林业科学  2004, Vol. 40 Issue (5): 16-24   PDF    
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文章信息

Li Fengri
李凤日
Modeling Crown Profile of Larix olgensis Trees
长白落叶松人工林树冠形状的模拟
Scientia Silvae Sinicae, 2004, 40(5): 16-24.
林业科学, 2004, 40(5): 16-24.

文章历史

Received date: 2003-11-24

作者相关文章

李凤日

长白落叶松人工林树冠形状的模拟
李凤日     
东北林业大学林学院 哈尔滨 150040
摘要:以长白山地区26 a生长白落叶松人工林为研究对象,采用枝解析的方法,测定了25株林木(直径10.5~24.9 cm)的树冠变量,并建立了预测树冠外侧形状的冠形模型。基于枝条着枝深度(DINC)和林木变量所建立的树冠形状模型包括:基径、枝长、着枝角度和弦长等预估模型。对于大小相同树木的主要枝条来讲,这些树冠变量是随着DINC的增加而增大;而林木的胸径(DBH)和树高(HT)变量很好地反映了不同大小树木的冠形变化。冠形预测模型预测效果良好,充分体现了树冠结构的变化趋势:树冠形状在树冠的中上部呈抛物线体,而在树冠的下部则为近圆柱体。文中所建模型,可以合理地描述长白落叶松人工林的树冠形状及其变化规律。
关键词长白落叶松    基径    枝长    着枝角度    树冠形状    枝解析    
Modeling Crown Profile of Larix olgensis Trees
Li Fengri     
Forestry College, Northeast Forestry University Harbin 150040
Abstract: Models for estimating crown profile of primary branches within live crown were developed in a 26-year-old Larix olgensis plantation in Changbai Mountain. Data of crown attributes were collected from 25 sample trees ranging in diameter at breast height (DBH) from 10.5 cm to 24.9 cm using the method of branch analysis. Crown profile models, including branch diameter, branch length, angle of origin, and branch chord length, were developed from branch attributes and tree variables. These crown attributes of primary branches for a specified tree increase with the depth into crown (DINC) increasing. DBH and total height (HT) were adequate variables of tree for summarizing the effects of external conditions on crown profile. The predicting models of crown profile were logical and perform well to represent the trends in crown structure. All crown shapes are parabolic in the upper and middle part of crown, and follow approximately cylinder at lower part of crown. Overall, the models were suitable in describing the trends and inherent variability of crown profile for Larix olgensis plantation.
Key words: Larix olgensis    Branch diameter    Branch length    Branch angle    Crown profile    Branch analysis    
1 Introduction

Photosynthesis, respiration, and transpiration of a tree mainly take place in the crown. Crown dimensions of a tree determine its foliage distribution and consequently affect its photosynthetic capacity. The relations between crown dimensions and foliage biomass may be also of considerable interest to better understand tree growth variations and relate them to tree status and environmental conditions. Thus an accurate description of crown profile that recognize to highly influence tree growth and stand dynamics is an important component of process-based models (Deleuze et al., 1996). The most detailed individual growth models incorporate crown variables to improve growth predictions (Mitchell, 1975; Cole et al., 1994).

Information on crown structure and branch characteristics is also important for estimating wood product quality and log grading (Maguire et al., 1994; Doruska et al. 1994; Roeh et al., 1997; M kinen et al., 1998). However, managers have had few quantitative tools available for linking silvicultural practices (e. g. initial spacing, thinning regime, harvest age) to branch development (Maguire et al., 1999). Hence, the quantitative crown profile models need to be studied for accurate description of crowns.

Crown shape has been widely studied in recent years for conifer species. Kajihara (1976) described morphology and dimensions of tree crowns in a Cryptomeria japonica plantation as consisting of an 'xposed' conical portion above lateral crown contact and a cylindrical portion below. Similar results were found in Larix olgensis plantations (Li et al., 1996) and in Black Spruce stands (Raulier et al., 1996). Hashimoto (1990) researched changes of crown morphology according to the branch characteristics in young Sugi stands and concluded that the light environment was the primary determinant of crown morphology and structure. Other researchers have been using regression equations that directly predict crown radius (or crown width) from various tree attributes (Mitchell, 1975; Raulier et al., 1996; Li et al., 1996; Baldwin et al., 1997; Hann, 1999; Gilmore, 2001), but they do not truly examine branch size (branch diameter).

