Interpolation and assimilation methods for European scale air quality assessment and mapping. Part I: Review and recommendations

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1 Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons December 2005 Fnal draft Bruce Denby, Jan Horálek, Sam Erk Walker, Kryštof Eben and Jaroslav Fala The European Topc Centre on Ar and Clmate Change (ETC/ACC) s a consortum of European nsttutes under contract of the European Envronmental Agency RIVM UBA-B UBA-V IIASA NILU AEAT AUTh CHMI DNMI NTUA ÖKO IEP TNO UEA

2 Front page pcture Combned rural and urban concentraton map of the annual average PM 0 for the year 200 based on spatal nterpolated concentraton felds and measured values at the ndcated measurng ponts (unts n µg.m -3.days). (The fgure s taken from ETC/ACC Techncal Paper 2005/8, Annex Fgure 6A. It represents the combned map as created by mergng the nterpolated rural map (combnaton of measured values wth EMEP model, alttude and sunshne duraton, usng lnear regresson and ordnary krgng of resduals) and the nterpolated urban map (usng nterpolaton of urban ncrement Delta by ordnary krgng). Countres wth nterpolaton based on addtonal data only: BG, GR, HR, HU, RO. Countres wth mssng populaton densty nformaton and therefore excluded from the mappng calculatons: AD, AL, BA, CH, CS, CY, IS, LI, MK, NO, TR.) DISCLAIMER Ths ETC/ACC Techncal Paper has not been subjected to European Envronment Agency (EEA) member state revew. It does not represent the formal vews of the EEA. Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 2

3 Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons Contrbutng authors Bruce Denby and Sam Erk Walker, Norwegan Insttute of Ar Research (NILU), Oslo Jan Horálek, Czech Hydrometeorologcal Insttute, Praha Kryštof Eben, Academy of Scences of the Czech Republc, Praha (EEA project manager: Jaroslav Fala) Fnal Draft December

4 Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 4

5 TABLE OF CONTENTS Introducton Interpolaton methods usng montorng data only Smple nterpolaton methods Inverse dstance weghtng (IDW) Methods of radal bass functons (RBF) Beer's ntutve method Geostatstcal methods Interpolaton methods usng montorng and supplementary data Regresson based on emsson felds Regresson based on land use Regresson based on alttude Regresson based on other data Use of statons measurng alternatve ar qualty data Satellte data Interpolaton methods usng montorng and modellng data Combnaton wth models usng nterpolaton of resdual felds Combnaton wth ftted models usng nterpolaton of resdual felds Other methods for combnng models and montorng Assmlaton methods usng montorng and models Optmal Interpolaton Varatonal methods Kalman flter methods Urban scale nterpolaton wthn the regonal scale The cluster method IDW wth dfferental weghtng Bounded nterpolaton No montorng data avalable Superposton of urban and rural nterpolated felds Objectve quantfcaton of nterpolaton qualty Conclusons and recommendatons Conclusons from the revew Selecton of methodologes for testng Summary of rural nterpolaton studes Summary of urban nterpolaton studes Summary and future recommendatons References

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7 Preamble The ETC/ACC manages the ar qualty database ArBase for the EEA, whch contans ar qualty data reported through the Exchange of Informaton Decson. ArBase covers thousands of montorng statons across Europe, but the densty of the network vares across regons. For both publc nformaton (e.g., the EEA In Your Neghbourhood project) and for ar qualty mpact assessments, the stuaton between statons must also be known. Ths paper s ntended to examne and recommend methods that can be used to nterpolate between statons, to arrve at a spatal representaton of ar qualty nformaton. 7

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9 Introducton The mportant task of an ar qualty nformaton system s to offer the most complete nformaton about the ar qualty n a gven regon. Informaton concernng the regonal dstrbuton of ar pollutants can be obtaned ether through modellng, whch gves good spatal coverage but poor quanttatve certanty, or by measurements, whch have lmted spatal coverage but are assumed to be more certan. Tradtonally assessment s based on montorng data but when nformaton s requred between statons, some form of nterpolaton s requred. It s clear that the number of measurng statons avalable throughout Europe s lmted. E.g. the EMEP network conssts of just over 00 statons, dependent on pollutant, whlst the AIRBASE network has just over 440 rural ozone and 205 rural PM 0 statons regstered. These statons are assumed to be representatve of a regon wth a radus of approxmately 50 km, but the average dstance between statons s much larger than ths. Any spatally dstrbuted analyss relant on ar qualty data, e.g. ecosystem or health studes, wll naturally requre nformaton between the statons. Ths leads drectly to the need for effcent and accurate nterpolaton methods. The overall am of ths report s to recommend sutable methodologes for the nterpolaton of regonal scale montorng data for the purposes of health and ecosystem exposure analyss as well as for publc nformaton on a European wde scale. There are a large varety of nterpolaton methods, developed for many vared applcatons, that use ndvdual pont datasets and create spatally dstrbuted felds from these. Methods such as krgng, blnear nterpolaton and nverse dstance weghtng can, and have, been used for nterpolatng ar qualty data. Methods that only use observatonal data fall nto two categores, geostatstcal and determnstc. The dfference between these two s that geostatstcal methods explctly utlze the spatal statstcal structure of the data. In addton to observed ar qualty t s also possble to ntroduce supplementary data, wth better spatal coverage, to mprove the nterpolaton. Typcally such supplementary data should be ether representatve of the data to be nterpolated, e.g. the use of PM0 to represent the dstrbuton of PM 2.5, or should reflect correlaton between the physcal processes that lead to the spatal dstrbuton of the data to be nterpolated, e.g. elevaton and precptaton. Whatever the supplementary data represents t must have a sgnfcant correlaton wth the data to be nterpolated for t to be applcable. There s currently no standard method for nterpolatng montorng data for ar qualty assessment on the regonal, or any, scale. Models can also provde complex spatal nformaton of a regon as they reflect the physcal and chemcal processes nvolved. However, a large number of uncertantes n models, such as emssons and model parametersatons, mean that they requre valdaton aganst montorng. Dfferent models used for regonal scale assessment do not gve the same results for the varous pollutants (van Loon et al, 2004). In general the results are consdered less certan than the measured data, when the models are evaluated at measurement ponts. The combnaton of model and montorng data n an optmsed way has the potental to decrease uncertanty n the resultng spatal felds. The fnal draft of the EU gudance (EC workng group, 2000) dstngushes among the eght dfferent ways of combnng measurements wth models, fgure.. Of these possbltes the most extreme forms, (a) and (h), cannot be used for ar qualty assessment as solated measurements wthout any further nterpretaton gve ncomplete nformaton and non valdated models cannot be consdered relable. Most modellng actvtes rely on valdaton, (f) and (g), to lend credblty to model results. 9

10 00% measurement measurement modellng a. Measurement, no nterpretaton b. Measurement + nterpretaton c. Measurement + nterpolaton d. Measurement + model ftted to measurements e. Data assmlaton f. Model valdated by measurements n the same zone g. Model valdated elsewhere h. Unvaldated model 00% modellng Fgure.. Degrees of combnng measurements and models, taken from EC workng group (2000) Ths revew wll cover the methods (c) Measurement and nterpolaton; (d) measurement and model ftted to measurement; and (e) data assmlaton. Most emphass wll be on the frst two areas, as these are the most accessble. In ths revew we dstngush between 4 dfferent methodologes for combnng montorng and modellng, that le wthn the methods (c-e) gven above. These are: Chapter 2. Interpolaton methods usng montorng data only - E.g. IDW nterpolaton, krgng, etc. Chapter 3. Interpolaton methods usng montorng and other supplementary data - Use of emsson data, land use, alttude, populaton, clmate etc. to mprove nterpolaton Chapter 4. Interpolaton methods usng montorng and models - Use of chemcal transport model output concentraton felds to mprove nterpolaton Chapter 5. Data assmlaton methods - Incluson of observatons n the model prognoss In addton to these methodologes one specal area of nterest, how to nclude smaller scale features such as urban conurbatons n regonal scale nterpolaton, wll be dscussed, chapter 6. Ths s of partcular mportance for populaton exposure calculatons based on nterpolated data. Chapter 7 wll specfy the best method for assessng the qualty of the nterpolaton methodologes and then fnally, n chapter 8, recommendatons on the best nterpolaton method(s) wll be made. The bass for the fnal recommendatons wll be bult upon both the avalable lterature but also to a large extent on studes carred out n Part II of ths report. In Part II the testng, assessment and applcaton of a number of selected nterpolaton methods s descrbed. The ndcators chosen for the studes are ndcatve of both exposure and eco-system effects, these beng the ozone ndcators AOT40, SOMO0 and SOMO35 as well as the ndcators for PM0, annual mean and 36 th hghest daly average concentratons. The studes are carred out wthn a common framework ensurng comparablty between dfferent methods. Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 0

