Table 2 Data sources and sampling designs for REDD monitoring

Field plots

Simple random sampling

$\overline{\text{y}}\text{=}\frac{{\displaystyle \sum _{\text{i}=1}^{\text{n}}{\text{y}}_{\text{i}}}}{\text{n}};\text{v(}\overline{\text{y}}\text{)=}\frac{{\displaystyle \sum _{\text{i}=1}^{\text{n}}{({\text{y}}_{\text{i}}-\overline{\text{y}})}^2}}{\text{n}(\text{n}-1)}$

Systematic sampling

Field plots & full coverage remote sensing imagery

Stratified sampling

Example: field plots and Spot satellite imagery

${\overline{\text{y}}}_{\text{st}}={\displaystyle \sum _{\text{h}=\text{1}}^{\text{L}}\frac{{\text{A}}_{\text{h}}}{\text{A}}\frac{{\displaystyle \sum _{\text{i}=\text{1}}^{{\text{n}}_{\text{h}}}{\text{y}}_{\text{hi}}}}{{\text{n}}_{\text{h}}}}$

$\text{v}\left({\overline{\text{y}}}_{\text{st}}\right)={\displaystyle \sum _{\text{h}=\text{1}}^{\text{L}}\frac{{{\left(\frac{{\text{A}}_{\text{h}}}{\text{A}}\right)}^{\text{2}}}^{}{\text{s}}_{\text{h}}^{\text{2}}}{{\text{n}}_{\text{h}}}}$;

Regression estimators

Example: field plots and TerraSarX (radar)

${\overline{\text{y}}}_{\text{lr}}=\overline{\text{y}}+\text{b(}\overline{\text{X}}-\overline{\text{x}}\text{)}$

$\text{v}\left({\overline{\text{y}}}_{\text{lr}}\right)\approx \frac{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\left[\left({\text{y}}_{\text{i}}-\overline{\text{y}}\right)-\text{b}\left({\text{x}}_{\text{i}}-\overline{\text{x}}\right)\right]}^{\text{2}}}}{\text{n(n}-\text{2)}}$

Field plots & partial coverage remote sensing imagery

2-phase sampling for stratification (double sampling for stratification)

Example: field plots & aerial photography

${\overline{\mathsf{\text{y}}}}_{\mathsf{\text{ds}}}=\sum _{\mathsf{\text{h}}=1}^{\mathsf{\text{L}}}\frac{{\mathsf{\text{n}}}_{\mathsf{\text{h}}}^{\prime }}{\mathsf{\text{n’}}}{\overline{\mathsf{\text{y}}}}_{\mathsf{\text{h}}}=\sum _{\mathsf{\text{h}}=1}^{\mathsf{\text{L}}}{\mathsf{\text{w}}}_{\mathsf{\text{h}}}{\overline{\mathsf{\text{y}}}}_{\mathsf{\text{h}}}$

$v\left({\overline{\mathsf{\text{y}}}}_{\mathsf{\text{ds}}}\right)=\frac{N-1}{N}\sum _{\mathsf{\text{h = 1}}}^{\mathsf{\text{L}}}\left(\frac{{\mathsf{\text{n}}}_{\mathsf{\text{h}}}-1}{\mathsf{\text{n}}\prime \mathsf{\text{ - 1}}}-\frac{{\mathsf{\text{n}}}_{\mathsf{\text{L}}}^{\prime }-1}{\mathsf{\text{N}}-1}\right)\frac{{\mathsf{\text{W}}}_{\mathsf{\text{h}}}^{\prime }-{\mathsf{\text{s}}}_{\mathsf{\text{h}}}^2}{\mathsf{\text{n}}\prime }+\frac{\mathsf{\text{N - n}}\prime }{\mathsf{\text{N(n}}\prime -1\mathsf{\text{)}}}\sum _{\mathsf{\text{h = 1}}}^{\mathsf{\text{L}}}{\mathsf{\text{W}}}_{\mathsf{\text{h}}\left({\overline{\mathsf{\text{y}}}}_{\mathsf{\text{h}}}-{\overline{\mathsf{\text{y}}}}_{\mathsf{\text{ds}}}\right)}$

where

$s_h^2=\frac{1}{n_h^{\prime }}\sum _{h=1}^L{\left(y_{hi}-{\overline{y}}_{\mathsf{\text{h}}}\right)}^2$

2-phase sampling with regression estimators

Example: field plots & LiDAR

${\overline{\text{y}}}_{\text{lr}}=\overline{\text{y}}+\text{b(}\overline{\text{x}}\text{'}-\overline{\text{x}}\text{)}$

$v\left({\overline{y}}_{lr}\right)=s_{y.x}^2\left[\frac{1}{n}+\frac{{\left(\overline{x}\text{'}-\overline{x}\right)}^2}{{\displaystyle \sum {\left(\overline{x}\text{'}-\overline{x}\right)}^2}}\right]+\frac{s_y^2-s_{y.x}^2}{n\text{'}}-\frac{s_y^2}{N}$

with $s_{y.x}^2=\frac{1}{n-2}\left[{\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2-b^2}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}\right]$