Some studies have used the indirect method to develop equations for predicting branch attributes, such as branch diameter (BD), branch length (BL), and branch angle, through regression analysis and then indirectly computing crown radius from trigonometric relationships (Hann, 1999). Based on cross-sectional data from temporary plots, numerous crown profile models for many tree species have been developed for estimating branch characteristics from tree dimensions (Cluzeau et al., 1994; Doruska et al., 1994; Maguire et al., 1994; 1999; Deleuze et al., 1996; Roeh et al., 1997; Ishii et al., 1999) and stand attributes (Mäkinen et al., 1998). In China, crown shape of larch plantation was fitted using crown taper of power function (Li et al., 1994; 1996) and crown structure of poplar was characterized by relative growth model (Zhu et al., 2000).

Because direct measurement of crown profile of stand-grown trees presents numerous difficulties, the objective of the present study was to develop crown profile models of primary branches in Changbai Larch (L. olgensis) plantation using method of branch analysis. The crown profile models that were developed from branch attributes and tree dimensions consisted of predicting models of branch diameter, branch length, branch angle, and chord length. These models that could accept tree diameter at breast height (DBH), height, and crown length as input variables could predict crown radius at any given depth into crown and could be applied to describe crown development for L. olgensis trees.

2 Materials and methods 2.1 Data collection

Sample trees were collected in a 26-year-old unthinned L. olgensis plantation from Songjianghe Forest Bureau, Western part of Changbai Mountain, which is located in Fusong County of Jilin Province, northeastern China, ranging across 127°12′~127°50′ E and 41°44′~42°21′ N. One plot with area of 0.12 hm2 was chosen in this stand. Stem diameter at breast height (DBH), total height (HT), height to crown base (HCB), crown width (CW), and the coordinate of x and y were recorded for each tree in the plot. Mean DBH, mean height and the number of trees per hectare of the plot were 17.4 cm, 16.2 m, and 1 025 stems, respectively. A circular subsample plot (radius 8.1m) for branch analysis was set up in the plot. In total, 25 trees within the subsample plot were selected as sample trees for stem analysis and branch analysis.

The sample trees were felled as carefully as possible to minimize damage to their crowns. The crown base was defined as the pseudo-whorl with at least one living branch with green needles. The discs were taken from the stem at a height of 1.3 m, stump, and then at 1 m interval above stump following Smalian's method of stem analysis. Each section at 1 m interval within the live crown is called "Layer". Every branch in each crown layer was then numbered. The descriptive variables of branch and tree characteristics were measured for each branch within the crown layer of sample trees. Branch radius (CR) was derived from branch chord length (BC) and branch angle (θ). In total 1966 branches on 25 trees were measured (see Table 1).

Tab.1 Summary of sample tree and branch attributes for the Larix olgensis plantation

As in Raulier et al. (1996), and Roah et al. (1997), crown profile is here defined as the curve connecting the tips of the largest branches within each pseudo-whorl (main branches); hence, there is biological appeal in modeling crown form as a function of primary branching structure. Therefore, data to describe crown profile were obtained through the selected primary branches within live crown. Two primary branches were chosen in each living crown layer (one per 0.5 m interval) because the mean annual increment of height was about 0.5 m for larch plantation. The sampled primary branches were the longest and the thickest ones in each pseudo-whorl (main living branches) that making up of basic crown framework in all directions for larch plantation. The number of sampled primary branches per tree ranged from 20 to 34 depending on the crown length (CL), that is 23% of total number of branches (Table 1).

Because the dead branches were a very minor component of all live crowns, occurring predominantly at the lower part of crown near crown base, only live branches were used in this study. Sample tree and branch characteristics were summarized in Table 1.

2.2 Crown profile models 2.2.1 Branch diameter(BD)

Graphic analysis of primary branch characteristics showed that the BD within crown was related mainly to depth into crown (DINC) for each tree of L. olgensis. BD increased continuously with increasing DINC and approached to an asymptotic size at crown base. Several basic model forms appropriate for describing sigmoid functions with initial value equal 0 were compared, such as the modified Weibull function, Chapman-Richards function, Korf equation and the Mitscherlich equation (Zeide, 1993). As the results, the Mitscherlich equation was selected as the basic model for developing the branch diameter model.

