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Optimal redistribution of the background ozonemonitoring stations over France

Lin Wu, Marc Bocquet

To cite this version:Lin Wu, Marc Bocquet. Optimal redistribution of the background ozone monitor-ing stations over France. Atmospheric environment, Elsevier, 2011, 45 (3), pp.772–783.�10.1016/j.atmosenv.2010.08.038�. �inria-00582492�

Optimal redistribution of the background ozone monitoring stations over France

Lin Wua,b, Marc Bocqueta,b

aUniversite Paris-Est, CEREA, joint laboratoryEcole des Ponts ParisTech - EDF R&D, Marne la Vallee, FrancebINRIA, Paris-Rocquencourt research center, France

Abstract

Ozone is a harmful air pollutant at ground level, and its concentrations are routinely measured with monitoring net-works. The network design problem aims at determining the optimal positioning of the monitoring stations. In thisstudy, the background stations of the French routine pollution monitoring network (BDQA) are partially redistributedover France under a set of design objectives. These background stations report ozone variations at large spatial scalecomparable with that of a chemistry-transport model (CTM).The design criterion needs to be defined on a regulargrid that covers France, where in general no ozone observations are available for validation. Geostatistical ozone es-timation methods are used to extrapolate concentrations tothese grid nodes. The geostatistical criteria are introducedto minimize the theoretical error of those geostatistical extrapolations. A physical criterion is also introduced to mea-sure the ability of a network to represent a physical ozone field retrieved from CTM simulations using geostatisticalextrapolation methods. A third type of criteria of geometrical nature, e.g. a maximal coverage of the design domain,are based uniquely on the distance between the network stations. To complete the network design methodology, astochastic optimization method, simulated annealing, is employed in the algorithm to select optimally the stations.

Significant improvement with all the proposed criteria has been found for the optimally redistributed networkagainst the original background BDQA network. For instance, the relative improvements in the physical criterionvalue range from 21% to 32% compared to randomly relocated networks. Different design criteria lead to differentoptimally relocated networks. The optimal networks under physical criteria are the most heterogeneously distributed.More background stations are displaced to the coast, frontiers, and large urban agglomerations, e.g. Paris and Mar-seilles. The ozone heterogeneous fields are not as well reconstructed from optimal networks under geostatistical orgeometrical criteria as from the optimal network obtained with the physical criterion. The values of the physical crite-rion for the geostatistically and geometrically optimal networks show deteriorations of about 8% and 17% respectivelycompared to that of the physically optimal network.

Keywords: Air quality, ozone monitoring, network design

1. Introduction

Ozone is a harmful pollutant at ground level. Its highconcentrations potentially injure human health, damagevegetation and material (Seinfeld, 1988; Pleijel et al.,2007). Ground monitoring networks have been de-ployed to evaluate the ozone concentrations. Due toits complex chemical mechanism (Meng et al., 1997)and the forcing of atmospheric transport, the ozone fieldis heterogeneous. This heterogeneity indicates that thespatial distribution of the monitoring stations could beoptimized. The optimal positioning of the ozone moni-toring stations is referred to as the ozone network designproblem.

Email address:Lin.Wu@cerea.enpc.fr (Lin Wu)

A typical methodology for an ozone network designproblem involves a design criterion, an ozone concen-tration estimation method, and an algorithm for the se-lection of monitoring sites. The definition of the net-work design problem is nevertheless problem-specific.For example, a dense redundant network might be re-duced to save maintenance cost (Nychka and Saltzman,1998; Fuentes et al., 2007; Wu et al., 2010). Converselya sparse network is often sought to be redistributedor augmented (Nychka and Saltzman, 1998; Rayner,2004).

A network station has its own spatial scale represen-tativeness of the underlying ozone field. Backgroundstations, which observe large spatial scale ozone con-centrations comparable with ozone fields from the sim-

Preprint submitted to Atmospheric Environment September 24, 2010

ulations of chemistry-transport models (CTM), are em-ployed in many applications especially for the valida-tion of CTMs. In this paper, we will examine how a se-lection of the background stations in the French BDQA(Base de Donnees sur la Qualite de l’Air) network couldbe redistributed to a regular grid over France for a bet-ter performance. This is of practical concern at Frenchoperational centers1.

In our redistribution methodology, we use a geosta-tistical ozone estimation method (also called kriging)and a station selection algorithm based on simulated an-nealing. Both were developed and described in detail inWu et al. (2010). The main difficulty is that the designcriteria have to be defined for each regular grid cell ofthe design domain where no observations are availablefor validation. The design criteria are rather subjective(Muller, 2007; Abida et al., 2008), although some maybe favored by the physical context. Different criteriamay lead to different optimal networks.

The simplest criterion is a measure of the distance be-tween stations in the network, upon which geometricalcriteria, e.g. the space filling design criteria, can be de-fined for a better geometrical coverage of the domain(Nychka and Saltzman, 1998).

Complex criteria can be defined using geostatisticalestimation methods, which are essentially based on thespatial correlation structure, to extrapolate concentra-tions to those regular grid cells. Geostatistical criteria,that seek to minimize the theoretical extrapolation error,make use of this correlation structure. This is closelyrelated to the optimum experimental design theory (Fe-dorov and Hackl, 1994; Muller, 2007). Another set ofpopular criteria is defined with the notion of entropy,in which the reduction of uncertainty given the networkobservations is maximized (see Le and Zidek (2006) fora review).

In the above criteria, the dynamics of the chemistryand transport of the ozone field are not considered ex-plicitly. In this regard, a third set of criteria, the physi-cal criteria, can be introduced to assess the ability of anetwork to reconstruct a reference physical ozone fieldfrom the observations sampled at the network sites. Thisreference ozone field can be generated, for instance, asthe simulation output of a chemistry-transport model.

It is of both theoretical and practical value to comparethe optimal networks under different criteria. For in-stance, Johnson et al. (1990) demonstrate that the spacefilling design of minimax type is identical to the geosta-tistical criterion that minimizes the maximum extrapo-

1private communication from L. Rouil and B. Besagnet at INERIS

lation error, if the correlations are supposed to be in-dependent. Under Gaussian assumptions, Lee and Ellis(1997) show that the kriging and maximum entropy esti-mators are equivalent, which implies that in some casesthe entropy criterion is identical to the geostatistical cri-terion.

