大兴安岭白桦削度方程

doi:10.11707/j.1001-7488.20201109

摘要/Abstract

摘要：

目的: 确定预测东北地区白桦不同高度直径和材积的最优削度方程，以弥补该地区没有白桦削度方程的不足。方法: 以伊勒呼里山北坡西北部立地亚区253株白桦伐倒木3 795对直径/高度数据为基础，基于林业上广泛应用的8个削度方程，利用SAS软件的非线性回归SUR法对方程进行拟合。使用一阶连续自回归误差结构模拟方程误差项并解释空间自相关，采用条件数评价方程多重共线性，选择确定系数(R²)、均方根误差(RMSE)、平均误差绝对值(MAB)和相对误差绝对值(MPB)作为方程评价指标，运用拟合统计量、直径和材积残差分布箱式图及检验统计量进行削度方程的综合比较。结果: 1) 从各削度方程拟合统计量来看，Kozak (2004)-2、Fang等(2000)和Max等(1976)方程排在前3位，Sharma等(2001)方程表现最差；2)基于直径和材积残差分布箱式图，Kozak (2004)-2、Fang等(2000)、Max等(1976)和Bi(2000)方程在预测直径和材积时误差较小且具有相似的等方差分布，Sharma等(2001)、Sharma等(2004)、Sharma等(2009)和Kozak (2004)-1方程具有较强的方差异质性；3)模型检验证实Kozak (2004)-2、Fang等(2000)和Max等(1976)方程表现较好，Kozak (2004)-2削度方程在预测直径和材积方面表现出一致性，且优于其他削度方程。结论: 根据模型拟合和检验统计量、图形分析和条件数，Kozak (2004)-2方程被推荐用于预测东北地区白桦不同高度的直径、总材积和商品材积。

关键词: 白桦, 削度, 材积, 自相关, 多重共线性

Abstract:

Objective: Stem taper functions are important components in forest management and planning systems. Currently,there is no taper function for Betula platyphylla in northeast China,therefore,it is necessary to develop the taper function for this species. Eight commonly used taper functions in forestry were compared to evaluate which would provide a better prediction for diameter at a specific height and total volume for B. platyphylla in northeast China. Method: The data used in this study were collected from 253 destructively felled sample trees with 3 795 diameter/height measurements in the northwest of the northern slope of Yilehuli Mountains of northeast China. A first-order continuous autoregressive error structure was used to model the error term and account for autocorrelation. Multicollinearity was also evaluated with condition number. Coefficient of determination (R²),mean absolute bias (MAB),root mean square error (RMSE) and mean percentage of bias (MPB) were selected as evaluation criteria of models. Comparison of the taper models was carried out using goodness-of-fit statistics,box plots of diameter and volume residual distributions and validation statistics. Result: 1) In terms of model fitting statistics, the models of Kozak (2004)-2, Fang et al. (2000) and Max et al. (1976) were the top three models. The model of Sharma et al. (2001) showed the poorest performance. 2) Based on the box plots of diameter and volume residuals, the models of Bi (2000), Max et al. (1976), Kozak (2004)-2 and Fang et al. (2000) were more accurate in diameter and volume prediction with smaller errors and almost similar residual diameter and volume distribution. The models of Sharma et al. (2001), Sharma et al. (2004), Sharma et al. (2009) and Kozak (2004)-1 had non homogeneous distribution of the diameter residuals along different sections of the stem. 3) Model validation also confirmed that Max et al. (1976), Kozak (2004)-2 and Fang et al. (2000) showed better performances. In general, the model of Kozak (2004)-2 showed consistent performances and was superior to other taper models in predicting diameter and volume. Conclusion: Based on the evaluation statistics of fitting and validation,graphic analysis and condition number,the model of Kozak (2004)-2 was recommended for estimating diameter at a specific height,total volume and merchantable volume for B. platyphylla in northeast China.

