Segmented taper models form for Manchurian fir and Korean spruce in northeastern China

HUSSAIN, AMNA; SHAHZAD, MUHAMMAD KHURRAM; JIANG, LICHUN; LI, FENGRI

doi:10.1590/01047760202127012659

ABSTRACT

Background:

Manchurian fir (Abies nephrolepis Maxim) and Korean spruce (Picea koraiensis Nakai) are primary conifer species of economic and ecological importance in northeast China. Several taper studies have analyzed for fir and spruce species in the countries harboring the boreal forests. However, taper models do not exist for Manchurian fir and Korean spruce in China or abroad. This study aimed to develop stem taper models for these species. A dataset of 188 destructively sampled trees (Manchurian fir 123 and Korean spruce 65) was used to evaluate eight well-known taper models. These models were fitted with generalized non-linear least squares by using 3,570 diameter and height measurements. We incorporated a first-order continuous-time error structure to adjust the inherent autocorrelation.

Results:

The form-class segmented model of Clark et al. (1991) best predicted the diameter, merchantable volume, and stem volume of the species when the upper stem diameter at 5.3 m was available or predicted.

Conclusion:

When diameter measurements at 5.3 m were not available, the Kozak (2004) and Max and Burkhart (1976) models were superior to other models in estimating the diameter of both species and volume of Korean spruce. For Manchurian fir, the Fang et al. (2000) model was more accurate in volume estimates.

Keywords:
Form-class; Stem diameter; Merchantable volume; Variable form model

HIGHLIGHTS

Eight taper models were evaluated for natural stands of two conifer species (Manchurian fir and Korean spruce) in NE China.

The Clark et al. (1991) model was superior to other models in estimating diameter, merchantable volume, and total volume.

The prediction method used in Clark et al. (1991) model did not affect its overall superiority.

The models of Kozak (2004), Max and Burkhart (1976), and Fang et al. (2000) performed well when diameter measurements at 5.3 m were not available.

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Species	Variable*	n	Min	Mean	Max	SD
Manchurian Fir	D (cm)	123	17.1	25.6	37.7	3.8
	H (m)	123	10.1	17.2	23.6	1.7
	d (cm)	2339	0.5	19.5	46.9	8.1
	h (m)	2339	0.3	7.5	23.6	5.4
Korean Spruce	D (cm)	65	10.0	25.9	42.0	5.6
	H (m)	65	9.1	17.3	21.4	2.1
	d (cm)	1231	0.2	19.8	59.0	8.8
	h (m)	1231	0.3	7.4	21.4	5.5

