Environ. Sci. Technol. 1998, 32, 2018-2024 Condensing Complex Atmospheric Chemistry Mechanisms. 1. The Direct Constrained Approximate Lumping (DCAL) Method Applied to Alkane Photochemistry S . W . W AN G AN D P . G . G E O R G O P O U LO S * En viron m en tal an d Occu pation al Health Scien ces In stitu te, 170 Frelin gh u ysen Road, Piscataw ay, New Jersey 08855 G . LI AN D H . R AB I T Z Departm en t of Ch em istry, Prin ceton Un iversity, Prin ceton , New Jersey 08544-1009 Atmospheric chemistry mechanisms are the most computationally intensive components of photochemical air quality simulation models (PAQSM s). The development of a photochemical mechanism, that accurately describes atmospheric chemistry w hile being computationally efficient for use in PAQSM s, is a difficult undertaking that has traditionally been pursued through semiempirical (“ diagnostic” ) lumping approaches. The limitations of these diagnostic approaches are often associated w ith inaccuracies due to the fact that the lumped mechanisms have typically been optimized to fit the concentration profile of a specific species. Formal mathematical methods for model reduction have the potential (demonstrated through past applications in other areas) to provide very effective solutions to the need for computational efficiency combined w ith accuracy. Such methods, that can be used to “ condense” a chemical mechanism, include “ kinetic lumping” and “ domain separation” . An application of the kinetic lumping method, using the direct constrained approximate lumping (DCAL) approach, to the atmospheric photochemistry of alkanes is presented in this w ork. It is show n that the lumped mechanism generated through the application of the DCAL method has the potential to overcome the limitations of existing semiempirical approaches, especially in relation to the consistent and accurate calculation of the timeconcentration profiles of multiple species. gen eration s of PAQSMs in corp orate m ore detailed con sideration s of h eterogen eou s ch em istry cou p led with th e gasp h ase p rocesses. Th is com p u tation al bu rden is p artly du e to th e fact th at atm osp h eric ch em ical kin etic system s are very “stiff”, i.e., th ey in clu de reaction s ran gin g from very fast to very slow an d so in volve ch an ges associated with a very disp arate ran ge of tim e scales; th is requ ires th e u se of elaborate n u m erical in tegration sch em es (“stiff in tegrators”) (6, 7). It is tru e th at rap id advan ces in com p u ter tech n ology an d in th e availability of cost-effective com p u tation al p latform s (su ch as lin ked server “clu sters”) (8-10) as well as im p rovem en ts in n u m erical m eth ods [e.g., th e sp arse m atrix vectorized Gear m ethod (7)], con tin ue to reduce the sim ulation tim e for existin g ap p lication s. However, th e dem an ds arisin g from th e con tin u ou sly evolvin g redefin ition of m odelin g p roblem s (which, in the case of PAQSMs, are driven by both scien tific an d regu latory requ irem en ts) easily ou tp ace th e above advan ces. In deed, in recen t years, th e ap p lication of PAQSMs h as rap idly evolved from u rban scale to region al scale, to n ested m u ltiscale (11). At th e sam e tim e, con sideration s of u n certain ty an d risk in develop in g defen sible air qu ality m an agem en t strategies requ ire th e p erform an ce of m u ltip le sim u lation s in order to iden tify an d select a robu st con trol ap p roach . Fu rth erm ore, tasks su ch as ap p lyin g region al scale m odels to th e m axim u m resolu tion of em ission in ven tories, u n dertakin g com p reh en sive sen sitivity an d u n certain ty assessm en ts for PAQSMs, etc., con stitu te p roblem s th at in p ractice exceed available com putation al resources. Thus, there is a stron g n eed for accurate an d validated bu t com p u tation ally efficien t version s of PAQSMs that can be used as acceptable approxim ation s (with kn own levels of accu racy) of th e “stan dard” m odels. In deed, efforts h ave focu sed recen tly on develop in g su ch “fast” an d “en gin eerin g” version s of regu latory an d related PAQSMs. Th ese efforts in clu de th e u se of th e “fitted” GRS m ech an ism (12) an d of a p rop rietary fast n u m erical solu tion sch em e (13). Altern atively, “form al” m ath em atical m eth ods for m odel redu ction h ave th e p oten tial (dem on strated th rou gh p ast ap p lication s in a n u m ber of areas) to p