÞ ¼ (3) n1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn yi yi =ˆyi Þ2 i¼1 ðˆ S:D:yˆ ð%Þ ¼ (4) n1 where b (cm) is the bias, S:D:yˆ (cm) the standard error, S:D:yˆ (%) the relative standard error, yˆ the estimated bark thickness or under bark volume, y the measured bark thickness or under bark volume, and n is the number of observations. 4. Results 4.1. Variation of bark thickness Average double bark thickness was greatest at the base (1% relative height) 2.02 cm and decreased uniformly towards the top (90% relative height) 0.45 cm. Bark proportion (from the diameter) decreased from the base to the 20% relative height being 7.5% at its minimum (Fig. 1). After that the bark proportion increased first slowly then rapidly being finally 15% at a relative height of 90%.
Fig. 1. Mean values of double bark thickness (cm) and proportional bark thickness (%) as a function of relative height.
If we look at the proportion of bark from the basal area, the bark proportion varies from 14 to 27% being at its minimum at a relative height of 20% (Table 5). Standard deviation, on the other hand, is at its minimum at the relative height of 40%. The general form of bark curves at different breast height diameter classes (Fig. 2) is highly similar and therefore we can assume that the general form of a bark curve does not depend on tree size. No clear indications of the occurrence of different bark types were found although some signs of rinding were found in most barren sites (CT). The correlation between bark thickness at different relative heights (Table 6) was found to be clearly lesser than that of over bark diameter at different relative heights, while the correlation between bark thickness and stem diameter was rather strong. The correlation between bark thickness values at adjoining relative heights is, however, higher than the correlation between bark thickness and stem diameter at a given relative height except at the upper part of the stem. When comparing the over bark and under bark stem curves, it was found that tapering at the base of the stem is greater in the over bark stem curve, because the proportion of bark decreases (Fig. 1; Table 5). Whereas at relative heights greater than 60% tapering of the under bark stem curve is greater. The difference is largest at a relative height of 20%. Dependency of bark thickness between different tree, site class, and stand variables was studied at breast height bark thickness. Bark thickness at breast height was found to be strongly correlated with breast height diameter (0.76), tree height (0.61), and tree age
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
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Fig. 2. Double bark thickness (cm) as a function of relative tree height by breast height diameter classes.
(0.61). In order to estimate the dependency of bark thickness on other variables we removed the effect of breast height diameter, tree height, and age to bark thickness by using covariance analyse. The model which is used in the analyses is b1:3ij ¼ m þ ai þ b1 d1:3 þ b2 h þ b3 t þ eij , where b1:3ij is breast height bark thickness in class i and in tree j, m is the general average, ai is the value of a quality variable in class i, b1, b2, b3 are regression coefficients of continuous variables, b1.3, h, t are continuous variables (covariants), and eij is the random error. It was found out that bark thickness is dependent on climate zone (the length of growth period, see classes in Table 2), tree form (d1.3 d6), and development class. The difference was most significant in climate zone 2. Site class (forest type) was not significant so that there is no dependency between bark thickness and forest type
(Table 7). Bark proportion is greater in the northern parts of the country. Climate zone was the most significant quality variable. 4.2. Bark thickness at the top of a tree Bark thickness was found to increase from the top to the base of the stem. The thinnest bark at the top of tree was however in the third and fourth annual shoots. In the two most upper annual shoots bark is somehow folded and therefore bark thickness seems to decrease in the third annual shoot (Table 8; Fig. 3). 4.3. Bark models Regression models (1) predicting bark thickness at a given relative height were derived for each of
Table 5 Bark percentage of over bark diameter and basal area at different relative heights Relative height
1 2.5 5 7.5 10 15 20 30 40 50 60 70 80 90
Proportion of bark (%) (diameter)
Proportion of bark (%) (basal area)
Mean
S.D.
Minimum
Maximum
Mean
S.D.
Minimum
Maximum
8.8 8.7 8.5 8.1 7.7 7.5 7.4 7.5 6.9 8.1 8.7 9.6 11.0 14.8
2.9 3.0 3.1 3.1 2.9 2.9 2.7 2.6 2.3 2.7 2.9 3.2 4.0 6.0
2.8 3.2 2.7 2.5 2.5 2.7 2.5 2.5 2.8 2.7 3.0 3.0 3.6 4.7
23.6 22.5 24.0 23.8 22.6 22.0 22.8 23.4 19.3 23.1 26.7 35.3 40.0 57.1
16.7 16.6 16.1 15.4 14.6 14.3 14.1 14.4 14.7 15.4 16.5 18.2 20.5 27.0
5.2 5.4 5.6 5.6 5.3 5.2 5.0 4.8 4.6 4.8 5.1 5.7 6.9 9.5
5.4 6.3 5.3 4.9 5.0 5.3 4.9 5.0 6.0 5.4 5.8 6.0 7.1 9.1
41.6 39.9 42.2 42.0 40.1 39.1 40.4 41.3 38.8 40.8 46.2 58.1 64.0 81.6
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Table 6 Correlation between bark thickness and stem diameter at fixed relative heights
d1 d2.5 d5 d7.5 d10 d15 d20 d30 d40 d50 d60 d70 d80 d90
b1 b2.5 b5 b7.5 b10 b15 b20 b30 b40 b50 b60 b70 b80 b90
b1 b2.5 b5 b7.5 b10 b15 b20 b30 b40 b50 b60 b70 b80 b90
d1
d2.5
d5
d7.5
d10
d15
d20
d30
d40
d50
d60
d70
d80
d90
1.00 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.95 0.93 0.92 0.88
1.00 0.99 0.99 0.99 0.99 0.98 0.98 0.97 0.96 0.95 0.94 0.92 0.88
1.00 0.998 0.996 0.99 0.99 0.99 0.98 0.97 0.96 0.95 0.93 0.89
1.00 0.999 0.997 0.99 0.99 0.98 0.98 0.97 0.95 0.93 0.89
1.00 0.998 0.997 0.99 0.99 0.98 0.97 0.95 0.93 0.89
1.00 0.999 0.996 0.99 0.98 0.97 0.96 0.94 0.90
1.00 0.998 0.99 0.99 0.98 0.96 0.94 0.90
1.00 0.997 0.99 0.98 0.97 0.95 0.91
1.00 0.997 0.99 0.98 0.96 0.92
1.00 0.996 0.99 0.97 0.93
1.00 0.99 0.98 0.94
1.00 0.99 0.95
1.00 0.97
1.00
b1
b2.5
b5
b7.5
b10
b15
b20
b30
b40
b50
b60
b70
b80
b90
1.00 0.92 0.88 0.85 0.83 0.82 0.81 0.80 0.78 0.76 0.73 0.71 0.64 0.52
1.00 0.94 0.91 0.89 0.86 0.85 0.82 0.80 0.76 0.73 0.69 0.62 0.49
1.00 0.95 0.92 0.89 0.86 0.83 0.81 0.77 0.73 0.69 0.62 0.48
1.00 0.95 0.91 0.89 0.84 0.81 0.77 0.73 0.69 0.61 0.47
1.00 0.94 0.90 0.86 0.83 0.78 0.74 0.70 0.63 0.49
1.00 0.95 0.91 0.88 0.84 0.79 0.75 0.67 0.54
1.00 0.94 0.92 0.87 0.83 0.78 0.71 0.57
1.00 0.95 0.91 0.87 0.82 0.74 0.61
1.00 0.94 0.90 0.85 0.77 0.64
1.00 0.93 0.88 0.81 0.68
1.00 0.91 0.84 0.71
1.00 0.88 0.75
1.00 0.83
1.00
d1
d2.5
d5
d7.5
d10
d15
d20
d30
d40
d50
d60
d70
d80
d90
0.73 0.69 0.66 0.65 0.66 0.70 0.72 0.75 0.77 0.78 0.77 0.79 0.76 0.71
0.72 0.69 0.66 0.65 0.66 0.70 0.72 0.75 0.77 0.78 0.77 0.79 0.76 0.71
0.71 0.68 0.65 0.64 0.65 0.69 0.72 0.75 0.77 0.78 0.77 0.79 0.77 0.72
0.71 0.67 0.64 0.64 0.65 0.69 0.72 0.75 0.77 0.78 0.77 0.79 0.77 0.72
0.70 0.67 0.64 0.63 0.64 0.68 0.71 0.74 0.76 0.78 0.77 0.79 0.77 0.72
0.70 0.66 0.63 0.62 0.63 0.68 0.71 0.74 0.76 0.77 0.77 0.79 0.77 0.72
0.69 0.65 0.62 0.61 0.63 0.67 0.70 0.73 0.76 0.77 0.77 0.79 0.77 0.72
0.68 0.64 0.61 0.60 0.61 0.66 0.69 0.73 0.75 0.77 0.77 0.79 0.77 0.72
0.67 0.63 0.60 0.59 0.60 0.65 0.69 0.72 0.75 0.77 0.77 0.79 0.77 0.73
0.66 0.62 0.59 0.57 0.59 0.64 0.68 0.71 0.74 0.76 0.77 0.79 0.78 0.73
0.65 0.61 0.58 0.57 0.58 0.63 0.67 0.71 0.73 0.76 0.77 0.79 0.78 0.73
0.65 0.60 0.57 0.56 0.58 0.62 0.66 0.69 0.72 0.75 0.76 0.79 0.78 0.74
0.64 0.60 0.57 0.55 0.57 0.62 0.65 0.69 0.72 0.74 0.75 0.78 0.79 0.74
0.63 0.60 0.57 0.56 0.58 0.62 0.65 0.68 0.71 0.73 0.74 0.77 0.77 0.75
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
43
Table 7 Significance and F-values of covariance analyses Variable
F-value
Significance
d1.3 h t Site (forest type) Climatic zone Development class Stem form
4161.5 258.1 586.1 1.7 6.3 3.5 2.6
*** *** *** – *** *** **
(–) No significant; (*) 0.05 (quite significant); (**) 0.01 (significant); (***) 0.001 (very significant). Fig. 3. Cross-sectional disks taken from the middle of the three most resent annual height growth shoots.