(1)

Where BD is the primary branch diameter of outside bark; DINC is depth into crown; A, k are parameters to be estimated from the data.

2.2.2 Branch length (BL)

The trend in BL through the crown was little different with BD. For a given tree, BL increased rapidly with DINC that was approximated a conic or parabolic-shape in the effective crown (Li et al., 1996) and then approached to an asymptotic size of BL with a cylinder shape in lower part of crown. The basic model for primary BL was also expressed this variable as a function of DINC. Several equations to represent primary BL were carefully compared to adequately fit the observation, to start from 0 and increase with DINC in upper crown, and to approach maximum length in lower crown. After comparing overall RSS and residual plots of the modified Weibull function, Chapman-Richards function, and the Mitscherlich equation that were fitted to the observed data, the modified Weibull function (Zeide, 1993) was chosen as the basic model to developing model of BL.

(2)

Where BL is the branch length; DINC is depth into crown; A, k and c are parameters to be estimated from the data.

To investigate the effect of tree variables on the BD and BL, the equation (1) and (2) were fitted for individual tree. The estimated parameter values of each tree were graphed against tree variables and correlation analysis and stepwise regressions were performed to determine final predicting models of the BD and BL that added various combination of tree variables into basic models. Tree level variable included DBH, HT, and CL. Criteria for the best combination of predictor variables were amount of variation accounted for ease of variable measurement.

The parameter estimates of nonlinear branch diameter and length models were performed using DUD method implemented in SAS 6.12 software (SAS Institute Inc., 1990). Goodness-of-fit of the alternative models were assessed on the basis of the statistics of following:

Residual Sum of Squarer (RSS):

Standard error of estimate (Sy.x):

Adjusted coefficient of determination (Ra2):

Where: yi is observed value, is predicted value, n is the number of observation, and p is number of parameters.

Assessment of each equation was also conducted visually by plotting residuals against the predicted values and residual analysis was used to compare candidate models.

2.2.3 Branch angle(θ)

θ is often directly related to amount of light (Hashimoto, 1990). θ became progressively larger (branches more horizontal) with increasing DINC probably because of the weight of the branch, light attenuation by mutual shading, or mechanical constraint (Deleuze et al., 1996). Cochrance et al. (1978) found that a simple linear equation based on whorl order from tree top adequately described branch angle for young plantation Sitka spruce (Picea sitchensis). Deleuze et al. (1996) also developed liner relationship between branch angle and whorl order for Norway spruce (Picea abies) for branches free of any contact (functional branches). However, Collin et al. (1992) found that curvilinear trend between θ and DINC for Picea abies. The smoothed graph of θ on DINC revealed that the average trend in θ down the stem was curvilinear with an asymptote of approximately 90 (Roeh et al., 1997). In contrast, Cluzeau et al. (1994) concluded that there was no relationship between θ and BL, and the variability of θ may change from branch by branch for tree. They used an average θ for all trees and branches.

From the observations of primary branches for L. olgensis, it was found that θ of primary branches was closely related to DINC and was slightly related to DBH and HT. The simple relative coefficient between θ with DINC, DBH, and HT were 0.45, -0.07, and -0.08, respectively. Therefore, multiple regression method was performed to develop the θ model. Independent variables considered in regression were DINC, DBH, and HT.

2.2.4 Branch chord length(BCL)

Many researches showed a strong linear relationship of slope λ between BCL and BL (Cluzeau et al., 1994; Deleuze et al., 1996):

(3)
2.2.5 Crown radius(CR)

CR was calculated from branch angle (θ) and branch chord length (BCL):

(4)
2.3 Validation

The independent validation data set includes 5 sample trees and the attributes were summarized in Table 1.

The independent validation procedures were performed to examine crown profile models using the following statistical measures (Mayer et al., 1993). The validation statistics were calculated for each model in the validation data. The measure and predicted values were transformed to the original scale before calculation of the error statistics.

Mean Error:

Mean Absolute Error:

Mean Percent Error (M%E):

Mean Absolute Percent Error (MA%E):

Precision Estimation: ; where

Where: yi is observed value, is predicted value, t0.05 is t value at 5% probability level, is standard error of mean estimate, n is the number of observation, and p is number of parameters.