rovide very effective solu tion s to th e n eed for a com p u tation al ap p roxim ation of a ch em ical m ech an ism th at com bin es efficien cy with accu racy. Th e “form al” m eth ods can in fact be u sed to overcom e som e of th e lim itation s associated with sem iem p irical (“diagn ostic”) ap p roaches. For exam ple, in diagn ostic lum pin g, in accuracies often arise from optim izin g the lum ped m ech an ism to fit th e con cen tration p rofile of a sp ecific sp ecies. So, if a m ech an ism h as been op tim ized sp ecifically to p redict ozon e con cen tration s, its ability to accu rately p redict p rofiles of oth er sp ecies m igh t h ave been com p rom ised. Introduction Alth ough th e ch em istry m echan ism s in corp orated in existin g com p reh en sive p hotochem ical air quality sim ulation m odels [e.g., CBM-IV(1), th e SAPRC93 m ech an ism s (2), an d RADMCM (3), th at is th e ch em ical m ech an ism u sed in th e region al acid dep osition m odel (4) an d SAQM (5)] are already sim plified represen tation s of the actual atm ospheric chem ical p rocesses, th ey are still th e m ost com p u tation ally in ten sive com p on en ts of th ese m odels. In deed, ch em ical kin etics calcu lation s typ ically con su m e over 90% of th e CPU tim e in sim u lation s th at em p loy th e ch em istry-tran sp ort m odu les of com p reh en sive p h otoch em ical m odelin g system s. Th is situ ation is on ly goin g to becom e m ore severe as n ew * To wh om all corresp on den ce sh ou ld be addressed. 2018 9 ENVIRONM ENTAL SCIENCE & TECHNOLOGY / VOL. 32, NO. 13, 1998 Background Gas-phase reaction m echan ism s for the atm ospheric organ ic/ NO x / O 3 system h ave reach ed an advan ced state of develop m en t. Available com p reh en sive ch em ical m ech an ism s can be classified as Explicit (or Detailed or “Referen ce”) an d as Redu ced (or Lu m ped). Exp licit m ech an ism s aim to accou n t for th e detailed ch em istry of all sp ecies an d in term ediates in volved in th e atm osp h eric p ollu tion system u n der con sideration . Typ ically, th ey con sist of h u n dreds of reaction step s an d are too lon g to be in corp orated in atm osp h eric m odels in ten ded for rou tin e u se. For th is reason , redu ced or lum p ed m echan ism s, gen erally in volvin g fewer than about a h u n dred reaction s, h ave been develop ed as system atic ap p roxim ation s of th e detailed ch em istry th at is described S0013-936X(97)00967-X CCC: $15.00  1998 Am erican Chem ical Society Published on Web 05/27/1998 FIGURE 1. Semiempirical (diagnostic) and mathematical (formal) approaches for “condensing” chemical kinetic systems. by th e exp licit m ech an ism s. Com m on ly, th e in organ ic ch em istry is retain ed in exp licit form , wh ereas th e ch em istry of organ ics is sim p lified by “lu m p in g” togeth er a n u m ber of reaction s an d/ or chem ical sp ecies. This m echan ism reduction h as been p u rsu ed in th e air qu ality m odelin g field th rough sem iem p irical (“diagn ostic”) ap p roaches that defin e a n u m ber of “sp ecies su bstitu tion s” for th e fu ll set of organ ics of con cern . In th ese ap p roach es, p h en om en ological rate con stan ts for th e reduced set are determ in ed through a fittin g p rocess th at adju sts calcu lation s p erform ed with th e lu m p ed m ech an ism so as to m atch observation s from sm og-cham ber exp erim en ts. Two m ajor ap p roach es for p erform in g diagn ostic lu m p in g can be iden tified: (1) Lum ped Molecule. The organ ics are grouped together in classes accordin g to th eir ch em ical ch aracter (i.e., as alkan es, olefin s, arom atics, aldeh ydes, etc.). Th en , eith er a gen eralized (“h yp oth etical”) sp ecies or a su rrogate (“actu al”) sp ecies is u sed to rep resen t th e ch em istry of each lu m p ed class. Exam p les of m ech an ism s derived th rou gh th is ap p roach are th e RADM (3) an d SAPRC (14) an d SAPRC93 (2) sch em es. (2) Lum ped Structure. Th e organ ics are grou p ed n ot accordin g to classical typ es of com p ou n ds, as in th e lu m p ed m olecu le m eth od, bu t accordin g to stru ctu re an d reactivity ch aracteristics. For exam p le, carbon atom s m ay be grou p ed, based on th eir bon din g, as sin gle bon ded, fast dou bly bon ded, slow dou bly bon ded, an d carbon yl atom s. Th e carbon bon d m ech an ism s (1) are th e p rim ary exam p le of em p loyin g th is ap p roach of lu m p in g. A gen eral altern ative