Table 1 List of adjustment functions used in recent methods to link models of different resolutions

Qin and Cao ([2006])

Proportional yield

1

${\tilde{p}}_{2,i}={\widehat{p}}_{2i}\left(\frac{s{\widehat{N}}_2}{{\displaystyle {\sum }_j{\widehat{p}}_{2j}}}\right)$

2

${\tilde{d}}_{2i}^2={\widehat{d}}_{2i}^2\left(\frac{s{\widehat{B}}_2/K}{{\displaystyle {\sum }_j{\widehat{p}}_{2j}{\widehat{d}}_{2j}^2}}\right)$

3

${\tilde{h}}_{2i}={\widehat{h}}_{2i}\left(\frac{s\left({\widehat{V}}_2-a{\widehat{N}}_2\right)}{b{\displaystyle {\sum }_j{\widehat{p}}_{2j}{\widehat{d}}_{2j}^2{\widehat{h}}_{2j}}}\right)$

Proportional growth

4

${\tilde{p}}_{2i}=\frac{{\widehat{p}}_{2i}}{{\widehat{p}}_{2i}+m_p\left(1-{\widehat{p}}_{2i}\right)}$

5

${\tilde{d}}_{2i}^2=d_{1i}^2+\left(\frac{s{\widehat{B}}_2/K-{\displaystyle {\sum }_j{\tilde{p}}_{2j}{d}_{1j}^2}}{{\displaystyle {\sum }_j{\tilde{p}}_{2j}\left({\widehat{d}}_{2j}^2-d_{1j}^2\right)}}\right)\left({\widehat{d}}_{2i}^2-d_{1i}^2\right)$

6

${\tilde{h}}_{2i}=h_{1i}+\left(\frac{s\left({\widehat{V}}_2-a{\widehat{N}}_2\right)-b{{\displaystyle {\sum }_j\tilde{p}}}_{2j}{\tilde{d}}_{2j}^2{h}_{1j}}{b{{\displaystyle {\sum }_j\tilde{p}}}_{2j}{\tilde{d}}_{2j}^2\left({\widehat{h}}_{2j}-h_{1i}\right)}\right)\left({\widehat{h}}_{2i}-h_{1i}\right)$

Constrained least squares

7

${\tilde{p}}_{2i}=\left({\widehat{p}}_{2i}+s{\widehat{N}}_2-{{\displaystyle \sum \widehat{p}}}_{2j}\right)/n$

8

${\tilde{d}}_{2i}^2={\widehat{d}}_{2j}^2-{\tilde{p}}_{2i}\left(\frac{{\displaystyle {\sum }_j}{\tilde{p}}_{2j}{\widehat{d}}_{2j}^2-s{\widehat{B}}_2/K}{{\displaystyle {\sum }_j{\tilde{p}}_{2j}^2}}\right)$

9

${\tilde{h}}_{2i}={\widehat{h}}_{2i}-{\tilde{p}}_{2i}{\tilde{d}}_{2i}^2\left(\frac{{\displaystyle {\sum }_j{\tilde{p}}_{2j}{\tilde{d}}_{2j}^2{\widehat{h}}_{2j}+s\left(a{\widehat{N}}_2-{\widehat{V}}_2\right)/b}}{{\displaystyle {\sum }_j{\tilde{p}}_{2j}^2{\tilde{d}}_{2j}^4}}\right)$

Coefficient adjustment

10

${\tilde{p}}_{2i}=p_{1i}/\left(1+\mathit{exp}\left[{\alpha }_0+{\alpha }_1{H}_1+{\alpha }_2{m}_p\left(d_{1i}/D{q}_1\right)\right]\right)$

11

${\tilde{d}}_{2i}=d_{1i}\left\{1+\mathit{exp}\left[{\beta }_0+{\beta }_1\ln B_1+{\beta }_2{A}_1+{\beta }_3\ln H_1+{\beta }_4{m}_d\left(\frac{d_{1i}}{D{q}_1}\right)+{\beta }_5\ln h_{1i}\right]\right\}$

12

${\tilde{h}}_{2i}=h_{1i}\left\{1+\mathit{exp}\left[{\gamma }_0+{\gamma }_1\ln B_1+{\gamma }_2{A}_1+{\gamma }_3\ln H_1+{\gamma }_4{m}_h\left(\frac{d_{1i}}{D{q}_1}\right)+{\gamma }_5\left(\frac{h_{1i}}{H_1}\right)+{\gamma }_6\ln d_{1i}\right]\right\}$

Cao ([2006])

Disaggregation

13

${\tilde{p}}_{2i}={\widehat{p}}_{2i}^{m_p}$

14

${\tilde{d}}_{2i}^2=d_{1i}^2+\left(\frac{s{\widehat{B}}_2/K-{\displaystyle {\sum }_j{\tilde{p}}_{2j}{d}_{1j}^2}}{{\displaystyle {\sum }_j{\tilde{p}}_{2j}\left({\widehat{d}}_{2j}^2-d_{1j}^2\right)}}\right)\left({\widehat{d}}_{2i}^2-d_{1i}^2\right)$