The main contribution of this paper is to comparethree types of criteria and their impact on the redistri-bution results, as well as to assess the performance ofthe optimal networks for ozone nowcasting. For in-stance, one could wonder whether geometrical criteriaare good substitutes for physical or geostatistical crite-ria (Nychka and Saltzman, 1998), and whether geosta-tistical criteria are good substitutes for physical criteria.Hopefully, practical instructions for the redistributionofBDQA background stations can also be learnt from thetheoretical optimization results.

The paper is organized as follows. The network redis-tribution methodology is presented in Sec. 2, where thedesign criteria, the geostatistical ozone estimator, andthe selection algorithm of the background stations aredetailed. Section 3 presents the setup of the redistri-bution experiment. The redistribution results are com-pared and discussed in Sec. 4. Conclusions are providedin Sec. 5.

2. Methodology

2.1. Background stationsAn ozone monitoring station provides instantaneous

or averaged observations on ozone concentrations at themeasuring location. According to the local scenarios,e.g. the emission rates and the meteorological condi-tions, the monitoring stations have their specific rep-resentativeness of a certain spatial scale. This can beroughly described by the typology of the stations. Forexample, in general, rural stations record ozone varia-tions with a spatial scale larger than that of urban sta-tions. By contrast, an industrial station may be highlyinfluenced by the local emissions, thus its scale repre-sentativeness may be less than one kilometer. Simulta-neous treatments of observations from stations of multi-ple spatial scales are far from straightforward (Malherbeet al., 2008). Nevertheless, it is improper to strictly re-late the scale representativeness to the station typology.For instance, an industrial station at rural region maybehave more like a rural station, when the industrial sitearound is less productive or simply shut down.

A typical Eulerian chemistry-transport model com-putes the concentrations of a set of chemical species bysolving a system of advection-diffusion-reaction equa-tions (see Sportisse (2007) for a review). The evolution

2

of the species concentrations depends on many factors,e.g. the meteorological conditions, emission and depo-sition rates, and the chemical reactions among species.In general, the concentrations are computed at a regulargrid defined in a three-dimensional model domain.

For regional applications, the model grid interval isusually from 5 to 50 km. For a successful fusion ofthe information from both the CTM model and observa-tions, the monitoring stations are expected to have com-parable scale representativeness. In other words, we areinterested in a subset of network stations, which mon-itor large scale ozone field and will be referred to asbackground stations in this paper.

Yet there is no clear definition for the backgroundstation in the literature. We will select the backgroundstations among the BDQA stations by their accordancewith the CTM simulations over France. This accor-dance can be evaluated by the root mean square errorand/or the correlation between the station observationsand the CTM simulations. Note that the model error,which addresses the model deficiency due to the dis-cretization (thus with unresolved smaller scales), is notexplicitly taken into account.

2.2. Spatial interpolationA geostatistical ozone estimation method is em-

ployed in our network redistribution problem. The al-gorithmic details can be found in Wu et al. (2010). Theozone concentrations are regarded as realizations of aspatiotemporal random fieldZ with given means and aknown spatial correlation structure.

Let G = {s1, . . . , sp} be the set of gauged lo-cations of p monitoring stations, and letU =

{sp+1, sp+2, . . . , sn} be the set of ungauged locations ofgiven regular grid points targeted for redistribution inthe French territory. The ozone concentration vector

x =[Z(sp+1),Z(sp+2), . . . ,Z(sn)

]Tat ungauged sitesU

can be estimated by a best linear unbiased estimator(BLUE) x, which is a linear combination of the observa-

tion vectory =[Z(s1),Z(s2), . . . ,Z(sp)

]Tat gauged sites

G:x = E[x] + ΣxyΣ

−1yy (y − E[y]) , (1)

where

Σxy = E

[(x − E[x])(y − E

[y])T], (2)

Σyy = E

[(y − E[y])(y − E

[y])T], (3)

are the covariance matrices. This estimator minimizesthe total variance of the unbiased estimation errorǫ =x − x. The covariance matrix for the estimation error is

Σǫǫ = E

[ǫǫ

T]= Σxx − ΣxyΣ

−1yyΣyx . (4)

The spatial correlations determine how the infor-mation from observationy is dispatched in the do-main by the BLUE analysis. The covariance matricesΣxx,Σyy,Σxy,Σyx are computed based on an isotropicdiurnal nested model:

C (h) = c0,b + σ2be−

hLb , b ∈ Ib , (5)

whereh is the distance between two sites,C (h) is thecovariance of ozone concentrations between the twosites,Ib = {0, . . . ,nb − 1} is the set of indices of binswhich partition the 24 hours of a day into several inter-vals of equal time length, andnb is the total number ofbins. Here for each bin indexed byb, Lb is the correla-tion length,c0,b is the background correlation for longdistances, andσ2

b is the a priori variance of the errorfield with subtraction of the background influence. Thevariance for the field is thusc0,b + σ

2b. For hourly bins,

nb equals to 24. Note that in this formula, the ozonefield is assumed to be daily stationary, but time-varyingduring the day. The parameters of the covariance modelshould be calibrated accordingly.

2.3. Three types of redistribution criteria

The network redistribution problem can be formal-ized by:

ξ∗ = argminξ

Ψ(ξ) , (6)

whereξ is a potential network configuration, andΨ is acertain scalar criterion. Again, denoteG = {s1, . . . , sp}

as the set of locations ofp monitoring stations, andU = {sp+1, sp+2, . . . , sn} the set of locations of the un-gauged grid points targeted for redistribution. LetB bethe binary set{0,1}, ξ ∈ B

n is then the vector that de-scribes the network configuration. Thei−th componentξi of the configuration vector is 1 if sitesi is included inthe displaced network, otherwiseξi = 0. Let r < p bethe number of stations to be displaced, the configurationξ must satisfy the following constraints:

n∑

i=1

ξi = p ,n∑

i=p+1

ξi = r . (7)

Let V1 = {si l |l = 1, . . . , r; i l ∈ {1, . . . , p}} the set ofstation locations to be displaced, and letV2 = {s j l |l =1, . . . , r; j l ∈ {p + 1, . . . ,n}} the set of targeted sitesfor redistribution, then the observationy for spatial in-terpolations is defined atp points from the setB ≡(G\V1)∪V2, and the estimatorx is defined atn− p− rpoints from the setA ≡ U\V2.

