Estimating Surface/Internal Sources and Sinks

Transcription

1 R. G. Prinn, 2.86/.57: Amspheric Physics & Chemisry, April 27, 26 Esiming Surce/Inernl Surces nd Sins Figure by MI OCW.

2 Mesuremen Equin In Lgrngin rmewr: (, Observing Sin s',', j s,, l, ime v(s',' Bc rjecry Psiin Velciy Figure by MI OCW. Chnge in mle rcin ( = ( ( ys, y*s, y*, = s xd ' ys, ( rm is iniil cndiin y* (, is given by x ds' = ds' v = v d ' where x is he ne chemicl prducin (ignring mleculr diusin nd ssuming perec deiniin bc rjecry Rel mesuremens y hve errrs ε s x y ( s, = ds' +ε( s, v In discree rm: y = h x + ε i ij j j i rue vlue i perec mdel Rel mesuremen Mesuremen errr In vecr-mrix rm r muliple bserving sies i y = Hx + ε mesuremen equin where H= h ij is he pril derivive (r bservin mrix, nd

3 h y y i i ij = = xj xj cmpue in mdel where ( y,x i j denes n esime ( i j y,x. Eulerin mdels: Here x reers grid pins in he Eulerin mdel rher hn pins lng bc rjecry. uilize he bve equins we mus deine: x = x grid pin x re,grid pin ( ( ( ( y = y grid pin y re,grid pin where y(re is he mle rcin cmpued in reerence run using bes vilble esimes x re he se vecr prir esiming x. ( he mesuremen equin expresses n ppren liner relin beween he bservin vecr y nd he unnwns cnined in he se vecr x. I is n expressin he rwrd prblem. he hery he Liner Klmn Filer llws us perrm he inverse prblem in which we pimlly esime x given mdel nd ime series bservins y ne r mre sies. Opiml esimin he cs uncin (J J See he esime x x h minimizes J (i.e. = x (Nin: prbbiliy disribuin uncin p ( = ( ( N nwledge p( x, p( y J = sum squres dierences beween predicins ( y Hx ( y Hx ( y Hx (b ( = nd bservins ( y = (clled les squres minimizin Knw p y nd hence R expecin hen see esime x ( ( J= y Hx R y Hx ( = εε x h mximizes he cndiinl prbbiliy p( y x : R weighs ech squred dierence by inverse vrince ech relevn bservin (clled weighed les squres minimizin r mximum lielihd Knw p y nd p x (c ( ( hen see x h mximizes p( x y which is deined rm Byes herem:

4 p(y x p(x p(x y = p(y (clled he minimum vrince Byes esime see belw r he deiniin J in his cse r he Klmn iler KALMAN FILER Suppse we hve n piml esime he se vecr vilble prir cnsiderin he h mesuremen in d series nd we wish bin new piml esime prpse in generl x x nd is errr =Κ x +Κ y y ν using his mesuremen. We cn hus (3 ν (4 = Κ ν + Κε nd see speciy he mrices Κ nd Κ. Using he mesuremen equin (5 y = Hx + ε deine y =Ηx, he deiniin x = x +ν, he demnd h he rndm mesuremen errrs hve zer men( Ε[ ε ] =, nd inlly, he demnd h esimins re unbised ( Εν [ ] =, we cn shw h Κ =ΙΚ Η. Hence he new esime is wih n errr x = x + Κ [y Η x ] ν = ( ΙΚΗ ν +Κε (5 (6 nd esimin errr cvrince mrix Ρ = Ε ν ν Τ [ ( ] (7 Subsiuing (6 in (7, using he deiniin he mesuremen errr cvrince mrix R =Εεε ] [, nd demnding h mesuremen errrs nd se errrs re Τ uncrreled (s h Ενε [ ] =Εε [ ( ν ] = we bin Ρ = ( Ι Κ Η Ρ ( Ι Κ Η + Κ R Κ (8 We nw use he crierin pimliy deermine Κ. Since we will ssume we nw p(y nd p(x, we will chse vlue r Κ which minimizes he cs uncin J (equin 2 r he minimum vrince Byes esime. Speciiclly J =Ε[( ν ν ] = rce [ Ρ ] (9