11 2 Interpolaton methods usng montorng data only These methods are based solely on measured data (wthout any addtonal nformaton concernng the spatal form of the data) collected at observatonal statons. Smlar problems whch arse n the mappng of ar polluton concentratons, also appear n meteorology and clmatology - especally n the mappng of precptaton and temperature felds. It can be useful to draw nspraton from these areas for use n ar qualty mappng. From a hstorcal perspectve, n the begnnng of the 20th century clmatology used the so-called Thessen method for the estmaton of the long-term means n the regons wth no measurements. Snce the mdsxtes geostatstcal methods, especally dfferent types of krgng, have been appled. These methods are based ether on the tme seres of measured quanttes (optmum nterpolaton), or they consder only one realzaton of the feld of measured quantty (krgng). Both optmum nterpolaton and the dfferent types of krgng are used n ar polluton mappng. Apart from the geostatstcal methods, whch utlze knowledge of the statstcal structure of the concentraton feld, several other smpler methods can be used, such as the mnmum curvature method, dfferent types of splnes or the IDW method (nterpolaton usng nverse dstance weghtng). 2. Smple nterpolaton methods 2.. Thessen polygon (nearest neghbour method) One of the smplest methods s the Thessen polygon or the nearest neghbour method. Ths method defnes ndvdual areas of nfluence around each of a set of ponts. These areas correspond to the area around a pont that s closest to that pont. They are mathematcally defned by the perpendcular bsectors of the lnes between all the ponts. Havng defned such a polygon the area wthn s gven the same value as the pont t contans Blnear nterpolaton (trangulaton) Gven a random feld of measurement ponts t s possble to lnearly nterpolate between ponts to produce a concentraton feld. Ths can be done by generatng Delaunay trangles, trangles that defne the closest 3 ponts to each other. From these ponts the equaton of a plane can be derved and used to determne concentratons wthn the trangle, or alternatvely to drectly produce contour maps, as s appled n a number of GIS systems (Burrough, 986). Ths form of nterpolaton ntroduces dscontnutes n the concentraton gradents at the edges of the nterpolatng trangles. Its smplcty though makes t attractve for smple applcatons. Some other smple nterpolaton methods can be found n SURFER (994). 2.2 Inverse dstance weghtng (IDW) The Inverse Dstance Weghtng (IDW) method mproves on the Thessen polygon method by weghtng the lnear combnaton of values measured at all ponts, where the weght s an nverse functon of the dstance accordng to the equaton: where Zˆ ( s ) 0 Z ( s ) n β = d 0 = n β = d 0 $Z ( s 0 ) s the nterpolated value of the concentraton n the pont s0, (2.)

12 ( ) Z s s the measured value of the concentraton n the -th pont, n s the number of surroundng statons from whch the nterpolaton s computed, d 0 s the dstance between the nterpolated pont and the -th staton, β s the weghtng power. In practce the weghtng power s mostly equal to two, then the name of the method s "nverse dstance squared". Ths method s used e.g. by Dttmann et al. (999). Accordng to the EC workng group (2000) the method IDW (when β = 4) has been used for constructng ar polluton maps n Belgum. Nalder and Wen (998) ntroduce a developed IDW method, whch addtonally tres to remove the nfluence of gradent has been termed gradent-plus-nverse dstance squared method: n Z ( s ) + ( x x ). c + ( y y ). c + ( v v ). 0 x = d y 2 = d 0 Z( s 0 ) = (2.2) n where x and y are the relevant geographcal coordnates, v s the alttude, x 0, y 0 and v 0 are the coordnates of the nterpolaton pont c x, c y and c v are the regresson coeffcents for x, y and v. 0 c v In the Czech Republc the so-called modfed verson of IDW, Fala et al. (2000), s routnely used. For each staton there a radus of representatveness s set and eventually ts weght for ar qualty assessment. Ths allows both urban and rural statons to be nterpolated together (dfferent types of statons have dfferent representatveness). The nterpolaton s computed by: ( s ) = ; d0 r 0 = n w d = 0 ( s ) n wz d Zˆ, (2.3) 0 where s,,s n s the set of statons nearest to s 0, whch satsfy the condton d 0 <r d 0 s the dstance between the nterpolated pont and the -th pont of measurement, r s the radus of representatveness of the -th staton (ndvdually assessed for each staton), n s the gven number of statons, from whch the nterpolaton s computed, w s the weght of -th staton (expressng ts credblty). Falke and Husar (998a) also extended the IDW method to account for groups of montorng statons assocated wth urban areas. They suggest a method for clusterng such urban groups by reducng the area of nfluence of clusters of statons so that they do not adversely nfluence regons far from the cty, gvng extra weght to rural statons that have an area of representatveness that s larger. They suggest a declusterng weght CW gven by Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 2

13 CW j = D j + D m k = j d jk m p (2.4) where s the nterpolated pont ndex, j s the montorng ste ndex, m s the number of stes wthn D j of the montorng ste (ncludng the montorng ste) D j s the dstance between the montorng ste and the nterpolaton pont k s an ndex of the (n-) neghbourng stes around the montorng ste d jk are the dstances between the montorng ste and ts neghbourng stes, p s the weghtng power. Ths declusterng weght s then multpled wth the normal ndvdual IDW to produce a total weght. The resultng nterpolaton then becomes Zˆ ( s ) j n CW = = n = j d Z d β j ( s ) β j They appled ths methodology for the nterpolaton of ozone maps n the USA. They state that by usng ths methodology the decluster weghtng wll adjust the total weghtng so that a cluster of statons 50 km from the estmaton pont wll have approxmately the same weght as a sngle staton 50 km away. By usng cross-valdaton they reported an mprovement n the resultng felds, partcularly n rural areas. (2.5) 2.3 Methods of radal bass functons (RBF) Radal bass functons methods nterpolate the measured values whle mnmzng the total curvature of the surface, so that the resultant surface s contnuous and soft. Identcally, as n the case of IDW, the nterpolaton goes exactly through the measured values. The nterpolaton s descrbed by n ( s 0 ) = w. Φ( d ) Zˆ + w (2.6) = 0 n+ where Φ(x) s a concrete RBF functon, d 0 s a dstance of nterpolated pont from -th staton, w,, w n+ are the weght parameters, n s a number of surroundng statons from whch the nterpolaton s computed. From the formal aspect the calculaton of the radal basc functons and the estmaton of ther parameters s rather complcated, whle from the computatonal aspect t s qute smple and fast. The parameters w,, w n+ are obtaned from the system of equatons gven by ( d ) + w Z( s ) n w j j n =, =,..., n, (2.7) j= n j= w j = w n+ 3