Models
Max and Burkhart (1976MAX, T.A.; BURKHART, H.E. Segmented polynomial regression applied to taper equations. Forest Science , v. 22, n. 3, p. 283-289, 1976.)
$d = D {[b_{1} (Z - 1) + b_{2} (Z^{2} - 1) + b_{3} {(a_{1} - Z)}^{2} I_{1} + b_{4} {(a_{2} - Z)}^{2} I_{2}]}^{0.5}$	[1]
$\{\begin{matrix} I_{1} = 1, i f Z \leq a_{1}; 0 o t h e r w i s e \\ I_{2} = 1, i f Z \leq a_{2}; 0 o t h e r w i s e \end{matrix}$	[1.1]
Clark et al. (1991CLARK, A.; SOUTER, R.A.; SCHLAEGEL, B.E. Stem profile equations for southern tree species (Research Paper SE-282). USDA Forest Service: 113, 1991.)
$d = {\{\begin{matrix} I_{S} [D^{2} (1 + (b_{2} + b_{3} / D^{3}) ({(1 - Z)}^{b_{1}} - {(1 - 1.3 / H)}^{b_{1}}) / (1 - {(1 - 1.3 / H)}^{b_{1}}))] \\ + I_{B} [D^{2} - (D^{2} - F^{2}) ({(1 - 1.3 / H)}^{b_{4}} - {(1 - Z)}^{b_{4}}) / {(1 - 1.3 / H)}^{b_{4}} - {(1 - 5.3 / H)}^{b_{4}}] \\ + I_{T} [F^{2} (b_{6} {((h - 5.3 / H - 5.3) - 1)}^{2} + I_{M} (1 - b_{6} / b_{5}^{2}) {(b_{5} - (h - 5.3 / H - 5.3))}^{2})] \end{matrix}\}}^{0.5}$	[2]
$I_{S} = \{\begin{matrix} \begin{matrix} 1 & h < 1.3 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{matrix}$	[2.1]
$I_{B} = \{\begin{matrix} 1 1.3 \leq h < 5.3 \\ 0 o t h e r w i s e \end{matrix}$	[2.2]
$I_{T} = \{\begin{matrix} 1 h > 5.3 \\ 0 o t h e r w i s e \end{matrix}$	[2.3]
$F : F = D (a_{1} + a_{2} {(5.3 / H)}^{2})$	[3]**
Fang et al. (2000FANG, Z.; BORDERS, B.E.; BAILEY, R.L. Compatible volume-taper models for loblolly and slash pine based on a system with segmented-stem form factors. Forest Science , v. 46, n. 1, p. 1-12, 2000.)
$d = c_{1} \sqrt{H^{(k - b_{1}) / b_{1}} {(1 - Z)}^{(k - b) / b} {q_{1}}^{I_{1} + I_{2}} {q_{2}}^{I_{2}}}$	[4]
$c_{1} = \sqrt{a_{0} D^{a_{1}} H^{(a_{2} - k) / b_{1}} / b_{1} (t_{0} - t_{1}) + b_{2} (t_{1} - q_{1} t_{2}) + b_{3} q_{1} t_{2}}$	[4.1]
$\{\begin{matrix} q_{1} = {(1 - p_{1})}^{(b_{2} - b_{1}) k / b_{1} b_{2}} \\ q_{2} = {(1 - p_{2})}^{(b_{3} - b_{2}) k / b_{2} b_{3}} \end{matrix}$	[4.2]
$\{\begin{matrix} t_{1} = {(1 - p_{1})}^{k / b_{1}} \\ t_{2} = {(1 - p_{2})}^{k / b_{2}} \end{matrix}$	[4.3]
$t_{0} = 1$	[4.4]
$\{\begin{matrix} I_{1} = 1, i f p_{2} \geq Z \geq p_{1}; 0 o t h e r w i s e \\ I_{2} = 1, i f 1 \geq Z \geq p_{2}; 0 o t h e r w i s e \end{matrix}$	[4.5]
$b = {b_{1}}^{1 - (I_{1} + I_{2})} {b_{2}}^{I_{1}} {b_{3}}^{I_{2}}, k = 0.0000785, Z = h / H$	[4.6]
Bi (2000BI, H. Trigonometric variable-form taper equations for Australian Eucalyptus Forest Science, v. 46, n. 3, p. 397-409, 2000.)
$d = D {[l n s i n (\frac{π}{2} Z) / l n s i n (\frac{π}{2} t)]}^{b_{0} + b_{1} s i n (\frac{π}{2} Z) + b_{2} c o s (\frac{3 π}{2} Z) + b_{3} s i n (\frac{π}{2} Z) / Z + b_{4} D + b_{5} Z \sqrt{D} + b_{6} Z \sqrt{H}}$	[5]
Lee et al. (2003LEE, W.K.; SEO, J.H.; SON, Y.M.; LEE, K.H.; VON GADOW, K. Modeling stem profiles for Pinus densiflora in Korea. Forest Ecology and Management , v. 172, n. 1, p. 69-77, 2003.)
$d = b_{1} D^{b_{2}} {(1 - Z)}^{b_{3} Z^{2} + b_{4} Z + b_{5}}$	[6]
Kozak (2004KOZAK, A. My last words on taper equations. The Forestry Chronicle , v. 80, n. 4, p. 507-515, 2004.)
$d = b_{0} D^{b_{1}} H^{b_{2}} x^{b_{3} Z^{4} + b_{4} (1 / e^{D / H}) + b_{5} x^{0.1} + b_{6} (1 / D) + b_{7} H^{w} + b_{8} x}$	[7]
$x = w / (1 - {(1.3 / H)}^{1 / 3}), w = 1 - Z^{1 / 3}$	[7.1]
Sharma and Parton (2009SHARMA, M.; PARTON, J. Modeling stand density effects on taper for Jack Pine and Black Spruce plantations using dimensional analysis. Forest Science , v. 55, n. 3, p. 268-282, 2009.)
$d = D [b_{0} (\frac{H - h}{H - 1.3}) {(\frac{h}{1.3})}^{b_{1} + b_{2} Z + b_{3} Z^{2}}]$	[8]

