A necessary distinction between spatial representativeness of an air quality monitoring station and the delimitation of exceedance areas

Abstract

The European legislation on ambient air quality introduces the concepts of spatial representativeness of a monitoring station and spatial extent of an exceedance zone. Spatial representativeness is an essential macro-scale siting criterion which should be evaluated before the setting-up and during the life of a monitoring point. As for the exceedance area, it has to be defined each time an environmental objective is exceeded in an assessment zone. No specific approach is prescribed to delimit such areas. A probabilistic methodology is presented, based on a preliminary kriging estimation of atmospheric concentrations at each point of the domain. It is applied to NO₂ pollution on the urban scale. In the proposed approach, a point belongs to the area of representativeness of a station if its concentration differs from the station measurement by less than a given threshold. To take the estimation uncertainty into account, the standard deviation of the kriging error is used in a probabilistic framework. The choice of the criteria used to deal with overlapping areas is first tested on NO₂ annual mean concentration maps of France, built by combining surface monitoring observations and outputs from the CHIMERE chemistry transport model. At the local scale, data from passive sampling surveys and high -resolution auxiliary variables are used to provide a more precise estimation of the background pollution in different French cities. The traffic-related pollution can also be accounted for in the map by additional predictors such as distance to the road, and traffic-related NO_x emissions. Similarly, the proposed approach is implemented to identify the points, at a given statistical risk, where the NO₂ concentration is above the annual limit value.

Appendices

Appendix 1: Relaxation of the Gaussian assumption on background estimation error: the unimodal distribution

In Chiles and Delfiner (2012), some general results are given for a variable A following a continuous and unimodal distribution:

$$ \forall \ \alpha \ \mathrm{P}\left( \left|A-\alpha \right|\ge t\, \sqrt{E\left( \left( A-\alpha \right)^{2} \right)} \right) \left\{\begin{array}{l} \le {\frac{4}{9t^{2}}}\ \text{if} \ t\ge \sqrt{\frac{8}{3}} \\ \le \frac{4}{3t^{2}}-\frac{1}{3}\ \text{if} \ t\le \sqrt{\frac{8}{3}} \end{array}\right. $$

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Thus, for a centered variable Y, and with α = 0, Eq. 25 becomes:

$$\begin{array}{@{}rcl@{}} \forall \ \alpha \ \mathrm{P}\left( \left|Y \right|\ge t\sigma \right) \left\{\begin{array}{l} \le {\frac{4}{9t^{2}}}\ \text{if} \ t\ge \sqrt{\frac{8}{3}} \\ \le \frac{4}{3t^{2}}-\frac{1}{3}\ \text{if} \ t\le \sqrt{\frac{8}{3}} \end{array}\right. \end{array} $$

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For $t > \sqrt [{}]{{\frac {8}{3}} }$ ≈ 1.63, this upper boundary is lower than Bienaymé-Tchebychev (${\frac {4}{9}} <1$) ∀t > 1. For t = 1, the two upper boundaries are equal. Last, for t < 1, the upper boundary is again greater than 1 (and even larger than Bienaymé-Tchebychev). The results under this unimodal assumption gives (Fig.13).

Fig. 13

Unimodal and Gaussian assumption on the error of kriging-based estimations and their related background representativeness areas in Montpellier (2007) and Troyes (2009). a Montpellier (unimodal error). b Montpellier (Gaussian error). c Troyes (unimodal error). d Troyes (Gaussian error)

Let find t so that:

$$\begin{array}{@{}rcl@{}} \mathrm{P}\left( \frac{\left|Y\right|}{\sigma} >t\right) \le \eta \end{array} $$

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• 1. if $t\ge \sqrt [{}]{{\frac {8}{3}} }$, a sufficient condition is ${\frac {4}{9t^{2} }} \le \eta$, i.e., $t\ge {\frac {2}{3\sqrt [{}]{\eta } }}$.

  • (a) if $\sqrt [{}]{{\frac {8}{3}} } \ge {\frac {2}{3\sqrt [{}]{\eta } }}$, i.e if $\eta \ge \frac {1}{6}$, the sufficient condition is also satisfied ;

  • (b) if $\sqrt [{}]{{\frac {8}{3}} } \le {\frac {2}{3\sqrt [{}]{\eta } }}$, i.e., if $\eta \le \frac {1}{6}$, the sufficient condition is $t\ge {\frac {2}{3\sqrt [{}]{\eta } }}$.