14 Some of the radal bass functons are: completely regularzed splne, splne wth tenson, thn-plate splne, multquadrc functon or quadrc beta-splnes. A more detaled descrpton of some radal basc functons are gven n Johnston et al. (200). These methods do not use knowledge of the covarance structure of data, but they nclude ths nformaton mplctly. It s used e.g. by Coyle et al. (2002). 2.4 Beer's ntutve method Beer and Doppelfeld (992, 999) ntroduced ther own methodology of spatal nterpolaton, whch s based on an ntutve approach to the problem. The general relaton for estmatng the concentraton Zˆ( ) at the locaton s0 s gven by N = ( s ) w0 Z = Zˆ ( s0 ) = (2.8) N w 0 The local weghtng functon w s defned as d 0 d 0 w 0 = when < (2.9) R( s ) R( s ) d 0 = 0 when R ( s ) where d 0 s the dstance between the pont s 0 and the measurng staton s, R(s ) s the radus of nfluence assgned to the measurng staton s and the relevant pollutant. The radus of nfluence s assessed ntutvely for the ndvdual pollutants and types of statons (traffc, ndustral, urban background and rural) - e.g. for NO x t s assessed as 0 km for the urban background statons and 20 km for the rural statons. Then the resultant nterpolaton s constructed dependent upon the number of statons around the pont X n the followng way: s 0 N = ( s ) w0 Z = Zˆ ( s0 ) = f d N 0 R(s ) for more than one staton s, w 0 ( sk ) w k + Z ( R) ( w k Zˆ ( s0 ) = Z 0 0 ) f d 0 R(s ) for exactly one staton s k, ( R) Z ˆ( s0 ) = Z f d 0 > R(s ) for all measurng statons s, (2.0) N where Z ( R) = N = Z( 2.5 Geostatstcal methods s ) s the arthmetc mean of the rural statons n the whole regon. These methods are based on knowledge of the statstcal structure of a random feld Z(X) (where X = X,X 2,...,X k. are the ponts n an Eucldean space R a ). They respect the statstcal relatonshp between the ndvdual ponts of the feld. A number of dfferent nterpolaton methods can be used for spatal predcton where optmum nterpolaton and krgng belong to these basc methods. They both assume the Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 4

15 homogenety of a random feld and defne an estmate of the unknown value of the feld Z at the pont x 0 as a lnear combnaton of Z measured at the ponts x = x, x 2,..., x M. Ths s formulated as M 0 ) = Z( x ) = Z ˆ ( x λ (2.) where the accuracy of the estmaton Zˆ( x 0 ) s characterzed by the mean squared error 2 Q$ ( ) EZx [ ( ) Zx $ ( )] 2 = (2.2) Z x0 0 0 where E represents the mean operator. However, when calculatng the weghts, λ, n equaton 2. optmum nterpolaton uses mostly the tme seres of the measurements, whlst krgng n ts basc form mproves only one value (consdered as the realzaton of a random feld). However, despte the dfferent orgn of these two methods, Cresse (993) states the prncples of optmum nterpolaton and krgng are dentcal Krgng and ts varants Krgng n ts basc form s used when only spatal, but not temporal, data are taken nto consderaton. To descrbe the concentraton feld varablty on the bass of measurements at N statons a semvarogram s constructed. Ths approach was ntroduced by a French statstcan G. Matheron, Matheron (963). He named ths method of optmal spatal lnear predcton after the South Afrcan mnng engneer D.G. Krge, who appled ths method to geostatstcal problems n geology (Krge, 95). A detaled descrpton of ths methodology s gven by Cresse (993). The semvarogram γ(h), resp. varogram 2γ(h), s a measure of the relaton between pars of measurng statons s and s 2, under the condton var(( Z( s ) Z( s2 )) = 2γ ( s s2 ), for all s, s 2, (2.3) where var s the varance and h s the two-dmensonal dstance (whch can be expressed by dstance and drecton). If the varogram s the same for all drectons,.e. γ(h)= γ( h ), then t s called sotropc. The emprcal semvarogram γ v s calculated by the equaton: n 2 γ v (h) = { Z(s ) Z(s + h) }, (2.4) 2 n = where Z(s ) and Z (s + h) are the measurements n the ponts s and s + h, n s the number of dstnct pars of ponts, the dstance of whch s h±δ, and δ s the tolerance. The emprcal semvarogram s ftted by an analytcal functon (model) - e.g. sphercal, exponental, Gaussan. (The basc parameters of a semvarogram are called nugget, sll and range. When the ftted curve does not pass through the orgn, t denotes the exstence of the so-called nugget effect). The estmated semvarogram s consequently used n the nterpolaton. The parameters used to establsh the semvarogram are shown n fgure 2.. 5

16 γ(h) Model functon Observatons Sll Nugget Range Dstance (h) Fgure 2.. Dagram showng the mportant parameters that descrbe the semvarogram, γ(h), used n krgng. Ordnary krgng performs spatal nterpolaton under two assumptons. These are: the model assumpton Z( s) = µ + ε( s), s D, (2.5) where µ represents the constant mean structure of the concentraton feld, ε(s) s a smooth varaton plus measurement error (both zero-mean) D s the examnng area and the nterpolator assumpton ) Z( s λ n 0 ) = Z( s ) = n λ =, =, (2.6) where ( ) ( ) $Z s s the nterpolated value of concentraton n the pont s 0 0 Z s s the measured value of concentraton n the -th pont, =,..,n n s the number of surroundng statons from whch the nterpolaton s computed λ,, λ n are the weghts assumed at the bass of varogram The weghts λ are derved from the semvarogram n order to mnmze the mean-square-error; n s the number of surroundng measurng stes from whch the nterpolaton s computed. The explct calculaton s acheved by the system of equatons n ( ) γ( ) λγ j s sj + s s m = n j= 0 0, =,..., n, λ =. (2.7) = Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 6

17 By ths system of equatons the parameters λ,..., λ n can be calculated (as well as m, the co-called Lagrange multpler, that ensures ( ) s 0 concentraton Ẑ. n λ = ). By ther substtuton nto equaton 2.6 we obtan the estmaton of = Krgng s an often used standard method, whch has been used e.g. n the materals of EMEP for constructng maps of background concentratons (Hjellbrekke, 2000). The area of Europe s n that case dvded nto nne regons, whch are processed separately (some poorly representatve statons are not taken nto consderaton). For each regon the varogram s constructed and the nterpolaton usng krgng s performed. Consequently the results from the regons are unted nto a complete map. An applcaton of krgng s presented by van Leeuwen et al. (994). They also document the fttng of varograms. Krgng s used also by e.g. Dttman et al. (999), Atkns and Lee (995), Lefohn et al. (988) and Casado et al. (994). To apply ordnary krgng t s necessary that some essental assumptons be fulflled, especally statonarty. (.e. the assumptons gven n equatons 2.3 and 2.5 are fulflled). As Federov (989) shows, ths s often not true. (He therefore presents an alternatve to krgng called the method of movng least squares). The stuaton, where the spatal trend exsts, s dealt wth by detrended krgng, whch s mentoned by Nalder and Wen (998), and especally by unversal krgng, the descrpton of whch s gven by Cresse (993). In unversal krgng the followng model assumpton s consdered: Z(s) = µ(s) + ε(s), where µ(s) s a trend. Unversal krgng has been used e.g. Blonck (985) for the nterpolaton of sulphate deposton data. (2.8) Şen (995) tres to solve the problem of nonstatonarty n a dfferent way. He constructs a so-called cumulatve semvarogram: k ( d k ) = ( ( 2 ~τ cum Z( s ) Z( s j )) (2.9) 2 l= where d k are the dstances ranked n ascendng order between the measurng statons for all combnatons of the pars of the statons, k =, 2,, n(n-)/2, Z(s ) and Z(s j ) are the concentratons measured at the statons s and s, the dstance of whch s d l. The graph of the cumulatve semvarogram enables the frst analyss of the relatons among the statons and enables the determnaton of the local dependences. The values of the cumulatve semvarogram can become the weghtng factors n the nterpolatng methods. The estmaton of the unknown concentraton Zˆ( s 0 ) n the pont s0 s calculated by the equaton Zˆ( s ) 0 n = = n λ Z( s ) = λ where d 0 s the dstance between the measurng staton s and the estmaton pont s 0, λ,, λ n are the weghts computed at the bass of cumulatve semvarogram. (2.20) Şen (998) apples ths method to estmate the SO 2 and SPM concentraton felds n the Istanbul urban area. The last method to be mentoned n ths secton s cokrgng, whch adds the property of correlaton between the varables. Sometmes t s used for nterconnecton of dfferent pollutants, otherwse for spatotemporal analyss of the one pollutant. Its theoretcal descrpton s also gven by Cresse (993). Cokrgng 7

