inverse air pollution modelling of urban-scale carbon

8/14/2019 Inverse Air Pollution Modelling of Urban-scale Carbon

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49 8 M. MULHOLLAND and J. H. SEINFELDlish emission rates at every point in a solution grid.Keidser and Rosb jerg (1991 ) consider four inverseapproac hes for simultaneously identifying the steady-state water flow pattern in aquifers, and the sourcesand diffusivity o f contaminants in this flow. Para-meters w ere adjusted to minimise the sum of squarederrors between observations of pressure and concen-tration, and mode1 predictions. Wagner (1992 ) treatsthe same problem by maximising a likelihood func-tion, instead of minimising the sum of squared errorsdirectly.

The least-squares fitting procedures above attemptto account for all measurements, including time vari-ations, simultaneously. There exists another class ofmethod s where the measurement data set is broken upinto relatively manag eable pieces , by sepa rate fittingat a series of times. Use o f a Kalman filter allows oneto carry a portion of the information from each fitforward to the next fit, so that the rates of variation ofthe parameters and th e goodness of fit can be con-trolled. This is especially useful where param eters areslowly varying, as in the case of atmospheric pollutantsources.

Mulholland (1989 ) inferred SO , emissions at 9pow er stations from concentrations at 8 receptors inthe Eastern Transvaal Highveld. A pseudosp ectraladvection-diffusion scheme allowed explicit calcu-lation of a matrix for the contribution of each sourceto each receptor. A discrete K alman filter was thenused to invert this calculation, so that the individualsources could be estimated, based on the time series ofconcentration measurements. Hartley and Prinn(1993 ) used Golombek and Prinns (1986 ) global,three-dimensional model for the transport and reac-tion of CFCI,, CF ,Cl,, CH ,CCl,, Ccl,, and N,O , asthe basis for a similar K alman filter inversion. CFCl,was used as a test tracer, since its sources are relativelywell known. Here the model equations could not besubstituted directly into the filter, but rather a Jaco-bian matrix, of the variation at each receptor for anincrement in each source, was evaluated in parallel ateach time-step, to provide the necessary linear model.Brown (1993 ) used Tungs (198 2) global, two-dimen-sional model (latitude, time) for the transport andreaction of CFC -11, methylchloroform and methane,to perform a simple inversion procedure in whichcurrent sou rce estimates are incremented one-by-oneby a fixed amount, and the mode1 is run forw ard for afixed time period for each such source chan ge. Forequal numbers of receptors and sources, the resultantJacobian matrix is square, and can be inverted andmultiplied by the desired changes in concentrationpredictions, to give changes in the sources that willmatch new concentration observations.

The above metho ds of Mulholland (1989 ), Hartleyand Prinn (1993 ) and Brown (1993 ) all have a funda-mental difficulty in dealing with the distributed natureof atmosph eric transport. Mulholland simply ignoresit-the time-variant plumes from the various sourcesare superpo sed (using filter weights) to match observa-

tions at any instant. This is not th e same as altering thesource rate at the same time, as such a change wouldrequire time to propag ate through the field. Hartleyand Prinn (1993 ) appear to choo se the time-step fortheir solution a little longer than the longest source-receptor lag (3 months), so that a non-zero factor willbe available in the Jacobian. Thus the receptor willjust be beginning to respond to the source at the end ofthis step. Thoug h the proportionality will be correctfor this precise transport time, filter gains from suchterm s are likely to be high , risking instability. Eventhough such a term m ay suggest a change in theemission rate, the change, when implemented, will be 3months late. Strangely, Brown s (1993 ) simple proced-ure need not suffer this disadvantage, as one canalways iterate until the specified sources cause thepresent concentration prediction to match the presentobservation. How ever, it is not just the source valuesduring the last time-step that determine present con-centrations, but values during all preceding steps.Thus th e immediate past source values can be ex-pected to be going through cycles of correction forprevious values. Although Hartley and Prinn (19 93)estimate their CFCl, sources as time variable, they arereally seeking convergence, since these emissions arebelieved to be steady.

