## Approach 1 variables and equations

This section covers the basis for the statistical calculation methods used in Approach 1 as complementary information to Section 3.2.3.1, Approach 1: Propagation of Error, and Table 3.2, Approach 1 Uncertainty Calculation. Key variables and equations used for the calculation are defined in this section.

### Explanation of the variables

Cx, = Value of an entry in Column C and row x, emissions or removals of each category of base year inventory 2 C = Sum of emissions and removals over all categories (rows) of the base year inventory Dx, = Value of an entry in Column D and row x, emissions or removals of each category of the year of inventory t 2 D, = Sum of emissions and removals over all categories (rows) of the year of inventory t

Column A-F

Input data of emissions and removals, activity data and emission factor uncertainties of each category Column G

Combined uncertainty using error propagation equation. See Equation 3.1 in Section 3.2.3.1.

Column H

Contribution to uncertainty. See also Equation 3.2 in Section 3.2.3.1. H _ (G x • D x )2

The total emission uncertainty is obtained using the error propagation equation:

Column I

Entries in Column I show how the difference in emissions between the base year and the year t changes in response to a 1 percent increase in emissions of category x emissions in the base year and year t. This shows the sensitivity of the trend in emissions to a systematic uncertainty in the emission estimate - i.e., one that is correlated between the base year and year t. This sensitivity is described as Type A sensitivity.

Ix = percentage trend if category x is increased by 1 percent in both years - percentage trend without increase = 0.01 • Dx +2D, - (0.01 • Cx +2 C,) • 100 2 Di -2 Ci • 100

Column J

Entries in Column J show how the difference in emissions between the base year and year t changes in response to a 1 percent increase in the emissions of category x in year t only. This shows the sensitivity of the trend in emissions to random uncertainty error in the emissions estimate - i.e., one that is not correlated between the base year and year Y. This sensitivity is described as Type B sensitivity.

Jx = percentage trend if category x is increased by 1 percent in year t - percentage trend without increase

Column K

Under the assumption that the same emission factor is used in both years and the actual emission factors are fully correlated, the percent error introduced by it is equal in both years. Therefore the formula for the uncertainty introduced on the trend by the emission factor is:

K X = sensitivity A • uncertainty of emission factor = IX • Fx

In case no correlation between emission factors is assumed, sensitivity B should be used and the result needs to be increased by V2 for the reason given below in the main derivation for Column L:

K X = sensitivity B • uncertainty of emission factor • 42 = J X • Fx

### Column L

The trend is the difference between the emissions in the base year and in the year t. Therefore the uncertainty of the activity data of the base year and year t has to be taken into account. The two uncertainties combined using the error propagation equation and the assumption that the uncertainty is the same in the base year and year t is:

= ^ (uncertainty (activity data, base year))2 + (uncertainty (activity data, year t))2 (uncertainty (activity data, year t))2 • 2

Since activity data in both years are assumed to be independent, Column L equals: LX = sensitivity B • combined uncertainty of activity data of both years

In case correlation between activity data is assumed, sensitivity A should be used and the V2 factor does not apply.

Column M

Uncertainty introduced on the trend by the uncertainty in the activity data and the emissions factor.

The entries M,- in Column M are combined to obtain the total uncertainty of the trend using the error propagation equation as follows:

Total uncertainty of the trend = 