Table 8 The number of measurements for each annual shoot, and the mean and the standard deviation of bark thickness in the middle of each annual shoots Consecutive annual height growth
n
Bark thickness in the middle of the annual shoot (mm)
Models to estimate bark thickness at breast height were made by using the following logarithmic functions:
Mean
S.D.
Model 2
1 2 3 4 5
19 19 19 18 16
3.01 3.15 2.55 2.85 3.22
0.94 0.65 0.65 0.50 0.93
lnðb1:3 Þ ¼ 1:66 þ 0:65 ln ðd1:3 Þ; R2 ¼ 0:66
Model 3
lnðb1:3 Þ ¼ 1:99 þ 1:00 ln ðd1:3 Þ
(5)
0:57 ln ðhÞ þ 0:20 ln ðtÞ; R2 ¼ 0:77
Annual shoots are numbered consecutively starting from the latest annual shoot.
(6)
T-values of all coefficients were very high (between 18.7 and 61.5). The breast height diameter (b1.3) explains 66% of the variations of bark thickness at breast height. By adding tree height and age to the function, the coefficient of determination is 11% higher.
the 14 relative heights. The regression coefficients and degrees of determination of these models are presented in Table 9.
Table 9 Regression model coefficients and degrees of determination (R2) Dependent variables
b1 b2.5 b5 b7.5 b10 b15 b20 b30 b40 b50 b60 b70 b80 b90 a
Coefficients.
R2
Independent variables a
a
bi 1
0.10320 0.01240 0.02120 0.04960 0.03270 0.04220 0.00841 0.02120 0.01140 0.01950 0.01090 0.03690 0.00839 0.06170
– 0.35870 0.34770 0.44780 0.50530 0.66300 0.53920 0.37910 0.50860 0.45450 0.53200 0.43270 0.47100 0.51840
bi + 1
a
0.96550 0.62120 0.62920 0.51450 0.44570 0.57960 0.41840 0.58410 0.42580 0.47860 0.42390 0.43940 0.49240 –
d1.3
a
0.01640 0.00292 0.00250 0.00271 0.00271 0.00329 0.00137 0.00347 0.00165 0.00385 0.00021 0.00439 0.00162 0.00144
a
a
h
t
0.00208 0.00301 0.00321 0.00457 0.00380 0.00380 0.00016 0.00304 0.00027 0.00201 0.00142 0.00102 0.00006 0.00447
0.00004 0.00056 0.00035 0.00009 0.00014 0.00027 0.00020 0.00024 0.00023 0.00009 0.00008 0.00011 0.00010 0.00009
0.86350 0.92180 0.93530 0.93980 0.92920 0.93870 0.93910 0.93320 0.93520 0.92170 0.90160 0.87570 0.84960 0.71530
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Table 10 Average and standard error of observed bark thickness (cm) and average of predicted bark thickness (cm) at each relative height Measured (bi)
Measuring point (%)
1 2.5 5 7.5 10 15 20 30 40 50 60 70 80 90 a
2.02 1.70 1.48 1.34 1.23 1.16 1.11 1.05 0.98 0.91 0.82 0.72 0.59 0.45
S:D:measuredðbi Þ 0.81 0.68 0.59 0.51 0.47 0.44 0.42 0.41 0.38 0.35 0.32 0.28 0.24 0.18
The estimation method Measured b1.3a
b1.3 = f(d1.3)a
b1.3 = f(d1.3, h, t)a
2.01 1.69 1.47 1.33 1.23 1.16 1.11 1.05 0.98 0.91 0.82 0.72 0.59 0.45
2.01 1.70 1.47 1.33 1.22 1.15 1.10 1.04 0.98 0.90 0.82 0.72 0.59 0.45
2.00 1.69 1.47 1.33 1.23 1.16 1.11 1.05 0.98 0.91 0.82 0.72 0.59 0.45
Estimated bark thickness (bi).
Table 11 Bias (b, dm3) and standard deviation (S.D., dm3) values of the difference of predicted and observed bark volumes and under bark stem volumes by the studied models b (dm3)
Model 1 Model 2 Model 3
0.16 0.10 0.20
S.D. (dm3)
Volume of bark
5.29 7.79 7.26
Volume without bark 3
S.D. (%)
Estimated (dm )
S.D. (%)
Estimated (dm3)
15.7 23.0 21.5
33.6 33.7 33.6
2.3 3.4 3.1
231.3 231.3 231.4
33.8a a
231.2a
Measured value.
Table 12 Bias (b, cm), absolute standard error (S.D., cm), and relative standard error (S.D. (%)) of bark thickness at the relative heights by the studied models Measuring point (%)
1 2.5 5 7.5 10 15 20 30 40 50 60 70 80 90
Estimated b1.3(d1.3)
Measured b1.3
Estimated b1.3(d1.3, h, t)
b (cm)
S.D. (cm)
S.D. (%)
b (cm)
S.D. (cm)
S.D. (%)
b (cm)
S.D. (cm)
S.D. (%)
0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.42 0.30 0.22 0.17 0.16 0.17 0.18 0.19 0.18 0.18 0.18 0.16 0.15 0.12
20.96 18.21 15.89 14.11 13.70 14.22 15.13 16.89 18.08 19.39 20.95 21.84 25.02 27.74
0.01 0.01 0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00
0.52 0.44 0.39 0.36 0.32 0.28 0.25 0.23 0.21 0.20 0.19 0.17 0.15 0.12
25.17 25.15 25.45 25.81 25.68 24.08 23.36 21.99 21.52 21.60 22.60 22.92 25.56 27.82
0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.49 0.40 0.34 0.30 0.27 0.24 0.23 0.22 0.21 0.20 0.19 0.16 0.15 0.12
23.91 23.24 22.72 22.26 21.78 20.74 20.77 20.77 21.19 21.65 22.86 23.13 25.50 27.86
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Average and standard error of observed bark thickness and average of predicted bark thickness at each studied relative height were calculated and tabulated (Table 10).
4.4. Reliability of the derived bark models The most accurate estimates within the equation group (Eq. (1)) were achieved as expected by model 1, which incorporated the measured bark thickness at breast height to the nearest relative height (Tables 11 and 12). In this case the absolute standard error of under bark stem volume was 5.29 dm3, which represents a relative standard error of 2.2%. The standard error of bark thickness at the studied relative heights decreased from 4.2 mm at the base to 1.2 mm at the top. The most accurate estimate was achieved at a relative height of 10% (Table 12). Prediction accuracy decreases both toward the base and toward the top of the stem. The result is quite understandable as the 10% relative height is closest to breast height in most cases. The relative standard error was 13.7% at its minimum. The relative standard errors of under bark stem volume in models 2 and 3 were more than one per cent higher than that of model 1 being, however, below 3.5%. The relative standard errors of bark thickness at the studied relative heights were also higher in models 2 and 3. The bark models were found to be unbiased. Models 1 and 3 produce nonetheless very slight underestimates at the base of the stem.
5. Discussion The purpose of this study was developed bark thickness models for predicting bark thickness at arbitrary stem heights of Norway spruce. In analyzing the factors, which can be used to estimate bark thickness, bark thickness models were developed, which can be combined to present taper curve models. The reliability of the models is influenced by the sampling error of the study material, errors related to the measuring of bark thickness, and the procedure implemented in order to eliminate the apparent bias in these measurements. The material used in this study is a representative sample of the whole Norway spruce population of Finland.