The goals of the present study are twofold. First, wedevelop a theoretical appro ach to inverse air pollutionmodelling that, while similar to that of Hartley andPrinn, and B rown, d iffers in that w e can control theextent to which the estimated emission distribution isallowed to deviate from a base-case. In contrast,Hartley and Prinn (1 993) only dampened the rate ofvariation of their estimate. Regression for adjustmentfactors in the present work, rather than absoluteemissions, automatically normalises parameters, sim-plifying the tuning of the filter. As a second goal, weapply the technique of inverse modelling to one ofthe major urban-scale air pollution models to estimatethe spatial and temporal distribution of CO sourcesin the South Coast Air Basin of California. The spatialand temporal variations of CO emissions in majorurban areas are provided by appropriate air pollutioncontrol agencies in the format of a mobile sourceinventory. When performing inverse modelling, thisCO emission inventory can be considered as thestarting point from wh ich to determine how the COsource distribution needs to be altered s o that ob-served and predicted C O concentrations at the moni-toring sites are matched as closely as possible. Indeed,there is evidence that CO emissions from motorvehicles, particularly in the South Coas t Air Basin,have been historically underestimated (Ingalls et al.,1989; Pierson et al., 1990; Fujita et al., 1992). Theinverse modelling approa ch provides a way to addressthe question-How should the CO emission inventorybe adjusted to provide the best match of observedand ptedicted CO concentrations? The requiredadjustment then is indicative of the degree of under-(or over-) estimation inherent in the inventory.


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Inverse air pollution modelling 499

2. RECURSIVE ESTIMATION OF THE GROUND-LEVELSOIJRCE DISTRIBUTION

Consider an Z x J horizontal grid over which weare able to compu te the solution for atmospherictransport of a pollutant released at ground level, for atime-varying source distribution pattern. W e wish toestimate th e spatia.lly and temporally varying sou rcedistribution that optimally accounts fo r time-variantconcentration observations at a limited number ofpoints (m onitoring stations) in the grid. We considerhere a non-reactive, conserved pollutant.

Divide the ground-level horizontal domain of thesolution volume up into D - 2 separate domains thattogether comprise the complete region. In principle,the largest number of separate source domains issimply Z x J, the total num ber of horizontal grid cellsused for the air pollution model. In practice, it is usefulto define a somew hat smaller number of grid domainsover which the source emissions are to be estimated.The D - 2 domains are chosen such that each encom-passes a particular distribution of ground-level sour-ces. Consider that we exercise the model individuallyfor each of the D - 2 source domains using a zero initialcondition, and zerfo boundary conditions. No w findone extra solution starting with the initial conditionalone and zero emissions, and another extra solutionfor the time-variant boundary condition alone andzero emissions. For a conservative species, wheresuperposition is valid, we can then add th ese Dsolutions together at any time, to build the completeconcentration distribution.

Now consider that th e solution is to be calculatedover a total of T time-steps. If we consider the emis-sion rate in each source domain to be constant duringeach time-interval, we are also able to discretise th esolution with respect to time. For a nominal releaserate distribution qi j dl, over the area of domain d duringinterval t, and zero emission at all other times, thesingle source dom.ain d produces the concentrationCdrPijr t ground position (i, j) at observation time t.Let the actual emlission for domain d during intervalt be a factor& . multiplying the nominal distribution inthis domain at this time. Then the concentration atpoint (i,j) at time t arising from all emissions will be

If x, is a vecto r containing all grid concentrations attime t, and f a vector of all d = 1,2, . . . D domainfactors for all emission times t = 1,2, . . . . T, then amore convenient notation for equation (1) is

Xt = C,f.Now consider that observations of concentration at alimited number of grid points A4 are available in ameasurement vector xrnt. The equivalent A4 predic-tions can be selected from xt using a matrix B (of Osand ls). Thus the error between observations and

predictions at time t is given bye, = xmr - BC ,f.

