The main purpose of a physical act of measurement is to enable decisions to be made. In case of a groundwater body, the measurements of physico-chemical parameters of groundwater are an indispensable first step in assessing the chemical status of GWB. The reliability of the decision whether the given GWB is in good or poor status heavily depends on knowing the uncertainty of the measurement results. If the uncertainty of measurements is underestimated, for example because the sampling process is not taken into account, then erroneous decisions can be made that can have substantial financial consequences. Therefore, it is essential that effective procedures are available for estimating the uncertainties arising from all parts of the measurement process, including sampling. Judgment on whether the contribution to the measurement uncertainty arising from the analytical procedure in the laboratory is acceptable, can only be made with knowledge of the uncertainty originating in the remaining steps of the entire measurement procedure (Eurachem, 2006).
Abundant literature exists on the theory of sampling and measurement process as well as on its practical implications for the assessment of uncertainty associated with the physical act of measurement. In addition to publications in scientific journals (e.g. AMC, 2004; de Zorzi et al., 2002; Kurfurst et al., 2004; Love, 2002; Ramsey, 1998, 2002, 2004; Ramsey et al., 1992, 1995, Thompson, 1998, 1999; Thompson and Maguire, 1993; Thompson et al., 2002; van der Veen and Alink, 1998), also several guides and international ISO regulations on this subject have been published (e.g. Codex, 2004, 2006; de Zorzi et al., 2003; Ellison et al., 2000; Eurachem, 2006; EPA, 2002; Gron et al., 2005; NORDTEST, 2006; IAEA, 2004; ISO, 1993; ISO 5667-14, 1998; ISO/IEC 17025, 2005; ISO 5725-1-1994/Cor 1, 1998; ISO 5725-2-1994/Cor 1, 2002; ISO 5725-3-1994/Cor 1, 2001; ISO 5725-4, 1994; ISO 5725-5-1998/Cor 1, 2005; ISO/TS 21748, 2004; Konieczka et al., 2004; Magnusson et al., 2004; Prichard, 2004; Quevauviller ed., 1995;
Taylor and Kuyatt, 1994). The remaining part of this chapter follows the concepts and approaches recommended in those public ations.
3.1 Fundamental concepts
Uncertainty of measurement is defined in metrological terminology as “A parameter,
associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand.”(ISO, 1993, Eurachem, 2006). The
‘parameter’ may be, for example, a range, a standard deviation, an interval (like a confidence interval) or other measures of dispersion such as relative standard deviation. When
measurement uncertainty is expressed as a standard deviation, the parameter is known as
‘standard uncertainty’, usually given the symbol u. Uncertainty is associated with each measurement result. A complete measurement result typically includes an indication of the uncertainty in the form x ± ?u, where x is the measurement result and u an indication of the uncertainty. This form of expressing a result is an indication to the end-user of the result that, with a reasonable confidence, the result implies that the value of the measurand is within this interval. The ‘measurand’ stands for quantity, such as a length, mass, or concentration of a material, which is being measured (Eurachem, 2006).
Although uncertainty is related to other concepts such as accuracy, error, trueness, bias and precision, there are important differences between them (Eurachem, 2006):
− Uncertainty is a range of values attributable on the basis of the measurement result and other known effects, whereas error is a single difference between a result and a ‘true (or reference) value’;
− Whereas uncertainty includes allowances for all effects which may influence the result (i.e.
both random and systematic errors), precision only includes the effects which vary during the observations (i.e. only some random errors);
− Uncertainty is valid for correct application of measurement and sampling procedures but it is not intended to make allowance for gross errors caused by failure of measuring
equipment and/or mistakes of the operator.
Figure D16.4 illustrates the influence of systematic and random effects on the measurement uncertainty.
Figure D16.4. Random and systematic effects on analytical results and measurement uncertainty (after NORDTEST, 2006).These effects are illustrated by the performance of someone practicing at a target – the reference value or true value. Each point represents a reported analytical result. The two circles are illustrating different requirements on analytical quality. In the lower left target requirement 1 is fulfilled and requirement 2 is fulfilled in all cases except the upper right. The upper left target represents a typical situation for most laboratories.
3.2 Sampling as a source of uncertainty of measurement
The act of taking a sample introduces uncertainty into the reported measurement result wherever the objective of the measurement is defined in term of the analyte concentration in the sampling target and not simply in the laboratory sample. This is the case of groundwater quality
monitoring.
Sampling protocols are never perfect in that they can never describe the action required for every possible eventuality that may arise in the real world in which sampling occurs (Eurachem, 2006). Awareness of these sources of uncertainty is important in the design and implementation of methods for the estimation of uncertainty. Similar arguments can be made for the uncertainty that arises in the process of physical treatment of a sample (e.g. preservation, transportation, storage) preceding treatment undertaken in the laboratory. The methods employed for sampling
should aim to reduce these errors to a minimum. Moreover, adequate procedures are required to estimate the uncertainty of the final measurement result arising from all of these steps.
Heterogeneity of the sampling target, in this case a groundwater body, will always lead to uncertainty of the analyzed quantity (Codex, 2006; Eurachem, 2006; Ramsey, 1998; Ramsey et al., 1995). This heterogeneity can be quantified in a separate experiment (see below), but if the aim is to estimate the average concentration of the given quantity characterizing the entire GWB, this heterogeneity is just one cause of the measurement uncertainty.
