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Table 2 Summary of methods used for crown biomass estimation in this study

From: Evaluation of sampling strategies to estimate crown biomass

Methods Equations for total crown biomass Selection probability Inclusion probability
Simple random sampling
SRS \( {\widehat{\tau}}_i=N{B}_i\left(\frac{1}{n}{\displaystyle \sum_{j=1}^n}{y}_{ij}\right) \) \( {\left(\frac{1}{N{B}_i}\right)}^n \) \( \frac{n}{N{B}_i} \)
SRS-RAT \( {\widehat{\tau}}_i=\frac{{\displaystyle {\sum}_{j=1}^n}{y}_{ij}}{{\displaystyle {\sum}_{j=1}^n}B{D}_{ij}^2}{\displaystyle \sum_{j=1}^{N{B}_i}}B{D}_{ij}^2 \) \( {\left(\frac{1}{N{B}_i}\right)}^n \) \( \frac{n}{N{B}_i} \)
PPS \( {\widehat{\tau}}_i={\displaystyle \sum_{j\in S}}\frac{y_{ij}}{\pi_{ij}^{PPS}} \) \( {\pi}_{ij}=\frac{B{D}_{ij}^2}{{\displaystyle {\sum}_{j=1}^{NB}}B{D}_{ij}^2} \) \( {\pi}_{ij}^{PPS}=1-{\left(1-{\pi}_{ij}\right)}^n \)
Systematic sampling
SYS \( {\widehat{\tau}}_i=N{B}_i\left(\frac{1}{n}{\displaystyle \sum_{j=1}^n}{y}_{ij}\right) \) \( \frac{n}{N{B}_i} \) \( \frac{n}{N{B}_i} \)
SYS-RAT \( {\widehat{\tau}}_i=\frac{{\displaystyle {\sum}_{j=1}^n}{y}_{ij}}{{\displaystyle {\sum}_{j=1}^n}B{D}_{ij}^2}{\displaystyle \sum_{j=1}^{N{B}_i}}B{D}_{ij}^2 \) \( \frac{n}{N{B}_i} \) \( \frac{n}{N{B}_i} \)
Stratified sampling
STR \( {\widehat{\tau}}_i={\displaystyle \sum_{h=1}^H}{\displaystyle \sum_{j=1}^{n_h}}\frac{N_{ih}}{n_h}{y}_{ijh} \) \( \frac{B{D}_{ij}^2}{{\displaystyle {\sum}_{j=1}^{NB}}B{D}_{ij}^2} \) \( \frac{n_h}{N_{ih}} \)
STR-RAT \( {\widehat{\tau}}_i=\frac{{\displaystyle {\sum}_{h=1}^H}{\displaystyle {\sum}_{j=1}^{n_h}}\frac{N_{ih}}{n_h}{y}_{ijh}}{{\displaystyle {\sum}_{h=1}^H}{\displaystyle {\sum}_{j=1}^{n_h}}\frac{N_{ih}}{n_h}B{D}_{ijh}^2} \) \( \frac{B{D}_{ij}^2}{{\displaystyle {\sum}_{j=1}^{NB}}B{D}_{ij}^2} \) \( \frac{n_h}{N_{ih}} \)
STR-PPS \( {\widehat{\tau}}_{i\left(STR-PPS\right)}={\displaystyle \sum_{h=1}^H}{\displaystyle \sum_{j\in {S}_h}}\frac{y_{ijh}}{\pi_{ijh}^{\left(STR-PPS\right)}} \) \( \frac{B{D}_{ijh}^2}{{\displaystyle {\sum}_{j=1}^{N{B}_{ih}}}B{D}_{ijh}^2} \) \( {\pi}_{ij}^{STR-PPS}=1-{\left(1-{\pi}_{ij}\right)}^n \)
  1. Notation: \( {\widehat{\tau}}_i \) is the estimated total crown biomass for i th tree; y ij is the oven dry weight of j th branch on i th tree; NB i is the number of branches on i th tree; n is number of branches sampled; N ih is the number of branches in h th stratum on i th tree; and n h is the number of branches sampled in h th stratum.