This section outlines methods for generating statistically based uncertainty estimates. These differ from those above in that they provide quantitative, or numerical, estimates of the error associated with emission estimates.

If the information used to assign parameter values within the emissions inventory is of sufficient quality or is otherwise sufficiently well defined, it is possible to specify data values as statistical quantities in addition to average or best estimate values. In these cases, it may be possible to construct parameter distribution functions (pdf) that describe either the

Three main categories of quantitative methods may be defined: expert estimation, error propagation and direct simulation.

A1.3.1 Expert Estimation Method

The method of performing a quantitative analysis of parameter uncertainty which is most immediate in character involves eliciting key parameter distribution statistics (

One approach is the highly formalised Delphi method [48,49] in which the opinion of a panel of experts working

Expert judgement may also be utilised outside a formal Delphi framework. Here, one or more experts make judgements as to the values of specific distributional parameters for a number of sources. Horie [50] uses graphical techniques to estimate confidence limits once upper and lower limits of emissions have been elicited, but any one of a number of techniques have been conceived, ranging from the relatively simplistic to the highly structured.

Most of these methods require the assumptions of independence of the emission factors and the activity rates. Also, the methods make the implicit assumption that emissions data are normally (or lognormally) distributed. An important consequence of the violation of any of these basic assumptions is that the uncertainty estimates that result are typically biased low.

On the positive side, however, a strength of these methods is their relatively low implementation cost when compared with the next two methods in this section.

Using widely accepted methodologies [51] it is possible in many cases to predict the manner in which uncertainties (here interpreted as 'errors') would be propagated during the arithmetical operations inherent in calculating the inventories of greenhouse gases. In this way the joint action of a number of individual factors, each with its own uncertainty, can be evaluated. The method presumes that information is available, or at least some measure of agreement can be reached, on the pdfs of individual parameters. As was made clear earlier, this will represent a significant restriction on its use.

The propagation methods are based upon the following assumptions:

- Emission estimates are formulated as the product of a series of parameters

- Each of these parameters is independent (
*i.e.*no spatial or temporal correlations)

One method of error propagation involves representing the equation describing the variance of a series of products in the form of a Taylor's series [52]. In this case the assumption of independence allows the variance of multiplicative products to be expressed in terms of the individual variances. There is general agreement that the uncertainty evaluated using this method is underestimated because of the necessary assumption of the independence of the input parameters.

Another method, advocated by the IPCC [53], involves combining uncertainty ranges for the emission factors and activity rates using a standard algorithm. Assuming that the uncertainty ranges correspond to the 95 percent confidence interval of a normal distribution, the method may be used where the individual uncertainty ranges do not exceed ±60% of the respective best estimates. The uncertainty in each component is first established using a classical statistical analysis, probabilistic modelling (see next sub-section), or the formal expert assessment methods outlined earlier. Hence the appropriate measure of the overall percentage uncertainty in the emissions estimate,

For individual uncertainties greater than 60% all that can be done is to combine limiting values to define an overall range. This will produce upper and lower limiting values which are asymmetrical about the central estimate [6].

If the estimated total emission for each gas is given by S

where

These comprise statistical methods in which the uncertainty and confidence limits in emission estimates are computed directly from the component parameter distributions (means, standard deviations, or ranges, as available) using statistical procedures such as Monte Carlo [54,55] and Latin Hypercube (LHS) approaches [56,57 ]. Bootstrap and resampling techniques [58] allow the analyst to refine the data further. A major advantage of these methods over those using error propagation is that correlations between the values of individual parameters can be allowed for as part of the statistical formulation of the problem. This general category of technique is favoured where sufficient information exists on the emissions inventory (for example, in the comprehensive study currently in progress in the US [59]) to enable uncertainties in the component parameters to be quantified.

More recently, the developing calculus of fuzzy numbers or fuzzy sets has allowed researchers [60] to formulate a method of analysing parameter uncertainty without repeated sampling of distribution functions. This approach will not be discussed further here other than to note that within it, the similarity between fuzzy sets and parameter distribution functions is used to formulate a way of 'transmitting' parameter uncertainty through to the output.

The

As originally described [63],

Both Monte Carlo and Latin Hypercube sampling methodologies are able to be modified [65] to induce restricted pairing between the values of the input parameters, thus introducing inter-parameter correlations. If such correlations are known to be present within the dataset of an emissions inventory but are discounted within a statistical analysis, the total uncertainty in the output may be significantly underestimated. Both methodologies are limited somewhat in that experts are most often required to specify both distribution types and ranges for each of the parameters. If a full set of such information is not available, the analyst need to be aware that default assumptions (

Both of the above approaches may be implemented using widely available software tools. In this study it is intended to use the @RISKä program [68], for which AEA Technology has a licence. The tool has the advantage of running seamlessly with Microsoft Excelä, and is therefore available to interface directly with the stored emissions databases stored in the latter.

Sampling methodologies such as

In almost every statistical data analysis, on the basis of a dataset

A particularly useful application of the technique relates to estimating a range of statistics for truncated samples (