In order to understand the differences and the pros and cons of the different aggregation methods, several estimation parameters are considered within this section. These parameters are unknown but may be interpreted as the true parameter which has to be estimated by the aggregation method. Ideally, the result of the aggregation and the
estimation parameter are equal, but in practise there is always a difference, i.e. an estimation error. This is due to the limited number of monitoring points.
1 Technical Report No 1: The EU Water Framework Directive: Statistical aspects of the identification of groundwater pollution trends, and aggregation of monitoring results. 2001
Estimation parameters
The following different estimation parameters may be considered:
1. What is the percentage (extent) of the GW-body (volume) that is not exceeding a quality standard or TV?
2. What is the true average concentration in a GW-body (average over the volume)?
3. What is the geometric mean in the GW-body of the annual mean concentrations throughout the GW-body? (= antilog of the average of the logged annual mean concentrations)?
4. What is the true concentration value that is not exceeded by the annual mean of x % (50 %, 70 %, 90 %) of the monitoring points in a GW-body?
5. What is the maximum annual mean concentration in the GW-body?
These estimation parameters can be derived from the distribution function F(x)
F(x) = (volume of the GW-body where concentration is below x) / (total volume of GW-body) F(x) can be used to calculate the ratio share of the GW-body below any concentration x. Let x=TV=threshold: F(TV) represents the share (extent) of the GW-body at which the
concentration is below the threshold value TV, i.e. F(TV) represents the percentage of the volume which is not exceeding a TV. Not only the percentage volume not exceeding, but also mean, median, 90 percentile and geometric mean can be derived from F(x) mathematically or graphically, see Figure 1.
Apparently the different estimation parameters may be interpreted as different aspects of the distribution function F(x). It is a political question, which aspects and which estimation
parameters shall be considered.
The percentage volume of the GW-body which is exceeding a standard or TV reflects the extent of pollution. However, it does not reflect the pollution in terms of concentration.
Percentiles reflect the overall status of the GW-body fair to good if the variation coefficient of monitoring point mean values is not too high but they do neither reflect outliers nor the impact of uneven distribution of pollution caused by local or diffuse sources, observed at some monitoring points in a GW-body which show higher concentrations than the rest of the GW-body.
The mean reflects the overall status of the GW-body very well if there is no extreme local pollution. The geometric mean is less often in use. It is very sensitive to small concentrations and less sensitive to higher concentrations.
Figure 1: Distribution function for nitrate in a GW-body. Read: 50 mg/L is not exceeded in approx. 45% of the GW-body. Or: in 70% of the GW-body the concentration is below 70 mg/L.
Nitrate distribution of annual mean concentration throughout the GW-body (Marchfeld, 2005)
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In the following the aggregation methods are described and discussed in relation to the corresponding estimation parameter. For some of these aggregation methods also the statistical confidence intervals are available. These are referred to as confidence limit (CL).
Percentage of sites / volume ration (percentage of volume) not exceeding a standard or TV
A straightforward measure is the percentage of monitoring points in a GW-body not exceeding standards or TVs. Instead of using the un-weighted percentage of sites the weighted percentage of points could be applied, where the different sites are weighted with regard to the assigned volume or area in the GW-body.
A third option is to replace the yes/no (1/0) response for each monitoring point by a statistical estimate for the percentage of the area assigned to this monitoring point, which is not
exceeding standards or TVs. This measure is referred to as volume ratio of non
exceedance. If the annual mean concentration is far below a standard or TV, this percentage will be close to zero, whereas for values clearly above a standard or TV the percentage will be close to 1. If the annual mean concentration equals a standard or TV, the percentage of the area assigned to the monitoring point will be 0.5.
The calculation of weights of the annual mean concentrations for the calculation of the volume ratio is derived from the normal distribution with a standard deviation of 25% which means that for values which are 2 times the standard deviation above a standard or TV the weight is nearly 1 (i.e. 0.98) (Example see Figure 3).
Both the un-weighted and the weighted percentage of sites not exceeding a standard or TV provide an estimate of the percentage of the GW-body not exceeding a standard or TV. The un-weighted percentage may be biased in case of inhomogeneous network or unequal share of each monitoring point. The weighted percentage corrects for this bias. However, both percentages are not very precise with regard to random error: in some years the percentage
will be too small, in others too large. This is especially true when the number of monitoring points is small and the annual mean concentrations are close to a standard or TV. This is demonstrated in the following example. The following Figure 2 represents the outcome of two aggregation methods for estimating the percentage of a GW-body not exceeding a standard or TV based on a time series of 1993–2005 and three monitoring points within a GW-body. In this example the standard or TV was selected to be 50. One monitoring point shows values far beyond 50 and two monitoring points show values around the standard or TV of 50; in some years the measured values are slightly exceeding 50 in others they are slightly below.
It is apparent that due to this random fluctuation the result of the percentage of sites not exceeding (violet line) is varying considerably, whereas the volume ratio (blue line) is apparently less affected by random effects.
Figure 2: Two aggregation methods for calculating the percentage of monitoring points not exceeding a quality standard or TV.