3

The network redistribution problem is closely relatedto the network augmentation problem, in which the net-work configurationξ is defined only forU since theexisting networkG is fixed.

Three types of criteria can be defined for redistribu-tion at locations where no observations are available forvalidation. These criteria are of geometrical, geostatis-tical, or physical nature. They are described in the fol-lowing.

2.3.1. Geometrical criteriaWhen there is little a priori knowledge available on

the statistics and the physical properties of the ozonefield, space filling design methods can be used (Nychkaand Saltzman, 1998). The criteria are uniquely basedon geometrical consideration. For instance, the optimalnetwork may be expected to uniformly cover the designdomain.

For our network redistribution problem, one geomet-rically optimal configuration could be

ξ∗ = argmaxξ

mins1, s2 ∈ A

s3, s4 ∈ B

{d(s1, s2),d(s3, s4)}

, (8)

where d(·, ·) is the great circle distance between twopoints. The minimal distance of the displaced networkis to be maximized, thus the resulting network wouldexpand and cover the whole design domain.

2.3.2. Minimization of kriging errorThe design criteria, that needs to be defined on the

ungauged sites of the regular grid, can be based on thegeostatistical estimation error Eq. (4) (Cressie, 1993).In our case, the error covariance matrixΣǫǫ for spa-tial interpolation depends on the network configuration.Then one demands that the estimation error covarianceΣǫǫ beminimal for the optimal displaced network, thatis,

ξ∗ = argminξ

f (Σǫǫ(ξ)) , (9)

where f is a function that maps a matrix into a scalar.This function f has different forms under different op-timality conditions. We adopt the terminology fromthe optimum experimental design theory (Silvey, 1980;Muller, 2007), in which the the covariance matrix (in-formation matrix) for the estimated parameters is re-ferred to. Note that in our case, the covariance matrixΣǫǫ is defined for the estimation error as in Berliner et al.(1999).

We test the following optimality conditions:

• A-optimality: the total estimation variance is to beminimized, that is,f = Tr (Σǫǫ).

• D-optimality: the confidence region of the estima-tion is to be minimized. This confidence region(ellipsoid under Gaussian assumptions) can be de-scribed by the principal directions ofΣǫǫ . The vol-ume of that ellipsoid is proportional to the determi-nant ofΣǫǫ . In this case, one can choosef =

∏iλi

whereλi is thei−th eigenvalue ofΣǫǫ .

• E-optimality: similar to D-optimality but the max-imal direction of the confidence ellipsoid is tobe minimized, that is,f = maxλ, λ ∈ {λi , i =1, . . . ,n− p− r}.

It is easy to adaptf to the diurnal covariance model. Forexample for A-optimality,f =

∑b

Tr(Σǫǫ,b)

with b ∈ Ib.

Note that Lee and Ellis (1997) show that certain max-imum entropy criterion, e.g. the variability-absorptiondesign, is identical to the D-optimality criterion un-der Gaussian assumptions. Thus the criterion of D-optimality defined above is a representative of the classof entropy-based criteria.

The kriging methods are statistical in nature, andare not truly faithful to the ozone variability owning tochemistry and physics. The physical properties of ozonefield are accounted for approximately by using diurnalcovariance models. The correlations are approximatedby empirical (calibrated) models. If errors in covarianceparameters are important, the criteria based on krigingerror (variance or covariance) is limited and may resultin antithetical networks compared to those generatedunder the criteria aiming at estimating covariance pa-rameters (Zimmerman, 2006). However, in the contextof ozone estimation, it has been found (Fig. 6 in Wuet al. (2010)) that the kriging is quite robust with dif-ferent calibrated covariance parameters. Therefore thelimitation of geostatistical criteria is less constrainingfor our ozone network design.

2.3.3. Performance of the ozone field reconstructionSimulations of chemistry-transport models can pro-

vide forecast ozone concentration on the regular un-gauged station grid (Reynolds et al., 1973; Russell andDennis, 2000), thanks to the numerical modeling of thepollutant chemistry and physics.

It has been found in Wu et al. (2010) that the CTMsimulations (of 0.25◦ horizontal resolution) are of lim-ited use for kriging with observations from the completeBDQA network. This is due to the mismatch of thespatial scales between the CTM and the observations.

4

When observations are taken only from background sta-tions, such limitation could be reduced.

Given an ozone field taken as a reference, one can re-construct that reference ozone field from sampled ozoneconcentrations (of the field) at background stations us-ing spatial interpolation methods. It is expected that abetter network of redistributed background stations hasbetter ability in representing the reference ozone field.Then, one redistribution criterion can be defined as the(hourly) root mean square error () between the ref-erence ozone field and the reconstructed ozone field:

ξ∗ = argminξ

1

|T | × |A|

∑

k∈T ,s∈A

(xk(s) − zk(s))2

12

,

(10)where| · | denotes the set cardinality,k is the time indexin a given periodT , zk(s) is the concentration of the ref-erence ozone field at timek and sites ∈ A, and xk(s)is the corresponding ozone estimation using the spatialinterpolation formula Eq. (1). The values of the obser-vation y for kriging take the reference realizationsz(s)at monitoring sites inB.

This criterion is consistant with statistical indicators() used in air quality modeling and it is related toseveral European air quality standards.

In addition to the CTM simulations, concentrationobservations can be obtained by direct measuring ofthe ozone field. However, the utility of the observa-tions are limited, because the measurements are in gen-eral erroneous and conducted at a given spatiotemporalscale. Data assimilation algorithms merge the informa-tion from both the CTMs and the observations aimingat a better account of the true ozone field (Elbern andSchmidt, 2001; Wu et al., 2008). Thus, the assimila-tion results can also serve as a reference ozone field forreconstruction.

A third possible reference ozone field can be gener-ated with the kriging results on the sites ofU using ob-servation from the location setG of the background sta-tions.

In summary, we will test seven design criteria in thispaper. They are of different nature and can be arrangedin three sets listed in Tab. 1.