3 Results and discussion 3.1 Branch diameter

In order to account for systematic variation in branch diameter (BD) trends among trees of different size, additional models were explored in which the parameters A and k of model (1) were considered as functions of other tree attributes, including DBH, HT and CL. From the estimated parameters A and k of model (1) for each tree, it was found that the parameter A was closely related to DBH and the parameter k was not related to DBH, HT and CL. The simple relative coefficient between parameter A with DBH, HT and CL was 0.67, 0.27, and 0.37, respectively, and the parameter k was -0.30, 0.14, and -0.35, respectively. Therefore, independent variables considered in BD predicting model were DINC and DBH.

After considering parameter A in model (1) as function of many transformations of DBH, such as linear, power and exponential function, the following two-variable equation was found to be the best model for describing the trend in primary branch diameter over DINC in L. olgensis plantation trees.

(5)

where BD is the branch diameter of outside bark (cm); DBH is the diameter at breast height; DINC is the depth into crown; a0~a2 are parameters to be estimated from the data.

Parameter estimates and fit statistics of the branch diameter model (5) for L. olgensis plantation using the method of nonlinear regression were presented in Table 2. Fig. 1A illustrated the effects of DINC and tree size (DBH) on diameter of primary branches. The residuals were plotted against the predicted BD as shown in Fig. 2A. The residuals are positively related to predicted branch diameter for modeling data. But, the validation data do not suggest any problem with model and no trends are apparent (Fig. 5a). Parameter estimates, fit statistics, and predicted trends in BD over DINC all suggested biologically reasonable behavior and an adequate fit to the data.

Tab.2 Parameter estimates and fit statistics for predicting model of branch diameter fitted to the fitting data set (n=530)
Fig.1 Predicted trends in primary BD (A) and BL (B) for three sample trees of different size. Predictions of A and B are from equation (5) and equation (6).
Fig.2 Residuals for predicting model of BD (Eq. (5) A) and BL (Eq. (6) B)

The predicting model (5) of branch diameter was monotonic increasing function related to DINC with no inflection point. For a given DINC, branch diameter was positively related to tree size (Fig. 1A).

3.2 Branch length

The modified Weibull function (2) was selected as the best for describing the primary branch length and the parameters of the equation (2) were expressed as functions of DBH and HT. The following equation was selected as the best model for predicting primary branch length (BL) of L. olgensis trees.

(6)

where b0~b4 are parameters to be estimated from the data.

Parameter estimates and fit statistics of the branch length model (6) for larch plantation using the method of nonlinear regression were listed in Table 3. The residual variation about predicted BL were presented in Fig. 2B and there is little bias in predicted branch length for modeling data. But, all residuals from the validation data are well within the range of the data set and no trends are apparent (Fig. 5b). The effects of tree size on length of primary branches were illustrated in Fig. 1B. All of these results did not indicate any problematic behavior.

Tab.3 Parameter estimates and fit statistics for predicting model of BL fitted to the fitting data set (n=530)

The equation (6) is typically sigmoid with an asymptotic line. For a tree of given DBH and HT, BL become progressively larger with DINC increasing. Moreover, BL at a given DINC increased with increasing tree size (Fig. 1B).

3.3 Branch angle

The branch angle (θ) model of primary branches was developed using multiple regression method and the following model was selected:

(7)

where c0~c3 are parameters to be estimated from the data.

The model form is linear, reflecting the monotonic increase in branch angle with increasing DINC and decrease slightly with increasing DBH and total height (HT). Parameter estimates and fit statistics of the primary branch angle model (7) for larch plantation were presented in Table 4 and residual plot in Fig. 3.

Tab.4 Parameter estimates and fit statistics of θ model (7) for L. olgensis
Fig.3 Residuals for model predicting θ of L. olgensis (Eq. (7)).
3.4 Branch chord length

The chord length (BCL) model of primary branches, equation (3), was fitted to the fitting data by linear least square, and the following equation was obtained:

(8)

The fitting graph of equation (8) was plotted in Fig. 4 and the validation result for the validation data was presented in Table 5.