Constraining individual-tree model with diameter-class attributes

15

$\left\{\begin{array}{l}{\widehat{p}}_{2i}=1/\left(1+\mathit{exp}\left[{\alpha }_0+{\alpha }_1{N}_1+{\alpha }_2{B}_1+{\alpha }_3{d}_{1i}\right]\right)\hfill \\ {\widehat{n}}_{2,k}={\displaystyle {\sum }_{i=1}^{n_{1,k}}{\widehat{p}}_{2i}}\hfill \end{array}\right.$

16

$\left\{\begin{array}{l}{\widehat{d}}_{2i}=d_{1i}+{\beta }_1{\left(\frac{A_2}{A_1}\right)}^{{\beta }_2}{H}_1^{{\beta }_3}{B}_1^{{\beta }_4}{d}_1^{{\beta }_5}\hfill \\ {\widehat{b}}_{2,k}=K{\displaystyle {\sum }_{i=1}^{n_{1,k}}{\widehat{p}}_{2i}}{\widehat{d}}_{2i}^2\hfill \end{array}\right.$

Constraining individual-tree model with stand attributes

17

$\left\{\begin{array}{l}{\widehat{p}}_{2i}=1/\left(1+\mathit{exp}\left[{\alpha }_0+{\alpha }_1{N}_1+{\alpha }_2{B}_1+{\alpha }_3{d}_{1i}\right]\right)\hfill \\ {\widehat{N}}_2={\displaystyle {\sum }_i{\widehat{p}}_{2i}/s}\hfill \end{array}\right.$

18

$\left\{\begin{array}{l}{\widehat{d}}_{2i}=d_{1i}+{\beta }_1{\left(\frac{A_2}{A_1}\right)}^{{\beta }_2}{H}_1^{{\beta }_3}{B}_1^{{\beta }_4}{d}_1^{{\beta }_5}\hfill \\ {\widehat{B}}_2=\left(\frac{K}{s}\right){\displaystyle {\sum }_i{\widehat{p}}_{2i}{\widehat{d}}_{2i}^2}\hfill \end{array}\right.$

Yue et al. ([2008])

Combined estimator

19

${\tilde{B}}_{\mathit{2}}\mathit{=}w{\widehat{B}}_{\mathit{2}T}\mathit{+}\left(1\mathit{-}w\right){\widehat{B}}_{\mathit{2}S}$ ,

where w is selected to minimize the variance of ${\tilde{B}}_2$.

Zhang et al. ([2010])

Combined estimator

20

${\tilde{B}}_2=w_1{\widehat{B}}_{2T}+w_2{\widehat{B}}_{2S}+w_3{\widehat{B}}_{2D}$,

where w _k is selected to minimize ${\displaystyle \sum }{\left(B_2-{\tilde{B}}_2\right)}^2$, and ${\displaystyle {\sum }_k^3{w}_k}=1$.

Cao ([2010])

1

21

${\tilde{p}}_{2i}={\widehat{p}}_{2i}^m$

Tree survival

2

22

${\tilde{p}}_{2i}=\frac{{\widehat{p}}_{2i}}{{\widehat{p}}_{2i}+m_p\left(1-{\widehat{p}}_{2i}\right)}$

3

23

${\tilde{p}}_{2i}=1/\left(1+\mathit{exp}\left[m_p{\alpha }_0+{\alpha }_3{d}_{1i}\right]\right)$

4

24

${\tilde{p}}_{2i}=1/\left(1+\mathit{exp}\left[{\alpha }_0+{\alpha }_1{N}_1+{\alpha }_2{B}_1+m_p{d}_{1i}\right]\right)$

5

25

${\tilde{p}}_{2i}={\widehat{p}}_{2i}+\left(\frac{s{\widehat{N}}_2-{\displaystyle {\sum }_j{\widehat{p}}_{2j}}}{s{N}_1-{\displaystyle {\sum }_j{\widehat{p}}_{2j}}}\right)\left(1-{\widehat{p}}_{2i}\right)$

Cao ([2010])

1

26

${\widehat{d}}_{2i}=d_{1i}+m_d{d}_1^{{\beta }_5}$

Tree diameter growth

2

27

${\widehat{d}}_{2i}=d_{1i}+{\beta }_1{\left(\frac{A_2}{A_1}\right)}^{{\beta }_2}{H}_1^{{\beta }_3}{B}_1^{{\beta }_4}{d}_1^{m_d}$

3

28

${\tilde{d}}_{2i}^2=d_{1i}^2+\left(\frac{s{\widehat{B}}_2/K-{\displaystyle {\sum }_j{\tilde{p}}_{2j}{d}_{1j}^2}}{{\displaystyle {\sum }_j{\tilde{p}}_{2j}\left({\widehat{d}}_{2j}^2-d_{1j}^2\right)}}\right)\left({\widehat{d}}_{2i}^2-d_{1i}^2\right)$