2.4. Optimization using simulated annealing

In the network design context, the combinatorial de-sign problem Eqs. (6, 7) can be solved by the simulatedannealing algorithm (van Groenigen and Stein, 1998;Abida et al., 2008; Abida and Bocquet, 2009; Wu et al.,2010). In a classical annealing, the optimization process

can escape from a local minimumξ(i) to a new configu-rationξ(i+1) with an acceptance probability (Metropoliset al., 1953; Kirkpatrick et al., 1983):

P(ξ(i), ξ(i+1), τ) = min

1,e−Ψ(ξ(i+1)) − Ψ(ξ(i))

τ

, (11)

whereτ is a global parameter which is an analog oftemperature. Oftenτ is initially high, and the iterativeprocess probes large-scale variation ofΨ. Whenτ de-creases according to certain cooling schedule, the iter-ations search for finer variations. By carefully choos-ing the cooling schedule, the global minimum can beapproached to some precision which can be arbitrarilysmall. We employ the geometric cooling schedule,

τ(k+1) = ατ(k), (12)

whereα (0 < α < 1) is a decreasing factor, and usuallyk coincides withi.

The new candidates are chosen from the neighbor-hood ofξ(i) via a flipping procedure. The configurationξ(i) is divided into two parts: the firstp components forthe monitoring stations, and the lastn − p componentsfor the targeted points. For each part, we randomly flipone component’s value from one to zero, and randomlyflip another component’s value from zero to one. Bythis way, we firstly randomly choose one different sta-tion to be displaced, then randomly move this stationto a different targeted point. Note that the constraintEq. (7) is automatically satisfied. In practice, the tuningof the parameters values, especially for the initial andfinal temperatures, is necessary to obtain a satisfactorysolution.

3. Experiment setup

3.1. CTM simulation

For this study, the P/P3D model(Boutahar et al., 2004; Mallet et al., 2007; Sartelet et al.,2007) is employed. The configurations of the model aredescribed as follows:

1. raw meteorological data: MM52 fields (resolutionof 12 km× 12 km or 36 km× 36 km, 29 verticallevels, time step of 3 hours);

2. land use coverage: USGS and GLCF3 land covermap (14 categories, 1 km Lambert);

2PSU/NCAR mesoscale model3Global Land Cover Facility

5

3. chemical mechanism: RACM (Stockwell et al.,1997);

4. aerosol: SIREAM (a SIze REsolved AerosolModel, Debry et al. (2007));

5. emissions: the EMEP4 inventory, converted ac-cording to Middleton et al. (1990);

6. biogenic emissions: computed as proposed inSimpson et al. (1999);

7. deposition velocities: the revised parameterizationfrom Zhang et al. (2003);

8. vertical diffusion: the Troen and Mahrt parameter-ization (Troen and Mahrt, 1986) (in the unstableboundary layer);

9. boundary conditions: gas and aerosol concentra-tions from the general circulation model LMDz(Hourdin et al., 2006);

10. numerical schemes: a first-order operator split-ting, the sequence being advection–diffusion–chemistry; a direct space-time third-order advec-tion scheme with a Koren-Sweby flux-limiter; asecond-order order Rosenbrock method for diffu-sion and chemistry (Verwer et al., 2002).

The model domain covers France (metropolitan area,[5.25◦W,41.75◦N] × [9.25◦E,52.25◦N]) with a 58× 43grid of 0.25◦ horizontal resolution (see Fig. 1). The al-titude is divided into 9 vertical layers. The tops of thevertical layers are 30 m, 150 m, 350 m, 630 m, 975 m,1360 m, 1800 m, 2270 m and 2780 m respectively. Thetop layer is high enough to enclose the planetary bound-ary layer. A time step of 300 s is used.

Ozone concentrations peak during summer, which isthe most risky scenario for human health and crop pro-duction. The CTM simulations covers summer 2005from 20 May at 0000 UTC to 1 September at 0000 UTC.The initial conditions are interpolated based on thecoarser LMDz simulations. The first 12 days are thespin-up period, and the simulations after spin-up areused in the network redistribution problem.

3.2. Background observations

The BDQA (Base de Donnees sur la Qualite de l’Air)network is dense for regional applications (informationavailable athttp://www.atmonet.org). There are678 stations within France. These stations (see Fig. 1)are located in typologically different areas, such as ur-ban districts and regional areas of cities, industry sites,and heavy traffic roads.

4Co-operative Programme for Monitoring and Evaluation of theLong-range Transmission of Air Pollutants in Europe

Figure 1: Map of the background BDQA Network. The circles indi-cate the locations of the background stations, and the crosssigns showthe locations of the BDQA stations that are not retained as backgroundstations. The regular tiny points are the targeted candidate locationsfor redistribution. These points are the P/P3D modelgrid-cell centers within the French territory.

The background stations are observers for the largescale chemistry-transport phenomena comparable withthe CTM simulations (see Sec. 2.1). For each BDQAstation, we collect its observations of hourly ozonemean concentrations from 1 June at 0000 UTC to 1September at 0000 UTC. Then the correlations and between those observations and the P-/P3D simulations for this given period arecomputed. In this paper, background stations are de-fined as those with correlation larger than 0.70 and with smaller than 25µg m−3. According to this defini-tion, for summer 2005, there are 190 background sta-tions (shown in Fig 1). Compared with the full BDQAnetwork, most of the rural stations are picked up asbackground stations. The majority of the retained sitesare rural and urban stations. There are also five indus-trial and one traffic stations in the set of background sta-tions. The stations in south-east and the France-Spainborder tend to be excluded in the network of backgroundstations, because at these mountainous areas (Alpineterritory), it is difficult to represent a large scale ozonefield. These statistics are consistent with the choice ofBDQA background stations from INERIS5.

3.3. Assimilation using optimal interpolationThe information from the CTM simulations and

BDQA observations can be combined by data assimila-tion algorithms to generate a more realistic ozone field.

