## 5.3   Estimating total emissions/removals and its uncertainty    In general terms, estimates of emissions and removals of carbon dioxide are made by summing differences in carbon density, multiplied by the area in which the change in carbon occurred. Generically one is dealing with terms of the type:
• Change in carbon between time t1 and t2 = (area of a given stratum) x (change in carbon density of the stratum)
or
• Change in carbon between time t1 and t2= (area transferred between two strata) x (change in carbon density between the strata)
Both areas and carbon densities have uncertainties which need to be combined with each other when estimating emissions or removals of carbon associated with each of the relevant pools (i.e. biomass, dead organic matter, litter and soil carbon). Similarly uncertainties for estimates of non-CO2 greenhouse gas emissions are calculated by combining component emission/removal factors and activity data uncertainties.
The calculation for the uncertainty of area and change in area is described in Section 5.1.5 and is expressed as the variance in the estimate of the mean, denoted by . The calculation for the uncertainty in carbon density change is described in Section 5.2.5, and is given by .
The corresponding emissions, , are calculated as the product of the area and the emissions factor,
If the units of are in carbon, then the conversion to CO2 is straightforwardly achieved by multiplying by 44/12.
Section 5.2.2.1 of GPG2003 cross-references section 6.3 of GPG2000 (1) which describes Rule B for combining uncertainties when quantities are multiplied together, as in Equation 35. Rule B states the percentage uncertainty of the product is the square root of the sum of squares of the percentage uncertainties estimated for each of the quantities being multiplied. This rule is often used to calculate the variance of the product of two random, independent (i.e. uncorrelated) variables, it is an approximation. Goodman, (1960) derived an exact expression for the variance that requires no additional information to calculate, and which is given by:
Equation 36 requires are the estimates of the mean and the variance of the mean for area (A) and the emissions/removals factor. An example of the calculation of total emissions from activity data and emissions factors for a single stratum is given in Box 28.
Often the required emissions estimate is one that combines N separate stratum-level estimates, to give a total estimate for all strata combined. In this case the total emissions is the sum of the total emissions for each stratum, ( i = 1..N), with the variance of the estimate equal to For a given sampling density, the uncertainties associated with degradation, or removals as the result of forest growth in either MNF or planted forests, will be greater than those associated with deforestation estimates. If the uncertainty in biomass estimation exceeds the difference in carbon densities between the two sub-strata, the uncertainty of the degradation estimate will exceed 100%; in other words although the central estimate will remain that degradation in forest carbon stocks has occurred, there will be some possibility that there has actually been a gain.
Non-CO2 greenhouse gas emissions associated with fire are estimated by multiplying emission/removal factors appropriate to the type of fire together with areas burnt and the amount of fuel combusted per unit area. Areas are estimated either by remote sensing from burn scars and have associated uncertainties, or from ground surveys. Emission/removal factors and uncertainty ranges are provided in Table 2.5 referenced in volume 1, section 2.4 of 2006GL (2). The combined uncertainty associated with these emissions can be estimated using the equations for combining uncertainties given in Equation 30 and in Section 5.2.6