Key words: Betula platyphylla, taper, volume, autocorrelation, multicollinearity

中图分类号:

S757

Shahzad Muhammad Khurram,Hussain Amna,何培,姜立春. 大兴安岭白桦削度方程[J]. 林业科学, 2020, 56(11): 87-96.

Muhammad Khurram Shahzad,Amna Hussain,Pei He,Lichun Jiang. Stem Taper Functions for Betula platyphylla in Daxing'anling[J]. Scientia Silvae Sinicae, 2020, 56(11): 87-96.

图/表 8

Table 1

Table 2

Analyzed taper functions①"

Model	Expression
Bi(2000)	$ d=D\left(\frac{\ln \sin \left(\frac{\pi}{2} q\right)}{\ln \sin \left(\frac{\pi}{2} t\right)}\right)^{b_{0}+b_{1} \sin \left(\frac{\pi}{2} q\right)+b_{2} \cos \left(\frac{3 \pi}{2} q\right)+b_{3} \sin \left(\frac{\pi}{2} q\right) / q+b_{4} D+b_{5} q \sqrt{D}+b_{6} q \sqrt{H}}$
Sharma et al. (2001)	${d = \sqrt {{D^2}{{\left({\frac{h}{{1.3}}} \right)}^{2 - {b_1}}}X} } $
Sharma et al.(2004)	$ {d = D\sqrt {{b_0}{{\left({\frac{h}{{1.3}}} \right)}^{2 - \left({{b_1} + {b_2}z + {b_3}{z^2}} \right)}}\left({\frac{{H - h}}{{H - 1.3}}} \right)} }$
Sharma et al.(2009)	$ {d = D\left[ {{b_0}\left({\frac{{H - h}}{{H - 1.3}}} \right){{\left({\frac{h}{{1.3}}} \right)}^{{b_1} + {b_2}z + {b_3}{z^2}}}} \right]}$
Max et al.(1976)	$d=\sqrt{D^{2}\left[b_{1}(q-1)+b_{2}\left(q^{2}-1\right)+b_{3}\left(a_{1}-q\right)^{2} I_{1}+b_{4}\left(a_{2}-q\right)^{2} I_{2}\right]} $ I₁=1, if q ≤ a₁; 0 otherwiseI₂ =1, if q ≤ a₂; 0 otherwise
Kozak (2004)-1	$d = {b_0}{D^{{b_1}}}{\left[ {\frac{{1 - {q^{1/4}}}}{{1 - {m^{1/4}}}}} \right]^{\left[ {{b_2} + {b_3}\left({1/{{\rm{e}}^{D/H}}} \right) + {b_4}D\left({\frac{{1 - {q^{1/4}}}}{{1 - {m^{1/4}}}}} \right) + {b_5}{{\left({\frac{{1 - {q^{1/4}}}}{{1 - {m^{1/4}}}}} \right)}^{D/H}}} \right]}} $
Kozak (2004)-2	${d = {b_0}{D^b}_1{H^{{b_2}}}{{\left[ {\frac{{1 - {q^{1/3}}}}{{1 - {t^{1/3}}}}} \right]}^{\left[ {{b_3}{q^4} + {b_4}\left({1/{{\rm{e}}^{D/H}}} \right) + {b_5}{{\left({\frac{{1 - {q^{1/3}}}}{{1 - {t^{1/3}}}}} \right)}^{0.1}} + {b_6}(1/D) + {b_7}{H^{1 - {q^{1/3}}}} + {b_8}\left({\frac{{1 - {q^{1/3}}}}{{1 - {t^{1/3}}}}} \right)} \right]}}} $
Fang et al.(2000)	$ \begin{array}{l}\begin{array}{*{20}{l}}{d = {c_1}\sqrt {{H^{\left({k - {b_1}} \right)/{b_1}}}{{(1 - z)}^{(k - b)/b}}q_1^{{I_1} + {I_2}}q_2^{{I_2}}} }\\{{c_1} = \sqrt {{a_0}{D^a}1{H^a}{2^{ - k/b}}1/\left[ {{b_1}\left({{t_0} - {t_1}} \right) + {b_2}\left({{t_1} - {q_1}{t_2}} \right) + {b_3}{q_1}{t_2}} \right]} }\end{array}\\{q_1} = {\left({1 - {p_1}} \right)^{\left({{b_2} - {b_1}} \right)k/{b_1}{b_2}}}\;\;{q_2} = {\left({1 - {p_2}} \right)^{\left({{b_3} - {b_2}} \right)k/{b_2}{b_3}}}\;\;\quad {t_0} = 1\quad {t_1} = {\left({1 - {p_1}} \right)^{k/{b_1}}}\\{t_2} = {\left({1 - {p_2}} \right)^{k/{b_2}}}\quad z = h/H\;\;\quad b = b_1^{1 - \left({{I_1} + {I_2}} \right)}b_2^{{I_1}}{b_3}^{{I_2}}\quad k = 0.000078539\\{\rm{ }}{I_1} = 1, \;{\rm{if }}~~{p_2} \ge z \ge {p_1}, 0{\rm{ }}\;\;{\rm{otherwise }}\;\;\;\;\;\;{I_2} = 1, {\rm{ if }}1 \ge z \ge {p_2}, 0{\rm{ }}\;\;{\rm{otherwise}}\end{array}$