18 s n practce used e.g. by Pardo-Iguzquza (998) for the constructon of precptaton maps usng both precptaton measurements and the alttude of the measurng statons. It s general form can be wrtten as Zˆ n m ( x ) = Z( x ) + η Y ( 0 = λ x ) j= j j (2.2) where Z(s ) are the prmary data at a measurement pont (n ths case ranfall), Y(s ) are the secondary data (n ths case alttude), λ and η j are the weghts, whch are based on knowledge of the varograms and the crossvarogram. The crossvarogram s determned usng the covarance of the two quanttes Z and Y n a smlar manner to the use of the varance, equaton 2.3, to determne the semvarogram. Bytnerowcz et al. (2002) nvestgated a large number of nterpolaton methods, ncludng both smple and geostatstcal methods, for the nterpolaton of ozone measurements n the Carpathan mountans of Eastern Europe. In total 28 dfferent methods were evaluated, all of whch were avalable n standard ArcGIS software (Johnston et al., 200). Results from that study ndcate a number of sutable nterpolaton methods but a sphercal model of cokrgng, wth alttude as the secondary varable, was fnally recommended for use n that analyss Optmum nterpolaton The theory of optmum nterpolaton was ntroduced by Gandn (963) and further developed by e.g. Alaka et al. (972). In the area of ar qualty t has been used by e.g. Hrdà (98). The optmum nterpolaton assumes that the assumpton of homogenety for the feld s fulflled:.e. the mean of the feld must be constant and the correlaton functon must depend only on the vector d j between the relevant ponts,.e. [ Z( X )] = µ E (2.24) K X, X ) = E[ Z(X ). Z(X )] = B r ( h ) (2.25) Z ( j j Z j In most cases fulflment of the assumpton of sotropy s also requred,.e. the correlaton functon of the concentraton feld must depend only on the sze of a vector d j (ndependent of the drecton and orentaton). If these assumptons are fulflled, ths method of nterpolaton s really objectve, because t mnmzes the sze of the mean squared error. The weghts λ ι = λ,, λ M, whch are necessary for equaton 2., are calculated from the equaton M = λ rj = r0 j j =, 2,, M (2.26) where r 0 and r j are the values of the correlaton functon for the consequent dstances between the statons. Beer and Dopperfeld (992) proceed from optmum nterpolaton, they construct the so-called modfed structure functon (MOST): MOST ( x x ) j k = c ( tk ) > h ( tk ) ( t ) 2 N c j, = (2.27) M c k where c (t k ) and c j (t k ) are the tme seres of the measured concentratons at x and x j, k=, 2,,N M s the number of measurements n the tme seres, h s the chosen postve constant. Ths functon s consequently used n the calculaton of nterpolaton Other spato-temporal methods Spato-temporal mappng ncludes many problems, as Rouhan et al. (992) shows. Among these problems are e.g. non-statonarty of the spatal means, mbalance between the avalablty of spatal and temporal Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 8

19 data, the presence of perodc and non-perodc temporal trends (e.g. seasonal trends) n data. Rouhan et al. (992) ntroduce a multvarate geostatstc model. Ths model s based on a mult-scale temporal approach (and t s n many ways analogous to an optmum nterpolaton). The values, whch are measured at the ndvdual statons, are consdered as realzatons of separate, but correlated random varables. The collecton of one-dmensonal random varables can be consdered as a set of correlated random functons. The only dsadvantage of that approach s a great number of varograms and covarance functons, whch need to be estmated. Let's consder spato-temporal data set {z ι (t α ); =,, N; α =,, T}, measured at N statons at T tme ntervals. These varables can be vewed as a realzaton of a set of one-dmensonal random functons {Z ι (t); =,, N}. It s useful to postulate the hypothess that the ncrements, Z ι (t α ) Ζ ι (t α +τ), are secondorder statonary: E[Z ι (t) Ζ ι (t+τ) ] = 0 (2.28) E[(Z ι (t) Ζ ι (t+τ). (Z ϕ (t) Ζ ϕ (t+τ))] = 2γ ιϕ (τ) (2.29) where 2γ ιϕ (τ) s defned as the cross varogram. In the partcular case where the varables themselves, Ζ ι and Ζ ϕ, can be consdered as second-order statonary and uncorrelated for large tme lags then γ ιϕ (τ) σ ιϕ for τ where σ ιϕ s the covarance of Ζ ι and Ζ ϕ. The value of ths varogram can be calculated drectly as T k γ ( ) = { ( ) ( + )}{ ( ) ( + j τ k z tα z tα τ z j tα z j tα τ )} (2.30) 2Tk α = where τ' s the tme lag belongng to a class of lags τ κ, T κ s the number of ncrement pars n such a class. Equaton 2.28 s based on the assumpton, that the ncrements are second-order statonary. If the assumpton s not fulflled, there are non-perodc trends n the set. On the bass of ths theory dfferent forms of nterpolatons can be performed, whch are based on cokrgng of the measured values. Z N T λ α j= α = j ( t ) = z ( t ) 0 j α (2.3) where j λ α s the weght of the observed value at the j-th staton at the α-th tme nterval. Chrstakos and Vyas (998) developed a method, whch s also based on a spatotemporal random feld. For ther method statonarty s not necessary (ths s mportant, because the assumpton of statonarty often s not fulflled, see secton 2.5.). They use ths method e.g. for constructng maps of short-term ozone concentratons or the annual wet deposton of sulphur. Another method ntroduced by Chrstakos and Serre (2000) s the approach of Bayesan maxmum entropy (when the hard and soft data are dstngushed). On the bass of ths method they construct the map of shortterm PM 0 concentratons n Northern Carolna. 9

20 Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 20

21 3 Interpolaton methods usng montorng and supplementary data In secton 2 the nterpolaton between montorng statons was carred out purely on the bass of the montorng data themselves. However, there can be well correlated physcal relatonshps between concentratons and other dstrbuted felds, whch may be better known. A typcal example s gven n the prevous secton n regard to cokrgng, where ranfall has a strong relatonshp to alttude. When dscussng supplementary, or addtonal, data we defne two types. Ancllary data s nformaton such as land use, clmatology and elevaton that are relevant for the processes that determne concentratons and surrogate (or proxy) data s data that can be used to represent the assumed dstrbuton of the feld, e.g. usng PM 0 as a surrogate for PM 2.5 or usng emsson felds, through a regresson relatonshp, to represent the dstrbuton of a concentraton feld. Relatonshps may exst between measured concentratons and spatally dstrbuted felds of: emssons, land use, populaton, road network dstrbuton, alttude, ranfall, lattude, clmatology, other observed ar qualty data, etc. Use can be made of these spatal relatonshps to mprove the nterpolaton between measured concentratons ponts. In ths chapter examples of some of these methods are revewed. They generally nvolve the use of regresson relatonshps between known spatal felds and lmted montorng data. The regresson relatonshp, or model as t s often referred to, can then be used to create a spatal map of data, based on the ancllary data feld. When referrng to ancllary data, ths nfers extra data that s useful for the nterpolaton. When referrng to surrogate data ths mples data that can effectvely replace, gven some converson functon (regresson model), the montored data. When the ancllary data has a spatal resoluton analogous to the desred nterpolaton feld, e.g. satellte or land use data, then the dfference between usng the ancllary data as an nterpolaton feld and usng the montored data as a converson functon (surrogate data) for the feld data becomes less well defned. 3. Regresson based on emsson felds The combnaton of emsson and observed ar qualty data has, for a long tme, been appled by Stedman (998). Hs approach, used for many pollutants (SO 2, NO 2, NO x, PM 0, CO, benzene,.3-butaden, CO, lead), s based on a x km grd resoluton emsson nventory. The nventory s used to calculate the contrbuton from sources wthn a 25 km 2 regon around the relevant measurng pont for local ar qualty (Stedman states, that ths area gves the most robust estmaton). Thus there s a dstncton between the contrbuton of large dstant sources (such as power plants or agglomeratons) and the contrbuton of local emsson sources. At frst the annual mean background feld of concentratons s calculated on the bass of the rural background statons (usng krgng or blnear nterpolaton). Subsequently, the dfference (df) between the ambent pollutant concentratons measured at urban montorng stes (not roadsde or ndustral stes) and the underlyng rural concentraton feld s calculated where montorng data are avalable. The relaton between ths dfference (df) and the estmaton of emssons (em) n the vcnty of the montorng statons s then examned through regresson analyss: df = k.em (3.) Then, wth the ad of the coeffcent k, the map of annual average concentratons s constructed by combnng the emsson database and the background map obtaned by nterpolatng data from rural background statons: where em k P Z(s 0 ) = P(s 0 ) + k. em(s 0 ) (3.2) s the sum of emssons n the area 25 km 2 around pont s 0 (n klotons) s the regresson coeffcent s the rural background feld. 2