to diagn ostic lu m p in g ap p roach es is offered by form al m ath em atical m eth odologies of system (or m odel) redu ction (see Figu re 1), th at aim to derive lower dim en sion al form u lation s th at m atch th e resp on se of th e origin al m odel (see, e.g., ref 15). In gen eral, su ch m eth ods can be classified in to (i) aggregation or lu m p in g m eth ods, wh ich are based on th e aggregation of state variables of th e origin al system ; th ese m eth ods will be fu rth er discu ssed in th e followin g section s; (ii) dom ain sep aration m eth ods (th e m ost im p ortan t bein g th e sin gu lar p ertu rbation m eth od), wh ich , for a kin etic system th at evolves in tim e, con sider tem p oral dom ain s an d are based u p on sp littin g th e system in to “fast” an d “slow” m odes (15, 16); variou s com p u tation al im p lem en tation s of su ch m eth ods h ave been p resen ted in th e literatu re (e.g.; 17-21); (iii) p artial realization m eth ods, wh ich are based u p on m atch in g certain m om en ts of th e origin al an d redu ced system s (22, 23); an d (iv) fittin g or op tim u m ap p roxim ation or resp on se m eth ods, wh ich are based upon m in im izin g the differen ce between the respon ses of th e origin al system an d an ap p rop riately selected redu ced rep resen tation , su ch as a sm aller system of differen tial equ ation s or a m u ltivariate su rface p rovided by an algebraic exp ression (24-26). In p rin cip le, th e selection of a ch em ical m ech an ism redu ction m eth od sh ou ld be based on th ree criteria: (a) th e reduced m odel m ust satisfy m ass con servation , (b) the steadystate resp on se of th e redu ced m odel sh ou ld be iden tical to (or very closely ap p roxim ate) th at of th e origin al m odel, an d (c) it sh ou ld be p ossible to m ath em atically relate th e p aram eters of th e redu ced m odel to th ose of th e origin al m odel. Form al m eth ods th at actu ally satisfy all th ese criteria are the m athem atical kin etic lum pin g an d dom ain separation m eth ods. However, m eth ods from categories iii an d iv above h ave th e p oten tial of p rovidin g, th ou gh at som e loss of gen erality, th e m ost “h igh ly con den sed” rep resen tation s of th e origin al m odel. Th is cou ld be of p articu lar im p ortan ce in som e ap p lication s, su ch as u n certain ty p rop agation calcu lation s for en viron m en tal im p act assessm en t an d risk ch aracterization . In su ch ap p lication s, an d if th e lim itation s of th e restricted rep resen tation are taken in to accou n t, th e loss of accu racy in in dividu al sim u lation s m ay be su fficien tly com p en sated by th e fact th at it wou ld be p ossible to p erform h u n dreds or th ou san ds of sim u lation s in a Mon te Carlo or sim ilar fram ework. Th e detailed descrip tion of th e kin etic lu m p in g m eth od an d its ap p lication to alkan e p h otoch em istry is given in th e n ext section . Th e gen eral backgrou n d of th e m eth ods com p risin g categories ii-iv above will be p resen ted, togeth er with dem on stration s of th e ap p licability of th ese m eth ods to atm osp h eric ch em istry p roblem s, in u p com in g articles. VOL. 32, NO. 13, 1998 / ENVIRONM ENTAL SCIENCE & TECHNOLOGY 9 2019 Kinetic-Lumping Approach: The DCAL Method Th e gen eral aggregation or lu m p in g m eth od for m odel redu ction was form alized by Aoki (27) in th e late 1960s. (Th e term “aggregation ” is used m ostly in the electrical en gin eerin g literatu re, wh ereas lu m p in g is u sed m ore exten sively in ch em ical an d com bu stion en gin eerin g ap p lication s; h ere, we u se th e term lu m p in g sp ecifically for ch em ical reaction system s: th is im p lies certain gen eral con strain ts, su ch as m ass con servation .) It h as also been sh own (28) th at several earlier m odel redu ction m eth ods (29-31) are sp ecial cases of lu m p in g. In th e late 1960s, Wei an d Kuo (32, 33) develop ed th e basic th eory for ch em ical kin etic lu m p in g of a relatively sim p le n com p on en t, reversible first-order (lin ear) reaction system ; Li an d Rabitz (34, 35) h ave sin ce exten ded th e th eory to n on lin ear reaction system s wh ile oth er research h as focused on the lum pin g of con tin uous reaction m ixtures (3639). Th e work p resen ted in th is p ap er focu ses on th e ap p lication of th e lin ear kin etic-lu m p in g ap p roach . Th is ap p roach is based on a form al m