45
In this study, bark thickness was measured with bark gauges which have been found to over-estimate bark thickness on the average by 0.25 mm (von Essen, 1974). Measurements were performed during the growth season and therefore the latest annual ring was yet soft (Laasasenaho and Sevola, 1972). This is one reason for the overestimation. Errors due to the time of measurement were corrected with separately derived regression models. The latest annual ring was soft for a longer period of time in the lower parts of the stem than in the upper parts and therefore the corrections were made for different time periods depending on measurement height. It seems that lignification was ended for the whole stem by mid-September. Errors related to the person carrying out the measurements were not taken into account. Bark proportion from diameter decreased from the base to the 20% relative height being 7.5% at its minimum. After that bark proportion increased being finally 15% at a relative height of 90%. These results are consistent with those presented by Hakkila (1967). Measurements of bark thickness at the top of trees showed that bark was thicker at the two uppermost shoots than in the two next shoots. Otherwise bark increased towards the base of the stem. Regression models predicting bark thickness for the 14 relative heights were derived. Bark thickness at each relative height was predicted by both endogenous variables (bi 1, bi + 1) and exogenous variables (d1.3, h, t). The coefficient of determination of these models was between 71.5 (90% height) and 94% (7.5% height). Models were combined into an equation group and solved by using the Thomas algorithm. Predicted bark thickness values were finally joined with spline functions to form a bark curve model. The derived models give unbiased estimates for bark thickness and bark volume. The absolute standard error for bark thickness ranged depending on the studied relative height from 5.2 to 1.2 mm. The respective relative standard error was 13.7% at its minimum at a relative height of 10 and 27.9% at its most, at the top of the stem. The standard error of bark volume varied depending on the model, from 5.3 to 7.8 dm3, which represents a 2.2–3.4% standard error of volume without bark. The standard error calculated from average bark volume was 15.7–23.0% for the models 1–3. Model 1, in which bark thickness was
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
measured at breast height, gave the best estimates for bark volume. Standard error was then 15.7%. Model 3, where bark thickness at breast height was estimated with the help of diameter, height and age variables, gave a standard error of 21.5% for bark volume. The best estimates could be obtained for trees on moist sites in Southern Finland having a breast height diameter of 25 cm or more. The analyses of the factors effecting bark thickness of Norway spruce proved that correlation between bark thickness and stem diameter is strong (Table 7, formula 5, see also Wredlind, 1917; Petrini, 1921). Bark thickness also depends on tree height, tree age, and tree form. Bark thickness varies also according to climate zones and development classes. When bark thickness models took into account tree height, diameter, and age, forest type was not a significant classify variable. Climate zone was no longer a significant variable either, except in the case of climate zone 2, in which the systematic error of bark volume was 2.3 dm3. The differences in other climate zones were not significant. Bark proportion is greater in the northern parts of the country. These findings are consistent with those presented in earlier studies ¨ stlin, 1963b; Ilvessalo, 1965). (O The accuracy of the models can be improved a little by deriving correction coefficients for different climatic zones or by deriving separate models for each zone. The better solution is that the length of the growth season, or thermal temperature sum, is taken account as an exogenous variable in the models. The use of e.g. thermal temperature sum as an indicator of geographical location is also possible with this technique. When comparing the over bark and under bark stem curves it was found that tapering at the base of the stem is greater in the over bark stem curve, because the proportion of bark decreases (Fig. 1; Table 5). Whereas at relative heights greater than 60% tapering of the under bark stem curve is greater. The difference is largest at a relative height of 20%. The results found in this study concerning tapering of the over bark and under bark stem curves confirm those found in previous studies (e.g. Aro, 1935). The derived bark models are applicable for both research and practical forest inventory purposes. They can be used to predict under bark diameters at any desired stem height and subsequently bark volumes of
any desired portion of the stem. The models can also be incorporated with existing stem curve models (Laasasenaho, 1982). The derived models require information on tree height, breast height diameter, and tree age as input data. Because the measurement of tree age is cumbersome, models for the estimation of tree age should be made, or models can be derived without the tree age variable. By incorporating the equation group models derived in this study with existing stem curve software, under bark stem volumes can be estimated by stem height, breast height diameter, and tree age. Under bark stem curves can also be derived and consequently automatic bucking can be improved and enhanced. The resulting models can also be applied in order to improve growth and increment predictions. Respective equation group bark models for two other major tree species occurring in Finland; Scots pine and Silver birch, have been derived by Laasasenaho and Melkas and so far unpublished research manuscripts drawn up.
Acknowledgements This research has been conducted at the Department of Forest Resource Management with funding from Metsa¨ miesten sa¨ a¨ tio¨ . The checking of data was carried out by Sari Alde´ n. Timo Melkas has done his Master of Science of the bark models of Picea abies and graduated in year 2000. The study has been supervised by Prof. Jouko Laasasenaho. The computing facilities have been provided by the University of Helsinki (Department of Forest Resource Management). The translation of the article was done by Mark-Leo Waite with Ms. Nina Garlo proof reading the text. Sampo Keskinen provided us with helpful comments on the text. We would also like to thank Metsa¨ miesten sa¨ a¨ tio¨ and all others who have made this work possible. References Ahonen, O.-P., 1996. Puun muodon ennustaminen runkoka¨ yrilla¨ ja simuloimalla (Predicting stem form by using taper curves and simulation). Metsa¨ tehon katsaus 4, 8 (in Finnish). Aro, P., 1935. Tutkimuksia rinnankorkeus- ja katkaisula¨ pimitan vaikutuksesta ka¨ ytto¨ puun ja hakkuuta¨ hteen ma¨ a¨ ra¨ a¨ n. Referat: Untersuchungen u¨ ber den Einfluss des Brustho¨ hen- und Mini-
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47 maldurchmessers auf die Menge des Gebrauchsholzes und der Hiebreste (Studies on the breast height- and cutting diameter’s impact on the amount of commercial timber and waste wood). Metsa¨ ntutkimuslaitoksen julkaisuja 20 (4), 1–159 (in Finnish). von Essen, C., 1974. Metning av barktjockleken genom electric avka¨ nning (Measuring bark thickness with electronic recognition). Skogsho¨ gskolan, Insitutionen fo¨ r skogsuppskatning och inledning. Raporter och uppsatser Nr. 3. Stockholm. Hakkila, P., 1967. Vaihtelumalleja kuoren painosta ja painoprosentista (Variation patterns of bark weight and bark percentage by weight). Metsa¨ ntutkimuslaitoksen julkaisuja 62 (5), 1–37 (in Finnish). Hakkila, P., Kalaja, H., Saranpa¨ a¨ , P., 1995. Etela¨ -Suomen ensiharvennusma¨ nniko¨ t kuitu- ja energianla¨ hteena¨ (Pine stands as a source of the pulpwood and energy in the first thinning in southern Finland). Metsa¨ ntutkimuslaitoksen tiedonantoja 582, 1–100 (in Finnish). Heikurainen, M., 1984. Ma¨ nnyn kuoren paksuuden vaihtelu ja mittaustarkkuus (Variation of bark thickness and measuring accuracy in pine trees). Master Thesis. University of Helsinki. Department of Forest Resource Management. 84 p. (in Finnish). Heiskanen, V., Rikkonen, P., 1976. Havusahatukkien kuoren ma¨ a¨ ra¨ ja siihen vaikuttavat tekija¨ t (The amount of bark in softwood logs and the factors influencing it). Folia For. 250, 1–67 (in Finnish). Ilvessalo, Y., 1948. Pystypuiden kuutioimistaulukot. Summary: volume tables for standing trees. Metsa¨ tieteellisen tutkimuslaitoksen julkaisuja 34 (4), 1–140. Ilvessalo, Y., 1965. Metsa¨ narvioiminen (Forest Mensuration). WSOY. Porvoo., 400 p. (in Finnish). Jonson, T., 1912. Afsmalnings- och Tillva¨ xttabeller fo¨ r Tra¨ duppskattning (Volume and Yield Tables for Tree Estimation) Stockholm. , 22 p. (in Swedish). Jonsson, L., Nylinder, M., 1990. Tallbarkens tjocklek la¨ ngs stammen – Funktioner fo¨ r aptering (Bark thickness of Pine in the direction of the stem – functions of cross-cutting). The Swedish University of Agricultural Sciences Department of Forest Products. Report Nr. 212, 1–41. Ka¨ rki, T., Eerika¨ inen, K., Heinonen, J., Korhonen, K.T., 1999. Harmaalepa¨ n (Alnus incana) tilavuustaulukot (Volume tables for Gray alder (Alnus incana)). Folia For. (Metsa¨ tieteen aikakauskirja) 1, 39–49 (in Finnish). Kilkki, P., Varmola, M., 1981. Taper curve models for Scots pine and their applications. Acta For. Fenn. 174, 1–60 (Helsinki). Kuusela, K., Salminen, S., 1969. The 5th National Forest Inventory in Finland. General design, instructions for field and data processing. Commun. Inst. For. Fenn. 69 (4), 1–72. Laasasenaho, J., 1982. Taper curve and volume functions for pine, spruce and birch. Seloste: Ma¨ nnyn, kuusen ja koivun
47
runkoka¨ yra¨ - ja tilavuusyhta¨ lo¨ t. Commun. Inst. For. Fenn. 108, 1–74. Laasasenaho, J., Sevola, Y., 1972. Havutukkien latvamuotolukujen vaihtelu (Variation of top form factor for the softwood logs). Folia For. 164, 1–20 (in Finnish). Lahtinen, A., 1988. On the construction of monotony preserving taper curves. Acta For. Fenn. 203 (Helsinki). Lahtinen, A., Laasasenaho, J., 1979. On construction of taper curves by using spline functions. Seloste: runkoka¨ yra¨ n muostaminen spline – funktiolla. Commun. Inst. For. Fenn. 95 (8), 1–63. Lappi, J., 1986. Mixed linear models for analyzing and predicting stem form variation of Scots pine. Seloste: ma¨ nnyn runkomuodon vaihtelun analysointi ja ennustaminen lineaarisen sekamallin avulla. Commun. Inst. For. Fenn. 134, 1–69. Mielika¨ inen, K., 1985. Koivusekoituksen vaikutus kuusikon rakenteeseen ja kehitykseen. Summary: effect of an admixture of birch on the structure and development of Norway spruce stands. Commun. Inst. For. Fenn. 133, 1–79. Nylinder, P., 1973. Virkesma¨ tning. Skogsho¨ gskolan. Inst. f. Virkela¨ ra¨ . (Wood measurement). Kompendium Nr. 5, (in Swedish). Ojansuu, R., 1993. Prediction of Scots pine increment using a multivariate variance component model. Tiivistelma¨ : ma¨ nnyn kasvun ennustaminen monimuutuja- ja varianssikomponenttimallilla. Acta For. Fenn. 239, 72. ¨ stlin, E., 1963a. Barkuppgifter fo¨ r tall, gran, bjo¨ rk m. fl. Del. 1 O Barkuppgifter fo¨ r la¨ n, regioner. Bark data for pine, spruce, birch etc. Part 1. Bark data for provinces and regions. Skogsho¨ gskolan. Institutionen fo¨ r skogstaxering. Rapporter och uppsatser Nr. 5, 56 p. ¨ stlin, E., 1963b. Barkuppgifter fo¨ r tall, gran, bjo¨ rk m. fl. Del. 2. O Barkuppgifter fo¨ r bonitets- och a˚ lderklasser och fo¨ r olika sortiment. Bark data for pine, spruce, birch etc. Part 2. Bark data for site- and ageclasses for sawlogs and pulpwood. Skogsho¨ gskolan. Institutionen fo¨ r skogstaxering. Rapporter och uppsatser Nr. 6, 37 p. Pa¨ ivinen, R., 1978. Kapenemis- ja kuorimallit ma¨ nnylle, kuuselle ja koivulle. Summary: taper and bark thickness models for pine, spruce and birch. Folia For. 353, 1–32. Petrini, S., 1921. Stamformsunderso¨ kningar. En sammanfattande analys om nordiska tallmaterial med avseende pa˚ faktorer som bestemmer nogranheten vid aptering pa˚ rot. Summary: Stem Form Investigations. Meddelanden fra˚ n statens skogsfo¨ rso¨ ksanstalt. Ha¨ fte 18, Stockholm. Saikku, O., Rikkonen, P., 1976. Kuitupuun kuoren ma¨ a¨ ra¨ ja siihen vaikuttavat tekija¨ t. Summary: bark amount of pulpwood and factors affecting it. Folia For 262, 1–21. Wredlind, E., 1917. Om tallens och granens bark (Studies on the Bark of Pine and Spruce Trees),. Skogsva˚ rdsfo¨ reningens Tidskrift, Stockholm (in Swedish).