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Time series of 3 monitoring points
year 1 2 3 4 5 6 7 8 9 10 11 12 13
Point 1 1.8 1.0 1.8 3.3 5.0 4.1 3.6 3.3 3.3 2.7 2.7 2.0 1.7 Point 2 44.2 46.3 48.4 53.9 55.4 51.1 46.8 48.6 44.6 49.7 47.7 51.0 39.8 Point 3 40.8 40.6 42.5 61.2 46.1 54.2 47.1 50.5 52.9 51.9 54.0 57.2 57.6
Figure 3: Weights of annual mean values of monitoring points for volume ratio calculation.
Weighting of Volume Ratio
Percentiles (50 (=Median), 70 or 90 Percentile)
The percentiles or quantiles (50 %, 70 %, 90 %) of the annual arithmetic mean
concentrations of monitoring points represent the annual mean concentration which is not exceeded at more than x % (50 %, 70 %, 90 %) of the monitoring points of the area (=GW-body).
The estimation is good in case of evenly distributed sites, but may be misleading in case of unevenly distributed sites. However it is possible to derive weighted percentiles in order to correct for different sub-bodies with different site density.
Methodologies of percentiles would require at least 10 sites for statistically sound results. In several countries small GW-bodies exist, which are monitored by less than 10 sites and cannot be grouped due to different hydrogeological conditions.
Arithmetic Mean (AM) and Confidence Limit (CLAM)
The arithmetic mean of the annual arithmetic mean concentrations over all monitoring points sampled represents simply the average over these sites.
Confidence limits (CL) for the mean are an interval estimate for the mean. Instead of a single estimate for the mean, a confidence interval generates a lower and upper limit for the mean.
The interval estimate gives an indication of how much uncertainty there is in the estimate of the true mean. The narrower the interval, the more precise is the estimate.
The upper confidence limit depends on the variability of the concentration level within the GW-body and on the number of monitoring points. The CL decreases with an increasing number of monitoring points within the GW-body or a decreasing variability of concentration levels. The use of the CL allows for reducing the number of monitoring points in GW-bodies with levels far below any standard or TV and enforces a higher number of monitoring points in GW-bodies with levels close to a standard or TV.
The arithmetic mean reflects the overall status of the GW-body very well in case of evenly distributed sites. The outlier sensitivity of the arithmetic mean is poor and its reflection of the impact of uneven distribution of pollution caused by local or diffuse sources, observed at some points in a GW-body which show higher concentrations than the rest of the points in a GW-body, is fair to good. The replacement of values below LOQ by substitute values may introduce some bias. All measurements are assumed to be stochastically independent and identically distributed.
Compliance with good groundwater chemical status at a given level of confidence can be demonstrated with a statistical test for the null hypothesis
H0: "GW-body is not in good status, i.e. true mean level exceeding a standard or TV"
and the alternative hypothesis
H1: "GW-body is in good status, i.e. true mean level below a standard or TV"
H1 may be considered as statistically proven at significance level alpha/2, if the
corresponding upper CL at confidence level (1-alpha) (e.g. 95 %) is below the limit. Value alpha denotes the probability of making a wrong decision for a good status (although the true, unknown mean exceeds a standard or TV); alpha might vary for different parameters.
Weighted Arithmetic Mean (wAM) and Confidence Limit (CLwAM)
The weighted arithmetic mean (wAM) is another estimator for the average annual mean concentration and is introduced for GW-bodies which can be divided into sub-bodies. It takes into account the share of the sub-bodies and the corresponding arithmetic means.
In order to be consistent with the AM, the calculation of the CLwAM is performed under the same model assumption as with CLAM, i.e. all measurements are assumed to be
stochastically independent and identically distributed. Hence it is assumed that there is no spatial correlation at all, and under this assumption calculation leads to the approximate upper confidence limit. If the sites are evenly distributed, calculation shows that CLwAM = CLAM.
Kriging Mean (KM) and Confidence Limit (CLKM)
The Kriging mean represents the average concentration in the area of a GW-body and takes regard of a heterogeneous distribution of monitoring points.
Both Kriging mean (KM) and arithmetic mean (AM) are methods that can formally be written w1y1 + w2y2 + ... + wnyn
where yi denotes the level of monitoring point i and wi the corresponding weight. For the AM the weights of sites are equal, wi = 1/n. For the KM the weights can be quite different:
monitoring points representing large areas have higher weights than monitoring points representing small areas. For evenly distributed sites these areas and hence the corresponding weights are similar.
The Kriging analysis can also be used for calculating the upper confidence limit for the Kriging mean.
Kriging reflects the spatial structure of the GW-quality data and also to a certain extent the impact of factors affecting the concentration level within an area as there are e.g. land use, hydrogeological conditions, etc. if these are spatially correlated. Anyway, the extension of the model to explicitly include hydrogeological information, etc. requires a much more
complicated statistical algorithm.
Mean based on the log-normal distribution
The mean based on the log-normal distribution equals the geometric mean. This is equivalent with the antilog of the average of the logged annual mean concentrations.
The estimation is less sensitive with regard to outliers, but much more sensitive with regard to the treatment of measurements below the LOQ.
Minimum, Maximum
Minimum and maximum reflect the impact of uneven distribution of pollution caused by local or diffuse sources, the overall status of the GW-body is poorly reflected. Furthermore, the method is highly outlier sensitive.