5personal communication from L. Malherbe

6

At one time step, the ozone concentrations computed bythe CTM simulations (denoted ascb) are adjusted by theobservation vectoro, so that the error variance of theozone estimationc is minimized. Under the Gaussianassumptions, a BLUE formula for optimal interpolation(OI) reads:

c = cb + BHT(HBHT + R)−1(o − H(cb)) , (13)

whereH is the observation operator that maps the ozoneconcentration vector to the observation vector,H is thelinear form ofH, B is the covariance matrix of the er-rors in cb, andR is the (assumed diagonal) covariancematrix of the errors ino. The matrixB is parameter-ized in Balgovind form, that is, the error correlationsare isotropic and described by the Balgovind function:

f (h) = (1 + hLB

)e−h

LB σ2B, whereh is the distance be-

tween two locations,LB is a characteristic length andσ2

B a background error variance. In this study, the pa-rametersLB and σ2

B are set to a priori values as 1◦

and 400 (µg m−3)2 respectively. The observational er-ror variance takes value of 100 (µg m−3)2.

It has been shown in Wu et al. (2008) that, althoughrelatively simple compared with other advanced assimi-lation algorithms, OI is an effective assimilation methodfor ozone estimation. In this study, only isotropic cor-relations are considered. Anisotropic considerationscould bring a positive impact, but the improvements arenot significant (Fig. 7 in Blond et al. (2003)). Further-more, only observations from background stations areused in this paper, which lessens the need to introducean anisotropic correlation model.

3.4. Kriging over the full domain

A reference ozone field can also be generated by krig-ing over the full domainU using Eq. (1). We use aconstant mean. The covariance model in use is withone hour bins and calibrated to the BDQA backgroundstation observations (detailed in Sec. 4.1). The periodfor kriging is also taken from 1 June to 1 September in2005.

4. Results and discussions

4.1. Covariance models and kriging performance

For a given time windowT , the covariance for theozone concentrations at two sitessi , s j within the timeperiod of a given bin can be estimated by

C(si , s j) =1Nt

Nt∑

t=1

(Zt(si) − Z(si)

) (Zt(s j) − Z(s j)

),

(14)

with

Z(si) =1Ni

Ni∑

t=1

Zt(si) , (15)

whereNi is the number of observations at sitesi , Nt isthe number of mutually available observation pairs forsite pair (si , s j), andZt(si) denotes the random variablefor the ozone field at time indext and sitesi .

Whensi , s j run on all the available sites, a cloud ofcovariance values against the distance can be obtained.The covariance cloud can then be averaged within con-tinuous regionsT(hi) = [hi − LT/2,hi + LT/2], forhi = i × LT − LT/2, i ∈ N. HereLT is set to 30 km inthis paper for the regional application. By this way, thecurves of the regionalized covariances can be plotted.We omit the figures about the covariance clouds and theregionalized covariance curves, because they are verysimilar to Fig. 2 and Fig. 3 in Wu et al. (2008).

Let Θ = [c0,b, σ2b, Lb] be the vector of unknown

parameters for the nested covariance functionC(·) inEq. (5). The parameterΘ is determined by solving theordinary least-square fitting problem

Θ∗ = argminΘ

Nh∑

i=1

(C(hi) −C(hi)

)2(16)

whereNh is the total number of the tolerance regions,hi is the center of thei−th region, andC(hi) is the cor-responding regionalized covariance. For each bin, theparameter vectorΘ is calibrated using the Levenberg-Marquardt algorithm. For observations at availablebackground BDQA stations during summer 2005, thefitting results for one-hour bins are shown in Fig. 2.The correlation lengthL and the variance for the fieldc0 + σ

2 are smaller at night and peak at noon. This isalso similar to the fitting results with observations fromall BDQA stations but during a shorter period of onemonth (Wu et al., 2010).

WhenZt(si) is sampled from the CTM simulations orthe data assimilation results, covariance clouds and re-gionalized covariances can be obtained in a similar way.Accordingly the covariance models can be calibrated forhourly bins. The fitted parameters for these hourly co-variance models are shown in Fig. 3. These parametersclearly demonstrate a diurnal cycle. The parameters ofthe covariance model for analyzed ozone field is a com-promise between the covariance models for BDQA ob-servations and CTM simulations. Its correlation lengthshows a similar pattern to that of the covariance modelfitted to BDQA observations, and its variance is a fusionof the two other covariance models.

7

Figure 2: The calibrated parameter values with respect to binindicesfor one-hour bins, using the ozone concentrations from the back-ground stations. The unit for the background correlationc0 and thea priori varianceσ2 is in (µg m−3)2, and the unit for the correlationlengthL is in km.

We evaluate the reconstruction Eq. (10) withreference ozone field set to the CTM simulation (cri-terion PHY-S in Tab. 1) for a large set of randomlydisplaced networks. These networks are generated byrandomly displacing a given number of background sta-tions to the regular ungauged grid points within Frenchterritory. The reconstruction are shown in Fig. 4.The worst reconstruction is less than 5µg m−3.This is far inferior to the CTM model error for hourlyozone concentrations (typically more than 20µg m−3).This indicates that the kriging method is satisfactory inreconstructing the ozone field generated by CTM simu-lations. The covariance model calibrated to BDQA ob-servations is employed for the above evaluation. Twoother CTM related covariance models (see Fig. 3) arealso tested for the same set of random networks. It isfound that the reconstruction is not sensitive to thecovariance models. The maximal relative differ-ence is 2% for these random networks. For the back-ground BDQA network (see Fig. 1), the reconstructions for the three covariance models are 4.61 , 4.57and 4.57 µg m−3.

4.2. Results of a reference redistribution optimization

Before carrying out sensitivity studies, we first de-fine a reference redistribution optimization. In this ref-erence algorithmic setting, the parameters for the hourlycovariance model are fitted to the observations fromBDQA background stations. There are only 40 stationsto be displaced to the regular grid of ungauged sites over

Figure 3: The calibrated parameter values with respect to binindicesfor one-hour bins. The unit forc0, σ

2 is in (µg m−3)2, and the unitfor L is in km. The upper panel (a) shows the results using the refer-ence simulation field. The lower panel (b) shows the results using thereference assimilation field.

8

Figure 4: The reconstructions for the kriging with randomly andoptimally displaced networks. The reference ozone field is generatedby CTM simulations. For each given number of stations to be dis-placed, 100 random networks are generated for evaluation. The er-ror bar shows the standard deviation in reconstruction for theserandom networks. The centers of the error bar are the means of thereconstructions. The squares shows the reconstructions ofoptimal networks against the given number of stations to be displaced.The algorithmic setting for optimization is detailed in Sec. 4.2.