	Max and Burkhart (1976)		Clark et al. (1991) Model 1		Clark et al. (1991) Model 2		Fang et al. (2000)
	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error
a₀							2×10^-5	3.0E-6
a₁	0.752	0.050			0.858	0.004	2.075	0.048
a₂	0.047	0.001			0.555	0.044	1.133	0.069
b₁	-2.699	0.308	37.590	1.416	37.594	1.418	7.505	1.7E-7
b₂	1.152	0.175	0.671	0.016	0.671	0.016	3×10^-5	4.0E-7
b₃	-0.685	0.180	-1263.01	236.4	-1263.61	237.3	6×10^-5	7.9E-6
b₄	193.832	10.490	1.986	0.453	2.030	0.462
b₅			1.026	0.012	1.037	0.014
b₆			-10.126	4.931	-6.798	2.859
p₁							0.041	0.001
p₂							0.957	0.006
	Bi (2000)		Lee et al. (2003)		Kozak (2004)		Sharma and Parton (2009)
	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error
b₀	-0.714	0.138			1.060	0.116	1.055	0.004
b₁	0.034	0.083	1.072	0.071	0.948	0.033	-0.006	1×10^-4
b₂	0.032	0.011	1.050	0.020	0.049*	0.050	0.134	0.022
b₃	0.473	0.087	1.832	0.148	0.283	0.024	0.010	0.030
b₄	0.001	2×10^-4	-2.847	0.195	0.265*	0.203
b₅	0.017*	0.022	1.886	0.068	0.716	0.040
b₆	0.094	0.038			-6.561	1.289
b₇					0.002*	0.002
b₈					-0.131	0.029

	Max and Burkhart (1976)		Clark et al. (1991) Model 1		Clark et al. (1991) Model 2		Fang et al. (2000)
	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error
a₀							3×10^-5	7.7E-6
a₁	0.841	0.028			0.872	0.007	1.889	0.056
a₂	0.056	0.002			0.755	0.071	1.165	0.109
b₁	-4.965	0.856	30.39	1.883	30.46	1.888	9.417	3.043
b₂	2.416	0.461	0.666	0.016	0.666	0.016	3×10^-5	7.6E-7
b₃	-1.960	0.461	524.26	177.0	524.48	177.5	3×10^-5	8.1E-7
b₄	170.87	12.84	4.383	0.749	4.290	0.787
b₅			0.784	0.024	0.777	0.027
b₆			2.413	0.170	2.241	0.182
p₁							0.074	0.002
p₂							0.565	0.028
	Bi (2000)		Lee et al. (2003)		Kozak (2004)		Sharma and Parton (2009)
	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error	Estimate	Approx. std. error
b₀	-0.256	0.191			0.661	0.072	1.093	0.007
b₁	0.318	0.114	1.453	0.102	0.866	0.031	-0.006	3×10^-4
b₂	0.087	0.019	0.967	0.021	0.315	0.054	0.125	0.037
b₃	0.177*	0.119	3.095	0.251	0.433	0.032	0.109	0.049
b₄	-1×10^-4*	3×10^-4	-4.377	0.314	-0.512	0.128
b₅	0.097	0.021	2.371	0.102	0.592	0.033
b₆	-0.089*	0.047			-0.007	9×10^-4
b₇					0.661	0.072

Models*	Manchurian fir				Korean spruce
Models*	MPB	RMSE	FI	CN*	MPB	RMSE	FI	CN
Max and Burkhart (1976)	3.2697	0.9175	0.9871	108	3.5932	1.0965	0.9845	190
Clark et al. (1991) Model 1	3.1607	0.8932	0.9878	18	3.4374	1.0545	0.9857	4
Clark et al. (1991) Model 2	3.2370	0.9046	0.9875	12	3.6054	1.0808	0.9849	3
Fang et al. (2000)	3.2941	0.9387	0.9865	107	3.7924	1.1511	0.9830	109
Bi (2000)	3.4599	0.9952	0.9848	383	3.9337	1.2439	0.9801	321
Lee et al. (2003)	3.8953	1.1060	0.9813	53	4.1621	1.2420	0.9801	40
Kozak (2004)	3.2453	0.9080	0.9874	186	3.5697	1.1010	0.9844	109
Sharma and Parton (2009)	3.4265	0.9590	0.9859	5	3.7844	1.1593	0.9827	6

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