22 To estmate the annual rural background feld of lead and benzene concentratons (whch are measured n Brtan at much fewer statons than the basc pollutants) emprcal relatons of these pollutants wth ntrogen doxde are utlsed. For benzene Stedman states the relaton (for annual averages) BZN [ppb] = 0.03 * NO 2 [ppb]. (3.3) To estmate emsson felds of lead and benzene emprcal relatonshps wth other pollutants, n the case of traffc emssons, are utlzed. For lead emssons Stedman used NO x emssons. Orgnally he related ntrogen oxdes computatons to benzene emssons (Stedman et al., 997) but n further work (Stedman, 998) he states that the relatonshp between benzene and NO x dffers n dependency on the speed of cars the emssons of ntrogen oxdes from road transport are larger at hgher speed, whle the emssons of benzene are hgher at lower speeds (ths leads to overestmaton of benzene concentraton at hghways). As a better alternatve he states the relatonshp between benzene and VOC (benzene s estmated as.55 % of total VOC) and uses the emsson nventory of VOC for the mappng of benzene. Ths approach to create the benzene emsson feld s further mproved n Stedman et al. (998). To assess the contrbuton of emssons from traffc an emprcal model s constructed, whch utlzes the relatonshp between VOC emssons from traffc (obtaned from an emsson database) and the concentratons of benzene measured at traffc statons. Stedman states that the most complcated feld s the constructon of PM 0 maps. The man reason s the large number of polluton sources. He separates PM sources nto the prmary partcles emssons from traffc, prmary partcles from statonary combuston sources, prmary partcles from non-combuston sources (e.g. mnng, demolton, dust drfted n by wnd) and secondary partcles. To estmate the contrbuton of secondary partcles nto the background feld Stedman suggests the use of ground-level ozone or background measurements of SO 4, the second of whch he consequently prefers. The method for calculaton of annual averages of PM 0 and NO 2 s further developed n Stedman et al. (200), see fgure 3. and 3.2 The sum of emssons (em) n equaton 3.2 s consdered for a larger regon (225 km 2 ) and n addton a dsperson model s utlzed. The coeffcent k s computed separately for agglomeratons and the remanng area. Verfcaton of the PM 0 maps was carred out. For background concentratons a regresson dependency was detected wth the value of the coeffcent R 2 between 0.50 a The statstcal dependency at the traffc statons was notceably weaker. Lloyd and Atknson (2004) make a comparson of nterpolaton methods ncludng IDW, lnear regresson, ordnary krgng, krgng wth an external drft and smple krgng wth a local varyng mean. Emsson felds of ntrous oxdes are used as ancllary data n the nterpolatons n order to calculate NO 2 felds for the Unted Kngdom. The concluson from that study s that smple krgng wth a locally varyng mean gves the best results, the local mean beng determned from the ancllary emsson data. Ths has been converted from NO x emssons to NO 2 by way of a lnear regresson wth NO 2 observatons. Fgure 3.. The relatonshp between ambent NO 2 (left) and PM 0 (rght) concentratons aganst the weghted sum of local emsson (Stedman et al., 200). These data are used to establsh the regresson coeffcents appled to produce the ar qualty maps shown n fgure 3.2. Measurements for urban and elsewhere (rural) have been separated and gven dfferent regresson coeffcents. Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 22

23 Fgure 3.2. Estmated annual mean (999) of NO2 (left) and PM0 (rght) concentraton (ugm-3) usng the regresson model from Stedman et al. (200). The top maps show the concentraton felds calculated from the emsson felds n combnaton wth the regresson relatonshp. The bottom graphs show the comparson of measured and modelled annual means at rural stes (ugm-3). 3.2 Regresson based on land use Jerrett et al. (2003) apples a regresson model to measured concentratons and land use data. In the study descrbed, 95 passve samplers for NO 2 were placed n Toronto over a 2 week perod. Measured concentratons were tested aganst 85 dfferent land use parameters (e.g. traffc densty, dstance from roads, populaton densty, physcal geography, wnd drecton, etc.). The fnal regresson model used 8 dfferent parameters and gave a coeffcent of determnaton (r 2 ) of 0.69 wth the most postve assocatons from traffc counts, road length and dstance, numbers of dwellngs and land us (ndustral), as well as up- or 23

24 downwnd ndcators. A comparson of the land use regresson model wth ordnary krgng s gven n fgure 3.3. Smlarly Brggs et al. (2000) and Lebret et al. (2000) have carred out regresson studes n 4 European ctes as part of the SAVIAH project. Here a large number of passve samplers for NO 2 were dstrbuted throughout the ctes and the resultng montored concentratons were used to buld up a regresson model on the bass of land use data that ncluded traffc volume wthn 300 m, land use (ndustral, commercal, housng densty) and elevaton. The study from Brggs et al. (2000), that looked at 4 areas wthn London, acqured coeffcents of determnaton for the 3 regons that ranged from 0.5 to 0.76 for the mean annual concentratons. A method smlar to ths s the approach of Kopeckà et al. (995). It s based on the lnear regresson among the measured ar polluton and the emsson densty, the roughness and the percentage woodness. If ths regresson dependency s sgnfcant, t s used to calculate the concentratons at locatons wthout measurements. These studes all ndcate that the nterpolaton of montorng data can be sgnfcantly mproved by the ncluson of land use data. These studes have all been carred out n urban regons and ndcate the mportance of the emsson feld n mprovng nterpolaton. Fgure 3.3. Resultng maps of NO2 from the study by Jerrett et al. (2003). Left s the krged surface generated wth a sphercal model of NO2 usng 95 data ponts. Rght s the fnal operatonal map that uses 8 dfferent land use parameters, ncludng wnd drecton, for the regresson model. 3.3 Regresson based on alttude The Czech Hydrometeorologcal Insttute n Prague has developed a method, Fala et al. (995) and Kveton et al. (998), whch uses lnear regresson between the values measured at the statons and alttude. Regresson s calculated for all statons ndvdually, whle only the statons n the gven area ambent to the partcular staton are consdered n each calculaton. The resultng regresson coeffcents are nterpolated by the IDW method to the nodal ponts. In the nodal ponts relatng to the statons the absolute members of regresson equatons are calculated. They are also nterpolated by the IDW method. From the above parameters, a lnear regresson equaton of dependency of montorng value on the alttude s constructed for each nodal pont. The fnal values for each nodal pont are then calculated by substtuton of the alttude. A smlar methodology has been appled n hydrology for the nterpolaton of precptaton data, usng alttude as secondary data (Goovaerts, 2000). The lnear relaton between the model and measurements was estmated at the ponts of measurements, and then ordnary krgng was appled on the resduals. Ths s sometmes termed krgng wth varyng local means. Ths methodology s the same as one of the methods appled by Lloyd and Atknson (2004), see secton 3., but uses alttude data nstead of emsson data. Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 24