5 Evluing J / Κ = nd slving r he s-clled Klmn Gin mrix Κ we hve Κ = Ρ Η [ Η Ρ Η + R ] (2 Subsiuing (2 in (8 hen yields Ρ = [ Ι Κ Η ] Ρ (2 Finlly, using he se spce equin (7 x( = M(, x( + η(, we hen bin he esimes x needed in (5 nd Ρ needed in (2 x = M x (22 Ρ = Μ Ρ Μ + Q (23 where ] =Εη η Q [, nd x nd Ρ re he piml upus rm he previus ierin he iler. Frm ur erlier discussin (Secin 3, Q culd represen rndm rcing in he sysem mdel due rnspr mdel errrs. use he iler we mus prvide iniil ( priri esimes r x nd P. hen rm ny prir upu esimes (x, Ρ, we use mesuremen inrmin ( y,r nd mdel inrmin ( H,Q geher wih equins (22, (23, (2, (5, nd (2 prvide upus x nd Ρ r inpus he nex sep. he iler equins re summrized in ble. Sme inuiive cnceps regrding he DKF re useul in undersnding is perin. Firs, rm equin (2, he gin mrix Κ Η (is mximum vlue, Κ Ρ Η R s he mesuremen errr cvrince (nise mrix R nd (is minimum vlue s R. Since he upde in he se vecr x x vries linerly wih, i is cler h mesuremens nisy enugh s h R much exceeds Κ ΗΡΗ, will cnribue much less imprvemen he se vecr esimin. In his respec we cn useully cnsider Η ΡΗ s he errr cvrince mrix r he mesuremen esimes. his emphsizes he imprnce he weighing he d inheren in R nd he disrins creed i errneus R re used. Ne h R y cn include mdel errr, mismch errr, nd insrumenl errr s ned erlier. Secnd, using (2, nd recgnizing h he mximum vlue Κ H = Ι, we see Ρ Ρ wih equliy ccurring r ininiely nisy mesuremens. Hence, he errr cvrince mrix (whse dignl elemens re he vrinces he se Ρ vecr elemen esimes decreses by muns sensiively dependen n he mesuremen errrs.

6 hird, we ne rm (23, h rndm rcings η in he sysem (se-spce mdel [equin (7], which re represened here by Q, will increse he exrpled errr cvrince mrix Ρ by muns depending n he relive vlues nd he exrplin mrix Μ Ρ Μ Q in he bsence sysem (sespce mdel nise. he inclusin Q lessens he inluence (r memry previus ierins in he iler perin. In he exreme, suicienly lrge vlues Q will preven he cpbiliy even nn-nisy mesuremens decrese nd hence increse he cnidence in he se vecr esime. In her wrds excellen (nnnisy mesuremens re lile use i he sysem (se-spce mdel is very nisy (e.g., hrugh rndm vriins η inrduced by rndm rnspr errrs. Ρ

7 ble : Klmn Filer Equins * Deiniin Equin Mesuremen equin (mdel y = H x + ε ; y = H x Sysem (se equin (mdel x = M x +η Se upde x x = Κ (y y Errr Upde Ρ = ( ΚΗ Ρ Klmn gin upde Κ = Ρ Η ( Η Ρ Η + R Se ime exrplin x =Μ x Errr ime exrplin Ρ = Μ Ρ Μ + Q Sysem rndm rcing cvrince Mesuremen errr cvrince Q = Εηη ( R = Εεε ( Esimin errr cvrince Ρ = Ε ν ν ( Inpu mesuremen mrix = Η = y / x Inpu sysem rndm rcing cvrince =Q Inpu se exrplin =Μ Inpu mesuremen y Inpu mesuremen errr cvrince =R Filer ierin (, esime (, exrple (, *A superscrip r superscrip denes respecively he vlue bere ( r er ( n upde n esime using mesuremens, nd denes he mesuremen number. In generl, errrs re ssumed rndm wih zer men nd mesuremen nd esimin errrs re uncrreled.