The modelling route writes the result as a function of input quantities, assigns a standard uncertainty to each and propagates them. It is the approach of the original guide and is worked through for chemical examples in the Eurachem/CITAC guide (Ellison and Williams, 2012). The empirical route starts from the observed performance of the whole method. It combines a precision term obtained under conditions that vary everything likely to vary in routine use with a term for bias, and it was developed in detail by Ellison and Barwick (1998) and in the Nordtest handbook (Magnusson et al., 2017). ISO has since published standards on the use of validation and quality control data for the purpose (ISO, 2012; ISO, 2017b), and the Eurolab report on alternative approaches treats the routes as equally legitimate when properly applied (Eurolab, 2007).

For a multi-element routine method the empirical route has a practical advantage. Its inputs are produced every working day, so the estimate rests on many observations and can be kept current. Its weakness is that it describes the samples used to generate the data, and those samples may not represent every sample the laboratory receives.

An ICP-OES result passes through sampling of the laboratory portion, digestion or dilution, nebulisation, excitation in the plasma, measurement of line intensity above background, and calibration. Each stage contributes. Nebuliser and torch behaviour drift over a run and change with dissolved solids. High concentrations of sodium, calcium and other easily ionised elements alter excitation conditions and shift analyte signals, an effect that depends on plasma robustness and viewing mode (Todolí et al., 2002). Mermet (1991) proposed the ratio of a magnesium ionic line to a magnesium atomic line as a simple indicator of that robustness. Spectral overlap from matrix elements such as iron and aluminium biases results at weak analyte lines unless corrected. Standard methods for the technique describe these interferences and the controls expected, including internal standards and interference check solutions (US EPA, 1994; ISO, 2007). Figure 1 groups the sources under the terms used in the procedure below.

The laboratory states what is being measured: the element, the matrix group, and whether the result refers to total content or to the fraction released by a defined digestion, such as an aqua regia extract. The distinction decides what counts as bias later. The working range of each element is then divided where necessary, because uncertainty near the limit of quantification is roughly constant in concentration units while at higher levels it is roughly proportional to concentration. The Eurachem guide describes how the two regimes may be represented (Ellison and Williams, 2012). In practice a low range expressed in absolute terms and a high range expressed in relative terms is usually enough.

A cause-and-effect diagram of the kind shown in Figure 1 is drawn for the method, and each source is marked according to whether the laboratory's quality control data would capture it. A control sample introduced only at the instrument does not capture digestion. A synthetic control solution does not capture matrix effects. A control analysed once per batch does capture recalibration and day-to-day drift. The purpose of the map is to prevent both omission and double counting, and it is the one place where the reasoning of the modelling route is retained.

The precision term is the standard deviation of results for a stable control sample analysed over a long period, by different analysts, with fresh calibrations, and carried through digestion where the method includes it. Several months of data from the control chart are normally available. If the control sample is simpler than routine samples, a second contribution is added from duplicate analyses of real samples, which reflect inhomogeneity and matrix-dependent variation (Magnusson et al., 2017). Separate values are obtained for the low and high ranges defined in Step 1. Outlying control results that were investigated and attributed to an identified fault may be excluded; results that were merely inconvenient may not.

Bias is assessed against the best reference values available, and Table 1 compares the three usual sources. Certified reference materials of matching matrix give the most direct evidence. Proficiency test results extend the evidence to more matrices and levels, at the cost of a less certain assigned value. Recovery of a spike added before digestion covers elements and matrices for which nothing else exists, though an added solution may be recovered more easily than the native analyte (Barwick and Ellison, 1999).

The component is calculated in the manner of the Nordtest handbook. Where several reference materials or proficiency rounds are available, the root mean square of the observed relative biases is combined in quadrature with the mean standard uncertainty of the reference values. Where a single reference material is used, the observed bias, the standard deviation of its mean and the certificate uncertainty are combined in the same way. The harmonised validation guidelines advise that a bias found to be significant should be investigated and removed if possible (Thompson et al., 2002); a bias that is understood and stable may be corrected, and its contribution then falls to the uncertainty of the correction.

The combined standard uncertainty is the square root of the sum of the squared reproducibility and bias

2 + ubias

2)1/2. It is multiplied by a coverage factor of 2 to give an expanded components, uc = (uRw uncertainty with a level of confidence of approximately 95 per cent, on the assumption that the components rest on enough observations for a normal distribution to be reasonable (JCGM, 2008). The calculation is made for each element, matrix group and range. Elements that behave alike may be grouped under the largest value in the group to keep the reporting table manageable, provided the grouping is recorded.

An estimate should be tested before it is used. Three checks are proposed. The first compares the relative uncertainty with what interlaboratory experience predicts at that concentration. The Horwitz function (Horwitz, 1982), with the modification proposed by Thompson (2000) for low concentrations, gives a reproducibility standard deviation that a single laboratory should not normally exceed and is unlikely to better by a wide margin. The second examines the laboratory's own proficiency test record: if the claimed uncertainty were correct, the differences from the assigned values should mostly lie within it. The third applies the modelling route to one or two elements as a cross-check, for which the spreadsheet method of Kragten (1994) is convenient. Marked disagreement in any check sends the analyst back to Step 2.

The expanded uncertainty is reported with the result, in the same units, with the coverage factor stated. The laboratory records which data sets were used and over what period. The estimate is recalculated at a fixed interval, annually in most cases, and whenever the method changes in a way that Step 2 shows to be relevant: a new digestion procedure, a change of viewing mode, a different internal standard or a new sample type.