France. A geometric schedule is employed with anneal-ing rate set to 0.9999. The criterion for redistributionis chosen to be the reconstruction of the referenceozone field generated by CTM simulations (PHY-S inTab. 1).

The optimal reconstruction for the reference set-ting is 3.121µg m−3. A relative 28% improvement inthe reconstruction is obtained through optimal re-distribution compared with the case of the backgroundnetwork without redistribution. The resulting map ofthe redistributed network is shown in Fig. 5a. Theclustered stations tend to be moved to the regions withsparse network coverage, e.g. the south-east France andthe frontier area between France and Spain. In theseareas, there are lacks of background stations. Target lo-cations in these large regions are uniformly distributed.

4.3. Sensitivity of the design to algorithmic settings

4.3.1. Discrepancy between two networksThe discrepancy between two different networks

needs a quantitative definition, so that the interpretationof the optimization results could be better approached.A geometrical distance between two networks, sayBa

andBb, has been introduced by Nychka and Saltzman(1998) as

∆(Ba,Bb) = maxsa∈Ba

minsb∈Bb

d (sa, sb) . (17)

whered(·, ·) is the great circle distance. The distance∆(Ba,Bb) is a scalar that roughly measures a geometri-cal difference between two networksBa andBb.

Another global measure on the discrepancy betweenBa andBb for a given regionD is to compute the dif-ference of their coverage for each sites ∈ D (Saunieret al., 2009). The coverage of a networkB at s can beestimated by the number of the network stations locatedwithin a circle centered ats of radiusRD. Let us denotethis coverage number asλB(s). The coverage difference

δλBa,Bb(s) = λBa(s) − λBb(s), s ∈ D (18)

provides a smooth discrepancy between two networksBa andBb. The map of this smooth discrepancy de-scribes the spatial difference (local augmentation or de-pletion of stations) between networks. In our case,Dis the regular ungauged grid over France, and the proberadiusRD is set to 75 km. Eventually an average of theabsolute coverage difference

Λ(Ba,Bb) =1|D|

∑

s∈D

|δλBa,Bb(s)| (19)

serves as another scalar measure of the difference be-tween two networks.

4.3.2. Sensitivity to simulated annealing and covari-ance models

In this section, we test the sensitivity of the redis-tribution results to the algorithmic settings. Five ad-ditional redistribution experiments are performed; eachone with a single alternating parameter different fromthe reference algorithmic setting in Sec. 4.2. These al-ternative parameters are either about simulated anneal-ing or about the nested covariance model.

Figure 5 describes the setting details (read the cap-tion) and shows the resulting redistributions with bothreference and the five alternative algorithmic settings.The optimal reconstruction for the five alterna-tive algorithmic settings (b–f) are 3.139 , 3.159 , 3.126 ,3.152 and 3.132µg m−3 respectively. The maximal rel-ative difference in reconstruction for these alter-native algorithmic settings against the reference setting(3.121µg m−3) is 1.2%. The maps of the redistributednetworks with all the six algorithmic settings show avery similar pattern. The resulting redistribution is onlyslightly sensitive to the covariance model and the an-nealing setting. Some target locations for the displace-ment are even identical near country borders.

In the following, if not mentioned, the optimiza-tions are performed using the covariance model fittedto BDQA data (Fig. 2) with the slow annealing rate(0.9999).

9

Figure 5: The maps of the redistributed networks obtained with the reference and alternative algorithmic settings, in which 40 background stationsare displaced. Each alternative setting (b–f) has one changed factor compared with the reference setting (Sec. 4.2). Thesetting (b) employs a fasterannealing rate 0.9993. The setting (c) chooses a different seed for the random number generation needed in simulated annealing. The setting (d) hasa different initial network from which start the iterations. The parameters of the covariance model in the setting (e) are set to those fitted with theCTM simulations (Fig. 3a), and in the setting (f) the covariance model is fitted to the assimilation results (Fig. 3b). The circles show the remainingstations, and the cross signs show the locations of the stations chosen to be reallocated. The squares indicate the new locations of these chosenstations.

10

4.3.3. Sensitivity to the redistribution criteria

The maps of the redistributed networks under theseven criteria listed in Tab. 1 are shown in Fig. 6.The optimal network under the geometrical criterionMAXMIN is uniformly distributed to cover the Frenchcontinental territory. The optimal networks underthe geostatistical criteria are less uniform because ofsmaller correlations in between distant stations. Theoptimal networks generated under criteria STAT-A andSTAT-E are very similar. By contrast, the optimalnetwork under criterion STAT-D are more evenly dis-tributed than its two counterparts, since the volume ofthe confidence ellipsoid defined by the estimation errorcovariance matrixΣǫǫ are minimized, whereas the othertwo criteria assign more weights to the main directionof the confidence ellipsoid.

The optimal networks under the physical criteria aremore heterogeneous than the above two catalogues ofoptimal networks. It is reasonable because the refer-ence ozone field itself is heterogeneous. More stationsare allocated around coast, frontiers, and large urban ag-glomerations, e.g. Paris and Marseilles, so that the highand uncertain ozone concentrations at these regions canbe better represented. The assimilation of concentra-tion observations would render the reference ozone fieldmore realistic and increase the ozone variation aroundthe uncertain regions compared with the plain CTMsimulations. As a result, the clusterings of stations atthe uncertain regions are more evident in the optimalnetwork with assimilation results as reference field (cri-terion PHY-A; Fig. 6a) than that with CTM simulationsas reference field (criterion PHY-S; Fig. 5a).

When the reference ozone field is generated via krig-ing using ozone observations from all the backgroundstations (criterion PHY-K; Fig. 5b), the new locationsremain close to those of the original background sta-tions, which are the best candidates in reconstructingthis reference field.

We list the performance of the resulting optimal net-works under the seven criteria in Tab. 2. For any crite-rion, the network with the best performance is the op-timal network obtained under that criterion. The op-timal network under criterion PHY-K remains close tothe original background network and performs the worstunder other criteria. The geometrically optimal networkunder criterion MAXMIN is the most uniformly dis-tributed and performs second worst under other criteria.

The optimal networks for the other two physical cri-teria PHY-S and PHY-A perform similarly under othercriteria. The optimal networks for the three geostatisti-cal criteria also perform similarly.