3.3 Approaches to uncertainty estimation
There are two broad approaches to the estimation of uncertainty. One, described as ‘empirical’
or ‘top down’, uses replication of the whole measurement procedure as a way of direct estimation of the uncertainty of the final result of the measurement. The second approach, described as ‘modelling’, ‘theoretical’, or ‘bottom up’, aims to quantify all the sources of uncertainty individually, and then uses a model to combine them into overall uncertainty characterizing the final result. Both approaches can be used together to study the same measurement system (Eurachem, 2006; Ramsey, 1998, 2002).
The overall objective of any approach is to obtain a sufficiently reliable estimate of the overall uncertainty of measurement (Eurachem, 2006). This means that not necessarily all individual sources of uncertainty are to be quantified; only that the combined effect be assessed. If, however, the overall level of uncertainty is found to be unacceptably high, i.e. the
measurements are not fit for purpose, specific action must be taken to reduce the uncertainty.
3.3.1 Empirical approach
The empirical approach is intended to obtain a reliable estimate of the uncertainty, without necessarily knowing the sources of uncertainty individually (Eurachem, 2006; Ramsey, 1998, 2002; Thompson, 1998). It is possible to describe the general type of uncertainty sources, such as random or systematic effects, and to subdivide these as those arising from the sampling process or the analytical process. Estimates of the magnitude of each of these effects can be made separately from the properties of the measurement methods, such as sampling precision (for random effects arising from sampling) or analytical bias (for systematic effects arising from chemical analysis). These estimates can be combined to produce an estimate of the uncertainty in the measurement result.
The overall uncertainty of measurements arises from four broad classes of errors (Eurachem, 2006; Ramsey, 2002). These four classes are the random errors arising from the methods of both the sampling and analysis, and also the systematic errors arising from these methods. These errors have traditionally been quantified as the sampling precision, analytical precision,
sampling bias and the analytical bias, respectively (cf. Table D16.1). If errors belonging to these four classes are quantified, separately or in combinations, it is possible to estimate the
uncertainty of the measurements that these methods produce.
Table D16.1. Estimation of uncertainty contributions in the empirical approach (after Eurachem, 2006).
Effect class Process
Random (precision) Systematic (bias) Analysis e.g. duplicate analyses e.g. certified reference materials Sampling duplicate samples Reference Sampling Target
Inter-Organisational Sampling Trial Sampling and analytical precision can be estimated by duplication of a proportion (e.g. 10%) of the samples and analyses respectively. Analytical bias can be estimated by measuring the bias against certified reference materials, or by taking it directly from the validation of the analytical method. Procedures for estimating sampling bias include the use of a Reference Sampling Target or they utilize measurements from Inter-Organisational Sampling Trials in which the sampling bias potentially introduced by each participant is included in the estimate of
uncertainty based on the overall variability (Eurachem, 2006; Ramsey, 2002). Although some of the components of uncertainty associated with systematic effects may be difficult to estimate, it may be unnecessary to do so if there is evidence that systematic effects are small and under good control.
A statistical model describing the relationship between the measured and true values of analyte concentration is needed for estimation of uncertainty using the empirical approach. This random effects model considers a single measurement of analyte concentration (x), on one sample (composite or single), from one particular samplin g target (Eurachem, 2006):
x = Xtrue + εsampling + εanalysis
where Xtrue is the true value of the analyte concentration in the sampling target, i.e. equivalent to the value of the measurand. For instance, this could be the concentration of a given element or constituent in sampled groundwater. The total error due to sampling is εsampling, and the total analytical error is εanalysis.
In an investigation of a single sampling target, if the sources of variation are independent, the measurement variance is given by:
σ 2meas = σ2sampling + σ2analytical
where σ2samplingis the between-sample variance on one target and σ2analytical is the between-analysis variance on one sample.
If statistical estimates of variance (s2) are used to approximate these parameters, then we get:
s2meas = s2sampling + s2analytical
The standard uncertainty of measurement (u) can be then identified with the square root of s2meas, given by:
u ≡ smeas= ssampling2 +sanalytical2
In a survey across several sampling targets (several monitoring points in case of a GWB), the model needs to be extended to include the variance of the concentration between the targets. If the sources of variation are independent, the total variance σ2tota l is then given by:
σ2total = σ2between-targets + σ2sampling + σ2analytical
and its best estimate becomes:
s2total = s2between-targets + s2sampling + s2analytical
The empirical approach adopted for estimating uncertainty associated with monitoring of groundwater quality is discussed in detail in chapter 5 below.
3.3.2 Modelling approach
The modelling approach, also known as ‘bottom-up’ approach, has been described for
measurement methods in general and applied to analytical measurements (Ellison et al., 2000).
It initially identifies all of the sources of uncertainty, quantifies the contributions from each source, and then combines all of the contributions, as an uncertainty budget, to give an estimate of the combined standard uncertainty. In the process, the measurement method is separated into all of its individ ual steps. This can usefully take the form of a cause-and-effect, or ‘fish-bone’, diagram (de Zorzi et al., 2002; Ellison and Barwick, 1998; Ellison et al., 2000; Eurachem, 2006;
Kurfurst et al., 2004). The cause-and-effect diagram for groundwater sampling is shown in Figure D16.5.The uncertainty of measurement generated by each of these steps is estimated independently, either empirically or by other methods. The combined uncertainty is then calculated by summing the uncertainty from all of the steps using a model that adds the
variances representing individual sources of uncertainties. This approach is well established for analytical methods (Ellison et al., 2000) but has only recently been applied to the process of sampling (de Zorzi et al., 2002; Kurfurst et al., 2004).
Figure D16.5. Schematic cause-effect diagram for the assessment of uncertainties associated with monitoring of groundwater quality.