Evaluation of Sampling with Partial Replacement and Double Sampling in a Managed Forest in the Brazilian Amazon

Reategui-Betancourt, Jorge L.; Mazzei, Lucas; Rezende, Alba Valéria; Briceño, Guido; Oliveira, Carlos Magno Moreira de; Gaui, Tatiana Dias; Khan, Salman; Figueiredo, Axa Emanuelle S.

doi:10.1590/2179-8087-FLORAM-2023-0011

Jorge L. Reategui-Betancourt

investigation (equal)

resources (equal)

writing - original draft (equal)

writing - review & editing (equal)

Universidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

https://orcid.org/0000-0002-1999-3001

Lucas Mazzei

writing - original draft (support)

writing - review & editing (support)

formal analysis (support)

Universidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

Empresa Brasileira de Pesquisa Agropecuária (EMBRAPA) Amazônia Oriental, Belém, PA, Brasil.

https://orcid.org/0000-0002-1757-9472

Alba Valéria Rezende

writing - original draft (support)

writing - review & editing (support)

formal analysis (support)

Universidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

https://orcid.org/0000-0002-1167-9798

Guido Briceño

conceptualization (equal)

methodology (equal)

writing - review & editing (equal)

supervision (equal)

Centre International de Recherche Agronomique pour le Développement (CIRAD), The French National Centre for Scientific Research (CNRS), Montpellier, France.

https://orcid.org/0000-0003-2194-4716

Carlos Magno Moreira de Oliveira

conceptualization (equal)

methodology (equal)

writing - review & editing (equal)

formal analysis (equal)

supervision (equal)

Universidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

https://orcid.org/0000-0002-2694-7388

Tatiana Dias Gaui

investigation (support)

Universidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

https://orcid.org/0000-0002-1793-3275

Salman Khan

investigation (support)

Centre International de Recherche Agronomique pour le Développement (CIRAD), The French National Centre for Scientific Research (CNRS), Montpellier, France.

https://orcid.org/0000-0002-2460-8390

Axa Emanuelle S. Figueiredo

investigation (support)

Universidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

https://orcid.org/0000-0002-7276-905X

CORRESPONDENCE TO Jorge L. Reategui-Betancourt Campus Universitário Darcy Ribeiro, Asa Norte, 70910-900, Brasília, DF, Brasil. e-mail: jorgereategui91@gmail.com

Associate editor: Rafaella de Angeli Curto http://orcid.org/0000-0001-5509-4655

CFI		DS		SPR
First occasion	Second occasion	First occasion	Second occasion	First occasion	Second occasion
P1 (m)	P1 (m)	P1 (u)		P1 (u)
P2(m)	P2(m)	P2(u)		P2(u)
P3(m)	P3(m)	P3(u)		P3(u)
P4(m)	P4(m)	P4(u)		P4(u)
P5(m)	P5(m)	P5(u)		P5(u)
P6(m)	P6(m)	P6(u)		P6(u)
P7(m)	P7(m)	P7(u)		P7(u)
P8(m)	P8(m)	P8(u)		P8(u)
P9(m)	P9(m)	P9(u)		P9(m)	P9(m)
P10(m)	P10(m)	P10(u)		P10(m)	P10(m)
P11(m)	P11(m)	P11(m)	P11(m)	P11(m)	P11(m)
P12(m)	P12(m)	P12(m)	P12(m)	P12(m)	P12(m)
P13(m)	P13(m)	P13(m)	P13(m)	P13(m)	P13(m)
P14(m)	P14(m)	P14(m)	P14(m)		P14(n)
P15(m)	P15(m)	P15(m)	P15(m)		P15(n)
P16(m)	P16(m)	P16(m)	P16(m)		P16(n)
P17(m)	P17(m)	P17(m)	P17(m)		P17(n)
P18(m)	P18(m)	P18(m)	P18(m)		P18(n)