Generally B will have few rows (e.g. M = 30) andmany columns (e.g. Z x J = 2400 ). The m atrix C, (withZ x J rows and D x T columns) is never handled com-putationally on its own, but rather as BC ,, which h asa more manageable dimension (A4 rows x D x T col-umns). C, is only invoked once the overall concentra-tion distribution is required. O ne possible solution forthe adjustment factors f might arise by minimising theweighted sum of square errors:

@ = (x,, - BC,f)*W& - BC,f),where W is diagonal with chosen weights. If we havesome confidence in the nominal emission distribution,we may want to dampen deviations of elements of ffrom unity using an additional weighted penalisation:

@ = (A,, - BC,f)*W(X,r - Kf)+ (f - l)*V(f - l),

where V is diagonal and 1 is an appropriate vector of1s. The desired factors f are obtained as

f = (C;B*WBC , + V)-(C;B*WX,,,, t VI). (2)How ever, the dimension of x,,,~ s likely to be small(e.g. 30) whilst that of f is likely to be large (e.g. 30domains x 72 times, making V 2160 x 2160). There isreally too little information to fix f, which, after all,influences all prediction times. We bring m ore in-formation to bear by augmenting BC, as BC= BC,IBC21BC31... and x,,,~as xm =x~~Ix&~~~~~~using observations at all times. This increases the

dimension of W from A4 x M to (M x T) x (M x T), e.g.2160 x 2160.The spatially distributed nature of the system re-

quires one to consider every emission time and everyobservation time simultaneously. Notice that thisideal solution properly accounts for transport lagsbetween sources and receptors. For three-dimen-sional, time-varying air pollution systems, how ever,more computationally tractable m ethods have had tobe developed, involving a degree of comprom ise.

Let us start with the assumption that the importanttransport times in the system are small comp ared withthe time-scale of variation of the source domainfactors f. We are hoping that th e time taken for achange in the source rate to be felt at a receptor issmall, comp ared with the time periods over which w eneed to vary f significantly to correct the emissions. Ifthis is the case, we do not have to distinguish betweenthe time t at which the factor is applied to its domain,and the time t of an observation or prediction. Thegreat benefit of this is that we no longer need acomplete time-series solution, for each time-intervalsource pu lse, for each domain. Rather, we can simplyrun solutions for each source domain parallel in time,and superpose these distributions with appropriateweighting factors f at any time instant, where the


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500 M. MULHOLLAND and J. H. SEINFELDfactors are designed to achieve optimal fitting at theobservation points. Equation (1) becomes

Xi j r = 2 di jr fd td=l

and the appropriate form of equation (2) isf , = (C;BTWB C, + V) -l (CTB TW ~, , , t + Vl), (3 )

where we see that the D domain factors are updatedat each time-step without regard to preceding valuesof f. W will be M x M and V will be D x D. Fo rconstant f w e could augment B C , and xmt as suggestedabove.

Inherent in equation (3) is some loss of information.If we really believe that f varies slowly, then precedingcomparisons of predictions C,_ i, 2, 3 ,,, and observa-tions lm r_ i, 2, 3 ,,, could be used to contribute to thepresent estimate of fp We develop below a method ofrecursive least-squares estimation (RLS ) based on thediscrete Kalman filter (Assefi, 1979; Kalman, 1960 ), togive a smoothly varying f.

After initialising a covarian ce matrix P ,, usuallyas diagonal with small values on the diagonal, wecalculate a gain matrix at each time-step as

K, = P ,CTB T(BC,P ,C;BT + W- l )- . (4 )Consider that an adjustment of f arises at each timestep by applying this gain to the deviation betweenpresent observations and predictions,

f,+ 1 = f, + K,knt - BW. (5 )and the covariance matrix is updated as

P +l=(I-K,BC,)P,+V- (6 )after which equations (4)-(6) can be repeated for thenext time-step. K alman (1960 ) show s that if W-lcontains the covariance of errors in the measurementx,,,, and V-I the covariance of errors in f, to beexpected from modelling, then the estimate (5) of fat each time-step will be optimal (for normally distri-buted, zero-mean errors). Since the covariances be-tween different elements of these vectors are unknown,we approximate W - and V- as diagonal, with anestimate of the variance of error of each vector elementon the axis. Thus a low value in the first element ofW - causes the RLS filter (4)-(6) to track the firstmeasurement closely, because of the implied con-fidence in the measurement. This is analogous to theinverse high value in W , which would cause the least-squares fit in (3) to pay special attention to the firstmeasurement. Conversely, low values in V- cause theassociated element of f to vary less as the filterattempts to fit measurements.The relationship between the RL S procedure(4)-(6), and the LS procedu re (3), can be understood asfollows. If new values of P are estimated at each time-step by equation (6), using a zero matrix as theprevious covariance, and if the previous value of f istaken as 1 in equation (5) on each time-step, then