Table 2

Fig.1

Table 3

Table 4

Fig.2

Fig.3

Table 5

参考文献 0

	姜立春, 马英莉, 李耀翔. 大兴安岭不同区域兴安落叶松可变指数削度方程. 林业科学, 2016, 52 (2): 17- 25.
	Jiang L C , Ma Y L , Li Y X . Variable-exponent taper models for dahurian larch in different regions of Daxing'anling. Scientia Silvae Sinicae, 2016, 52 (2): 17- 25.
	Anta M B , Diéguez-Aranda U , Castedo-Dorado F , et al. Merchantable volume system for pedunculate oak in northwestern Spain. Annals of Forest Science, 2007, 64, 511- 520. doi: 10.1051/forest:2007028
	Belsley D A. 1991. Conditioning diagnostics: collinearity and weak data in regression. John Wiley & Sons, Inc., New York, 396.
	Bi H . Trigonometric variable-form taper equations for Australian eucalypts. Forest Science, 2000, 46, 397- 409.
	Biging G S . Taper equations for second-growth mixed conifers of Northern California. Forest Science, 1984, 30, 1103- 1117.
	Corral-Rivas J J , Diéguez-Aranda U , Rivas S C , et al. A merchantable volume system for major pine species in El Salto, Durango (Mexico). Forest Ecology and Management, 2007, 238, 118- 129. doi: 10.1016/j.foreco.2006.09.074
	Crecente-Campo F , Alboreca A R , Diéguez-Aranda U . A merchantable volume system for Pinus sylvestris L. in the major mountain ranges of Spain. Annals of Forest Science, 2009, 66, 1- 12.
	de-Miguel S , Meht talo L , Shater Z , et al. Evaluating marginal and conditional predictions of taper models in the absence of calibration data. Canadian Journal of Forest Research, 2012, 42, 1383- 1394. doi: 10.1139/x2012-090
	Demaerschalk J . Converting volume equations to compatible taper equations. Forest Science, 1972, 18, 241- 245. doi: 10.1093/forestscience/18.3.241
	Diéguez-Aranda U , Castedo-Dorado F , lvarez-González J G , et al. Compatible taper function for Scots pine plantations in northwestern Spain. Canadian Journal of Forest Research, 2006, 36, 1190- 1205. doi: 10.1139/x06-008
	Fang Z , Bailey R L . Compatible volume and taper models with coefficients for tropical species on Hainan Island in Southern China. Forest Science, 1999, 45, 85- 100.
	Fang Z , Borders E , Bailey L . Compatible volume-taper models for loblolly and slash pine based on a system with segmented-stem form factors. Forest Science, 2000, 46, 1- 12.
	Gómez-García E , Crecente-Campo F , Diéguez-Aranda U . Selection of mixed-effects parameters in a variable-exponent taper equation for birch trees in northwestern Spain. Annals of Forest Science, 2013, 70, 707- 715. doi: 10.1007/s13595-013-0313-9
	Heidarsson L , Pukkala T . Taper functions for lodgepole pine (Pinus contorta) and Siberian larch (Larix sibirica) in Iceland. Icelandic Agricultural Sciences, 2011, 24, 3- 11.
	Jiang L , Brooks J R , Wang J . Compatible taper and volume equations for yellow-poplar in West Virginia. Forest Ecology and Management, 2005, 213, 399- 409. doi: 10.1016/j.foreco.2005.04.006
	Kozak A . A variable-exponent taper equation. Canadian Journal of Forest Research, 1988, 18, 1363- 1368. doi: 10.1139/x88-213
	Kozak A . My last words on taper equations. The Forestry Chronicle, 2004, 80, 507- 515. doi: 10.5558/tfc80507-4
	Lee W K , Seo J H , Son Y M , et al. Modeling stem profiles for Pinus densiflora in Korea. Forest Ecology and Management, 2003, 172, 69- 77. doi: 10.1016/S0378-1127(02)00139-1