ath em atical an alysis of th e en tire reaction system , to obtain lu m p ed sp ecies th at can ap p roxim ate th e p rediction s for th e origin al or exp licit sp ecies. Th is ap p roach has the p oten tial to reveal sim ilarities in ch em ical sp ecies th at m ay n ot be ap p aren t in an an alysis solely based on con sideration s of ch em ical attribu tes an d can be u sed to ap p ly sim ilar weigh t factors on sim ilar sp ecies. A brief descrip tion of th e th eoretical basis of lin ear kin etic lu m p in g is given below. Let th e kin etics of an n com p on en t reaction system be described by dy ) f(y) dt (1) wh ere y is an n -dim en sion al com p osition vector, an d f(y) is an n -dim en sion al vector fu n ction th at describes th e reaction system . In lin ear kin etic lu m p in g, an n̂ -dim en sion al lu m p ed com p osition vector ŷ corresp on din g to y is obtain ed by ŷ ) My (2) wh ere M is a con stan t, n̂ × n dim en sion al lu m p in g m atrix. Th e n -dim en sion al system is con sidered to be exactly lu m p able if th ere exists a vector fu n ction f̂(ŷ) su ch th at dŷ ) f̂(ŷ) dt (3) Li an d Rabitz (34) p roved th at an exactly lu m p ed kin etic system is f̂(ŷ) ) Mf(M h ŷ) (4) wh ere M h is a gen eralized in verse of M. Th e m atrix M h is defin ed by MM h ) In̂ (5) wh ere In̂ is th e n̂ -dim en sion al iden tity m atrix. Fu rth erm ore, Li an d Rabitz (34) sh owed th at wh en th e lu m p in g is exact, th e su bsp ace sp an n ed by th e rows of M m u st be JT(y) in varian t, wh ere JT(y) is th e tran sp ose at Jacobian of f(y). A m eth od of determ in in g M was described based on exp an din g JT(y) accordin g to m JT(y) ) ∑ a (y)A k k (6) k )1 wh ere con stan t m atrices Ak are viewed as a set of basis m atrices of JT(y), an d fin din g th e n̂ -dim en sion al su bsp ace th at is sim u ltan eou sly in varian t with resp ect to all Ak. It was 2020 9 ENVIRONM ENTAL SCIENCE & TECHNOLOGY / VOL. 32, NO. 13, 1998 also sh own th at alth ou gh M h is n ot u n iqu e, a u n iqu e lu m p ed system is obtain ed u sin g an y M h , p rovided th e lu m p in g is exact. In p ractice, the lum p ed m odel is usually required to satisfy som e restriction s. For exam p le, it m ay be requ ired to keep certain ch em ical sp ecies of in terest u n lu m p ed. To accom m odate su ch restriction s, “con strain ed” lu m p in g m eth ods h ave been develop ed (35, 40): th e lu m p in g is con strain ed by a priori sp ecification of a p art of th e lu m p in g m atrix. In con strain ed lu m p in g, M is rep resen ted as M) ( ) MG MD (7) wh ere MG is given , an d MD is to be determ in ed. Un der th ese con strain ts, exact lum pin g schem es m ay n ot exist. Therefore, m eth ods for th e determ in ation of con strain ed ap p roxim ate lu m p in g sch em es are n ecessary. In ap p roxim ate lu m p in g, th e rows of th e M m atrix are n ot in varian t with resp ect to JT(y). Li an d Rabitz (35) develop ed a m eth od of determ in in g a globally op tim al M for ap p roxim ate lu m p in g, su ch th at th e sp ace sp an n ed by th e rows of M is “m ost n early” JT(y) in varian t for all y, an d sh owed th at th e op tim al ch oice of M h for an orth on orm al lu m p in g m atrix M is sim p ly MT. More recen tly, Li an d Rabitz (40, 41) h ave p rop osed a m ore straigh tforward m eth od for ap p roxim ate lum pin g, the direct con strain ed approxim ate lum pin g (DCAL) m eth od. Th e DCAL m eth od is based on determ in in g basis m atrices Ak’s in eq 6 for JT(y) n u m erically, by u sin g valu es of y in a region of n -dim en sion al com p osition sp ace th at is of in terest. It was sh own th at M can be determ in ed from () XTMT ) 0 (8) wh ere X is “m ost n early” orth ogon al to MG MGAT1 ... MG(AT1 )s 1-1 Z ) ... MG (9) MGATm ... MG(ATm )s m -1 wh ere sm is th e ran k of Ak or set to n an d can be determ in ed by th e followin g p rocedu re (35, 42): tran sform MG(ATk )i to an orth on orm al m atrix Q(G)Tk i u sin g Gram -Sch m idt orth ogosk-1 n alization , con stru ct a sym m etric m atrix Y ) ∑km)1 ∑i)0 QT (G)k iQ(G)k i an d determ in e th e eigen valu es an d eigen vectors of Y. Th e eigen vectors corresp on din g to th e sm allest n - n̂ eigen valu es of Ycom p rise th e Xm atrix, an d th e eigen vectors corresp on din g to th e largest n̂ eigen valu es of Ycom p rise th e rows of M. To force MG to corresp on d to th e eigen vectors of Y with th e largest eigen valu es, MG in Z is m u ltip lied by a large con