• 2. let $t\le \sqrt [{}]{{\frac {8}{3}} }$ ; the sufficient condition is ${\frac {4}{3t^{2}}} \, -{\frac {1}{3}} \, \le \eta$ is ${\frac {2}{\sqrt {1 + 3\eta } }} \le t$. This condition is satisfied if and only if ${\frac {2}{\sqrt {1 + 3\eta } }} \le \sqrt {{\frac {8}{3}} }$, i.e., if $\eta \ge {\frac {1}{6}}$.

Last,

• 1. if $t\ge \sqrt [{}]{{\frac {8}{3}} }$ and $\eta \ge \frac {1}{6}$, the sufficient condition is checked ;

• 2. if $t\ge \sqrt [{}]{{\frac {8}{3}} }$ and $\eta \le \frac {1}{6}$, the sufficient condition is $t\ge {\frac {2}{3\sqrt [{}]{\eta } }}$ ;

• 3. if $t\le \sqrt [{}]{{\frac {8}{3}} }$ and $\eta \ge {\frac {1}{6}}$, the sufficient condition is $t\ge {\frac {2}{\sqrt {1 + 3\eta } }}$ ;

• 4. the case $t\le \sqrt [{}]{{\frac {8}{3}} }$ and $\eta \le \frac {1}{6}$, corresponding to the lower boundary for t, is impossible.

Table 4 sums up the different situations.

Table 4 Statistical risk η and boundary for t

To sum up:

if $\eta \le \frac {1}{6},\ t\ge {\frac {2}{3\sqrt [{}]{\eta } }}$ and if $\ \eta \ge \frac {1}{6},\ t\ge {\frac {2}{\sqrt [{}]{1 + 3\eta } }}$. When $\eta =\frac {1}{6}$, the upper boundaries for t are the same.

If kriging with external drift is involved, the previous results become:

if $\ \eta \le \frac {1}{6},\ |{\Delta }_{\mathbf {x}-\mathbf {x}_{0}}^{\text {K}}| \le \delta -\frac {2\sigma _{\text {K}}(\mathbf {x})}{3\sqrt {\eta }}$ and if $\ \eta \ge \frac {1}{6},\ |{\Delta }_{\mathbf {x}-\mathbf {x}_{0}}^{\text {K}}| \le \delta -\frac {2\sigma _{\text {K}}(\mathbf {x})}{3\sqrt {\eta }}$

Appendix 2: Proportional effect

It is possible to correct the kriging standard deviation by considering the so-called proportional effect which is identified on the scatterplot between NO₂ local mean in a 500-m moving window and its related NO₂ local standard deviations (Fig. 14). Let γ_V be the local variogram in the neighborhood V. It can be obtained from the global variogram γ (i.e., the variogram calculated with all the sampling points) by introducing a multiplicative factor that enables some non-stationarity in the model. This factor is a function of the local mean m_V (Matheron 1974; Journel and Huijbregts 1978; Chiles and Delfiner 2012):

Fig. 14

Scatterplot between NO₂ local mean (μg m^− 3) and local standard deviations (μg m^− 3) computed within a 500-m moving window

$$\begin{array}{@{}rcl@{}} \gamma_{V}(h)=f(m_{V})\gamma(h) \end{array} $$

The function f is fitted by a curve f(m_V) = αm_V or $f(m_{V})=\alpha {m_{V}^{2}}$ (more generally $f(m_{V})=\alpha {m_{V}^{k}}$). Only the variogram sill is changed; thus, the kriging weights remain unchanged and the kriging standard deviations are corrected by the ratio f(m_V)/f(m_g) where m_g is the global mean.

This correction has a significant influence on the results (see Figs. 15 and 16): the kriging standard deviations are increased in the areas where high NO₂ levels are estimated and reduced for lower concentrations.

Fig. 15

Standard deviations, representativeness areas for background stations and probabilities of exceeding the threshold of 32 μg.m^− 3 in Montpellier without accounting for proportional effect

Fig. 16

Standard deviations, representativeness areas for background stations and probabilities of exceeding the threshold of 32 μg.m^− 3 in Montpellier when accounting for proportional effect

The representativeness area of the suburban station is now larger and covers the whole north and west part of the study field $\mathcal {D}$, which was left blank without the proportional effect (Fig. 5a). The representativeness area of the urban station is reduced downtown and also appears more broken up in the suburbs. The probabilities of exceeding the threshold of 32 μg m^− 3 in background configuration are also heavily modified: at a false detection risk β set to 34%, a small exceedance area is identified downtown while the status of the whole downtown was in the uncertainty area without accounting for the proportional effect.