25 A methodology was appled by Lobl et al. (2000) for the nterpolaton of ozone felds from montorng statons. Ths correcton was based on a functonalty of alttude and tme of day. The nterpolaton method s currently appled n the Austran Ozone Montorng Network- vsualsaton nterface. 3.4 Regresson based on other data Abraham and Comre (2004) mprove standard krgng of O 3 felds by usng a hybrd regressonnterpolaton methodology. The mappng of local patterns s enhanced wth pre-nterpolaton regresson modellng of local-scale devaton-from-mean varablty, preservng varaton n the montor data that s ubqutous across the modellng doman (.e., the areal mean). The model s traned on several years of devaton-from-mean hourly O 3 data, and predctor varables are developed usng theoretcally and emprcally derved proxy regresson varables. The regresson model explans a sgnfcant proporton of the varaton n the data (r 2 = 0.54), wth an average error of 7. ppb. When augmented wth the areal mean, the r 2 of the pre-nterpolaton model ncreases to Model resduals are then spatally nterpolated to the extent of the modellng doman. The authors report sgnfcant mprovements to the coeffcent of determnaton when applyng ths method 3.5 Use of statons measurng alternatve ar qualty data Falke and Husar (998b) make use of surrogate data to mprove nterpolaton. Apart from usng drectly measured concentratons of the speces to be nterpolated, other supplementary data can be ntroduced e.g. other pollutants that are measured n a denser montorng network. They ntroduce the followng formulaton, whch assumes a drect relatonshp between the two compounds (Z) and (Y), n Z( s ) wj. Y ( s ) Zˆ( s0 ) j= =. Y ( s n wj j= where Z(s ) s the measured value n measurng ponts s, =,, n Y(s ) s the supplementary value n the measurng ponts s Y(s 0 ) s the supplementary value n the estmated pont s 0 w j s the weght (whch depends on the dstance of ponts s, s j ). 0 ) The new supplementary staton values can consequently be used n any of the nterpolaton methods lsted n Chapter 2. To apply ths method to ar qualty problems the surrogate compound must undergo smlar physcal processes and have a smlar spatal emsson feld as the pollutant beng mapped. An example (Falke and Husar, 998b) are PM 2.5 maps for Eastern U.S. usng PM 2.5 (30 statons) as well as PM 0 (500 statons) and vsblty measurements (280 statons) as supplementary measurements. Ths methodology does not use regresson methods but drectly apples measured PM ratos and vsblty converson factors to determne a rato feld, usng IDW. The authors report an mprovement n the crossvaldaton regresson coeffcent (R 2 ) of 0.59 to 0.69 for the thrd quarter when applyng ths method, fgure 3.4. (3.4) 25

26 Fgure 3.4. Comparson of PM2.5 maps for Eastern Unted States based on montored PM2.5 data (left) and after the ncluson of PM0 measurements as surrogate data (rght).taken from Falke and Husar (998b). 3.6 Satellte data The use of satellte data, whch has a very good spatal coverage but generally poor temporal coverage, has also been suggested and appled n nterpolaton. In general the actual concentratons of pollutants cannot be drectly measured usng satellte remote sensng but other ndcatve parameters can be, for example optcal thckness for aerosol measurements. In an applcaton to Strasbourg (Weber et al., 200) mages from Landsat TM nstrument were used to nterpolate maps of PM 0 based on 3 montorng statons. The process there nvolved:. Identfyng pxels n the mage, usng channels TM-TM5, whch were smlar to the pxels correspondng to measurement stes. 2. Creatng vrtual statons at these smlar pxel postons and provdng these vrtual statons wth PM 0 concentratons based on a regresson relaton between the thermal nfrared Landsat channel TM6 and montored PM 0 values 3. These vrtual statons are then used to create a spatal map, based on the thn plate nterpolaton method. An nterpolaton method smlar to the quadratc beta splne method outlned n Secton 2.2. Ths method s based on prevous studes that ndcate a correlaton between TM6 data and PM measurements. The use and bass for the vrtual statons s not clearly defned. Sarganns et al. (2003) use ground based measurements of partcles and other gases to develop a regresson model for PM concentratons and satellte measured values of aerosol optcal thckness. The resultng map, based on the spatal AOT measurements can then be used as a surrogate for PM concentratons. Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 26

27 4 Interpolaton methods usng montorng and modellng data The correlaton wth ancllary data, such as emssons, precptaton, wnd speed etc. ndcates that there are a number of physcal and chemcal processes governng atmospherc concentratons. Ths s, of course, not surprsng and the next step to mprove nterpolaton s to use model generated concentraton felds that better reflect these processes. Ths can be done n several ways and some of the nterpolaton methods descrbed n chapter 2 have been utlzed n the applcaton of these methods. The use of model felds for nterpolaton reles on a certan amount of confdence n the models ablty to correctly reflect the physcal processes governng ar qualty and, n addton, that nput data such as emssons are representatve of the real world. The model confdence s reflected n an assessment of the uncertanty n model calculatons. Uncertanty n model results should be taken nto account when applyng nterpolaton methods. Ths s however only carred out n more advanced data assmlaton technques, see chapter Combnaton wth models usng nterpolaton of resdual felds To produce European wde maps of pollutant concentratons as accurately as possble, a methodology has been developed to combne both EMEP montorng and modellng data through spatal nterpolaton, to produce geographcally dstrbuted concentratons felds (Tarrason et al., 998). After an ntal selecton process that excludes nconsstent montorng data, a comparson s made between modelled and montored data. The dfference between the modelled and measured values at each measurement pont s taken and ths dfference feld s nterpolated usng radal bass functons, smlar to the quadratc beta-splne method descrbed n secton 2.3, to create a dfference feld. The model and dfference feld are then added together usng a dstance weghtng and a defned regon of nfluence, smlar to the Intutve method descrbed n secton 2.4. Ths nfluence dstance s compound dependent as t concdes to the spatal covarance. For NO 2, the example gven here n fgure 4., the nfluence dstance s 300 km but ths dstance can be as large as 750 km, dependng on the typcal tme scale of the pollutant. The methodology followed s gven below. Let m(x t,y t ), t=,,n be the measurement value at each measurement ste (x t,y t ) and mod(,j), =,, max, j=,,j max be the modelled values. For each measurement pont (x t,y t ), let the dfference between the measured and modelled value n the correspondng grd cell ( t,j t ) be dff(x t,y t ) = m(x t,y t ) - mod( t,j t ). (4.) The dfferences are nterpolated usng radal bass functons f(x,y) = n k k= c ϕ ( d ( x, y) ) k where d k (x,y) s the Eucldean dstance to measurement pont (x k,y k ) and φ(r) = (r). The parameters c k are calculated by solvng the lnear system of equatons obtaned by lettng dff(x t,y t ) = f(x t,y t ), t=,...,n. The functon f s a two dmensonal contnuous functon descrbng the dfference at any pont (x,y) wthn the modelled grd, and equals the dfference between observed and modelled values at all measurement ponts. The new optmal maps are produced by adjustng the model calculatons wth the nterpolated dfference. Close to measurements, larger weght s gven to observed values whereas n regons wth no observatons modelled results are preferred. The actual regon of nfluence of measured values depends on the type of component, n partcular, on ts characterstc transport dstance, and s determned by the range of spatal covarance used n krgng. The new concentraton felds N(x,y) are derved as follows: N(x,y) = f(x,y) + mod(x,y) d N(x,y) = f(x,y) (D comp -d)/(d comp -) + mod(x,y) < d D comp (4.2) N(x,y) = mod(x,y) d > D comp (4.3) 27

