Convergence of the Desroziers scheme and its relation to the lag innovation diagnostic
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1 Covergece of the Desrozers scheme ad ts relato to the lag ovato dagostc chard Méard Evromet Caada, Ar Qualty esearch Dvso World Weather Ope Scece Coferece Motreal, August 9, 04
2 o t t O x x x y x y Oservato space resduals OS O O provde formato aout the true error statstcs ut they are comed! What we would lke to kow, s how ca we partto the to oservato ad ackgroud error compoets? O O Desrozers ad colleagues came up wth the dea wth a smple practcal scheme for usg other OS O A O A O a a K x x K I x y K x y x y to do the parttog. ut to work t assumes kowledge aout the spatal statstcs, e.g. spatal correlatos of oth oservato ad ackgroud errors. Smlarly to the OI method.
3 Ojectve of the talk ecause the Desrozers method s wdely used practce Exame the mathematcal propertes of the scheme Itroduce aother way to do the parttog, usg the temporal correlato of the ovatos or the lag ovato covarace Daley 99 3 he Derozers method ad the Lag ovato method are fact complemetary. he Desrozers method medded to the Lag ovato method. New vew of the prolem 3
4 Desrozers teratve scheme usg oservato space resduals A No parametrc full covarace estmato Parametrc estmato Desrozers teratve scheme usg J o J Desrozers et al. QJMS 005 Desrozers et al. ECMWF 009 Desrozers ad Ivaov QJMS 00 O A O A O Varace scalg factors Correlato legth scale,,,, o o L L L L Parametrc estmato, a a o x J x J, ] ˆ [,, ] ˆ [ K K I E tr x J E tr x J a a o compare wth 4
5 Iovato Covarace Cosstecy [ICC] s whe O O Most of the tme, your frst guess estmate of error statstcs wll ot meet the ICC 0 0 O O A sgle step of the o parametrc Desrorers scheme produces ICC Oce the ICC s meet o further updates ca e made wth the Desrozers scheme 5
6 he codto for the estmates to e the truth Covergece to the truth ff,, K K Parametrc estmato of varace scalg factors, Expermet were we kow the true error covaraces, Ideal stuato where we have oservatos at each grd pot ad for each varales I 6
7 Fxed pot s for the estmato of oth oe stale fxed pot two ustale uphyscal ull varace fxed pots 3 meas the estmate s equal to the truth
8 Fxed pot s for the estmato of oth oe stale fxed pot two ustale uphyscal ull varace fxed pots 3 msspecfed ackgroud error correlatos
9 3 truth L L a Correctly specfed error correlatos 3 truth L L Pertured ackroud error correlato log lkelhood cotours 3 ovato varace codto
10 Estmatg oly Covergece to the truth ff wo fxed pots o o Physcal soluto Sale Ustale Null varace soluto Ustale Stale Expermet = I Each spectral compoet has ts ow ad depedet Desrozers teratve scheme 0
11 Case Estmatg the spatal correlato of frst guess 0 + true error estmates
12 Case Estmatg the spatal correlato of ad k k k all waveumers frst guess 0 + true error estmates
13 Case Estmatg the spatal correlato of ad k k k some waveumers frst guess 0 + true error estmates 3
14 Short coclusos he Desrozers scheme always coverge ut the covergece to the truth deped o the complemetary Iformato. Msspecfcato of the complemetary formato wll troduce a error the estmate 4
15 Optmalty of aalyss For a artrary Kalma ga, the aalyss error covarace A 0.3 I K I K K K t s mmum whe scalar prolem wth 0 o o Aalyss error varace - ormalzed KK o / f /...3. he aalyss error s mmum whe ot ecessarly whe the estmates are equal to the truth
16 Mmzg the aalyss error gves the true Kalma ga ut ot the true error statstcs. We eed a addtoal formato, such as the ovato covarace cosstecy Scalar case K o o Iformato aout the true Kalma ga gves formato aout what the true rato of the error varaces should e. Addg ovato cosstecy, gves formato aout the sum of error varaces 6
17 Lag ovatos Daley MW 99 he seral correlato of ovatos gves formato aout the optmalty of a assmlato system I KF framework f O O O O O O K K M K P M C So f the he covergece to the truth of the Desrozers scheme ca e motored usg the lag ovato covarace K K 0 C 7
18 Estmate. ackgroud error varace 0% larger tha truth C 8
19 Estmate. 0 ackgroud error varace % larger tha truth C 9
20 Estmate. 00 ackgroud error varace same as truth C 0
21 C we get Comg Lag ovatos ad Desrozers Cosder a fxed oservato etwork, ad suppose we have statc error covaraces, or that we eglect the effect of dyamcs etwee o the ackgroud error covarace. So from t ad t M K C C K K K Desrozers he dfferece etwee the Desrozers sgle step Estmate ad the truth s the lag ovato
22 A ew scheme may look lke ths. I the case where we ca assume that the oservato error ot correlated tme Sgle step of the Desrozers o parametrc to have ICC Lag Desrozers C C where the lag ovato covarace do the fal parttog etwee oservato ad ackgroud error covaraces
23 Coclusos Outle of the mathematcal propertes of the Desrozers method he Desrozers schemes coverges whe the ICC s reached Iaccurate complemetary formato creates a error the estmated error statstcs Uder the assumpto of serally ucorrelated oservato error, the lag ovato covarace provdes formato aout the true Kalma ga, ad whe used cojucto wth the Desrozers method provde a dagostc of covergece to the truth he Desrozers method s fact emedded to the Lag ovato dagostc Future work wll e o evaluatg the lag ovato covarace a few dfferet applcatos, ad lookg mplemetg the ew scheme 3
24 Extra sldes 4
25 0 Lagged-ovato varace - ormalzed / 5 o / f
26 Estmate
27 Estmate
28 If [ICC] the Desrozers Full covarace Parametrc OS Parametrc Var Coverged / Fxed pot p or p J m ut t does t mea that the estmates, has coverged to the truth, 8
29 Codtos of covergece whe estmatg oth Always coverge to somethg o a Fast covergece whe L L Coverge a sgle step the theoretcal case of estmatg oth full covaraces, Example s: estmatg oth oservato ad ackgroud error varaces requres that the error correlatos are correctly specfed estmatg error varace ad error correlatos for oth oservato ad ackgroud, requres that the correlato model e.g. Gaussa, SOA are correctly specfed 9
30 Iterato o oservato error varace Error varace / referece error varace 4 AMSU a varace rato for 63, ord 5 AMSU varace rato for 63, ord 4.5 chael umer chael umer utued tue tue tue 3 tue 4 Page 30
31 Comparso wth other estmates Page 3
32 ackgroud ad Os Error Varaces ad kalma Ga For C4, after smoothg Smoothed Varaces for MIPAS C4 Kalma Ga /+O for C4 after smoothg os. err ackgroud err Error varaces oservato ackgroud Kalma ga f K o f Page 3 Studet Verso of MALA Studet Verso of MALA
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