The relevance of the proportional effect is however questionable because the quality of the fitting made for f is suspicious. But even if the representativeness and exceedance areas are not the same with and without the use of this effect, they are not totally inconsistent. The key point is to keep the same approach when the computation of these areas is updated over the years.

Appendix 3: Conditional expectation

In the main results provided by the “Results” section, kriging with NO_x emissions used as external drift is involved to map the NO₂ annual mean. On such urban areas, that are quite small, so that the local estimation of the drift becomes useless, a residual kriging in which the regression between NO₂ concentrations and NO_x emissions is first estimated is also appropriate. Its main advantage would be to ease the use of a non-linear geostatistical framework, more adapted to deal with the problem of identifying exceedances.

The model for the process Z(x) is still:

$$ Z(\mathbf{x})= \mu(\mathbf{x})+R(\mathbf{x}) $$

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with μ(x) the regression of NO₂ on NO_x emissions and R(x) a second-order stationary residual, having no correlation with μ(x).

From now on, in the identification of the exceedances problem, the residuals $\hat {R}(\mathbf {x}_{\alpha })$ are transformed by the anamorphosis ϕ₁(.) to the new standard variable Q(x) (see, e.g., Rivoirard 1991):

$$ Q(\mathbf{x})= \phi_{1}\left[\hat{R}(\mathbf{x})\right] $$

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Thus, the definition for exceeding the threshold s becomes:

$$ Z(\mathbf{x})>s \Rightarrow Q(\mathbf{x})>\phi_{1}\left( s-\mu(\mathbf{x})\right) $$

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And for the representativeness, it is:

$$\begin{array}{@{}rcl@{}} |Z(\mathbf{x})\,-\,Z(\mathbf{x}_{0})|\!<\! \delta \!&\Longleftrightarrow&\! |\mu(\mathbf{x})+R(\mathbf{x})\\ &&\!-Z(\mathbf{x}_{0})| \!<\! \delta \end{array} $$

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$$\begin{array}{@{}rcl@{}} &\Rightarrow & |R(\mathbf{x})|<\delta\,-\,|\mu(\mathbf{x})\,-\,Z(\mathbf{x}_{0})| \\ & \Longleftrightarrow & Q(\mathbf{x})<\phi_{2} \\ &&\times\left( \delta-|\mu(\mathbf{x})-Z(\mathbf{x}_{0})| \right) \end{array} $$

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with Q(x) = ϕ₂(|R(x)|) denotes in this problem the anamorphosis of $|\hat {R}(\mathbf {x})|$.

How to estimate $\mathbf {1}_{Q(\mathbf {x})>\phi _{1}\left (s-\mu (\mathbf {x})\right )}$ and $\mathbf {1}_{Q(\mathbf {x})<\phi _{2} \left (\delta -|\mu (\mathbf {x})-Z(\mathbf {x}_{0})| \right )}$?

Conditional expectation (see e.g. Lajaunie, 1993) provides an estimation of the probability of exceeding a threshold s that is more consistent than simply introducing a conventional Gaussian assumption on the kriging error. In the specific case of a multi-Gaussian random function Q(x), i.e., the multivariate distribution of $\left (Q(\mathbf {x}),Q(\mathbf {x}_{\alpha })\right )$ is multi-Gaussian (meaning that any linear combination of these variables is Gaussian), the following decomposition is used:

$$ Q(\mathbf{x})=Q^{\text{KS}}(\mathbf{x})+\sigma_{\text{KS}}U(\mathbf{x}) $$

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with U the standard residual of the simple kriging Q^KS(x) with zero mean and kriging variance $\sigma _{\text {KS}}^{2}$. Q(x) is seen as the sum of a data-conditioned member Q(x_α) and a second term, independent from the available data. The distribution of Q(x), conditionally to the data, thus follows a Gaussian distribution with mean Q^KS and variance equals to σ_KS.