Figure 7: Map of smooth discrepancy between optimally redistributednetworks under criteria PHY-S and MAXMIN.

The optimal networks under geostatistical or geomet-rical criteria perform poorer for the representation of theheterogeneous ozone field. The values of the referencephysical criterion PHY-S for the geostatistically and ge-ometrically optimal networks increase by about 8% and17% respectively with respect to that of the physicallyoptimal network. Therefore, for complicated design ob-jectives, e.g. those related to the heterogeneity of theozone field, the physical criterion may be more appro-priate than the geostatistical or geometrical criteria.

The geometrical distance Eq. (17) and the averagecoverage difference Eq. (19) between these seven op-timal networks are computed and listed in Tab. 3. Theoptimal networks under criteria MAXMIN and PHY-Khave larger distances (or differences) with other optimalnetworks. This is reasonable since the two optimal net-works are either the most uniform or the most heteroge-neous among the seven optimal networks.

Figure 7 shows the map of the smooth discrepancyEq. (18) between the optimal networks under criteriaPHY-S and MAXMIN. It can be clearly observed thatmore stations are displaced to the south-east coast andthe frontier area near Spain for the optimal network un-der PHY-S. Note that there are slightly more stationsaround Paris for the optimal network under criterionMAXMIN. The number of stations to be displaced forthis case is set to 40, which is not large enough to breakthe station clustering around Paris.

4.3.4. Sensitivity to the number of stations to be dis-placed

In this section, we evaluate the impact of the numberof stations to be displaced on the design results. The cri-

11

Figure 6: The maps of the redistributed networks under different criteria. Here 40 background stations are displaced.The settings for the covariancemodel and simulated annealing are same as those in the referencealgorithmic setting (Sec. 4.2). The criteria (a–f) are detailed in Tab. 1 in Sec. 2.3.The circles show the remaining stations, and the cross signs show the locations of the stations chosen to be reallocated. The squares indicate thenew locations of these chosen stations.

12

terion is chosen to be PHY-S. The reconstructionsof the optimal networks against the number of displacedstations are shown in Fig. 4. The relative improvementsin reconstruction range from 21% to 32% as com-pared to the random networks. As expected, with morestations displaced, one observes improvement in the re-construction. However, there are no obvious im-provement beyond 100 displaced stations. On the con-trary the improvement seems to diminish.

This might be explained by the differences in thesites’ locations between the BDQA network from whichthe stations are drawn to be moved, and the regular gridof targeted sites for relocation. Since there is a minimaldistance between these grid points, there will also be aminimal kriging error when interpolating from gaugedgrid points to ungauged points on the grid. The errorcould be smaller than this minimal threshold if gaugedsites are set within grid cells. In fact, a significant frac-tion of the BDQA stations are located in the grid cells,and a few of them are crucial for the interpolation andmay have locations which are better candidates than thetargeted grid points for kriging. Beyond 100 relocatedstations, it seems that part of these crucial stations arebeing relocated to worse sites (grid points), which startsdegrading the global performance.

The maps of optimally displaced networks with dif-ferent number of displaced stations are shown in Fig. 8.When a few stations are displaced, the new locationstend to be in the regions of heterogeneous and uncertainozone concentrations, e.g. the coast, the frontiers, andthe large urban agglomerations. When more stations aredisplaced, the new locations result from a balance be-tween the ozone heterogeneity and the coverage of thedesign domain.

4.3.5. Population density as an additional constraintOther spatial constraints may also be taken into ac-

count in addition to the three sets of criteria introducedin this study, such as the local population density and thecost of the stations. This can be considered as a multi-objective network design, but detailed investigations onthis matter are beyond the scope of this paper. How-ever, we conduct a preliminary study by incorporatingthe local population density in our criteria.

The physical criterion Eq. (10) is adjusted to take intoaccount the local population density at any point in thedomainU:

ξ∗ = argminξ

1

|T | × |A| ×W

∑

k∈T ,s∈A

w(s) (xk(s) − zk(s))2

12

,

(20)

Figure 9: The maps of the optimally redistributed network under thereference criterion (reconstruction performance of the CTMsimula-tions) adjusted by the population density. The population density isinterpolated to the grid of the design domain. The dark regions havelarger populations. The maximal population density is 1.93 millionper grid cell at Paris. The logarithm of the population density dataare plotted for a clear contour shape. Here 40 background stations aredisplaced. The circles and the cross signs show the remainingand dis-placed stations respectively. The squares indicate the newlocations.

wherew(s) is the population density at grid points, andW is the total population in the complete design domainU or in the estimation domainA ⊂ U. With this newcriterion, the regions with large population are givenmore importance in the optimization. This addressesthe fact that the emissions are closely related to the hu-man activities and that the exposure to the air pollutionat populated regions would have more social and eco-nomic impact. Whether the population is summed overthe complete or the estimation domain makes little dif-ference since the size of the set of displaced stationsV1

is much smaller than the size of the set of estimationpointsA.

The map of the optimal redistributed networks underthis new criterion is shown in Fig. 9, in which the refer-ence ozone field is taken to be the CTM simulation. Itis clear that, under this criterion, the stations tend to bedisplaced to the regions with large population (Ile-de-France and Vallee-du-Rhone regions). Note that thereare still many stations moved to the regions with lowerpopulation density. This is necessary for a reasonablereconstruction of the reference ozone field.

5. Conclusion

We have developed a methodology for the reloca-tion of the BDQA background stations that monitor

13

Figure 8: Maps of optimally displaced networks. The number of displaced stations are 20, 60, 100, 140 respectively. The circles show the remainingstations, and the cross signs show the locations of the stations chosen to be reallocated. The squares indicate the new locations of these chosenstations.

14

large spatial scale ozone properties comparable with thescales probed by an Eulerian chemical and transportmodel. This methodology relies on kriging methods forozone estimations, on a set of redistribution criteria, andon a selection algorithm based on simulated annealing.The kriging performs very well in the reconstruction ofthe reference field generated by CTM simulations.

The monitoring network redistribution problem turnsout to be quite different from the reduction problemstudied by Wu et al. (2010) because it requires estimat-ing pollutant concentrations on sites where no observa-tions are available for validation.