Modelling bark thickness of Picea abies with taper curves Jouko Laasasenaho*, Timo Melkas, Sari Alde´n Department of Forest Resource Management, University of Helsinki, Faculty of Agriculture and Forestry, Latokartanonkaari 7 (PL 27), 00014 University of Helsinki, Finland Received 29 September 2003; received in revised form 22 July 2004; accepted 12 October 2004
Abstract Models predicting bark thickness at 14 different relative stem heights were derived, and factors affecting bark thickness were analyzed. The study material was collected from the tracts of the fifth National Forest Inventory (NFI) during the years 1968– 1971. It consists of sample tree data measured from 1864 Norway spruce (Picea abies) stems. Bark thickness was measured with bark gauges at the following 14 different relative heights: 1, 2.5, 5, 7.5, 10, 15, 20, 30, 40, 50, 60, 70, 80, and 90%. Linear regression models predicting bark thickness at each relative height were derived. These models were combined into an equation group in which double bark thickness at each relative height was predicted with both endogenous variables (bark thickness at neighboring relative heights) and exogenous variables (tree age, tree height, and breast height diameter). This equation group was solved with the Thomas algorithm. Predicted bark thickness values were finally joined with spline functions to form a bark curve model. The models gave unbiased estimates for both bark thickness and bark volume. The relative error for bark thickness varied from 13.7 to 27.9% and the standard error for stem volume under bark varied from 5.27 to 7.79 dm3 depending on the amount of input data. Bark thickness was found to correlate with diameter at breast height, tree height, tree age and stem tapering. Climatic zone also clearly affected the amount of bark. By combining the models derived in this study with existing stem curve models, it is possible to accurately calculate stem volume both under and over bark for any arbitrary portion of the stem. # 2004 Elsevier B.V. All rights reserved. Keywords: Bark; Bark model; Spruce bark; Polynomial model; Taper curve; Stem form; Spline function; Equation group
1. Introduction In Nordic countries, bark-related research has concentrated on the analysis of factors influencing * Corresponding author. Tel.: +358 9 191 58176; fax: +358 9 191 58159. E-mail address: [email protected] (J. Laasasenaho).
bark thickness and the derivation of models predicting bark thickness at breast height. The stem volumes over and under bark have been traditionally presented in a tabular form although nowadays the volumes are commonly calculated with volume functions or stem curve models (e.g. Laasasenaho, 1982; Ahonen, 1996). In bark-related research, emphasis has been on the prediction of the amount of bark found in harvested timber. Research on the
0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.10.058
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amount of bark in standing trees has been very limited. In Finland, the first comprehensive set of volume tables for domestic tree species was published by Ilvessalo (1948). In these tables, bark amount was presented as bark percentage of the total stem volume as a function of tree species, tree height, diameter at breast height, and stem form. They were derived by graphical smoothing. Jonson (1912) published a respective set of tables for conifers in Sweden a few decades before this. Studies on the amount of bark in harvested timber has been carried out by Aro (1935), Heiskanen and Rikkonen (1976), and Saikku and Rikkonen (1976). In research related to bark prediction models, emphasis has been put on models predicting bark thickness at breast height, since the prediction of stem growth has been based on breast height characteristics. Such models have been presented by Jonson (1912), ¨ stlin (1963a), Pa¨ ivinen (1978), and Mielika¨ inen O (1985). Variation of bark thickness along the stem has ¨ stlin (1963a), Hakkila (1967), and been studied by O Nylinder (1973). Hakkila (1967) derived models for bark taper as a function of stem height for Norway spruce, Scots pine, and silver birch. Ojansuu (1993) derived both over and under bark stem curve models for Scots pine. Within models, stem form has been depicted as multidimensional vectors of polar coordinates. Ka¨ rki et al. (1999) derived models predicting bark proportion for Gray alder based on mixed linear models (Lappi, 1986). In Sweden, Jonsson and Nylinder (1990) derived bark functions for predicting a bark curve for Scots pine. Hakkila et al. (1995) studied the effects of bark on the value of first thinning Scots pine timber and its suitability for various purposes. In Finland, the estimation of standing tree stem volumes is currently based on stem curve models and volume functions that include bark (Laasasenaho, 1982). In order to improve estimation of stem volume and stem growth, prediction of stem form, and stem bucking, bark amount should be predicted reliably. The purpose of this study is to examine and derive models for predicting bark thickness at various stem heights of Norway spruce and to consequently derive models and methods for estimating stem volumes over and under bark. Bark thickness is calculated with a method based on the use of spline functions.
2. Material 2.1. Characteristics of the data The research material consists of 1864 Norway spruce (Picea abies) stems measured from sample plots of the fifth National Forest Inventory (NFI), which was carried out during the years 1968–1971. The sample plots were laid out in a systematic manner representing the whole of Finland. The sample trees were selected from the plots with relascopes (BAF 2 m2/ha). Bark thickness was measured from each stem with bark gauges at 14 different relative heights. More information on the gathering and characteristics of the study material is presented in Kuusela and Salminen (1969) and in Laasasenaho (1982). The measured stem variables used in this study are: - diameter over bark at relative heights 1, 2.5, 5, 7.5, 10, 15, 20, 30, 40, 50, 60, 70, 80 and 90% (di); - bark thickness at the same relative heights (bi); - diameter over bark at breast height (d1.3); - bark thickness at breast height (b1.3); - upper diameter (d6.0); - tree height (h); - tree age (t); - date of measurements (date). Minimum and maximum values, means (x) and standard deviations (S.D.) of the central sample tree measurement characteristics (d1.3, b1.3, h, and t), and volumes (v, dm3) are presented in Table 1. In this study, bark amount is expressed either as double bark thickness or as the proportion of over bark diameter or volume. Forest type (site), climatic zone (length of the growth period), and development class were measured as stand variables. The classifications can be seen in Table 1 Characteristics of the data
d1.3 (cm) b1.3 (cm) h (m) t (years) v (dm3)
Minimum
Maximum
Mean
S.D.
1.5 0.2 1.8 4 1
61.9 3.8 32.7 299 3789
18.0 1.25 13.8 66 265
8.6 0.54 5.8 44 312
v, volume (dm3).
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
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Table 2 Classifications of forest type, development class, and climate zone Site class
Forest type
Class
Development class
Climatic zone
The length of growing period (days)
1 2 3 4 5 6 7 8
Very rich sites (OMaT) Rich sites (OMT) Damp sites (MT) Sub-dry sites (VT) Dry sites (CT) Barren sites (ClT) Rocky sites Hill top sites and fields
0 1 2 3 4 5 6 7 8
Forestry land other than forest land Open regeneration area and seed tree stand Seedling and sapling stands with standards Seedling and sapling stands Thinning stands Preparatory stands Mature stands Shelter wood stands Low yielding stands
1 2 3 4 5 6 7
<125 125–135 135–145 145–155 155–165 165–175 >175
(Kuusela and Salminen, 1969).
Table 2. Stem form is expressed with tapering, (d1.3 d6) and the classification of stem form is done by using 1-cm taper classes. 2.2. Correction of bark thickness values measured at relative heights and at breast height Bark thickness has been measured by using bark gauges. Laasasenaho and Sevola (1972) and Heikurainen (1984) have shown that the use of bark gauges may easily lead to biased measurement results, because the results depend on the power with which you strike the bark gauge to the surface of the tree. The authors have also found out that the results are dependent on the time of year of the measurement. Therefore, the influence of measurement date on measured bark thickness was examined in order to eliminate any time-related systematic errors in the measured bark thickness values. The material of this study was measured between May and October. Mean and standard deviation values for absolute and relative bark thickness at each relative height were calculated for each measurement month. In order to eliminate the effect of a tree’s size on the thickness of bark, the study was conducted by using relative bark thicknesses for each measurement month (Table 3). Bark thickness measurements were found to increase from May to July and in turn, decrease in October. It seems that relatively soft early growth allows the bark gauge to sink not only through the bark but also through the growing annual ring. These obvious systematic errors were corrected by using
linear regression models. First, all measurement days from the 1st of June to the end of August were numbered consecutively from 1 to 92. All other measurement days received a value zero. A regression model was then derived for each relative height. Relative bark thickness at the respective height (bi/di) was selected as the dependent variable. The independent variables included were diameter at the respective height (di), tree age (t), ratio of tree age to respective diameter (t/di), consecutive measurement date number (date), and the square of the consecutive measurement date number (date2). Next, bark thickness (best) for each day between 1.6 and 31.8 was calculated with the derived regression models. Another estimation for the same bark thickness (bzero) was calculated by setting date to zero. The measured bark thickness was finally corrected by subtracting the difference of best and bzero from the measured value. The correction was made only if the difference was positive. It was assumed that the difference of best and bzero would approach zero at both the beginning of June and the end of August. This did not, however, happen. At relative heights of 70, 80, and 90%, the difference turned negative already a week before the end of August. On the other hand, the differences were still clearly positive at the end of August at relative heights of 1–20%. Final corrections were made using time intervals beginning from June to mid-September at relative heights of 1–20% and the beginning of June to the end of August at relative heights of 30–90%. Bark thickness values measured at breast height were also corrected. Breast height was transferred to a
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Table 3 Mean and standard deviation values for relative bark thickness (bi/di) at each relative height Variables (%)
Relative height (%) 1
2.5
5
7.5
10
15
20
30
40
50
60
70
80
90
Mean
May (n = 392) Mean S.D.