Then, the probability of exceeding s is:

$$\begin{array}{@{}rcl@{}} \mathrm{E}\left[ \mathbf{1}_{Q(\mathbf{x}) \geq s} | Q(\mathbf{x}_{\alpha}) \right]\!&=&\!{\int}_{-\infty}^{+\infty} \mathbf{1}_{Q^{\text{KS}}(\mathbf{x})+\sigma_{\text{KS}}(\mathbf{x})u \geq s}g(u)du \\ \!&=&\!{\int}^{+\infty}_{\frac{s-Q^{\text{KS}}}{\sigma_{\text{KS}}}}g(u)du\\ \!&=&\!1-G\left( \frac{s-Q^{\text{KS}}(\mathbf{x})}{\sigma_{\text{KS}}(\mathbf{x})} \right) \end{array} $$

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with g(.) the probability density function and G(.) the cumulative density function of the standard Gaussian distribution.

By using the Hermite polynomials decomposition (see again Rivoirard 1991), of the indicator function 1_Q(x)≥s, the conditional expectation is computed as:

$$\begin{array}{@{}rcl@{}} \mathrm{E}\left[ \mathbf{1}_{Q(\mathbf{x}) \geq s} | Q(\mathbf{x}_{\alpha}) \right]\!&=&\!\mathrm{E}\left[ 1\,-\,G(s)\,-\,{\sum}_{i \geq 1}\frac{1}{\sqrt{i}}H_{i-1}\right.\\ &&\quad\!\left.\times(s)g(s)H_{i}\left[ Q(\mathbf{x})\right] | Q(\mathbf{x}_{\alpha}) \vphantom{{\sum}_{i \geq 1}\frac{1}{\sqrt{i}}H_{i-1}}\right] \\ \!&=&\!1-G\,+\,{\sum}_{i \geq 1}H_{i-1}(s)g(s)r^{i}\\ &&\!\times H_{i}\left( \frac{Q^{\text{KS}}(\mathbf{x})}{r}\right) \end{array} $$

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with H_i the i th Hermite polynomial and r the variance of Q^KS(x) that is easily obtained from the kriging variance:

$$\begin{array}{@{}rcl@{}} \sigma_{\text{KS}}^{2}&=&\text{Var}\left[Q(\mathbf{x})\right]-{\sum}_{\alpha} \lambda_{\alpha} C_{Q}(Q(\mathbf{x}_{\alpha}),Q(\mathbf{x}))\\ &=&\text{Var}\left[Q(\mathbf{x})\right]-r^{2} \end{array} $$

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with C_Q(.,.) the covariance function of the random function Q(x).

For the background concentration (thus replacing Z(x) by Y (x) in the previous calculations) in the urban area of Montpellier (2007), the representativeness areas and probabilities of exceedance obtained by conditional expectation are as follows (Fig. 17):

Fig. 17

Representativeness areas for background stations and probabilities of exceeding the threshold of 32 μg m^− 3 in Montpellier (2007) obtained by conditional expectation

Let us note that a simple kriging is used because after anamorphosis of Z(x), Q(x) is a standard Gaussian variable. However, the stationary assumption is required contrary to the ordinary kriging in which only the intrinsic hypothesis is required. As a consequence, the simple kriging system cannot be written with variograms.

The estimations made by conditional expectation show some similarities with the map accounting for the proportional effect under a conventional Gaussian assumption. Indeed, when only considering the conventional Gaussian assumption with no proportional effect, the kriging variance is not conditioned to the observations while the anamorphosis explicitly take them into account, meaning that in low polluted areas, the variance of the error is lower than in high polluted areas. This variance is:

$$\begin{array}{@{}rcl@{}} \text{Var}\left[ \mathbf{1}_{Q(\mathbf{x}) \geq s} | Q(\mathbf{x}_{\alpha}) \right]&=&\mathrm{E}\left[(\mathbf{1}_{Q(\mathbf{x}) \geq s})^{2} | Q(\mathbf{x}_{\alpha}) \right]\\ &&-\mathrm{E}\left[\mathbf{1}_{Q(\mathbf{x}) \geq s}) | Q(\mathbf{x}_{\alpha}) \right]^{2} \\ &=& \left[1-G\left( \frac{s-Q^{\text{KS}}(\mathbf{x})}{\sigma_{\text{KS}}(\mathbf{x})} \right) \right] \\ && \times G\left( \frac{s-Q^{\text{KS}}(\mathbf{x})}{\sigma_{\text{KS}}(\mathbf{x})}\right) \end{array} $$

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It is also what a proportional effect approach does.