	Li R , Weiskittel A , Dick A R , et al. Regional stem taper equations for eleven conifer species in the Acadian region of North America: development and assessment. Northern Journal of Applied Forestry, 2012, 29, 5- 14. doi: 10.5849/njaf.10-037
	Li R , Weiskittel A R . Comparison of model forms for estimating stem taper and volume in the primary conifer species of the North American Acadian Region. Annals of Forest Science, 2010, 67, 302- 317. doi: 10.1051/forest/2009109
	Lumbres C , Abino A C , Pampolina N M , et al. Comparison of stem taper models for the four tropical tree species in Mount Makiling, Philippines. Journal of Mountain Science, 2016, 13 (3): 536- 545. doi: 10.1007/s11629-015-3546-2
	Max T A , Burkhart H E . Segmented polynomial regression applied to taper equations. Forest Science, 1976, 22, 283- 289.
	Muhairwe C K . Taper equations for Eucalyptus pilularis and Eucalyptus grandis for the north coast in New South Wales, Australia. Forest Ecology and Management, 1999, 113, 251- 269. doi: 10.1016/S0378-1127(98)00431-9
	Muhairwe C K , LeMay V M , Kozak A . Effects of adding tree, stand, and site variables to Kozak's variable-exponent taper equation. Canadian Journal of Forest Research, 1994, 24, 252- 259. doi: 10.1139/x94-037
	Newton P F , Sharma M . Evaluation of sampling design on taper equation performance in plantation-grown Pinus banksiana. Scandinavian journal of forest research, 2008, 23, 358- 370. doi: 10.1080/02827580801995349
	Poudel K P , Cao Q V . Evaluation of methods to predict Weibull parameters for characterizing diameter distributions. Forest Science, 2013, 59, 243- 252. doi: 10.5849/forsci.12-001
	Reed D D , Green E J . Compatible stem taper and volume ratio equations. Forest Science, 1984, 30, 977- 990.
	Rojo A , Perales X , Sánchez-Rodríguez F , et al. Stem taper functions for maritime pine (Pinus pinaster Ait.) in Galicia (Northwestern Spain). European Journal of Forest Research, 2005, 124, 177- 186. doi: 10.1007/s10342-005-0066-6
	Sharma M , Oderwald R G . Dimensionally compatible volume and taper equations. Canadian Journal of Forest Research, 2001, 31, 797- 803. doi: 10.1139/x01-005
	Sharma M , Parton J . Modeling stand density effects on taper for jack pine and black spruce plantations using dimensional analysis. Forest Science, 2009, 55, 268- 282.
	Sharma M , Zhang S . Variable-exponent taper equations for jack pine, black spruce, and balsam fir in eastern Canada. Forest Ecology and Management, 2004, 198, 39- 53. doi: 10.1016/j.foreco.2004.03.035
	Thomas C E , Parresol B R . Simple, flexible, trigonometric taper equations. Canadian Journal of Forest Research, 1991, 21, 1132- 1137. doi: 10.1139/x91-157
	Wilhelmsson L . Two models for predicting the number of annual rings in cross-sections of tree stems. Scandinavian Journal of Forest Research, 2006, 21, 37- 47. doi: 10.1080/14004080500486799
	Zhang S , Liu C . Predicting the lumber volume recovery of Picea mariana using parametric and non-parametric regression methods. Scandinavian Journal of Forest Research, 2006, 21, 158- 166. doi: 10.1080/02827580500531791