stan t (40). Th is en su res th at th e u n lu m p ed sp ecies sp ecified by MG are p art of th e lu m p ed system . On ce th e eigen valu es an d eigen vectors of th e Y m atrix h ave been determ in ed, it is relatively straigh tforward to determ in e th e M m atrix requ ired to derive a lu m p ed system of arbitrary dim en sion n̂ < n . Wh en th e basis m atrices Ak of JT(y) are u sed, th e resu ltan t M is globally op tim al. In p ractice, it is often desired to ap p ly th e lu m p ed m odel to on ly a lim ited region of th e com p osition sp ace. In th is case, TABLE 1. Initial Conditions (ppm) for Chemical Species in the Explicit Model Mechanism no. name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 O3 NO2 NO NO3 N2O5 HNO3 HONO RNO3 N2 O2 H2O H2O2 CO CO2 O3P O1D HO2 OH RO2R description ozone nitrogen dioxide nitric oxide nitrogen trioxide dinitrogen pentoxide nitric acid nitrous acid alkyl nitrate nitrogen gas oxygen gas w ater hydrogen peroxide carbon m onoxide carbon dioxide oxygen atom (triplet) oxygen atom (singlet) hydroperoxy radical hydroxyl radical chem ical operator for NO to NO2 conversions w ith generation of HO2 conc no. name 7.08 × 10-2 1.09 × 10-3 0 0 0 0 0 0 7.81 × 10+5 2.09 × 10+5 1.04 × 10+4 0 0 0 0 0 0 0 0 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 NC4 ISC4 NC5 ISC5 NEC5 CLC5 NC6 2M C5 3M C5 22M B 23M B CLC6 M CL5 NC7 3M C6 24M P 23M P M CL6 NC8 butane isobutane pentane isopentane neo-pentane cyclopentane hexane 2-m ethylpentane 3-m ethylpentane 2,2-dim ethylbutane 2,3-dim ethylbutane cyclohexane m ethylcyclopentane n -heptane 3-m ethylhexane 2,4-dim ethylpentane 2,3-dim ethylpentane m ethylcyclohexane n -octane description 1.5 × 10-2 3 × 10-3 4.5 × 10-3 7.5 × 10-3 2.4 × 10-4 4.4 × 10-4 4.5 × 10-3 2.1 × 10-3 1.3 × 10-3 2.4 × 10-4 2.4 × 10-4 9.2 × 10-4 1.35 × 10-3 5.87 × 10-3 1.1 × 10-3 2.4 × 10-4 2.4 × 10-4 1.92 × 10-3 1.26 × 10-3 conc 4M C7 ISC8 ECL6 4-m ethylheptane 2,2,4-trim ethylpentane ethylcyclohexane 4.87 × 10-4 2.4 × 10-4 2.1 × 10-4 20 RO2N chem ical operator for NO to Nitrate conversions 0 44 45 46 21 R2O2 chem ical operator for NO to NO2 conversions 0 47 48 NC9 4EC7 n -nonane 4-ethylheptane 5.3 × 10-4 2 × 10-4 22 23 24 NC1 NC2 NC3 m ethane ethane propane 1.8 6 × 10-3 1.5 × 10-3 49 50 51 52 NC10 4PC7 NC11 NC12 n -decane 4-propylheptane n -undecane n -dodecane 3.92 × 10-4 2 × 10-4 1.06 × 10-3 5.1 × 10-4 Aks can be rep laced by JT(yk )s calcu lated at som e com p osition p oin ts yk (k ) 1, 2, 3, ...) in th e con sidered regim e. Application to Alkane Photochemistry An ap p lication of th e DCAL m eth od was p erform ed startin g from a “m odel” exp licit m ech an ism , wh ich con tain s 68 reaction s in volvin g 52 sp ecies. Th is “m odel” exp licit m ech an ism con siders detailed alkan e p h otoch em istry an d in organ ic ch em istry in th e trop osp h ere. Th e lu m p in g p rocess is restricted to th e 30 n on m eth an e alkan e sp ecies sh own in Table 1, wh ile th e 21 in organ ic com p ou n ds an d m eth an e sh own in Table 1 are con strain ed to rem ain as in dividu al sp ecies. Th e effects of tem p oral variation in tem p eratu re an d ligh t in ten sity on th e rate con stan ts of th is reaction m ech an ism were also con sidered. Dep en din g on th e n atu re of tem p eratu re dep en den ce, rate con stan ts can be sp ecified as differen t fun ction s of tem perature. In this “m odel” explicit m echan ism , three differen t groups of fun ction s are em ployed for the rate con stan ts. The first group of fun ction s is followin g th e Arrh en iu s form u la to sp ecify tem p eratu re dep en den ce for the rate con stan ts of in organ ic reaction s (except photolysis reaction s). Th e rate con stan ts of p h otolysis reaction s are in flu en ced by ligh t in ten sity an d are sp ecified by th e secon d grou p of fu n ction s obtain ed via a sp lin e in terp olation tech n iqu e, wh ich can gen erate th e fitted cu rves from valu es of the photolysis rate con stan ts at differen t observation tim es. Th e th ird grou p of fu n ction s is u sed to sp ecify tem p eratu re effects on rate con stan ts for alkan e reaction s. Sin ce th ere are n o Arrh en iu s kin etic data available for every sin gle n on m eth an e alkan e sp ecies, we also u se th e sp lin e in terp olation tech n iqu e to obtain exp licit fu n ction s by in terp olatin g m easu rem en t data of rate con stan ts at differen t tem p eratu res from literatu re. Th erefore, a total of 