28 where d s the dstance to the nearest measurement pont, n grd squares, and D comp defnes the regon of nfluence for a gven component. D comp s set to 2 grd squares (300km) for NO 2, whle for ntrate n precptaton D comp s set to 5 grd squares (EMEP grd squares of 50 x 50 km). No objectve assessment of the method, e.g. cross-valdaton, was carred out usng ths method but the authors conclude the results are mproved. Fgure 4.. Example of nterpolaton feld usng the methodology descrbed n the text above (yearly average concentraton of NO2, 999). Left: the fnal nterpolated feld where the nterpolated dfference feld has been added to the model feld. Rght: the nterpolated dfference feld calculated by subtractng the model feld from the observatons and nterpolatng usng radal bass functons and the ntutve approach. 4.2 Combnaton wth ftted models usng nterpolaton of resdual felds Ths methodology s smlar to the method descrbed n secton 4., but t frst calbrates the model. The model concentraton felds are ftted to measurement data, usng (mostly) lnear regresson between measured data and the model, at the ponts of measurement: M '( s) = c + a. M ( s) (4.4) where M (s) are the values of the ftted model at the pont of grd s, M(s) are the orgnal values of the model at the pont s, c and a are the parameters of lnear regresson. For every measurng pont s,, s N the dfferences are calculated dff ( s ) = Z( s ) M ' ( s ). (4.5) These dfferences are then nterpolated (usng dfferent approaches, see chapter 2) nto the feld of dfferences D. Ths nterpolaton feld s added to the ftted model M to produce the fnal nterpolated feld: ) Z s = M ' s + D (4.6) ( ) ( ) ( ) 0 0 s0 In the Czech Republc ths method s used for the routne mappng of ar-qualty under the name method of the weght feld, see Fala et al. (2000). The mapped area s dvded nto several regons (e.g. by dfferent clmatologcal regme). The ftted model (4.4) s computed for each of these regons separately to gve the lnear regresson parameters for each regon. The nterpolaton of dfferences from (4.5) s carred out usng the modfed verson of IDW (see secton 2..2). A smlar approach s appled by Stedman (2005), based on the methodology descrbed by Abbott and Vncent (999), for rural SO 2 data. They use ordnary krgng as the nterpolaton method. Results from Abbott and Vncent (999) show that the determned regresson factor c=3 ugm -3 and a=., ndcatng that Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 28

29 the model was generally capable of predctng SO 2 concentratons but tests aganst ndependent data sets dd not show a hgh correlaton between these and the resultng nterpolated felds. Horalek et al. (2003) also apply ths method usng krgng, IDW and radal bass functons as the nterpolaton methods for the Czech republc. Results from ths study are shown n fgures 4.2 and 4.3. PM 0 - measurement vs. model PM 0 - measur. vs. model (log. transform.) 80 5 measurement y = 6,44x + 24,28 R 2 = 0,2267 log (measurement) y = 0,2465x + 3,448 R 2 = 0, model log (model) Fgure 4.2. Comparson of modelled verses measured annual mean concentratons of PM 0 for all observatonal stes wthn the Czech Republc, Left: the lnear comparson, rght: the log comparson. The lognormal regresson relaton s used n the ftted model (Horalek et al., 2003). Note the model results are sgnfcantly lower than the observatons. a) b) c) Fgure 4.3. Annual mean PM 0 maps for 200 n the Czech Republc a) made by nterpolaton usng cokrgng wth alttude, b) from the pure unftted model, c) the fnal combned map usng ftted model and cokrgng of resduals wth alttude. Maps are taken from Horalek et al. (2003). 29

30 The major dfference between ths method and that descrbed n secton 4. s that the model s frst ftted, through lnear regresson, to the observatons and t s ths model feld that s used to determne the dfference, or resdual feld, that s nterpolated. Ths frst step can mprove the model feld n a general sense, ncludng overall effects of mssng emssons, ncorrect descrptons of boundary layer parameters or other chemcal and physcal processes. 4.3 Other methods for combnng models and montorng 4.3. FLADIS The German system "FLADIS" (Wegand and Degmann, 2000) computes both short- and long-term concentratons of basc pollutants (ncludng PM 0 ) by a combnaton of nterpolated measured data and the result of both statstcal and physcal models based on emsson, meteorologcal and topographcal data. The combnaton s carred out usng a lnear weghng factor. Ths factor s gven by the coeffcent of determnaton between the measured and modelled data at the measurng ponts such that when the coeffcent of determnaton s hgh, based on all the measurement stes, then the model feld s more heavly weghted. When the coeffcent of determnaton s low then the nterpolated observatonal feld s more heavly weghted. The system has been used to generate NO 2 and O 3 maps n a number of German states ArQUIS A now-cast system, developed by NILU and appled n Hafa, Israel, also combnes observatons and dsperson model calculatons to mprove concentraton felds that are produced and dsplayed on an hourly bass for the cty of Hafa (Denby and Flcsten, 2005). The methodology employed s dependent on the accuracy requrements defned by the user. In ths case calculated concentraton felds are consdered acceptable f the model concentratons, wthn a km radus of the montorng stes, are wthn ±25% of the observed values. Based on these requrements the assmlaton of observatons s carred out n a 3 step process.. For each ste the model concentraton that s closest to the measured value, wthn a km radus of the montorng ste, s chosen. 2. Ths value s used as bass for the model feld adjustment. Ths assumes that the model feld can be adjusted by a constant scalng factor that s derved by mnmsaton of the root mean square error of modelled and measured values. 3. A fnal local adjustment, based on a local weghtng around the montorng stes, s carred out to nsure complance wth the specfed acceptance crtera. The most mportant aspect of the assmlaton s step 2, whch adjusts the model felds accordng to the best ft wth observatons. The applcaton of ths method suggests that certan parameters n the model, or model nput data, have an overall effect on the model regon, whch s 20 x 20 km 2. Such factors as wnd speed, mxng heght and turbulent dffuson wll all have a general nfluence on the concentraton levels. The resultng felds for dspersed pollutants, e.g. traffc related compounds, are mproved wth assmlaton but the method s less successful when appled to plumes, e.g. SO 2, as the montorng network, even wth 5 statons, s not suffcently dense to detect and adjust plume concentratons Bayesan approach Brabec (2002) ntroduced an approach for combnng measured and modelled data, whch dstngushed hard (measured) and soft (modelled) data and works wth the spatal structure of both of them. Ths approach s more sophstcated and also more computatonal demandng than the other methods descrbed here. It s based on the Bayesan statstcs, see e.g. Gelman et al. (997). Ths methodology s used n order to jon the measured data (so-called pror) and the result of the model (apror nformaton) nto the updated verson of pror, so-called posteror. Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 30

31 4.3.4 Background estmaton, RIVM In the Netherlands (EC workng group, 2000) the unknown background contrbuton s estmated usng modelled and observed felds. The background feld of PM 0 s estmated by takng the dfference between the measured and modelled data at every rural background staton. These resduals, whch ndcate a mssng background contrbuton, are spatally nterpolated nto a 5x5 km grd. Ths nterpolated feld s then added nto the modelled feld, thereby gvng an mproved PM 0 feld. The urban statons are used for verfcaton of the model. 3

32 Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 32

33 5 Assmlaton methods usng montorng and models Data assmlaton, n order to provde spatal dstrbuton of pollutants, takes the nterpolaton of observatonal data wth model results to the next logcal level. That s that observatonal data s used to nfluence the model results, allowng for uncertantes n nput data and process representaton, to obtan the best physcal representaton of concentraton felds. The prevously descrbed assmlaton methods, chapters 2 to 4, are by no means consstent wth physcal prncples, e.g. ndependently nterpolated ozone and NO 2 felds may not be physcally realsable under meteorologcal and emsson condtons known to exst. Data assmlaton often requres the use of advanced models, especally on the regonal level where chemstry s most mportant, as well as substantal computer resources. Meteorology s one of the scentfc felds that has benefted enormously from the use of data assmlaton. The subject area of assmlaton n atmospherc chemstry modellng s large and ever ncreasng. In ths revew a summary of current methods and applcatons s provded. 5. Optmal Interpolaton The optmal nterpolaton (OI) method (Gandn, 963) has been qute popular for many years, see secton Appled to a regonal scale grd model the method s based on fndng an optmal weght-matrx W for whch the nnovaton vector y o H(x f ) (y o s the observatonal vector and H s the model operator) wll be multpled and added to the forecasted model state x f n order to form a new assmlated model state x a. The weghts used n the method are optmal n the sense that the predcton varance s mnmzed usng these weghts. The technque reles upon the user to specfy, n advance, covarances between the model calculated concentratons (model errors between dfferent grd cells), and to specfy the observatonal errors nvolved. The model error covarances are usually specfed n the method by usng one or more spatal correlaton or covarance functons, modellng the covarances horzontally and vertcally n the grd as a functon of dstance. The OI-method s formally equvalent to the 3D-Var and PSAS data assmlaton methods (see secton 5.3 below), both of whch are now consdered to be more modern and flexble than the OI approach (Kalnay, 2003). Several numercal weather predcton (NWP) centres have n recent years started to use these other methods nstead of OI due to ther greater flexblty and ease of operaton. However, the OI-method may be of use n order to perform regonal scale data assmlaton. Applcaton of the method s relatvely straghtforward snce, once the model error covarances have been determned, only a smple system of lnear equatons needs to be solved. The assmlaton of observed satellte aerosol depth nto regonal models of PM s one of the areas where optmal nterpolaton s currently used. Optmal nterpolaton of satellte data has been appled to the MATCH (Collns et al., 200) and TM (Verver et al., 2002) models. Both the MATCH and TM models nclude sulphate, sol dust, black and organc carbon and sea salt aerosols n ther partcle descrpton. The assmlaton procedure ntroduces extra unknown sources and snks of aerosol mass n the model. These are then used to adjust the model feld for an optmal ft to observatons. The scheme has 7 tuneable parameters. 5.2 Varatonal methods Varatonal methods are based on the mnmzaton of a cost functon for the dfference between model concentratons and observatons (Lorenc, 986). One of the dsadvantages of these technques s the requrement of developng a so-called adjont verson of the model, whch s a complcated task for chemstry transport models. In contrast to sequental methods the model uncertantes must be specfed beforehand, whch s not straghtforward. In addton, the varatonal methods are less flexble than ensemble methods, snce the model adjont model needs to be revsed every tme the model s changed or mproved. However, the computatonal effort s only a few tmes the normal effort to run the model, and thus much lower than usng the ensemble Kalman flterng technques, see secton

