## 5.2.6.2   Method 2: Calculation of emissions/removals factors from change-over-time    For the second method, the same strata are monitored through time, and if change occurs (such as clearing or degradation) then an emissions/removals factor can be calculated from the observed change,
Where and correspond to the carbon density of the forest before and after the change, respectively.
The calculation of the uncertainty of the emission/removal factor in this case depends upon the sampling design, and in particular whether there were permanent plots that were surveyed at both t1 and t2. In the simple case when there were no permanent plots and the carbon density estimates were obtained from independent samples at t1 and t2, then the overall uncertainty can be calculated in an analogous way to Equation 28,
In contrast, if the plots were permanently located and if all of the sample plots measured at t1 were re-measured at t2, then the samples are correlated, and this correlation should be taken into account. In this case the uncertainty in is given by,
where r is the correlation in biomass density from t1 to t2 across the sample plots.
When biomass density is positively correlated between t1 and t2, then the final term of Equation 31 acts to reduce the overall variance, and thus to increase the precision. More generally, Equation 30 and Equation 31 can also be used to determine the uncertainty of any change in measured biomass between two time periods, such as in the analysis of general forest monitoring. In this case and in the absence of significant disturbance the correlation is likely to be high, and typically greater than 0.8 (Köhl et al., 2006) especially if t1 and t2 are relatively close together in time (< 10 years apart).
Equation 30 and Equation 31 represent two extreme cases where either all of the original plots were re-surveyed Equation 31), or none of the original plots were re-surveyed (Equation 30). The intermediate situation occurs when only a fraction of the plots are permanent, with some plots only measured at t1, and some plots only measured at t2. This can occur if e.g. some plots were lost or destroyed after t1 but were replaced by other plots at t2, or if there were difficulties with re-locating plots in the field. Ensuring a mixture of both permanent and temporary plots can also be built into the survey design to provide some insurance against the situation where, over time, permanent plots become non-representative, thus potentially introducing bias.
A sampling design with a mixture of temporary and permanent plots is known as sampling with replacement (Loetsch & Haller, 1964), with the calculation of the estimate of being more complex than either of the extreme cases. Köhl et al., (2015) provide a more complete description of sampling with replacement in the context of REDD+, and also present the calculations required to estimate for this situation. These calculations use linear regression to update the mean carbon density at t1 based on information embedded within the t2 survey results, and therefore the estimate for the mean change in carbon density no longer equals the simple difference given in Equation 21. If this is considered undesirable, then an alternative estimate for under sampling with replacement (Päivinen & Yli-Kojola, 1989) can be used instead. The calculation is described in Box 27.
Reducing uncertainties
Uncertainties in the estimation of emission/removal factors can be reduced by:
• increasing sampling density without further sub-stratification
• further sub-stratification to focus sampling on forest areas likely to be affected by REDD+ activities, after as well as before the transfers between strata or land use change has occurred. If further stratification is adopted then the estimates for and may need to be calculated using the estimators appropriate for a stratified sampling design, as described under ‘Stratified estimators’ in Section 5.1.5. No more than 6-8 strata are generally recommended (Cochran 1977, p134).
• retaining the same stratification and sampling density but using auxiliary information to verify the direction of change. For example in the case of degradation, if the direction of transfer was consistent with advancing forest fragmentation, then increased forest carbon density would be unlikely and the probability distribution of the degradation estimate should be considered truncated so as to eliminate the possibility of increases.
• Increasing the number of permanent sample plots, if using Method 2 to estimate the change in carbon density over time.
Other uncertainties
The calculation for the uncertainty of the emissions factor given in Equation 28 and Equation 30 includes only error due to sampling, and although it is typically the most important source of error, there are a number of other error sources, such as measurement errors, errors associated with the use of allometric models used to predict tree biomass, or errors in expansion factors such as root:shoot ratios for estimating below-ground biomass. These additional errors can be considered independent from the sampling error, and thus the total error variance can estimated by adding them to . Of these additional error sources, uncertainty arising from the prediction of each individual from the biomass model, and uncertainty resulting from a choice of alternative suitable models, are likely the major additional terms that should be considered for inclusion. The former of these diminishes with increasing sample size, and hence its importance is partially a function of the total number of individuals estimated. The latter error source is independent of sample size, and thus cannot be reduced by increased field effort. If alternative allometric models are available for a given situation, then it is recommended that the uncertainty due to model choice be considered for inclusion in the total error estimate. Measurement errors, such as errors in the estimation of stem diameter, are generally small so long as standard forestry protocols have been used.
A wide range of different methods can be used to estimate these additional allometric model error terms, including analytical approximations (e.g. Lo 2005, Ståhl et al., 2014), Monte-Carlo methods (e.g. Molto et al., 2013, Picard et al., 2015) and hybrid approaches (e.g. Chave et al., 2004). These additional error sources can combined into a single variance term, , and added to to provide an estimate of total error.

### Box 27: Calculation of the uncertainty of emissions/removals factors under sampling with replacement

Sampling with replacement is a survey design where the measured change over time involves a combination of permanent and temporary sample plots. The estimate of the uncertainty of the difference between two time periods requires the following quantities:
n12: The number of ‘common’ or permanent plots across both t1 and t2.
n1: The total number of plots at t1.
n2: The total number of plots at t2.
n1- : The number of plots unique to t1.
n-2 : The number of plots unique to t2. : The variance of the measured carbon density at times t1 and t2.
r The plot-level correlation in carbon density between t1 and t2.
From this information two weighting parameters are calculated: