UMA METODOLOGIA PARA ESTIMAR O VOLUME E O AFILAMENTO DO TRONCO BASEADA EM SÓLIDOS DE REVOLUÇÃO

Soares, Carlos Pedro Boechat; Leite, Helio Garcia; Silva, Gilson Fernandes da

Carlos Pedro Boechat Soares ^**Corresponding author.

Conceptualization

Results analysis

discussion

Writing

review

editing

Universidade Federal de Viçosa, Departamento de Engenharia Florestal, Viçosa, Minas Gerais - Brazil

https://orcid.org/0000-0001-6475-3376

Helio Garcia Leite

Conceptualization

Results analysis

discussion

Writing

review

editing

Universidade Federal de Viçosa, Departamento de Engenharia Florestal, Viçosa, Minas Gerais - Brazil

https://orcid.org/0000-0002-6357-9174

Gilson Fernandes da Silva

Conceptualization

Results analysis

discussion

Writing

review

editing

Universidade Federal do Espírito Santo, Departamento de Ciências Florestais e da Madeira, Jerônimo Monteiro, Espírito Santo - Brazil

https://orcid.org/0000-0001-7853-6284

*Corresponding author.

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Figure 1
Solids of revolutions generated for sample tree #2 and observed diameters outside bark along the stem.

Figura 1
Sólidos de revoluções gerados para a árvore de amostra 2 e diâmetros observados com casca ao longo do tronco.

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Figure 2
Occurrence of different forms (neiloid, cone, and paraboloid) along the stems of sample trees.

Figura 2
Ocorrência de diferentes formas (neilóide, cone e parabolóide) ao longo dos troncos das árvores amostradas.

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Figure 3
Differences in percentage between the volumes of the sample trees obtained by the frustums of the cone (Vcone), paraboloid (Vpar), and neiloid (Vnei) formulas applied to all sections of the stems and for each section separately according to the classification of forms (Vsolids).

Figura 3
Diferenças percentuais entre os volumes das árvores amostrais obtidas pelas formulas dos troncos de cone (Vcone), parabolóide (Vpar) e neilóide (Vnei), aplicadas a todas as seções dos tronco e para cada seção separadamente de acordo com o classificação das formas (Vsolids).

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Figure 4
Distribution of the observed diameters outside bark and estimated by the taper models (A, B, C, and D) and by the proposed methodology (E) along the stem of the sample tree #2.

Figura 4
Distribuição dos diâmetros com casca observados e estimados pelos modelos de afilamento (A, B, C e D) e pela metodologia proposta (E) ao longo do tronco da árvore amostra 2.

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Figure 5
Trend curves between coefficients of forms (r) and relative heights (h/H) (A); heightdiameter relationship (B), and diameter at the base of the stems (db) and dbh (C).

Figura 5
Curvas de tendência entre coeficientes de formas (r) e alturas relativas (h/H) (A); relação altura-diâmetro (B), e diâmetro na base dos troncos (db) and DAP (C).

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Figure 6
Mean Absolute Deviation (MAD) (A) and Root Mean Square Error (RMSE) (B) for the sample trees calculated considering diameter estimates by the proposed methodology (Solids of revolution) and the taper models (Demaerschalk; Kozak 1988Kozak A. A variable exponent taper equation. Canadian Journal of Forest Research. 1988; 18(11):1363-1368. https://doi.org/10.1139/x88-213
https://doi.org/10.1139/x88-213... ; Kozak et al. 1969Kozak A, Munro DD, Smith JHG. Taper functions and their application in forest inventory. Forestry Chronicle. 1969; 45(4):278-283. doi:10.5558/tfc45278-4
https://doi.org/10.5558/tfc45278-4... ; and Ormerod).

Figura 6
Desvio Médio Absoluto (MAD) (A) e Raiz Quadrática do Erro Quadrático médio (RMSE) (B) para as árvores amostrais calculadas considerando estimativas de diâmetro pela metodologia proposta (Sólidos de revolução) e pelos modelos de afilamento (Demaerschalk; Kozak 1988Kozak A. A variable exponent taper equation. Canadian Journal of Forest Research. 1988; 18(11):1363-1368. https://doi.org/10.1139/x88-213
https://doi.org/10.1139/x88-213... ; Kozak e outros 1969; e Ormerod).

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Table 1
Range of the sample trees data use to develop the proposed methodology (diameter at breast height (dbh); total height (H); commercial height (Hc); minimum ou limit diameter along the stems (dl), defining Hc

Tabela 1
Faixa de dados das árvores amostrais utilizadas para desenvolver a metodologia proposta (diâmetro à altura do peito (DAP); altura total (H); altura comercial (Hc); diâmetro mínimo ou limite ao longo do tronco (dl), definindo Hc

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Table 2
Average lengths of the stem sections (

\bar{L}

) and representation in terms of the average percentage of total tree height (

\bar{% H}

) and volume (

\bar{% V}

)* by diameter classes and geometric forms

Tabela 2
Comprimentos médios das seções dos troncos (

\bar{L}

) e representação em termos da porcentagem média da altura total da árvore (

\bar{% H}

) e do volume (

\bar{% V}

)* por classes de diâmetro e formas geométricas

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Table 3
Number of sample trees by position on the stems (base, middle, and top) where the methodology or models were more accurate