68 fu n ction s were develop ed alon g with th is “m odel” exp licit m echan ism to accoun t for the tem perature an d light in ten sity effects on th e 68 rate con stan ts in th is reaction m ech an ism . Th ese 68 rate con stan ts are all fu n ction s of tem p eratu re or tim e. Th u s, on ce th e tem p eratu re p rofile is sp ecified as a fu n ction of tim e, th e dep en den t variable of th ese 68 fu n ction s is ju st tim e. Th erefore, th e rate con stan ts can be u p dated at each tim e step du rin g th e in tegration p rocess for solvin g th e differen tial equ ation s for ch em ical kin etics. Methodology. Th e p erform an ce of th e DCAL-derived m ech an ism s was evalu ated by sim u latin g a 24 h sm ogcham ber experim en t usin g these m echan ism s an d com parin g th e resu lts with sim u lation s of th e sam e exp erim en t u sin g the “m odel” explicit m echan ism , as well as the correspon din g alkan e p ortion of CB4 m ech an ism an d SAPRC93 m ech an ism . In itial con cen tration s sh own in Table 1 for th e sim u lated sm og-ch am ber exp erim en ts were typ ical of p ollu ted u rban air for th e sp ecies in clu ded in th e exp licit m ech an ism (43). Sim u lation s with th e “m odel” exp licit, DCAL-derived, th e alkan e p ortion s of CB4 an d SAPRC93 m ech an ism s were perform ed usin g a box m odel (44) with the LSODE (45) routin e to solve th e stiff ch em ical kin etic system . To ob tain th e lu m p in g m atrix M for derivin g th e con den sed m ech an ism s, th e sim u lation with th e “m odel” exp licit m ech an ism is perform ed first to evaluate the Jacobian m atrices of the kin etic equ ation s at differen t tim es. For th is “m odel” exp licit m ech an ism , 52 differen tial equ ation s are gen erated to describe th e kin etic beh avior of 52 exp licit ch em ical sp ecies. Th e evalu ation s of 52 × 52 Jacobian m atrices are recorded at 13 equ ally sp aced tim e p oin ts of th e sim u lation to serve as th e basis m atrices for fin din g th e sim u ltan eou s “m ostly in varian t” su bsp ace as th e lu m p in g m atrix. Sin ce 22 sp ecies in clu din g 21 in organ ic sp ecies an d m eth an e are kep t u n lu m p ed, MG is a 22 × 52 m atrix, th e first 22 colu m n s of wh ich are com p rised of orth ogon al u n it vectors, an d th e rest of th e colu m n s are n u ll vectors. With th e in form ation for the 13 basis m atrices an d MG, on e can calculate the sym m etric m atrix Y from eq 9, by settin g all sk ) n . Wh en th e eigen vectors of th e sym m etric m atrix Y are arran ged acVOL. 32, NO. 13, 1998 / ENVIRONM ENTAL SCIENCE & TECHNOLOGY 9 2021 FIGURE 2. Comparisons of concentration-time profiles for (a) O3, (b) NO, (c) NO2, and (d) OH predicted by the “model” explicit mechanism, the CB4 mechanism, the SAPRC93 mechanism and three DCAL-derived mechanisms for alkane photochemistry. cordin g to th e decreasin g order of th e m agn itu des of th eir eigen valu es, th e first n̂ eigen vectors are th e best con strain ed lu m p in g m atrix MT with dim en sion n̂ . Th erefore, th e eigen vector m atrix of Ysu p p lies all th e best lum p in g m atrices with differen t n̂ . Th e eigen valu es can be u sed as a relative m easu re of th e error of th e lu m p in g m atrices. If on e ch ooses th e first n̂ eigen vectors to be th e lu m p in g m atrix, th e su m of th e eigen valu es for oth er eigen vectors is th e relative m easu re of its error. Th erefore, wh en th e dim en sion of th e lu m p in g m atrix becom es h igh er, th e error becom es sm aller. Wh en th e eigen valu es of oth er eigen vectors are all zero, th e lu m p in g m atrix is exact. After determ in in g th e lu m p in g m atrix M, th e n̂ dim en sion al lu m p ed system can be con structed by applyin g eq 4, which was presen ted in the previous section . By settin g th e n̂ dim en sion al lu m p ed com p osition vector ŷ as th e dep en den t set of variables, th e origin al n dim en sion al com position vector y can be expressed as a lin ear com bin ation s of n̂ com p on en ts of th e vector ŷ th rou gh th e tran sform ation of M h ŷ. Th u s, th e n dim en sion al vector fu n ction f(M h ŷ) h as on ly n̂ dep en den t variables. Th erefore, th e n̂ dim en sion al vector fu n ction f̂(ŷ), wh ich describes th e 2022 9 ENVIRONM ENTAL SCIENCE & TECHNOLOGY / VOL. 32, NO. 13, 1998 ch em ical kin etics of th e lu m p ed system , can be obtain ed by Mf(M h ŷ). Th e m atrix m an ip u lation s requ ired to obtain