equation (3) results. Clearly equation (3) ignores allprevious behaviour. One benefit of using the RL Sprocedure, for example, is that the estimate of a sourcedomain factor is left at the previously supported value,if that domain loses correlation with receptors. Thisoccurs when a plume from the domain moves awayfrom receptors, and wo uld cause th e LS solution (3) torevert to f= 1.

Once f, is evaluated, we are able to generate animproved overall concentration distribution as

XI = C,f,and the improved emission distribution is

Qt=fTst>which transfers q, the nominal emissions at the groupsof grid points comprising each domain d, with appro-priate adjustment by an element off for each domain,into the adjusted emission grid Q. In solving for thelimited number of adjustment factors (D), rather thanfor all I x J em issions in Q, we have been able toreduce the size of the estimation problem, yet stillbenefit from the nominal (first estimate) spatial andtemporal structure within each domain.

3 . IN CLU S IO N O F TH E K ALMA N F ILTER IN TH E CITP H O T O C H E M I C A L A I R S H E D M O D E L F O R T H E

LO S A N G ELES BAS IN

In the Southern California Air Quality Study(SCA QS), ambient air quality data w ere collected overa 250 km area of the South Coast Air Basin, in anintensive 3-day campaign from 27 August to 29 Aug-ust 19 87 (Lawson , 1990 ). These data included continu-ous measurements of 9 species at 30 sites, detailedmeteorological measurements, and mobile andstationary source emissions data for construction ofthe source emissions inventories. The emission in-ventory for the South Coast Air Basin during the 1987SCAQ S study was provided by the State of CaliforniaAir Resources Boar d (Wagner and Allen, 1990). Mo-bile source em issions were based on a travel dem andmodel and the EMFA C 7E emission factor model(Yotter and Wa de, 1989 ). There is evidence, based onmeasurements in the Van Nuys Tunnel, that themobile source emissions of CO (and hydrocarbon s)have been underestimated (Ingalls et al., 1989; Piersonet al., 1990 ; Fujita et al., 1992). While efforts areunderway to improve the accuracy of mobile sou rceemission inventories (Coordinating Research Council,1992 ), an approa ch that h as not heretofore beenattempted is to estimate the emission inventory thatleads to the optimal fit of observed and p redictedambient concentrations. That is the goal of this work .The CIT photoch emical airshed m odel is an urban-scale model that ha s been applied to simulate airpollution primarily over the South C oast Air Basin(McRae et al., 1982a,b; McRae and Seinfeld, 1983;Harley et al., 1993 a, b). The m odel treats the transport,


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Inverse air pollution modelling 501diffusion, and reaction of 35 species based on solutionof the atmosph eric diffusion equation:

$+V.(uc,)=V.(KVc,)+R,(c, )..., c,), k=l,2, . ..) II.

The model is solved in an 80 (x: W-E) by 30 (y: S-N)grid with 5 km horizontal grid spacing. For modelinput, a divergence-free, three-dimensional wind-fieldis constructed, and this, together with all other ra wdata, is stored at 1 h intervals in two- or three dimen-sional grids as appropriate. Height-dependent para-meters are stored at the five unequal vertical intervalsof the solution grid, up to 1100 m above the localsurface (Harley et al., 1993a).

A conve x solution region is initially selected in the80 x 30 grid, then the model equations are solvedsimultaneously within it using the metho d of operatorsplitting. On each time-step, first x-advection/diffu-sion is applied, then y-advection/diffusion. The ver-tical advection-d iffusion is done in a comb ined in-tegration with reaction and emission input. Theadvection scheme employs a finite-element Galerkinmetho d, followed br nonlinear filtering.