34 The applcaton of varatonal technques n a chemstry transport model was poneered by Elbern et al. (997; 999; 2000). These authors mplemented a 4D-var smoother technque for the EURAD model. The system was used to assmlate ozone concentratons durng the frst hours of an epsode over central Europe. The assmlated state was used to perform forecast smulatons. The authors showed that the system was able to reduce the absolute dfference between observed and modelled concentraton felds n the assmlaton wndow consderably. The nfluence of the assmlated ntal state dmnshes after about hours. At that tme the dfference between the frst guess of the free runnng model, usng a background ozone concentraton feld as frst guess, and the forecast smulaton s very small. Hence, even the uncertantes n the model descrpton of a hgh-level chemstry transport model lke EURAD confne the perod n whch data assmlaton mproves forecasts to -2 days. 5.3 Kalman flter methods Kalman flter (KF) methods represent a large class of dfferent technques for performng data assmlaton. The man dea of the KF-methods s an automatc estmaton and updatng of the model error covarance matrx B from one tme step to the next usng the model operator (the regonal scale model) tself. The assumptons n the methods are essentally the same as n the varatonal methods (3D-Var, 4D-Var, PSAS), namely that the model state and observaton errors (uncertantes) nvolved must have a Gaussan dstrbuton. The orgnal Kalman flter (Kalman, 960) was proposed for a pure lnear system of evoluton. Later t was extended to non-lnear systems, and s then known as the Extended Kalman flter. The Kalman flter n ts orgnal form (whether lnear or extended) has too large a computatonal complexty to be mplemented n practce, so smplfcatons of the orgnal flter are necessary. The frst applcaton of Kalman flterng technques was n oceanography. The successful applcaton of ths methodology led to the ntroducton of smlar technques for ar polluton studes at the regonal scale. The Dutch research communty has been actve n the applcaton of Kalman flterng to ozone formaton. Several studes have been performed usng the regonal scale chemstry transport models LOTOS (Van Loon et al., 2000; Segers et al., 997, 2000; Segers, 2002), and EUROS (Hanea et al., 2004). Further, the number of applcatons of Kalman flterng technques s growng and applcatons to other models start to appear (Eben et al., 2005). There are a number of varatons on Kalman flters of whch the ensemble Kalman flter and the Reduced Rank Kalman flter have been the most wdely appled Ensemble Kalman flters The ensemble Kalman flter method (EnKF) was orgnally proposed n (Evensen, 994). The dea behnd the method s to apply an ensemble of N model states {x (), =,,N} to represent the model error covarance matrx. The method thus avods completely the complcatons nvolved n updatng ths matrx. Instead one must run the regonal scale model N tmes n order to propagate the ensemble of model states {x () } to the next tme step usng Monte Carlo random draw procedures to smulate model errors. The soluton converges towards the exact soluton of the Extended Kalman flter when N ncreases. Hanea et al. (2004) mplemented and compared two Kalman flterng technques, ensemble (EnKF) and Reduced Rank (RRSQRT-KF) to assmlate ozone concentratons over Europe. For ths purpose EMEP and ArBase observatons were dvded n assmlaton and valdaton montorng data. Kalman flterng requres applyng random nose to key model parameters to buld an ensemble of smulatons. In ths study, nose was appled to the emssons of NO x and VOC as well as the NO 2 photolyss rate. An extensve descrpton of these data assmlaton methods, the chemstry transport model and expermental setup can be found n Hanea et al. (2004) and the references theren. Interpolaton and assmlaton methods for European scale ar qualty assessment and mappng Part I: Revew and recommendatons 34

35 Fgure 5.. Comparson of the performance of the ensemble Kalman flter methods EnKF and RRSQRT-KF usng dfferent numbers of modes/ensembles. The average absolute resduals AARs (ug/m3) are averaged over all the valdaton statons n Europe. Taken from Hanea et al. (2004). The results of the smulatons, fgure 5., showed that the assmlaton system s able to sgnfcantly reduce the average absolute resdual (AAR), equvalent to MAE (see chapter 7), between the assmlated and observed concentratons at the valdaton statons. Fgure shows the dependence of the mprovement n AAR as a functon of the number of ensemble members, and therewth the ncrease n computatonal tme compared to a sngle smulaton. For the Ensemble Kalman flter the resduals decrease wth the number of ensemble members or modes. For the reduced rank square root flter (RRSQRT-KF), whch constantly reduces the number of modes to those wth the most nformaton, the mprovement s hgher than the EnKF setup wth a lmted number of modes. However, one stll needs a consderable number of modes that ncrease the computatonal tme consderably. The authors show that adjustment of the emssons strengths gave the best mprovement of the modelled ozone concentratons. Moreover, they also showed that the area of nfluence of an observaton staton was small, whch can be explaned by the local characterstcs of the photochemcal equlbrum Reduced Rank Kalman flters The dea behnd reduced rank Kalman flter methods (Verlaan and Heemnk, 997) s to use a low-rank square-root approxmaton of the model error covarance matrx, whch s easer to handle and update than the orgnal matrx. The low-rank approxmaton s based on representng only the N < n man egenvalues of the covarance matrx, and assume that the model s lnear enough so that the assocated egenvectors are suffcent to represent the probablty dstrbuton. Usng ths method the regonal scale model must be run N tmes nstead of ~n. In practcal applcatons ths represents a huge savng (as compared wth the Extended Kalman flter), snce N s usually much smaller than n. The mplementaton of the technque s, however, less straghtforward than the EnKF-method. Many dfferent varants of reduced rank Kalman flters exsts, such as the RRSQRT (Verlaan and Heemnk, 997), POEnKF and COFFEE (Heemnk et al., 200), SEEK and SEIK (Pham et al., 998) flters. Some of these are also combnatons wth the EnKF-method. They dffer bascally n the way the ensemble s beng created and mantaned. The RRSQRT Kalman flter has been appled n data assmlaton of tropospherc ozone n the European regonal scale atmospherc transport chemstry model LOTOS (Segers, 2002). Ozone observatons were used n order to reduce the uncertanty of emssons and depostons n ths model. In ths study the RRSQRT Kalman flter was found to work well and was also found to be somewhat faster than the (orgnal) Ensemble Kalman flter method EnKF. One of the man practcal advantages of the EnKF and the reduced rank Kalman flter methods s the separaton of the model mplementaton and flter algorthm. Nether a complcated tangent lnear model nor an adjont operator s requred (as n the 4D-Var approach) whch makes the codng of the method smpler. For all the ensemble based methods the choce of ensemble sze N s very mportant. It s dffcult to say n 35