th e DCAL derived m ech an ism were p erform ed u sin g a com bin ation of FORTRAN code an d MAPLE (46). Th e evalu ation of th e m eth od was con du cted by com p arin g th e ozon e, NO, NO 2, an d OH p rofiles p redicted by this m echan ism with those p redicted by th e “m odel” exp licit m ech an ism an d th e corresp on din g CB4 an d SAPRC93 m ech an ism s. Results and Discussion Accordin g to th e m agn itu des of th e eigen valu es, th e first 25 colu m n s of th e eigen vector m atrix of Y can com p ose an “alm ost exact” lu m p in g m atrix, sin ce th e first 25 eigen vectors h ave eigen valu es larger th an 0.1 an d th e rest of th e 27 eigen vectors corresp on d to eigen valu es sm aller th an 1.0 ×10-4. Th erefore, th e lu m p in g m atrix is obtain ed by takin g th e tran sp ose of th e first 25 colu m n s of th e resu ltan t eigen vector m atrix of Y. Sin ce 22 sp ecies h ave been kep t u n lu m p ed, th is in dicates th at th ree DCAL m ech an ism s can be form u lated with on e, two, or th ree lu m p ed sp ecies. Th rou gh th e tran sform ation of th e lu m p in g m atrix, th e 30 n on m eth an e alkan e sp ecies can be lu m p ed in to on e, two, or th ree lu m p ed sp ecies. Th e con cen tration -tim e p rofiles for ozon e, NO, NO 2, an d OH obtain ed from th e sim u lated sm og-cham ber experim en ts with the “m odel” explicit m echan ism , th ree DCAL-derived m ech an ism s, an d th e alkan e p ortion s of th e CB4 an d SAPRC93 m ech an ism s are sh own in Figu re 2. All th ree DCAL m ech an ism s sh ow sign ifican tly better agreem en t with th e p rediction s of th e fu ll m odel th an th e alkan e p ortion of th e CB4 m ech an ism . Th e DCAL m ech an ism with th ree lu m p ed sp ecies ap p ears to p rodu ce p rediction s alm ost iden tical to th at of th e fu ll m odel. For reaction tim es exceedin g 600 m in , th e alkan e p ortion of SAPRC93 m ech an ism sh ows better agreem en t with th e fu ll m odel in p redictin g ozon e p rofile th an th e corresp on din g CB4 m ech an ism as well as th e two DCAL m ech an ism s with on e an d two lu m p ed sp ecies resu lts. However, wh en p redictin g NO an d NO 2 p rofiles, th e SAPRC93 m ech an ism exh ibits larger deviation s from th e fu ll m odel calcu lation s th an th e CB4 m ech an ism an d th e th ree DCAL m ech an ism s. It sh ou ld be n oticed th at two differen t lu m p in g m atrices were gen erated by u sin g two differen t “stan dard” available software p rogram s (e.g. IMSL rou tin e EIGRS an d NETLIB rou tin e DSYEV) for fin din g eigen valu es an d eigen vectors of th e sam e sym m etric m atrix Y. Th e two differen t lu m p ed system s con stru cted from th ese two differen t lu m p in g m atrices h ave th e sam e con cen tration -tim e p rofiles for all th e sp ecies. Th is in dicates th at th e two su bsp aces sp an n ed by th e rows of th ese two lu m p in g m atrices m igh t be overlappin g. On e way to check this poin t here is to determ in e th e degree of coin ciden ce between two su bsp aces. We u se d c to rep resen t a qu an titative descrip tion of th e degree of coin ciden ce between two su bsp aces. Accordin g to th is geom etric con cep t, wh en on e of th e two subsp aces lies in side th e oth er on e, th e basis vectors of on e su bsp ace are lin ear com bin ation s of th ose in th e oth er su bsp ace. In th is case, d c is u n ity. Wh en th e two su bsp aces are orth ogon al to each oth er, d c is equ al to zero. In oth er cases, 0 < d c < 1. Let th e n × r an d n × r′ m atrices P(r) an d P(r′) be th e m atrix rep resen tation s of th e two su bsp aces with r′ e r. Th e degree of coin ciden ce d c of th e two su bsp aces is defin ed as follows to satisfy th e above requ irem en ts: 1 d c ) tr[P(r′)TP(r)P(r)TP(r′)] r′ We calcu lated th e d c between th e two differen t lu m p in g m atrices in th is stu dy, an d it is equ al to 0.999. Th u s, th e two subspaces span n ed by these two lum pin g m atrices are alm ost overlap p in g each oth er. Th erefore, th e two differen t lu m p ed system s h ave th e sam e resp on ses as th e fu ll m odel. Testing for Different Initial Conditions. Th e above resu lts were obtain ed by sim u latin g on ly on e set of in itial con dition s for ch em ical sp ecies con cen tration s. To test th e robu stn ess of th e lu m p ed m odel, variou s sets of in itial con dition s were also sim u lated for th e com p arison of resp on ses between th e lu m p ed m odel an d th e fu ll