In simulations of episode periods during theSCAQS study, Harley et al. (1993 a, b) considered anoverall tripling of motor vehicle CO and hyd rocarbonemissions to improve predictions of ozone. (Hydrocar-bons, of course, are by far the predominant contri-butor to ozone, when com pared to CO , so the triplingof emission values to improve ozone predictions isrelated almost entirely to the effect of the organicemissions.) The consumption of CO by photochem icalreactions over the one to two day time scale isrelatively small, so that CO is in itself an ideal tracer towhich to apply inverse modelling. In the work dis-cussed below, we use the CO m easurements to infernot just an overall inlventory adju stment, but one thatvaries with time and location.

The CIT model is structured such that the advec-tion/diffusion solutions for the 35 chemical speciesproceed in parallel, with concentrations only beingexchanged once each time-step for the purpose ofcalculating the reaction rates. Since chemistry is ab-sent in our CO transport study, the structure of themodel is available for up to 3 5 (or more if needed)pseduo-species, one for each of the D source domains.Thus th e domain solutions C, are all available at anytime t, as input to the Kalman filter, and for predictionof an adjusted concentration distribution by theweighted superposition

X, = C,f,

4 . T E S T I D E N T I F I C AT I O N O F T W O A R T I F I C I ALP O I N T S O U R C E S

The Kalman filter was integrated with the CITmodel code, drawing from the separate domain

solutions at each time-step to construct C,. An initialtest aimed to establish tuning p arameters, and showthat the filter could track variations in two artificialground-level point sources, located in the Los Angelesbasin, under the influence of the meteorology of 27-29August 1987 . For this test, 2 point sources (separatedby 50 km) and 10 receptors were randomly located inthe modelling region, with the receptors generallydownwind, spread over an 80 km x 80 km region. TheCIT m odel w as run w ith a +_50 % step in Source 1 at7 am on the first day, Thursday, and a - 50% step inSource 2 at 9 am, in order to generate measurementdata sets for Recepto rs l-10. The Kalman filter wasthen run using this measurement set, but with thenominal rate in each of these two dom ains set steadilyto the initial rate. In Fig. 1 we see factorsf,, and_&,attempting to follow the ideal + 50% and -50%steps at these times. High values on the axis of theW m atrix (W /V= lOO), will reduce deviationsfrom observations, but cause fi, and fi, to varysharply. Reducing the ratios to W/V =O.Ol givessmoo ther variations infit and fZt, but inferior trackingof observations.In these point source studies, it was found that thebest filter performance was achieved using adaptivetuning of the W weights. Thus if the observation atReceptor 1 dropped from 5 to 2 ppm, the correspond-ing weight wi 1would be increased from (4)) o ($, forV = I. (This means th at the corresponding error vari-ancein W- would be decreased from 5* to 2.) In thisway, the fractional deviations from observations aredistributed uniformly across all observations. Thesereal-time changes in W- had no effect on stability.This point-source identification is a severe test, owingto the high concentration gradients in the plumescrossing th e receptor sites. The subsequent appli-cations below all apply to more diffuse area sources,for which fixed weights W = I and V = I worked well.

5. A N A LY S IS O F BAS ECA S E CIT MO D EL P R ED ICTIO N SF O R C O

The original base-case ground-level CO emissioninventory referred to here is that provided by theCalifornia Air Resou rces Board (W agner and Allen,1990). Elevated sources of CO were ignored owing totheir minor contribution, as was the effect of consump-tion of CO by reaction, though we note that m odellinearity, and thus the ability to superpose source-domain solutions, would be preserved for first-orderreactions. Figure 2 show s the locations of some of theCO monitoring stations in the South C oast Air Basin,along with several landmarks.When the CIT model is run fo r the data set of 27-29August 1987 , it produces predictions which comparewith corresponding observations of CO as in Fig. 3.Note that virtually all CO measurements were re-ported at the discrete nearest values 0, 1,2,3,4,5 ppm.If predictions of the original observations were exact,


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50 2 M . MULHOLLA ND and J. H. SEINFELD1 7 5