### Box 28: Applying uncertainty analysis to deforestation

This example uses the results from the change in area due to deforestation calculation described in Box 24, and combines it with a hypothetical change in carbon density scenario.
Step 1. Change in area of deforested land.
The example in Box 24 gives the total area of forest loss as 21,158 ha, with a standard error of 3,142 ha. The required quantities for calculating total emissions are:  Step 2. Calculation of EF from the change in biomass density.
The carbon density of intact forest is assumed to be 250 tC/ha, with a standard error of 25 tC/ha (corresponding to an uncertainty of 10%). The carbon density of the post-clearing forest is assumed to be 30 tC/ha, with a standard error of 3 tC/ha (also corresponding to an uncertainty of 10%). The residual carbon in the post-clearing forest arises from e.g. slash residues, or patches of incomplete deforestation.
Assuming the field survey data underlying the carbon density estimates for pre- and post- deforestation involved independent sampling, then the calculation of change in biomass density and its uncertainty is, t CO2/ha [using Equation 28 from Section 5.2.6.2]
The constant 44/12 is used to convert carbon density into units of CO2
Step 3. Calculation of total emissions
The total emissions due to deforestation and its uncertainty are calculated using Equation 35 and Equation 36 respectively. t CO2 For this hypothetical example deforestation led to a loss of approximately 17.1 million tonnes of CO2, with a standard error of = 3.2 million tonnes of CO2. The 95% confidence interval (as used in IPCC guidance and guidelines) is calculated as the standard error multiplied by 1.96, which yields the final result of 17.1 ± 6.3 million tonnes of CO2, or a 95% confidence interval of 10.8 to 23.4 million tonnes CO2.
Uncertainties may also be combined using probabilistic simulation (Monte-Carlo Analysis)(3) The Monte Carlo analysis is suitable for detailed category-by-category assessment of uncertainty, particularly where uncertainties are large, distribution is non-normal, the algorithms are complex functions and/or there are correlations between some of the activity sets, emissions factors, or both. Monte Carlo simulation requires the analyst to specify probability distribution functions (Fishman, 1996) that reasonably represent each model input for which the uncertainty is quantified. The probably distribution functions may be obtained by a variety of methods including statistical analysis of data or expert elicitation. A key consideration is to develop the distributions for the input variables to the emission/removal calculation model so that they are based upon consistent underlying assumptions regarding averaging time, location, and other conditioning factors relevant to the particular assessment (e.g. climatic conditions influencing agricultural greenhouse gas emissions). Monte Carlo analysis can deal with probability density functions of any physically possible shape and width, as well as handling varying degrees of correlation (both in time and between source/sink categories)(4).
If emissions and removals are estimated using a fully integrated system (Chapter 3, Section 3.2), rather than the simple multiplication of activity data and emissions/removals factors, then Monte-Carlo analysis may be the only feasible approach for estimating the uncertainties. The input data are the same as for the simple method just described, and (if data are available) the approach can also take account of auto- and cross-correlations, which cannot readily be included in the simple method. IPCC has shown(5) that, with the same input data, the simple method and probabilistic simulation give similar results.
Countries may need to estimate the uncertainty associated with a difference between an emissions or removals outturn and a FREL or FREL. Box 29 presents a typical example of how to do this for deforestation, using the methods described in this section.

### Box 29: Uncertainty in the difference between a FREL/FRL and deforestation emissions during an assessment period

Suppose that to establish the FREL, a number N successive annual determinations of deforestation rate were made and that these had values ha/yr (i=1…N), and that using the methods outlined in Section 5.1.5, the uncertainty of each determination was estimated to be corresponding to the variance of the mean deforestation rate (see also Section 5.1.5). In this case, for the FREL the annual area deforested averaged over the N determinations is
And, the successive determinations are uncorrelated, the corresponding uncertainty is
Similarly if during the assessment period, M successive determinations of the deforestation rate are made with values ha/yr (j=1…M), each determination having an uncertainty of again using the methods set out in Section 5.1.5, the average annual deforestation rate during the assessment period is
and the corresponding uncertainty is
Now suppose that the emissions/removals factor (the carbon density per unit area) is tCO2 /ha with an uncertainty of . The methods for calculating emissions/removals factors and their uncertainties are given in Section 5.3, including the case where permanent plots (with correlated errors) are used in their calculation. Finally the mean annual difference in CO2 emissions between the FREL and the assessment period is calculated as the difference in area multiplied by the emissions/removals factor
with the uncertainty of given in Equation 44, consistent with Equation 36:
 (2) The method in section 3.2.1.4 of GPG2003 indexes non-CO2 emissions from fire to emissions from CO2 and does not provide default uncertainty ranges.
 (4) Volume 1, chapter 3, section 3.2.3 of 2006GL provides detailed guidance on Mote Carlo methods which are not repeated here.