Continuous Forest Inventory (CFI)
Estimators	First occasion	Second occasion	Growth
Sampling error	Absolute: $E_{a} = t \cdot S_{\bar{x}}$	Absolute: $E_{a} = t \cdot S_{\bar{y}}$	Absolute: $E_{a} = t \cdot S_{{\bar{d}}_{m}}$
Sampling error	Relative: $E_{r} = t \cdot \frac{S_{\bar{x}}}{\bar{x}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{\bar{y}}}{\bar{y}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{{\bar{d}}_{m}}}{{\bar{d}}_{m}} \cdot 100$
Standard error of the mean	$S_{\bar{x}} = \sqrt{{S^{2}}_{\bar{x}}}$	$S_{\bar{y}} = \sqrt{{S^{2}}_{\bar{y}}}$	$S_{{\bar{d}}_{m}} = \sqrt{{S^{2}}_{{\bar{d}}_{m}}}$
Variance of the Mean	${S^{2}}_{\bar{X}} = \frac{{S^{2}}_{X}}{n_{1}} \cdot f c;$	${S^{2}}_{\bar{y}} = \frac{{s^{2}}_{y}}{n_{1}} \cdot f c;$	${S^{2}}_{{\bar{d}}_{m}} = {S^{2}}_{\bar{x}} + {S^{2}}_{\bar{y}} - \frac{2 \cdot S_{x y}}{m}$
Variance of the Mean	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: S_xy = Covariance between the two occasions
Variance	${S^{2}}_{x} = \frac{\sum_{i = 1} n_{1} {(X i - \bar{X})}^{2}}{n_{1} - 1}$	${S^{2}}_{y} = \frac{\sum_{i = 1} n_{2} {(Y i - \bar{Y})}^{2}}{n_{2} - 1}$
Mean	${\bar{X}}_{m} = \frac{\sum_{i = 1} n_{1} X_{i}}{n_{1}}$	${\bar{Y}}_{m} = \frac{\sum_{i = 1} n_{2} Y_{i}}{n_{1}}$	${\bar{d}}_{m} = {\bar{Y}}_{m} - {\bar{X}}_{m}$
Double Sampling (DS)
Estimators	First occasion	Second occasion	Growth
Sampling error	Absolute: $E_{a} = t \cdot S_{{\bar{x}}_{1}}$	Absolute: $E_{a} = t \cdot S_{\bar{y}}$	Absolute: $E_{a} = t \cdot S_{{\bar{d}}_{r}}$
Sampling error	Relative: $E_{r} = t \cdot \frac{S_{{\bar{x}}_{1}}}{{\bar{x}}_{1}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{\bar{y}}}{\bar{v}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{{\bar{d}}_{r}}}{{\bar{d}}_{r}} \cdot 100$
Standard error of the mean	${S_{\bar{x}}}_{1} = \sqrt{{S^{2}}_{{\bar{x}}_{1}}}$	$S_{\bar{y}} = \sqrt{{S^{2}}_{\bar{y}}}$	$S_{{\bar{d}}_{r}} = \sqrt{{S^{2}}_{{\bar{d}}_{r}}}$
Variance of the Mean	${S^{2}}_{{\bar{x}}_{1}} = \frac{{S^{2}}_{x}}{n_{1}} \cdot f c;$	${S^{2}}_{\bar{y}} = \frac{{S^{2}}_{y} \cdot (1 - ρ^{2})}{n_{2}} + \frac{ρ^{2} \cdot {S^{2}}_{y}}{n_{1}}$	${S^{2}}_{{\bar{d}}_{r}} = {S^{2}}_{y} \cdot (\frac{{(1 - ρ)}^{2}}{n_{1}} + \frac{1 - ρ^{2}}{n_{2}})$
Variance of the Mean	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: $ρ = \frac{C o ν (x y)}{S_{x} S_{y}}$
Variance	${S^{2}}_{x_{1}} = \frac{\sum_{i = 1} n_{1} {(X_{i} - \bar{X})}^{2}}{n_{1} - 1}$	${S^{2}}_{y} = \frac{\sum_{i = 1} n_{2} {(Y_{i} - \bar{Y})}^{2}}{n_{2} - 1}$
Mean	${\bar{x}}_{1} = P u {\bar{x}}_{u} + P u {\bar{x}}_{m}$	$y_{r} = y_{m} + b \cdot (x_{1} - x_{m})$	${\bar{d}}_{r} = {\bar{y}}_{r} - {\bar{x}}_{1}$
		Where: b =
		b = angular coefficient of the regression