Fig.4 BCL of primary branch versus branch length with fitting the model (8)
Tab.5 Validation results of equation (5), (6), and (8) for independent data set (n=140)
3.5 Model validation

For the validation procedure, the performance evaluation criteria were computed with the predicting models of primary branch diameter, branch length, and chord length developed in above for the validation data set (Table 1). The result of statistical validation test was summarized in Table 5.

The results indicated that deviance measures were all fairly low. The primary branch diameters and length were slightly overestimated at all compared with branch chord length. In predicted diameter and length of primary branches, the mean absolute errors are 0.33 and 29.67, respectively, and the mean percent errors (M%E) are -11% and -13%, respectively. Thus the predictions for branch diameter and branch length are reasonably precise, but slightly biased. The model for predicting branch chord length was generally better fit with very small biases. The estimated precisions of three models for the validation data set were all greater than 95%.

The validation data likewise do not suggest any problems with the model (5) and (6). All differences of primary branch diameter and branch length from validation data are well and no strong biases were evident in predicting branch diameter, and branch length, depending on the depth into crown (Fig. 5).

Fig.5 Differences between observed and predicted BD (a) and BL (b) representing primary branches for the validation data
3.6 Crown profile model

Equations (6), (7), (8), and (4) were combined to obtain the crown profile model for L. olgensis plantation.

(9)

where CR is crown radius; BL and θ were estimated from equation (6) and (7).

The individual predicting model such as branch length (BL), branch angle (θ), and chord length (BC) fits well to the pooled data of the sample trees, as previously shown. However, it was important to analyze the adequacy of the combined crown profile model (9) at the tree level. To evaluate the crown profile model, the observed and predicted crown profile were compared individually for three sample trees of different size (Fig. 6).

Fig.6 Observed and predicted crown profiles for three sample trees of different size

The crown profile model (Eq. 9) is reasonably adequate for different tree size of L. olgensis plantation. The slightly overestimated values of crown radius in the upper part of crown of large tree are due to the relatively small development of the branches concerned (see Fig. 6).

All crown shapes are conical in the upper and middle part of crown, and follow approximately cylinder at lower part of crown. Relative to small tree, large trees are more round crown shape. The crown shapes of different tree size almost have same pattern in the upper crown that DINC ranged from 0 to 3 m, because of free from inter-trees competition (Fig. 6). These results are similar that were found in Sugi plantation (Kajihara, 1976) and in black spruce stands (Raulier et al., 1996).

4 Conclusion

Data from 25 trees were used to develop crown profile model in a L. olgensis plantation. The basal diameter (BD) and length (BL) of primary branches for a specified tree were mainly related to the depth into crown (DINC). DBH and HT were adequate variables of tree for summarizing the effects of external conditions on the branch diameter and length. The branch diameter predicting model (Eq. 5) and length predicting model (Eq. 6) presented in this study were logically constrained to pass through zero at the top of the tree, and do conform well to the growth trends in primary branch diameter and branch length for L. olgensis plantation. Branch diameter increases continuously with DINC increasing and approached to an asymptotic size at crown base (see Fig. 1A). For a given tree, BL increases rapidly with DINC that was approximated a parabolic-shape in upper half of crown (relative DINC= 0 to 0.6) and then approaches to an asymptotic size with a cylinder shape in lower part of crown (relative DINC > 0.6) (see Fig. 1B).

The crown profile models, including branch diameter, branch length, angle of origin, and branch chord length, were developed and evaluated at the tree level. These attributes of primary branches for a specified tree were mainly related to DINC. DBH and HT were adequate variables of tree for summarizing the effects of external conditions on the crown profile. The recommend models appear to perform well and consistently reflect the observed patterns of crown profile of primary branches for different tree size. All crown shapes are parabolic in the upper and middle part of crown, and follow approximately cylinder at lower part of crown. Relative to small tree, large trees are more round crown shape. The crown shapes of different tree size almost have same pattern in the upper crown that DINC ranged from 0 to 3m, because of free from inter-trees competition. Therefore, the crown forms of the exposed portion invariant among tree sizes in L. olgensis plantations.

The general development of crown for a given stand was illustrated in this study. In future work, as more trees will be sampled across a wider variety of stand conditions, opportunities will arise for introducing more stand and site-level variables into the crown profile model of branches for the L. olgensis plantation.

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