8.55 2.84
8.44 2.81
8.10 2.82
7.78 2.87
7.46 2.70
7.22 2.50
7.13 2.51
7.20 2.47
7.41 2.46
7.76 2.53
8.42 2.82
9.10 3.14
10.40 3.82
13.90 5.63
8.49 1.73
June (n = 378) Mean S.D.
9.37 3.02
9.27 3.07
8.86 3.19
8.49 3.14
8.07 2.90
7.73 2.69
7.62 2.69
7.69 2.75
7.88 2.61
8.43 2.92
8.94 3.11
9.90 3.70
11.50 4.53
14.90 6.81
9.19 1.88
July (n = 354) Mean S.D.
10.80 2.93
10.80 3.05
10.40 3.13
9.98 3.12
9.61 3.06
9.34 2.98
8.95 2.68
9.02 2.48
9.23 2.41
9.57 2.46
10.20 2.55
11.00 3.07
12.70 4.11
16.20 6.19
10.56 1.84
August (n = 292) Mean 10.90 S.D. 3.10
10.97 3.49
10.65 3.67
10.20 3.58
9.70 3.53
9.31 3.46
9.01 3.25
8.76 2.88
8.84 2.57
9.09 2.56
9.52 2.77
10.30 2.87
11.70 3.79
15.70 5.93
10.33 1.72
September (n = 275) Mean 9.45 S.D. 2.84
9.34 2.86
9.18 2.96
8.19 2.91
8.56 2.77
8.41 2.82
8.17 2.61
8.12 2.50
8.36 2.52
8.71 2.64
9.25 2.74
10.10 2.94
11.50 3.45
14.95 5.09
9.45 1.76
October (n = 173) Mean 7.60 S.D. 2.22
7.68 2.42
7.21 2.44
6.88 2.50
6.64 2.23
6.35 2.10
6.34 2.10
6.36 2.05
6.48 1.82
6.86 1.87
7.26 1.97
8.34 2.17
9.59 2.34
13.50 3.70
7.65 1.84
relative height by dividing 1.3 with tree height. The minimum relative breast height was 3.96% and the maximum was 72.2%. Each observation was assigned to the nearest fixed relative height. Bark thickness measured at breast height was corrected by using the respective regression model presented in Table 4.
2.3. Additional measurements The NFI material includes measured bark thickness data only up to the relative height of 90%. However, in order to derive complete under bark stem curves it was considered necessary to acquire some additional
Table 4 Coefficients and R2 values of the bark thickness correction models Dependent variables
b1/d1 b2.5/d2.5 b5/d5 b7.5/d7.5 b10/d10 b15/d15 b20/d20 b30/d30 b40/d40 b50/d50 b60/d60 b70/d70 b80/d80 b90/d90 a
Coefficients.
R2
Independent variables a
a
di
11.6012 11.6848 11.4768 11.4221 10.7146 10.2761 10.0127 9.5983 9.4071 9.9255 10.5445 11.5314 13.4232 18.8119
0.0163 0.0209 0.0257 0.0277 0.0278 0.0278 0.0277 0.0292 0.0319 0.0372 0.0462 0.0619 0.0950 0.2237
a
a
t
t/di
0.0175 0.0232 0.0256 0.0247 0.0255 0.0255 0.0256 0.0266 0.0281 0.0261 0.0254 0.0233 0.0192 0.0103
0.0022 0.0234 0.0203 0.0199 0.0694 0.0764 0.1016 0.2052 0.3244 0.4122 0.6180 1.0140 1.6555 4.1523
a
2a
Date
Date
0.0740 0.0744 0.0707 0.0652 0.0599 0.0559 0.0519 0.0545 0.0545 0.0537 0.0541 0.0502 0.0674 0.0546
0.0007 0.0007 0.0006 0.0006 0.0005 0.0005 0.0005 0.0006 0.0006 0.0006 0.0006 0.0006 0.0008 0.0006
0.4216 0.4539 0.4776 0.4792 0.4909 0.4879 0.4688 0.4494 0.4501 0.4209 0.3958 0.3488 0.3160 0.3215
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
measurements above this height. A total of 19 tops were measured at cutting sites in the Porvoo and Helsinki region (Southern Finland) in September 1997. Diameters under and over bark were measured at the middle of each annual height growth starting from the latest annual height growth. Measurements were taken from as many annual height growths as possible. The amount of measurements taken from a single top ranged between 4 and 10. The measurements were made by using a digital caliber gauge. Diameters under bark were measured after the bark was carefully carved away from the measurement point with a knife. Additionally, diameters over and under bark for five tops were measured with a year ring microscope having an accuracy of 0.01 mm. This was performed in order to see whether the usage of a knife caused any systematic error to the measurements. However, comparisons made between values measured manually and those measured with the microscope showed no systematic error.
3. Methods 3.1. Modelling of bark thickness Factors influencing bark thickness were studied by analyzing correlation matrices and scatter plots of various variables. Covariance analysis was used in the case of continuous and non-continuous variables such as site class and climatic zone. All statistical analyses were carried out with the SAS statistical software package. The bark thickness functions were created by deriving regression models depicting bark thickness at each relative height. The independent variables included in these models were bark thickness at the preceding and following relative heights, diameter over bark at breast height, tree height, and tree age. The independent variables included were selected by studying the correlation between the dependent and various potential independent variables and by comparing the R2 values of models including various combinations of the independent variables. The inclusion of each independent variable was verified with t-tests. Regression coefficients were solved by using the least sum of squares method. The general
39
form of the derived model is presented in Eq. (1). The main characteristics of the derived models are presented in Table 9. bi ¼ a þ b1 bi1 þ b2 biþ1 þ b3 d1:3 þ b4 h þ b5 t (1) where bi is the bark thickness at relative height i, bi 1 the bark thickness at relative height i 1, bi + 1 the bark thickness at relative height i + 1, d1.3 the diameter over bark at breast height, h the tree height, t the tree age, a the constant, and b1, . . ., b5 are the coefficients. The individual bark thickness functions were then included in an equation group (Laasasenaho, 1982, p. 26 and 33) in which bark thickness at each relative height is depicted with both endogenous and exogenous variables. The endogenous variables included in the equation group were bark thickness at the preceding and following relative heights, and the respective exogenous variables were diameter over bark at breast height, tree height, and tree age. The equation group model used in this study requires far less calculations than the model proposed by Kilkki and Varmola (1981), which includes bark thickness at all other relative heights as endogenous variables. The resulting equation group includes equations for estimating bark thickness at each fixed relative height. The equation group can be solved if the exogenous variables (d1.3, h, and t) in the equation group are known or estimated. In addition, we can get more accurate results if bark thickness for one or more of the fixed relative heights is known. However, in practice, bark thickness at any specific relative height is not known. Therefore, we used measured or estimated bark thickness at the breast height to estimate bark thickness at the nearest relative height. Estimation of bark thickness at the nearest relative height was done by multiplying bark thickness at breast height with the ratio di/d1.3. Diameter for the fixed relative height (di) that is closest to breast height was calculated by using Laasasenaho’s polynomial stem curve function. The equation group was solved using Thomas’s algorithm. A detailed description of this algorithm is presented by Lahtinen and Laasasenaho (1979, p. 12) and Laasasenaho (1982). Bark thickness curves were derived by using a cubic spline function of the bark thickness values estimated by the equation group. We tested this calculation system using either measured or estimated bark thickness at breast height in the following three different ways:
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Model 1 Model 2 Model 3
b1.3 bˆ 1:3 = f(d1.3) bˆ 1:3 = f(d1.3, h, t)
3.2. Model reliability The reliability of the bark thickness models was assessed by comparing the predicted bark thickness at relative height with the respective measured values. Bias (b, cm) and standard deviation (S.D., cm) were calculated for the observed differences at each fixed relative height. Standard errors for bark thickness and absolute and relative bark volumes were calculated by using different variable classes. Under bark stem volumes estimated with cubic splines (Lahtinen and Laasasenaho, 1979; Lahtinen, 1988) were regarded as measured under bark stem volumes. Bias and standard error values were calculated using the following formulas. Pn ðˆy yi Þ b ðcmÞ ¼ i¼n i (2) n1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn yi yi Þ2 i¼n ðˆ S:D:yˆ ðcmÞ ¼ (3) n1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn yi yi =ˆyi Þ2 i¼1 ðˆ S:D:yˆ ð%Þ ¼ (4) n1 where b (cm) is the bias, S:D:yˆ (cm) the standard error, S:D:yˆ (%) the relative standard error, yˆ the estimated bark thickness or under bark volume, y the measured bark thickness or under bark volume, and n is the number of observations. 4. Results 4.1. Variation of bark thickness Average double bark thickness was greatest at the base (1% relative height) 2.02 cm and decreased uniformly towards the top (90% relative height) 0.45 cm. Bark proportion (from the diameter) decreased from the base to the 20% relative height being 7.5% at its minimum (Fig. 1). After that the bark proportion increased first slowly then rapidly being finally 15% at a relative height of 90%.
Fig. 1. Mean values of double bark thickness (cm) and proportional bark thickness (%) as a function of relative height.