1 5 0

z1 2 5

I !' 0f 1 0 0

cr0; 7 5

I T

Ea l 5 0Et;37 25

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filter-identified f2

0 2 4 4 0 7 2Thursday Friday Saturday

Time [hrlFig. 1. Identification of a + 50% step in Source 1 at 7 am on Thursd ay and a - 50% step in Source2 at 9 am on Thursday.

we would generally expect symmetric positive andnegative deviations of predictions about each suchobservation value. How ever, for the reasons following,some skewn ess is expected in this scatter. The numberof observations corresponding to each discrete obser-vation value is indicated on the lower axis of Fig. 3.The number of predictions in each such category wasfound to be skewed across the range in the same wayas the observation number density. Thus direct aver-ages of the predictions within each category above1 ppm , e.g. obtained by summing the predictionscorresponding to all observations of 2 ppm, are likelyto be low ( < 2 ppm), and the average of predictionscorresponding to the observation class 0 ppm will behigh unavoidably, since no negative predictions arepossible. The graph shown, and all such comparisonspresented, has been corrected for the skew numberdensity by weighting predictions in each of the cat-egories l-5 pp m by the inverse observation numberdensity ac ross the category. A smoo th curve followingthe discrete observation density distribution gave thenumber of observations corresponding, to e.g. 1.5 ppm

(690) and 2.5 ppm (380). For the 2 ppm observation,predictions lying in the 1.5-2.5 ppm interval wereweighted linearly between l/690 and l/380, which alsoacted as saturation values for predictions falling out-side the range. In this way, the number bias in thepredictions of each category would be removed , p ro-vided the prediction number density w as similar to theobservation number density, which seems likely.For predictions at the observation 0 ppm , an as-sumption of symmetry in the prediction distributiongenerally fixes the mean at 0 ppm. How ever, in Fig. 3,the point of symmetry for predictions is fixed at0.2 ppm, this being the minimum achievable pre-diction due to a fixed boundary condition of 0.2 ppmin this original solution. For comparison, the uncor-rected m ean is also shown in such plots.

An alternative representation of the predicted vsobserved CO concentrations is given in Fig. 4. Here azero-mean, normally distributed random variable,with a standard deviation of approximately 0.4 ppm,has been added to every observation value, simulatingthe effect of errors resulting from representing the


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1011121314

AnaheimAzusaBurbankLos AngelesClaremontCosta MesaEl RioFontanaHawthorneHesperiaLa HabraLong BeachNorth LongLynwood

15 Pasadena16 Pica Rivera17 Palm Springsla Pomona19 Reseda20 Riverside21 Rubidoux22 Simi Valley23 San Bernardino24 El Toro25 Upland26 Whittier

Beach 27 West Los Angeles

Fig. 2. Soulth Coast Air Basin showing the locations of the 27 sites atmonitored. which ambient CO was

5 ,

4 --

0 1 2 3 4 5456 points 722 points 587 points 1 4 7 points 27 points 5 points

CO Observation [ppm]Fig. 3. Original CIT model predictions for CO concentrations vs observations.


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50 4 M. MULHOLLAN D and J. H. SEINFELD

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CO Observation (+random) [ppm]Fig. 4. Original CIT mod el predictions for CO vs observations. A zero-mean normal randomvariable (S D = 0.4 ppm) has been added to observations to reveal the liumber density of each discreteobservation 0, 1,2,3,4,5ppm.

observed concentrations only in discrete increments of1 ppm. The resulting horizontal expansion of theO-5 ppm observation lines reveals the number density.Quite clearly no predictions reach below 0.2 ppm.

A line following the mean prediction in Fig. 3 isnearly straight, w ith an intercept of 0.2-0.4ppm . Ifour view is that p redictive errors result from erroneoussource magnitudes within the modelled area, then acorrection factor of 3.0 seems suitable, particularly ifthe offset is first reduced.The boundary condition of0.2 ppm CO used by H arley et al. (,1993a), and in thepredictions shown in Figs 3 and 4, is largely respons-ible for this offset. Could it be that the inflow of COfrom outside the region has little impact? Upwindmeasurements on San Nicolas Island were actuallylower at 0.11 ppm CO during the period 27-29 August1987 (Lurmann and Main, 1992 ). For the purpose ofthis study, with measurements in the range O-5 ppm,little error w as incurred by ignoring this boundarycondition. It is nevertheless recognised that the 23% ofall measurements registered at 0 ppm may well repres-ent 0.2 ppm in reality, bearing in mind the limitedmeasurement accuracy.