Tabela 3
Número de árvores amostrais por posição no tronco (base, meio e topo) onde a metodologia ou modelos foram mais precisos

(Eq.1)

(Eq.2)

(Eq.3)

(Eq.4)

(Eq.5)

(Eq.6)

(Eq.7)

(Eq.8)

(Eq.9)

(Eq.10)

(Eq.11)

(Eq.12)

(Eq.13)

(Eq.14)

(Eq.15)

(Eq.16)

(Eq.17)

(Eq.18)

(Eq.19)

(Eq.20)

Table 1 Range of the sample trees data use to develop the proposed methodology (diameter at breast height (dbh); total height (H); commercial height (Hc); minimum ou limit diameter along the stems (dl), defining Hc

Variable	Minimum	Maximum
dbh (cm)	15.75	25.40
H (m)	24.30	31.80
Hc (m)	20.00	30.50
dl (cm)	2.03	5.84

Table 2 Average lengths of the stem sections (L¯) and representation in terms of the average percentage of total tree height (%H¯) and volume (%V¯)* by diameter classes and geometric forms

Diameter classes (cm)	$\bar{H}$ (m)	Neiloid			Cone			Paraboloid
Diameter classes (cm)	$\bar{H}$ (m)	$\bar{L}$ (m)	$\bar{% H}$	$\bar{% V}$	$\bar{L}$ (m)	$\bar{% H}$	$\bar{% V}$	$\bar{L}$ (m)	$\bar{% H}$	$\bar{% V}$
15.0-17.49	26.7	4.9	18.5	34.0	13.0	48.5	52.5	10.3	33.0	13.5
17.5-19.99	29.8	5.8	17.8	34.3	11,8	39.5	46.5	12.7	42.7	19.2
20.0-22.49	28.8	5.9	20.6	37.9	14.1	48.3	50.8	8.8	31.2	11.3
22.5-24.99	31.6	6.2	19.5	38.9	10.7	33.8	42.6	14.7	46.7	18,5
25.0-27.49	29.9	8.5	28.4	53.7	12.5	41.8	39.8	8.9	29.8	6.5
Average		5.7	19.0	39.8	12.4	42.6	46.4	11.6	38.4	13.8

Table 3 Number of sample trees by position on the stems (base, middle, and top) where the methodology or models were more accurate

	Mean Absolute Deviation (MAD)
Metodology/ Models	Base	Midle	Top	Base	Midle	Top
Solid of revolution	14	12	10	15	14	10
Kozak et al. (1969)Kozak A, Munro DD, Smith JHG. Taper functions and their application in forest inventory. Forestry Chronicle. 1969; 45(4):278-283. doi:10.5558/tfc45278-4 https://doi.org/10.5558/tfc45278-4...	3	7	3	1	6	3
Demaerschalk (1972)Demaerschalk JP 1972. Converting volume equations to compatible taper equations. Forest Science. 1972; 18(3):241-245. doi:10.1093/ forestscience/18.3.241 https://doi.org/10.1093/forestscience/18...	1	3	4	0	2	5
Ormerod (1973)Ormerod DW. A simple bole model. Forestry Chronicle. 1973;49(3):136-138. doi:10.5558/tfc49136-3 https://doi.org/10.5558/tfc49136-3...	2	1	2	3	0	2
Kozak (1988)Kozak A. A variable exponent taper equation. Canadian Journal of Forest Research. 1988; 18(11):1363-1368. https://doi.org/10.1139/x88-213 https://doi.org/10.1139/x88-213...	7	4	8	8	5	7
Total	27	27	27	27	27	27

\begin{aligned} \ln (d c c) = & \ln (α_{0}) + α_{1} \times \ln (d b h) + \ln (α_{2}) \times d b h \\ + β_{1} \times \ln (z) x^{2} \\ + & β_{2} \times \ln (z) \times \ln (x + 0.001) \\ + & β_{3} \times \ln (z) \times \sqrt{x} \\ + & β_{4} \times \ln (z) \times e^{x} \\ + & β_{5} \times \ln (z) \times (\frac{d b h}{H}) + ε \end{aligned}

\begin{matrix} {(\frac{d c c}{d b h})}^{2} = 10^{- 0.61114} \times d b h^{0.134054} \times (H - h)^{1.384254} \times H^{- 1.078298} \\ (- 9.16) (72.99) (125.87) (- 18.33) \\ {\bar{R}}^{2} = 0.9576; S_{y \cdot x} = \pm 0.06183 c m \end{matrix}

\begin{aligned} \ln (d c c) = - 1.84166 + 1.781777 \times \ln (d b h) \\ (- 4.40) (8.67) \\ - 0.03251 \times d b h - 0.46273 \times \ln (z) \times x^{2} \\ (- 3.19) (- 28.31) \\ + 0.364322 \times \ln (z) \times e^{x} \\ (68.41) \\ {\bar{R}}^{2} = 0.9740; S_{y x x} = \pm 0.0747 c m \end{aligned}

[1] *Corresponding author.

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UMA METODOLOGIA PARA ESTIMAR O VOLUME E O AFILAMENTO DO TRONCO BASEADA EM SÓLIDOS DE REVOLUÇÃO

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