If we look at the proportion of bark from the basal area, the bark proportion varies from 14 to 27% being at its minimum at a relative height of 20% (Table 5). Standard deviation, on the other hand, is at its minimum at the relative height of 40%. The general form of bark curves at different breast height diameter classes (Fig. 2) is highly similar and therefore we can assume that the general form of a bark curve does not depend on tree size. No clear indications of the occurrence of different bark types were found although some signs of rinding were found in most barren sites (CT). The correlation between bark thickness at different relative heights (Table 6) was found to be clearly lesser than that of over bark diameter at different relative heights, while the correlation between bark thickness and stem diameter was rather strong. The correlation between bark thickness values at adjoining relative heights is, however, higher than the correlation between bark thickness and stem diameter at a given relative height except at the upper part of the stem. When comparing the over bark and under bark stem curves, it was found that tapering at the base of the stem is greater in the over bark stem curve, because the proportion of bark decreases (Fig. 1; Table 5). Whereas at relative heights greater than 60% tapering of the under bark stem curve is greater. The difference is largest at a relative height of 20%. Dependency of bark thickness between different tree, site class, and stand variables was studied at breast height bark thickness. Bark thickness at breast height was found to be strongly correlated with breast height diameter (0.76), tree height (0.61), and tree age
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
41
Fig. 2. Double bark thickness (cm) as a function of relative tree height by breast height diameter classes.
(0.61). In order to estimate the dependency of bark thickness on other variables we removed the effect of breast height diameter, tree height, and age to bark thickness by using covariance analyse. The model which is used in the analyses is b1:3ij ¼ m þ ai þ b1 d1:3 þ b2 h þ b3 t þ eij , where b1:3ij is breast height bark thickness in class i and in tree j, m is the general average, ai is the value of a quality variable in class i, b1, b2, b3 are regression coefficients of continuous variables, b1.3, h, t are continuous variables (covariants), and eij is the random error. It was found out that bark thickness is dependent on climate zone (the length of growth period, see classes in Table 2), tree form (d1.3 d6), and development class. The difference was most significant in climate zone 2. Site class (forest type) was not significant so that there is no dependency between bark thickness and forest type
(Table 7). Bark proportion is greater in the northern parts of the country. Climate zone was the most significant quality variable. 4.2. Bark thickness at the top of a tree Bark thickness was found to increase from the top to the base of the stem. The thinnest bark at the top of tree was however in the third and fourth annual shoots. In the two most upper annual shoots bark is somehow folded and therefore bark thickness seems to decrease in the third annual shoot (Table 8; Fig. 3). 4.3. Bark models Regression models (1) predicting bark thickness at a given relative height were derived for each of
Table 5 Bark percentage of over bark diameter and basal area at different relative heights Relative height
1 2.5 5 7.5 10 15 20 30 40 50 60 70 80 90
Proportion of bark (%) (diameter)
Proportion of bark (%) (basal area)
Mean
S.D.
Minimum
Maximum
Mean
S.D.
Minimum
Maximum
8.8 8.7 8.5 8.1 7.7 7.5 7.4 7.5 6.9 8.1 8.7 9.6 11.0 14.8
2.9 3.0 3.1 3.1 2.9 2.9 2.7 2.6 2.3 2.7 2.9 3.2 4.0 6.0
2.8 3.2 2.7 2.5 2.5 2.7 2.5 2.5 2.8 2.7 3.0 3.0 3.6 4.7
23.6 22.5 24.0 23.8 22.6 22.0 22.8 23.4 19.3 23.1 26.7 35.3 40.0 57.1
16.7 16.6 16.1 15.4 14.6 14.3 14.1 14.4 14.7 15.4 16.5 18.2 20.5 27.0
5.2 5.4 5.6 5.6 5.3 5.2 5.0 4.8 4.6 4.8 5.1 5.7 6.9 9.5
5.4 6.3 5.3 4.9 5.0 5.3 4.9 5.0 6.0 5.4 5.8 6.0 7.1 9.1
41.6 39.9 42.2 42.0 40.1 39.1 40.4 41.3 38.8 40.8 46.2 58.1 64.0 81.6
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Table 6 Correlation between bark thickness and stem diameter at fixed relative heights
d1 d2.5 d5 d7.5 d10 d15 d20 d30 d40 d50 d60 d70 d80 d90
b1 b2.5 b5 b7.5 b10 b15 b20 b30 b40 b50 b60 b70 b80 b90
b1 b2.5 b5 b7.5 b10 b15 b20 b30 b40 b50 b60 b70 b80 b90
d1
d2.5
d5
d7.5
d10
d15
d20
d30
d40
d50
d60
d70
d80
d90
1.00 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.95 0.93 0.92 0.88
1.00 0.99 0.99 0.99 0.99 0.98 0.98 0.97 0.96 0.95 0.94 0.92 0.88
1.00 0.998 0.996 0.99 0.99 0.99 0.98 0.97 0.96 0.95 0.93 0.89
1.00 0.999 0.997 0.99 0.99 0.98 0.98 0.97 0.95 0.93 0.89
1.00 0.998 0.997 0.99 0.99 0.98 0.97 0.95 0.93 0.89
1.00 0.999 0.996 0.99 0.98 0.97 0.96 0.94 0.90
1.00 0.998 0.99 0.99 0.98 0.96 0.94 0.90
1.00 0.997 0.99 0.98 0.97 0.95 0.91
1.00 0.997 0.99 0.98 0.96 0.92
1.00 0.996 0.99 0.97 0.93
1.00 0.99 0.98 0.94
1.00 0.99 0.95
1.00 0.97
1.00
b1
b2.5
b5
b7.5
b10
b15
b20
b30
b40
b50
b60
b70
b80
b90
1.00 0.92 0.88 0.85 0.83 0.82 0.81 0.80 0.78 0.76 0.73 0.71 0.64 0.52
1.00 0.94 0.91 0.89 0.86 0.85 0.82 0.80 0.76 0.73 0.69 0.62 0.49
1.00 0.95 0.92 0.89 0.86 0.83 0.81 0.77 0.73 0.69 0.62 0.48
1.00 0.95 0.91 0.89 0.84 0.81 0.77 0.73 0.69 0.61 0.47
1.00 0.94 0.90 0.86 0.83 0.78 0.74 0.70 0.63 0.49
1.00 0.95 0.91 0.88 0.84 0.79 0.75 0.67 0.54
1.00 0.94 0.92 0.87 0.83 0.78 0.71 0.57
1.00 0.95 0.91 0.87 0.82 0.74 0.61
1.00 0.94 0.90 0.85 0.77 0.64
1.00 0.93 0.88 0.81 0.68
1.00 0.91 0.84 0.71
1.00 0.88 0.75
1.00 0.83
1.00
d1
d2.5
d5
d7.5
d10
d15
d20
d30
d40
d50
d60
d70
d80
d90
0.73 0.69 0.66 0.65 0.66 0.70 0.72 0.75 0.77 0.78 0.77 0.79 0.76 0.71
0.72 0.69 0.66 0.65 0.66 0.70 0.72 0.75 0.77 0.78 0.77 0.79 0.76 0.71
0.71 0.68 0.65 0.64 0.65 0.69 0.72 0.75 0.77 0.78 0.77 0.79 0.77 0.72
0.71 0.67 0.64 0.64 0.65 0.69 0.72 0.75 0.77 0.78 0.77 0.79 0.77 0.72
0.70 0.67 0.64 0.63 0.64 0.68 0.71 0.74 0.76 0.78 0.77 0.79 0.77 0.72
0.70 0.66 0.63 0.62 0.63 0.68 0.71 0.74 0.76 0.77 0.77 0.79 0.77 0.72
0.69 0.65 0.62 0.61 0.63 0.67 0.70 0.73 0.76 0.77 0.77 0.79 0.77 0.72
0.68 0.64 0.61 0.60 0.61 0.66 0.69 0.73 0.75 0.77 0.77 0.79 0.77 0.72
0.67 0.63 0.60 0.59 0.60 0.65 0.69 0.72 0.75 0.77 0.77 0.79 0.77 0.73
0.66 0.62 0.59 0.57 0.59 0.64 0.68 0.71 0.74 0.76 0.77 0.79 0.78 0.73
0.65 0.61 0.58 0.57 0.58 0.63 0.67 0.71 0.73 0.76 0.77 0.79 0.78 0.73
0.65 0.60 0.57 0.56 0.58 0.62 0.66 0.69 0.72 0.75 0.76 0.79 0.78 0.74
0.64 0.60 0.57 0.55 0.57 0.62 0.65 0.69 0.72 0.74 0.75 0.78 0.79 0.74
0.63 0.60 0.57 0.56 0.58 0.62 0.65 0.68 0.71 0.73 0.74 0.77 0.77 0.75
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
43
Table 7 Significance and F-values of covariance analyses Variable
F-value
Significance
d1.3 h t Site (forest type) Climatic zone Development class Stem form
4161.5 258.1 586.1 1.7 6.3 3.5 2.6
*** *** *** – *** *** **
(–) No significant; (*) 0.05 (quite significant); (**) 0.01 (significant); (***) 0.001 (very significant). Fig. 3. Cross-sectional disks taken from the middle of the three most resent annual height growth shoots.
Table 8 The number of measurements for each annual shoot, and the mean and the standard deviation of bark thickness in the middle of each annual shoots Consecutive annual height growth
n
Bark thickness in the middle of the annual shoot (mm)
Models to estimate bark thickness at breast height were made by using the following logarithmic functions:
Mean
S.D.
Model 2
1 2 3 4 5
19 19 19 18 16
3.01 3.15 2.55 2.85 3.22
0.94 0.65 0.65 0.50 0.93
lnðb1:3 Þ ¼ 1:66 þ 0:65 ln ðd1:3 Þ; R2 ¼ 0:66
Model 3
lnðb1:3 Þ ¼ 1:99 þ 1:00 ln ðd1:3 Þ
(5)
0:57 ln ðhÞ þ 0:20 ln ðtÞ; R2 ¼ 0:77
Annual shoots are numbered consecutively starting from the latest annual shoot.