In the formulation of the Kalman filter in Section 2,one extra adjustment factor was allowed to weight thesolution for the initial condition alone, and anoth er

extra factor was provided to weight the solution forthe boundary condition alone. These factors wereinitialised at 1 and 0 (effectively in vector l) , and fixedthere by choice of high penalties in the move suppres-sion matrix V. Thus the boundary condition contribu-tion w as forced to zero, whilst the initial conditioncould not b e altered. There is no justification foraltering an initial condition contribution as the solu-tion proceeds.

6 . I D E N T I F I C AT I O N O F A N I M P R O V E D C O E M I S S I O ND I S T R I B U T I O N F O R T H E S O U T H C O A S T A I R B AS I N

The CO concentration predictions above arise froman open-loop prediction using the base-case emissioninventory. The conditions during the period 27-29August 1987 have not resulted in high ambient COmeasurements, so that even rounding errors off0.5 ppm have become significant in comparisonwith the measurement range of O-5 ppm . Similarerrors in the representativeness of measurementsmight arise from mon itors being close to, or distantfrom, local sources like freeways. If we assume thatsuch errors are randomly distributed with zero mean,


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Inverse a.ir pollution modelling 505we nevertheless have a valid data set with which todemonstrate the recursive technique. Th e recursionitself entails closing the loop, so that p redicted COconcentrations can be forced to match measurements,by manipulating the CO source inventory. Uncer-tainty in the measurelments will, of course, give poorerresults, but suitable filter tuning achiev es an averagin geffect. Likewise, under m ore stable conditions, steepconcentration gradients w ould make the representat-iveness of positional measurements less certain, andwe would again rely on the damp ed response of theclosed loop to individual measurements. In this studyit will be seen that definite trends are identified despitesuch measurement uncertainty.

Figure 3 show s that the original C O predictions bythe CIT model are generally low, whilst Fig. 4 suggeststhat there is poor correlation between predictions andobservations. A fixed adjustment of CO emissions, sayusing three times the EM FAC 7E emission facto rs(Yotter and W ade, 1989 ), as suggested by Harley et al.(1993 a), seems approximately correct at high CO(3-5 ppm), but too high at low C O (O-2 ppm), evenwith the 0.2 ppm bou.ndary-condition offset removedin Fig. 3. Quite soon in the analysis of Kalman filteradjustments below, specific spatial and temporalpatterns emerge, which suggest that combustion

1 0

9

6

stoichiometry, alone, is not the best means to accountfor observations.

An early indication of the importance of emissiontiming, for any adjustment of the CO emission in-ventory, arose from a recursion with a single domainenclosing all sources. Figure 5 show s the single fittedadjustment factor over the 72 h period 27-29 August1987 . Sources are raised to nearly 3 times their originalvalues at midday on weekd ays, but actually need to bereduced in the evening. Although the factor on Satur-day reaches as high as 10.0, the predicted emissions forSaturday remain considerably lower than those forweekd ays, on account of the low value of the baseemission inventory. Figure 6 show s typical improve-ments of predictions at several sites. Comp arison ofFig. 7 and Fig. 3 reveals the g reat improvement inmost predictions achieved by this single factor.

The spatial dependence of source adjustment fac-tors was revealed by definition of 29 domains as inFig. 8. Each square in this 7 x 4 pattern encloses 16points of the original 80 x 30 source definition grid.The surrounding area, i.e. the balance of the 80 x 30grid, then becom es the 29th domain. A high weightwas set in the 29th diagonal element of V to preventdeviations of this factor from unity, since all significantCO so urces are enclosed by the 7 x 4 pattern. Figure 9

f

Thursday24

Friday Saturday72

Fig. 5. Single adjustment factor for all CO sources: time-variation to achieve best CIT model fitto CO concentration observations.