Data	Variable	Number of trees	Mean	Min.	Max.	SD
Fitting	D	191	18.18	5.0	43.8	8.26
Fitting	H	191	17.25	7.9	24.1	3.92
Validation	D	62	16.84	5.2	36.3	8.00
Validation	H	62	15.77	8.5	22.1	3.09

Models	b₀	b₁	b₂	b₃	b₄	b₅	b₆	b₇	b₈	a₀	a₁	a₂	p₁	p₂
Bi (2000)	0.339 4 (0.061 0)	0.339 4 (0.030 1)	0.339 4 (0.009 7)	0.339 4 (0.033 8)	0.339 4 (0.000 2)	—	—
Sharma et al. (2001)		2.037 6 (0.000 7)
Sharma et al. (2004)	1.011 2 (0.004 1)	2.033 4 (0.000 7)	—	0.363 7 (0.012 3)
Sharma et al. (2009)	1.010 8 (0.002 3)	-0.013 8 (0.000 3)	0.201 6 (0.014 7)	0.046 1 (0.021 4)
Max et al. (1976)		-3.919 8 (0.361 9)	1.878 8 (0.198 5)	-1.775 3 (0.176 7)	156.608 3 (12.162 7)						0.747 3 (0.027 6)	0.064 4 (0.002 4)
Kozak (2004)-1	1.353 7 (0.027 5)	0.945 8 (0.006 7)	0.346 9 (0.012 2)	0.357 4 (0.040 5)	0.008 3 (0.000 8)	-0.697 2 (0.062 3)
Kozak (2004)-2	0.892 2 (0.034 6)	0.967 2 (0.007 7)	0.076 6 (0.017 9)	0.432 4 (0.021 6)	0.154 0 (0.035 0)	0.365 9 (0.014 6)	—	0.011 8 (0.001 9)	-0.086 3 (0.025 3)
Fang et al. (2000)		6.62E-06 (1.8E-7)	0.000 035 (2.7E-7)	0.000 030 (6.0E-7)						0.000 048 (3.4E-6)	1.950 1 (0.015 1)	0.970 3 (0.034 4)	0.047 6 (0.001 3)	0.613 2 (0.034 1)

Models	RMSE	R²	Rank	CN
Bi (2000)	1.528 7	0.971 1	1.667	113.4 5
Sharma et al. (2001)	1.885 1	0.955 9	8.000	1.000 0
Sharma et al. (2004)	1.578 6	0.969 1	2.527	1.800 0
Sharma et al. (2009)	1.528 2	0.971 1	1.662	8.904 8
Max et al. (1976)	1.495 6	0.972 3	1.123	274.69
Kozak (2004)-1	1.630 6	0.967 1	2.905	30.220
Kozak (2004)-2	1.488 7	0.972 6	1.000	77.830
Fang et al. (2000)	1.492 4	0.972 4	1.074	83.083