(6)
T-values of all coefficients were very high (between 18.7 and 61.5). The breast height diameter (b1.3) explains 66% of the variations of bark thickness at breast height. By adding tree height and age to the function, the coefficient of determination is 11% higher.
the 14 relative heights. The regression coefficients and degrees of determination of these models are presented in Table 9.
Table 9 Regression model coefficients and degrees of determination (R2) Dependent variables
b1 b2.5 b5 b7.5 b10 b15 b20 b30 b40 b50 b60 b70 b80 b90 a
Coefficients.
R2
Independent variables a
a
bi 1
0.10320 0.01240 0.02120 0.04960 0.03270 0.04220 0.00841 0.02120 0.01140 0.01950 0.01090 0.03690 0.00839 0.06170
– 0.35870 0.34770 0.44780 0.50530 0.66300 0.53920 0.37910 0.50860 0.45450 0.53200 0.43270 0.47100 0.51840
bi + 1
a
0.96550 0.62120 0.62920 0.51450 0.44570 0.57960 0.41840 0.58410 0.42580 0.47860 0.42390 0.43940 0.49240 –
d1.3
a
0.01640 0.00292 0.00250 0.00271 0.00271 0.00329 0.00137 0.00347 0.00165 0.00385 0.00021 0.00439 0.00162 0.00144
a
a
h
t
0.00208 0.00301 0.00321 0.00457 0.00380 0.00380 0.00016 0.00304 0.00027 0.00201 0.00142 0.00102 0.00006 0.00447
0.00004 0.00056 0.00035 0.00009 0.00014 0.00027 0.00020 0.00024 0.00023 0.00009 0.00008 0.00011 0.00010 0.00009
0.86350 0.92180 0.93530 0.93980 0.92920 0.93870 0.93910 0.93320 0.93520 0.92170 0.90160 0.87570 0.84960 0.71530
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Table 10 Average and standard error of observed bark thickness (cm) and average of predicted bark thickness (cm) at each relative height Measured (bi)
Measuring point (%)
1 2.5 5 7.5 10 15 20 30 40 50 60 70 80 90 a
2.02 1.70 1.48 1.34 1.23 1.16 1.11 1.05 0.98 0.91 0.82 0.72 0.59 0.45
S:D:measuredðbi Þ 0.81 0.68 0.59 0.51 0.47 0.44 0.42 0.41 0.38 0.35 0.32 0.28 0.24 0.18
The estimation method Measured b1.3a
b1.3 = f(d1.3)a
b1.3 = f(d1.3, h, t)a
2.01 1.69 1.47 1.33 1.23 1.16 1.11 1.05 0.98 0.91 0.82 0.72 0.59 0.45
2.01 1.70 1.47 1.33 1.22 1.15 1.10 1.04 0.98 0.90 0.82 0.72 0.59 0.45
2.00 1.69 1.47 1.33 1.23 1.16 1.11 1.05 0.98 0.91 0.82 0.72 0.59 0.45
Estimated bark thickness (bi).
Table 11 Bias (b, dm3) and standard deviation (S.D., dm3) values of the difference of predicted and observed bark volumes and under bark stem volumes by the studied models b (dm3)
Model 1 Model 2 Model 3
0.16 0.10 0.20
S.D. (dm3)
Volume of bark
5.29 7.79 7.26
Volume without bark 3
S.D. (%)
Estimated (dm )
S.D. (%)
Estimated (dm3)
15.7 23.0 21.5
33.6 33.7 33.6
2.3 3.4 3.1
231.3 231.3 231.4
33.8a a
231.2a
Measured value.
Table 12 Bias (b, cm), absolute standard error (S.D., cm), and relative standard error (S.D. (%)) of bark thickness at the relative heights by the studied models Measuring point (%)
1 2.5 5 7.5 10 15 20 30 40 50 60 70 80 90
Estimated b1.3(d1.3)
Measured b1.3
Estimated b1.3(d1.3, h, t)
b (cm)
S.D. (cm)
S.D. (%)
b (cm)
S.D. (cm)
S.D. (%)
b (cm)
S.D. (cm)
S.D. (%)
0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.42 0.30 0.22 0.17 0.16 0.17 0.18 0.19 0.18 0.18 0.18 0.16 0.15 0.12
20.96 18.21 15.89 14.11 13.70 14.22 15.13 16.89 18.08 19.39 20.95 21.84 25.02 27.74
0.01 0.01 0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00
0.52 0.44 0.39 0.36 0.32 0.28 0.25 0.23 0.21 0.20 0.19 0.17 0.15 0.12
25.17 25.15 25.45 25.81 25.68 24.08 23.36 21.99 21.52 21.60 22.60 22.92 25.56 27.82
0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.49 0.40 0.34 0.30 0.27 0.24 0.23 0.22 0.21 0.20 0.19 0.16 0.15 0.12
23.91 23.24 22.72 22.26 21.78 20.74 20.77 20.77 21.19 21.65 22.86 23.13 25.50 27.86
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
Average and standard error of observed bark thickness and average of predicted bark thickness at each studied relative height were calculated and tabulated (Table 10).
4.4. Reliability of the derived bark models The most accurate estimates within the equation group (Eq. (1)) were achieved as expected by model 1, which incorporated the measured bark thickness at breast height to the nearest relative height (Tables 11 and 12). In this case the absolute standard error of under bark stem volume was 5.29 dm3, which represents a relative standard error of 2.2%. The standard error of bark thickness at the studied relative heights decreased from 4.2 mm at the base to 1.2 mm at the top. The most accurate estimate was achieved at a relative height of 10% (Table 12). Prediction accuracy decreases both toward the base and toward the top of the stem. The result is quite understandable as the 10% relative height is closest to breast height in most cases. The relative standard error was 13.7% at its minimum. The relative standard errors of under bark stem volume in models 2 and 3 were more than one per cent higher than that of model 1 being, however, below 3.5%. The relative standard errors of bark thickness at the studied relative heights were also higher in models 2 and 3. The bark models were found to be unbiased. Models 1 and 3 produce nonetheless very slight underestimates at the base of the stem.
5. Discussion The purpose of this study was developed bark thickness models for predicting bark thickness at arbitrary stem heights of Norway spruce. In analyzing the factors, which can be used to estimate bark thickness, bark thickness models were developed, which can be combined to present taper curve models. The reliability of the models is influenced by the sampling error of the study material, errors related to the measuring of bark thickness, and the procedure implemented in order to eliminate the apparent bias in these measurements. The material used in this study is a representative sample of the whole Norway spruce population of Finland.
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In this study, bark thickness was measured with bark gauges which have been found to over-estimate bark thickness on the average by 0.25 mm (von Essen, 1974). Measurements were performed during the growth season and therefore the latest annual ring was yet soft (Laasasenaho and Sevola, 1972). This is one reason for the overestimation. Errors due to the time of measurement were corrected with separately derived regression models. The latest annual ring was soft for a longer period of time in the lower parts of the stem than in the upper parts and therefore the corrections were made for different time periods depending on measurement height. It seems that lignification was ended for the whole stem by mid-September. Errors related to the person carrying out the measurements were not taken into account. Bark proportion from diameter decreased from the base to the 20% relative height being 7.5% at its minimum. After that bark proportion increased being finally 15% at a relative height of 90%. These results are consistent with those presented by Hakkila (1967). Measurements of bark thickness at the top of trees showed that bark was thicker at the two uppermost shoots than in the two next shoots. Otherwise bark increased towards the base of the stem. Regression models predicting bark thickness for the 14 relative heights were derived. Bark thickness at each relative height was predicted by both endogenous variables (bi 1, bi + 1) and exogenous variables (d1.3, h, t). The coefficient of determination of these models was between 71.5 (90% height) and 94% (7.5% height). Models were combined into an equation group and solved by using the Thomas algorithm. Predicted bark thickness values were finally joined with spline functions to form a bark curve model. The derived models give unbiased estimates for bark thickness and bark volume. The absolute standard error for bark thickness ranged depending on the studied relative height from 5.2 to 1.2 mm. The respective relative standard error was 13.7% at its minimum at a relative height of 10 and 27.9% at its most, at the top of the stem. The standard error of bark volume varied depending on the model, from 5.3 to 7.8 dm3, which represents a 2.2–3.4% standard error of volume without bark. The standard error calculated from average bark volume was 15.7–23.0% for the models 1–3. Model 1, in which bark thickness was
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J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47
measured at breast height, gave the best estimates for bark volume. Standard error was then 15.7%. Model 3, where bark thickness at breast height was estimated with the help of diameter, height and age variables, gave a standard error of 21.5% for bark volume. The best estimates could be obtained for trees on moist sites in Southern Finland having a breast height diameter of 25 cm or more. The analyses of the factors effecting bark thickness of Norway spruce proved that correlation between bark thickness and stem diameter is strong (Table 7, formula 5, see also Wredlind, 1917; Petrini, 1921). Bark thickness also depends on tree height, tree age, and tree form. Bark thickness varies also according to climate zones and development classes. When bark thickness models took into account tree height, diameter, and age, forest type was not a significant classify variable. Climate zone was no longer a significant variable either, except in the case of climate zone 2, in which the systematic error of bark volume was 2.3 dm3. The differences in other climate zones were not significant. Bark proportion is greater in the northern parts of the country. These findings are consistent with those presented in earlier studies ¨ stlin, 1963b; Ilvessalo, 1965). (O The accuracy of the models can be improved a little by deriving correction coefficients for different climatic zones or by deriving separate models for each zone. The better solution is that the length of the growth season, or thermal temperature sum, is taken account as an exogenous variable in the models. The use of e.g. thermal temperature sum as an indicator of geographical location is also possible with this technique. When comparing the over bark and under bark stem curves it was found that tapering at the base of the stem is greater in the over bark stem curve, because the proportion of bark decreases (Fig. 1; Table 5). Whereas at relative heights greater than 60% tapering of the under bark stem curve is greater. The difference is largest at a relative height of 20%. The results found in this study concerning tapering of the over bark and under bark stem curves confirm those found in previous studies (e.g. Aro, 1935). The derived bark models are applicable for both research and practical forest inventory purposes. They can be used to predict under bark diameters at any desired stem height and subsequently bark volumes of
any desired portion of the stem. The models can also be incorporated with existing stem curve models (Laasasenaho, 1982). The derived models require information on tree height, breast height diameter, and tree age as input data. Because the measurement of tree age is cumbersome, models for the estimation of tree age should be made, or models can be derived without the tree age variable. By incorporating the equation group models derived in this study with existing stem curve software, under bark stem volumes can be estimated by stem height, breast height diameter, and tree age. Under bark stem curves can also be derived and consequently automatic bucking can be improved and enhanced. The resulting models can also be applied in order to improve growth and increment predictions. Respective equation group bark models for two other major tree species occurring in Finland; Scots pine and Silver birch, have been derived by Laasasenaho and Melkas and so far unpublished research manuscripts drawn up.