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Inverse air pollution modelling 50 7

4

1Uncorrected

0 1 2 3 4 5456 points 722 points 587 points 147 points 27 points 5 points

CO Observation [ppm]Fig. 7. CIT m odel adjusted by a single time-varying factor (Fig. 5) applied to one solution for theentire source-domain: CO concentration predictions vs observations.

30

20

10

020 30 40 50 60

Fig. 8. South Coast Air Basin showing the definition of 29 source domains to reveal the spatialdependence of regressed source adjustment factors.


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f

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0 72 0 72 0 72 0 72 0 72 0 72 0 72t [hrl t [hrl t WI t Ml t [hrl t WI t [hrl

E2 E3 E4 E5 E6 E7 E8Fig. 9. R egressed weighting factors to be applied to the solutions for the 28 source domains in Fig. 8, giving an optimalfit to CO concentration observations over the period 27-2 9 August 1 987 .

Burbank

S a nBernardino

Fontana-Arrow

UplandARB

ClaremontCollege

0 24 48 0 24 48 72Thur Fri Sat Thur Fri sat

Time [hr] Time [hr]O r i g i n a l r e d i c t i o n s Predictions fitted to

obsewations by weightedconSnatkn of 28 sowce-&main sdutiom

Fig. 10. Com parison of predictions and observations (angular line) of CO concentrations atseveral sites, 27-29 August 198 7. The superposition of 7 x 4 sounx domain solutions (Fig. 8),adjusted with 2 8 recursed factors (Fig 9), gives a relatively close fit.


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Inverse air pollution modelling 509

UncorrectedMean

00 1 2 3 4 5

4% points 72 2 points 587 polntr 147 points 27 polntr 5 points

CO Observation [ppm]Fig. 11. CIT model adjusted by 7 x 4 factors applied to source-domain solutions corresponding toFigs 8 and 9: CO concentration predictions vs observation.

at 8 am and midday iare similar, but em issions doublethe midday values by 4 pm, and become far moreextensive.

Figure 15 show s the effect of the high facto rs alongthe coastline at 4 pm on Saturday. The high emissionzones are assumed 1.0 arise from recreational trafficnot identified in the original data, and should provemore continuous ifhigher-resolution source domainswere specified. The contribution of the regressedadjustment factors :in defining these high emissionzones is evident in Fig. 9.

7 . COMPE NSATION OF EMISSION DISTRIBUTIONS FORSOURCE-RECEPTOR LAGS

It was noted in Section 1 that th e methods ofMulholland (1989 ), Hartley and Prinn (1993 ) andBrown (1993) do not adequately address the problemof differing transport times between sources and re-ceptors. In the present technique, described in Sections2-6 above, a complete solution is generated for eachsource. domain, and these solutions are superpo sedwith the regressed f, weighting at any instant. Thus the

weightings do not represent an adjustment of theemission rates in a particular domain at that instant,except for receptors within or close to that domain.Fortunately, in this urban environment, the strongestcontributor to a receptor, and thus the largest term inthe row of BC,, usually arises from a close-by sourcedomain. How ever, this is not alway s th e case, as isevident in the degraded quality of predictions inFig. 16, and on the left-hand side of Fig. 18. Thesearise from a direct solution of the CIT m odel, usinga new emission inventory, constructed by adjustingthe 28 source dom ains by f, , at the same instant t thateach factor was evaluated. Figure 16 should be com-pared with Fig. 11, whe re the factors were used tocombine the source-domain solutions instead. Noticethat the mean prediction in Fig. 16 deviates fu rtherfrom the observation, and that the standard deviationof the prediction almost doub les in comparison withFig. 11.An ideal solution for all source-domain factors at alltimes is of the form (2 ), which accounts for thedistributed nature of the system. Since this form isintractable, we opt rather to post-process our approx-imate solution above, to correct the major phasing


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E9 Fig. 12. Emission adjustment factors in the domains of Fig. 8 at 4 pm on Thursday 27 August 1987.

Fig. 13. Emission adjustment factors in the domains of Fig. 8 at 4 pm on Saturday 29 August 1987.


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CO Observat ion [ppm ]Fig. 16. Direct C IT model solution using Fig. 8 source-domain emission adjusted by the 28factor histories of Fig. 9, with no compensation for source-receptor lags.

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