m odel. Th e in itial con dition s for fou r ch em ical sp ecies, N 2, O 2, H 2O, an d CH 4, were kep t th e sam e, corresp on din g to valu es of typ ical u rban air con cen tration s am on g variou s sets of in itial con dition s, sin ce th e con cen tration s of th ese fou r sp ecies are m u ch h igh er th an th ose of oth er sp ecies an d, th u s, cou ld be treated as con stan ts. Th e in itial con cen tration s of th e oth er sp ecies were taken from ran dom sam p les in th e ran ge between p ossible m in im u m an d m axim u m con cen tration valu es for each sp ecies. On e th ou san d sim u lation s were p erform ed by takin g 1000 ran dom sam p les of in itial con dition s to stu dy th e agreem en t between th e lu m p ed m odel an d th e fu ll m odel again st differen t in itial con dition s. For each of th ese 1000 sim u lation s, th e ran ges of th e absolu te errors an d relative errors for th e resp on ses (e.g., ozon e con cen tration s) between th e lu m p ed m odel an d th e fu ll m odel were also calcu lated. On th e basis of th e in form ation of th e absolu te an d relative error ran ges from th ese sim u lation s, th e m axim u m an d m in im u m valu es of th e absolu te an d relative errors were determ in ed as in dicators of th e robu stn ess of th e lu m p ed m odel. It was fou n d th at th e ran ge of th e absolu te error for p redicted ozon e con cen tration s between two m odels ran ges from -0.6 to 0.1 p p b, an d th e ran ge of th e relative error is from 0 to 0.6%. Th u s, it ap p ears th at th e lu m p ed m odel con stru cted from th e set of in itial con dition s typ ical of u rban air is fairly robu st with resp ect to variation in th ese con dition s. Typ ically, th e m ost robu st lu m p ed m odel sh ou ld be con stru cted based u p on con sideration of th e wh ole com p osition regim e of th e in itial sp ecies con cen tration s. Th e reason for requ irin g th e lu m p ed m odel to be so robu st is th at th e lu m p in g m atrix is con stru cted based on a sim ulation scen ario where the com position regim e is covered p rop erly by th e ch osen p oin ts, in wh ich th e Jacobian m atrices are servin g as th e basis m atrices. It m ay n ot be th e case th at a robu st m odel can be con stru cted based u p on on ly on e set of in itial con dition s wh en m ore com p licated reaction system th at in clu de oth er organ ic grou p s su ch as alken es, arom atics, an d carbon yls h ave to be con sidered. Testing for Com putational Efficiency. Th e CPU tim es for p erform in g 100 sim u lation s, corresp on din g to variou s in itial con dition s, on a SUN ULTRA SPARC-1 170 MHz workstation , for th e fu ll alkan e m ech an ism , for th e alkan e p ortion s of CB4 an d SAPRC93 m ech an ism s, an d for th e th ree DCAL m ech an ism s associated with on e, two, an d th ree lum ped species, are 246.4, 125.6, 134.3, 117.5, 122.7, an d 130.5 s, resp ectively. So, th e CB4 an d SAPRC93 m ech an ism s an d all th ree DCAL m ech an ism s ach ieve abou t 50% redu ction in com p u tin g tim e com p ared with th e fu ll m ech an ism ; fu rth erm ore, th e DCAL m ech an ism with th ree lu m p s also ach ieves excellen t accu racy. Th ese resu lts su ggest th at th e DCAL m eth od can be u sed to derive th eoretically op tim al lum ped representation m echanism s and to determ ine achievable error bou n ds for oth er lu m p ed m ech an ism s. In sigh ts obtain ed by applyin g this m ethod can also be used to im prove th e lu m p in g strategies em p loyed in existin g p h otoch em ical m ech an ism s. Acknow ledgments Th is work was con du cted u n der th e au sp ices of th e Ozon e Research Cen ter (ORC); base fu n din g for th e ORC is p rovided by th e New Jersey Dep artm en t of En viron m en tal Protection (NJDEP); addition al fu n din g was p rovided by U.S.EPA’s Nation al Exp osu re Research Laboratory (NERL) u n der Coop erative Agreem en t CR 823717, an d by a Du p on t Edu cation al Gran t. Supporting Information Available Fou r tables [th e first 25 eigen valu es of m atrix Y(Table 2), list of reaction s in th e exp licit m odel m ech an ism (Tables 3 an d 4), list of reaction s of th e alkan e p ortion of CB4 m ech an ism (Table 5), an d list of reaction s of th e alkan e p ortion of SAPRC93 m ech an ism (Table 6)] (4 p ages) will ap p ear followin g th ese p ages in th e m icrofilm edition of th is volu m e of th e jou rn al. 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