Acknowledgements This research has been conducted at the Department of Forest Resource Management with funding from Metsa¨ miesten sa¨ a¨ tio¨ . The checking of data was carried out by Sari Alde´ n. Timo Melkas has done his Master of Science of the bark models of Picea abies and graduated in year 2000. The study has been supervised by Prof. Jouko Laasasenaho. The computing facilities have been provided by the University of Helsinki (Department of Forest Resource Management). The translation of the article was done by Mark-Leo Waite with Ms. Nina Garlo proof reading the text. Sampo Keskinen provided us with helpful comments on the text. We would also like to thank Metsa¨ miesten sa¨ a¨ tio¨ and all others who have made this work possible. References Ahonen, O.-P., 1996. Puun muodon ennustaminen runkoka¨ yrilla¨ ja simuloimalla (Predicting stem form by using taper curves and simulation). Metsa¨ tehon katsaus 4, 8 (in Finnish). Aro, P., 1935. Tutkimuksia rinnankorkeus- ja katkaisula¨ pimitan vaikutuksesta ka¨ ytto¨ puun ja hakkuuta¨ hteen ma¨ a¨ ra¨ a¨ n. Referat: Untersuchungen u¨ ber den Einfluss des Brustho¨ hen- und Mini-
J. Laasasenaho et al. / Forest Ecology and Management 206 (2005) 35–47 maldurchmessers auf die Menge des Gebrauchsholzes und der Hiebreste (Studies on the breast height- and cutting diameter’s impact on the amount of commercial timber and waste wood). Metsa¨ ntutkimuslaitoksen julkaisuja 20 (4), 1–159 (in Finnish). von Essen, C., 1974. Metning av barktjockleken genom electric avka¨ nning (Measuring bark thickness with electronic recognition). Skogsho¨ gskolan, Insitutionen fo¨ r skogsuppskatning och inledning. Raporter och uppsatser Nr. 3. Stockholm. Hakkila, P., 1967. Vaihtelumalleja kuoren painosta ja painoprosentista (Variation patterns of bark weight and bark percentage by weight). Metsa¨ ntutkimuslaitoksen julkaisuja 62 (5), 1–37 (in Finnish). Hakkila, P., Kalaja, H., Saranpa¨ a¨ , P., 1995. Etela¨ -Suomen ensiharvennusma¨ nniko¨ t kuitu- ja energianla¨ hteena¨ (Pine stands as a source of the pulpwood and energy in the first thinning in southern Finland). Metsa¨ ntutkimuslaitoksen tiedonantoja 582, 1–100 (in Finnish). Heikurainen, M., 1984. Ma¨ nnyn kuoren paksuuden vaihtelu ja mittaustarkkuus (Variation of bark thickness and measuring accuracy in pine trees). Master Thesis. University of Helsinki. Department of Forest Resource Management. 84 p. (in Finnish). Heiskanen, V., Rikkonen, P., 1976. Havusahatukkien kuoren ma¨ a¨ ra¨ ja siihen vaikuttavat tekija¨ t (The amount of bark in softwood logs and the factors influencing it). Folia For. 250, 1–67 (in Finnish). Ilvessalo, Y., 1948. Pystypuiden kuutioimistaulukot. Summary: volume tables for standing trees. Metsa¨ tieteellisen tutkimuslaitoksen julkaisuja 34 (4), 1–140. Ilvessalo, Y., 1965. Metsa¨ narvioiminen (Forest Mensuration). WSOY. Porvoo., 400 p. (in Finnish). Jonson, T., 1912. Afsmalnings- och Tillva¨ xttabeller fo¨ r Tra¨ duppskattning (Volume and Yield Tables for Tree Estimation) Stockholm. , 22 p. (in Swedish). Jonsson, L., Nylinder, M., 1990. Tallbarkens tjocklek la¨ ngs stammen – Funktioner fo¨ r aptering (Bark thickness of Pine in the direction of the stem – functions of cross-cutting). The Swedish University of Agricultural Sciences Department of Forest Products. Report Nr. 212, 1–41. Ka¨ rki, T., Eerika¨ inen, K., Heinonen, J., Korhonen, K.T., 1999. Harmaalepa¨ n (Alnus incana) tilavuustaulukot (Volume tables for Gray alder (Alnus incana)). Folia For. (Metsa¨ tieteen aikakauskirja) 1, 39–49 (in Finnish). Kilkki, P., Varmola, M., 1981. Taper curve models for Scots pine and their applications. Acta For. Fenn. 174, 1–60 (Helsinki). Kuusela, K., Salminen, S., 1969. The 5th National Forest Inventory in Finland. General design, instructions for field and data processing. Commun. Inst. For. Fenn. 69 (4), 1–72. Laasasenaho, J., 1982. Taper curve and volume functions for pine, spruce and birch. Seloste: Ma¨ nnyn, kuusen ja koivun
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runkoka¨ yra¨ - ja tilavuusyhta¨ lo¨ t. Commun. Inst. For. Fenn. 108, 1–74. Laasasenaho, J., Sevola, Y., 1972. Havutukkien latvamuotolukujen vaihtelu (Variation of top form factor for the softwood logs). Folia For. 164, 1–20 (in Finnish). Lahtinen, A., 1988. On the construction of monotony preserving taper curves. Acta For. Fenn. 203 (Helsinki). Lahtinen, A., Laasasenaho, J., 1979. On construction of taper curves by using spline functions. Seloste: runkoka¨ yra¨ n muostaminen spline – funktiolla. Commun. Inst. For. Fenn. 95 (8), 1–63. Lappi, J., 1986. Mixed linear models for analyzing and predicting stem form variation of Scots pine. Seloste: ma¨ nnyn runkomuodon vaihtelun analysointi ja ennustaminen lineaarisen sekamallin avulla. Commun. Inst. For. Fenn. 134, 1–69. Mielika¨ inen, K., 1985. Koivusekoituksen vaikutus kuusikon rakenteeseen ja kehitykseen. Summary: effect of an admixture of birch on the structure and development of Norway spruce stands. Commun. Inst. For. Fenn. 133, 1–79. Nylinder, P., 1973. Virkesma¨ tning. Skogsho¨ gskolan. Inst. f. Virkela¨ ra¨ . (Wood measurement). Kompendium Nr. 5, (in Swedish). Ojansuu, R., 1993. Prediction of Scots pine increment using a multivariate variance component model. Tiivistelma¨ : ma¨ nnyn kasvun ennustaminen monimuutuja- ja varianssikomponenttimallilla. Acta For. Fenn. 239, 72. ¨ stlin, E., 1963a. Barkuppgifter fo¨ r tall, gran, bjo¨ rk m. fl. Del. 1 O Barkuppgifter fo¨ r la¨ n, regioner. Bark data for pine, spruce, birch etc. Part 1. Bark data for provinces and regions. Skogsho¨ gskolan. Institutionen fo¨ r skogstaxering. Rapporter och uppsatser Nr. 5, 56 p. ¨ stlin, E., 1963b. Barkuppgifter fo¨ r tall, gran, bjo¨ rk m. fl. Del. 2. O Barkuppgifter fo¨ r bonitets- och a˚ lderklasser och fo¨ r olika sortiment. Bark data for pine, spruce, birch etc. Part 2. Bark data for site- and ageclasses for sawlogs and pulpwood. Skogsho¨ gskolan. Institutionen fo¨ r skogstaxering. Rapporter och uppsatser Nr. 6, 37 p. Pa¨ ivinen, R., 1978. Kapenemis- ja kuorimallit ma¨ nnylle, kuuselle ja koivulle. Summary: taper and bark thickness models for pine, spruce and birch. Folia For. 353, 1–32. Petrini, S., 1921. Stamformsunderso¨ kningar. En sammanfattande analys om nordiska tallmaterial med avseende pa˚ faktorer som bestemmer nogranheten vid aptering pa˚ rot. Summary: Stem Form Investigations. Meddelanden fra˚ n statens skogsfo¨ rso¨ ksanstalt. Ha¨ fte 18, Stockholm. Saikku, O., Rikkonen, P., 1976. Kuitupuun kuoren ma¨ a¨ ra¨ ja siihen vaikuttavat tekija¨ t. Summary: bark amount of pulpwood and factors affecting it. Folia For 262, 1–21. Wredlind, E., 1917. Om tallens och granens bark (Studies on the Bark of Pine and Spruce Trees),. Skogsva˚ rdsfo¨ reningens Tidskrift, Stockholm (in Swedish).