Models	Diameter				Volume
Models	MPB	MAB	RMSE	Rank	MPB	MAB	RMSE	Rank
Bi (2000)	7.428 3	0.982 2	1.696 9	1.613	10.102 4	0.022 0	0.046 8	2.081
Sharma et al. (2001)	10.713 1	1.416 5	2.338 3	8.000	17.913 1	0.039 1	0.070 7	8.000
Sharma et al. (2004)	7.814 3	1.033 1	1.714 9	2.180	10.621 4	0.023 1	0.048 1	2.445
Sharma et al. (2009)	7.331 8	0.969 4	1.656 9	1.358	16.236 0	0.035 4	0.066 7	6.817
Max et al. (1976)	7.294 0	0.964 4	1.605 2	1.143	9.980 1	0.021 7	0.047 1	2.039
Kozak (2004)-1	8.463 2	1.119 0	1.811 4	3.342	8.771 7	0.019 1	0.041 8	1.000
Kozak (2004)-2	7.185 3	0.950 0	1.651 9	1.148	9.653 0	0.021 0	0.045 5	1.745
Fang et al. (2000)	7.249 5	0.958 5	1.634 9	1.179	10.078 3	0.022 0	0.046 4	2.043

[1]	刘中原,刘峥,徐颖,刘珊珊,田志兰,解庆军,高彩球. 白桦HSFA4转录因子的克隆及耐盐功能分析[J]. 林业科学, 2020, 56(5): 69-79.
[2]	沈钱勇,汤孟平. 浙江省毛竹竹秆材积模型[J]. 林业科学, 2020, 56(5): 89-96.
[3]	常建国. 油松树干横截面面积年增长量的垂直分布及与材积年增长量和叶量的关系[J]. 林业科学, 2019, 55(6): 55-64.
[4]	尤磊, 哈登龙, 谢明坤, 张晓鹏, 宋新宇, 庞勇, 唐守正. 基于三维激光点云与断面轮廓曲线的树干材积计算[J]. 林业科学, 2019, 55(11): 63-72.
[5]	张凌宇, 刘兆刚. 基于地理加权回归模型的大兴安岭中部天然次生林更新分布[J]. 林业科学, 2019, 55(11): 105-116.
[6]	马岩岩,姜立春. 基于非线性分位数回归的落叶松树干削度方程[J]. 林业科学, 2019, 55(10): 68-75.
[7]	金明月, 姜峰, 金光泽, 刘志理. 不同年龄白桦比叶面积的生长阶段变异及冠层差异[J]. 林业科学, 2018, 54(9): 18-26.
[8]	Shahzad Muhammad, 韩斐斐, 姜立春. 不同抽样方法对兴安落叶松立木材积方程预测精度的影响[J]. 林业科学, 2018, 54(8): 99-105.
[9]	马岩岩, 姜立春. 异速生长模型的误差结构和误差函数[J]. 林业科学, 2018, 54(2): 90-97.
[10]	娄明华, 张会儒, 雷相东, 李春明, 臧颢. 基于空间自相关的天然蒙古栎阔叶混交林林木胸径-树高模型[J]. 林业科学, 2017, 53(6): 67-76.
[11]	刘宇, 徐焕文, 尚福强, 焦宏, 张利民, 罗建新, 滕文华, 刘桂丰. 16年生白桦种源变异及区划[J]. 林业科学, 2016, 52(9): 48-56.
[12]	李贵雨, 卫星, 汤园园, 吕琳, 杜欣竹. 白桦不同轻基质容器苗生长及养分分析[J]. 林业科学, 2016, 52(7): 30-37.
[13]	李晖, 曾伟生. 不同区域落叶松二元立木材积表的检验及差异分析[J]. 林业科学, 2016, 52(6): 157-162.
[14]	姜立春, 马英莉, 李耀翔. 大兴安岭不同区域兴安落叶松可变指数削度方程[J]. 林业科学, 2016, 52(2): 17-25.
[15]	马岩, 许洪刚, 杨春梅, 许世祥. 小径木双面刨切单板的材积计算与效益分析[J]. 林业科学, 2016, 52(11): 142-147.