Three sets of design criteria have been defined in thedesign domain (a regular grid) where no observationsare available for validation. These criteria are of geo-statistical, physical or geometrical nature. Significantimprovements on the performance for all those crite-ria have been achieved through optimal redistribution ofthe original BDQA background stations. For instance,the relative improvements in the value of a referencephysical criterion range from 21% to 32% compared tothe randomly relocated networks. Moreover, we haveshown that this improvement saturates beyond 100 re-located stations. The optimal redistribution results havebeen found to be very little sensitive to the algorithmicsetting, e.g. the annealing rate and the parameters of thecovariance model.

The optimal redistributed networks under differentcriteria have been shown to be quite different. For ex-ample, the optimal networks for the physical criteriaare more heterogeneous (adapting to the heterogeneousozone reference field) than those for the other two setsof criteria. The values of the reference physical crite-rion (root mean square error of the reconstructed hourlyconcentrations) increase by about 8% for the geostatis-tically optimal networks and by 17% for the geometri-cally optimal networks, as compared to the minimum ofthis reference physical criterion obtained for the physi-cally optimal network. Therefore, for complicated de-sign objectives, e.g. those related to the heterogeneity ofthe ozone field, the physical criterion may be more ap-propriate than the geostatistical or geometrical criteria.Note that the design efficiency (computational aspect) isnot considered as a factor for choosing design criteria inthis paper.

For all the optimally redistributed networks, stationsfrom the clusters of background stations tend to bemoved to the regions with sparse network coverage.For the optimal networks under physical criteria, morebackground stations are displaced to the coast, frontiers,and large urban agglomerations, e.g. Paris and Mar-seilles. The optimal network under the geometrical cri-

terion is the most uniformly distributed. By contrast, theoptimal networks under geostatistical criteria are lessuniform because of small correlations in between dis-tant points.

In this study, errors of the model and of the observa-tion were not taken into account. However, by selectingthe background stations, we have reduced the represen-tativity error, which is part of model error. These errorterms are absent in our design criteria. An ongoing sub-ject is to incorporate the posterior error covariance ma-trix, which is a natural output of the data assimilationscheme (Wu et al., 2008), into the design criteria.

The topographic constraints in complex moun-tain/littoral areas make them difficult to be associatedwith large scale ozone patterns. This renders the inter-pretation of the redistribution of stations to these areasless transparent. The optimal target locations in theseareas could be better evaluated with nested higher res-olution CTM models. The use of observations in thesecomplex area will be a key point to better represent morecomplex ozone field at finer spatial scales.

We have also attempted to extend this study to otherpollutants, such as PM10, or NO2. However the absenceof clear long-range correlations, or that are at best notas prominent as for O3, gives little hope of a straightfor-ward generalization of the schemes presented here thatrely on geostatistical interpolation.

As a step forward, the augmentation of a sparse net-work is an affordable issue. The methodology of net-work redistribution can be adapted to this context, sinceboth need the criteria defined at locations where no ob-servations are available. The network design based onmultiple criteria is another topic worth further investi-gations (Reed et al., 2003). The network design in thiscase will result from a balance of many criteria, e.g. theproposed criteria in this paper, the constraint of popu-lation density, and the minimization of the cost of thenetwork construction or maintenance.

Acknowledgements.This research was supported bythe French Research Network on Sustainable Develop-ment (Reseau R2DSIle-de-France2007-21). The au-thors acknowledge the help from O. Saunier and O. Is-nard from IRSN with the population density data, anduseful discussions with scientists at INERIS and AIR-PARIF.

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Table 1: Brief description of the seven design criteria tested in this paper.

Name Type DescriptionPHY-S Physical Reconstruction with CTM simulations.PHY-A Physical Reconstruction with analyzed field.PHY-K Physical Reconstruction with kriged field.STAT-A Geostatistical Kriging error minimization under A-optimality condition.STAT-D Geostatistical/Entropy Kriging error minimization under D-optimality condition.STAT-E Geostatistical Kriging error minimization under E-optimality condition.MAXMIN Geometrical Space filling criterion of maxmin type.

Table 2: Performance of the resulting optimal networks under all the seven criteria listed in Tab. 1. These optimal networksare obtained undertheir corresponding criteria (plotted in Fig. 5a and Fig. 6 respectively). Each row shows the performance of all the resulting optimal networks withrespect to the criterion listed in the first column of that row.Note that the values of the criteria STAT-D and STAT-E are in exponential notation,and only the significand parts are kept in this table.

Physical Geostatistical GeometricalCriterion Net-PHY-S Net-PHY-A Net-PHY-K Net-STAT-A Net-STAT-D Net-STAT-E Net-MAXMINPHY-S 3.16 3.20 4.25 3.39 3.40 3.46 3.70PHY-A 3.49 3.45 4.77 3.76 3.70 3.85 4.10PHY-K 2.09 2.06 1.65 2.01 1.99 2.38 2.84STAT-A 7.84 7.86 8.44 7.75 7.83 7.86 8.11STAT-D 8.93 8.94 8.99 8.93 8.92 8.96 8.99STAT-E 1.68 1.66 10.2 1.16 1.87 1.11 2.79MAXMIN 29.3 27.8 13.4 48.2 37.7 39.3 62.2

Table 3: Discrepancy between the optimal networks under different criteria. The seven optimal networks are detailed in Tab. 2. The geometricaldistance Eq. (17) and the average coverage difference Eq. (19) are computed between networks.

Geometrical DistanceNet-PHY-A Net-PHY-K Net-STAT-A Net-STAT-D Net-STAT-E Net-MAXMIN

Net-PHY-S 66.02 151.2 53.18 61.44 54.02 63.40Net-PHY-A 132.7 53.18 56.09 53.18 68.90Net-PHY-K 50.09 61.44 52.01 63.81Net-STAT-A 53.25 51.51 63.40Net-STAT-D 58.87 63.40Net-STAT-E 63.40

Average Coverage DifferenceNet-PHY-S 0.808 1.337 0.959 0.865 1.155 1.241Net-PHY-A 1.317 1.044 0.863 1.163 1.259Net-PHY-K 1.099 1.098 1.283 1.157Net-STAT-A 0.915 0.876 1.214Net-STAT-D 1.098 1.050Net-STAT-E 1.132

17