Sampling with Partial Replacement (SPR)
Estimators	First occasion	Second occasion	Growth
Sampling error	Absolute: $E_{a} = t \cdot S_{\bar{x}}$	Absolute: $E_{a} = t \cdot S_{\bar{y}}$	Absolute: $E_{a} = t \cdot S_{{\bar{d}}_{p}}$
Sampling error	Relative: $E_{r} = t \cdot \frac{S_{\bar{x}}}{\bar{X}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{\bar{y}}}{\bar{Y}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{{\bar{d}}_{p}}}{{\bar{d}}_{p}} \cdot 100$
Standard error of the mean	$S_{\bar{x}} = \sqrt{{S^{2}}_{\bar{x}}}$	$S_{\bar{y}} = \sqrt{{S^{2}}_{\bar{y}}}$	$S_{{\bar{d}}_{p}} = \sqrt{{S^{2}}_{{\bar{d}}_{p}}}$
Variance of the Mean	${S^{2}}_{\bar{x}} = \frac{{s^{2}}_{x}}{n_{1}} \cdot f c;$	${S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}$	${S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}$
Variance of the Mean	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: $r = \frac{C o v (x y)}{S_{x} S_{y}}$	Where: $A = \frac{m}{n_{2} - P_{u} \cdot n \cdot r^{2}} + \frac{n \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{x}}{S_{y}} B = \frac{- m \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}} - \frac{n_{2} \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}}$
Variance	${S^{2}}_{x} = \frac{\sum_{i = 1} n 1 {(X_{i} - \bar{X})}^{2}}{n_{1} - 1}$	${S^{2}}_{y} = \frac{\sum_{i = 1} n 1 {(Y_{i} - \bar{Y})}^{2}}{n_{2} - 1}$
Mean	$\bar{X} = (1 - b) \cdot {\bar{x}}_{u} + b \cdot {\bar{x}}_{m} + c \cdot {\bar{y}}_{m} - c \cdot {\bar{y}}_{n};$	$\bar{Y} = a \cdot {\bar{x}}_{u} - a \cdot {\bar{x}}_{m} + c \cdot {\bar{y}}_{m} + (1 - c) \cdot {\bar{y}}_{n};$	${\bar{d}}_{p} = A \cdot {\bar{y}}_{m} + B \cdot {\bar{x}}_{m} + (1 - A) \cdot {\bar{y}}_{n} - (1 + B);$
Mean	Where: $b = \frac{n_{2} \cdot P_{m}}{n_{2}} - P_{u} \cdot n \cdot r^{2} c = \frac{- n \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}}$	Where: $c = \frac{m}{n_{2}} - P_{u} \cdot n \cdot r^{2} a = \frac{m \cdot P_{u}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}}$	Where: $A = \frac{m}{n_{2} - P_{u} \cdot n \cdot r^{2}} + \frac{n \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{x}}{S_{y}} B = \frac{- m \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}} - \frac{n_{2} \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}}$

{S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}

{S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}

[1] CORRESPONDENCE TO Jorge L. Reategui-Betancourt Campus Universitário Darcy Ribeiro, Asa Norte, 70910-900, Brasília, DF, Brasil. e-mail: jorgereategui91@gmail.com

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Evaluation of Sampling with Partial Replacement and Double Sampling in a Managed Forest in the Brazilian Amazon

Abstract