4DVAR, according to the name, is a four-dimensional variational method.

Transcription

1 4D-Varatnal Data Assmlatn (4D-Var) 4DVAR, accrdng t the name, s a fur-dmensnal varatnal methd. 4D-Var s actually a smple generalzatn f 3D-Var fr bservatns that are dstrbuted n tme. he equatns are the same, prvded the bservatn peratrs are generalzed t nclude a frecast mdel that wll allw a cmparsn between the mdel state and the bservatns at the apprprate tme. 4D-Var seeks the ntal cndtn such that the frecast best fts the bservatns wthn the assmlatn nterval.

2 Cst functn fr 4DVAR f a Let x ( t) = M x ( t ) represent the (nnlnear) mdel frecast that advances frm the prevus analyss tme t t the currentt. Assume the bservatns dstrbuted wthn a tme nterval ( t, t n ) wll be used. he cst functn ncludes a term measurng the dstance t the backgrund at the begnnng f the nterval, and a summatn ver tme f the cst functn fr each bservatnal ncrement cmputed wth respect t the mdel ntegrated t the tme f the bservatn: J( x( t )) ( x( t ) x ( t )) B ( x( t ) x ( t )) ( H( x ) y ) R ( H( x ) y ) (6.) N b b = = where N s the number f bservatnal vectrs y dstrbuted ver tme. he cntrl varable (the varable wth respect t whch the cst functn s mnmzed) s the ntal state f the x, whereas the analyss at the end f the nterval s gven by the mdel ntegratn mdel wth the tme nterval ( t ) frm the slutn ( t ) M ( t ) x x. n = In ths sense, the mdel s used as a strng cnstrant,.e., the analyss slutn has t satsfy the mdel equatns. In ther wrds, 4D-Var seeks the ntal cndtn such that the frecast best fts the bservatns wthn the assmlatn nterval. he fact that the 4D-Var methd assumes a perfect mdel s a dsadvantage snce (fr example) t wll gve the same credence t lder bservatns at the begnnng f the nterval as t newer bservatns at the end f the nterval. 2

3 he 4DVAR tres t use all bservatns n the assmlatn tme nterval as well as pssble. he fllwng s an example f hw nfrmatn s prpagated n tme by a smple advectn mdel, s that the bservatns can be cmpared wth the frst guess. 3

4 A varatn n the cst functn when the cntrl varable ( ) t x s changed by a small amunt δ x ( t ) s gven by J J J J δj = J[ x( t) + δx( t)] J( x( t) δx + δx δxn = δx( t) x x2 xn x( t) (6.2) J J where the gradent f the cst functn = x( t) xj ( t) j s a clumn vectr. Iteratve mnmzatn schemes requre the estmatn f the cst functn gradent. Smlar t the 3DVAR case, the steepest descent methd s the smplest scheme n whch the change n the J cntrl varable after each teratn s chsen t be ppste t the gradentδ x( t) = a J = a. x ( t ) x( t ) Other, mre effcent methds, such as cnjugate gradent r quas-newtn als requre the use f the gradent, slve ths mnmzatn prblem effcently, we need t be able t cmpute the gradent f J wth respect t the elements f the cntrl varable. 4

5 Calculatn f the gradent f 4DVAR cst functn As we saw earler, gven a symmetrc matrx A and a functnalj = xax, the gradent s gven by J = Ax. 2 x If J = y Ay, and y = yx, ( ) then (by applyng the chan rule) 2 J x y = x Ay (6.3) where y x kl, y = x k l s a matrx. We can wrte (6.) as J = Jb + J, and frm the rules dscussed abve, the gradent f the backgrund cmpnent f the cst functn ( ( ) b ( )) ( ( ) b Jb = x t x t B x t x ( t )) wth respect t x( t ) 2 s gven by J b x( t ) = B ( x( t ) x ( t )). (6.4) b hs s the same as n the 3DVAR case. 5

6 Befre we prceed t determne the gradent f the bservatn term, let s frst defne angent Lnear Mdel and Its Adjnt. A lnear tangent mdel (r tangent lnear mdel, LM as t s ften called) s btaned by lnearzng the mdel abut the nnlnear trajectry f the mdel between t and t, s that f we ntrduce a perturbatn n the ntal cndtns, the fnal perturbatn s gven by [ ] [ ] 2 x( t ) + δx( t ) = M x( t ) + δx( t ) = M x( t ) + L δx( t ) + O( δx ) (6.5) he lnear tangent mdel L s a matrx that transfrms an ntal perturbatn at t t the fnal perturbatn at t. he LM equatn s then δx( t ) = L δx ( t ). u u Fr example, the nnlnear advectn + u = s ntegrated usng the frward-n-tme centered-n-space fnte t x dfference scheme (ths scheme s actually unstable t s gven here fr llustratn purpse nly), n+ n t n n n u = u + u ( u+ u ). 2 x he LM mdel fr perturbatn u s t δu = δu + u ( δu δu ) + δ u ( u u ) = µ u δ u + + µ ( u u ) δ u + µ u δu 2 x n+ n n n n n n n n n n n n n n

7 where µ t. herefre 2 x δu δ u + µ ( u u ) µ u δu = = = Lδu n+ n n n n δ un µ un + µ ( un+ un ) δun n+ n 2 n n n n+ n n n n n n+ δ u2 µ u + µ ( u3 u) µ u3 δu2 n. If there are several steps n a tme nterval t -t, the tangent lnear mdel that advances a perturbatn frm t t t s gven by the prduct f the tangent lnear mdel matrces that advance t ver each step: L ( t, t ) = L ( t, t + ) = L = L L... L (6.6) j j j 2 j= j= he adjnt mdel, defned as the transpse f the lnearzed frward mdel, s gven by = j+ j = j = j= j= L ( t, t ) L ( t, t ) L LL... L (6.7) Eq. (6.7) shws that the adjnt mdel advances a perturbatn backwards n tme, frm the fnal t the ntal tme, because the rght mst L n the equatn s that f the fnal tme nterval (frm t t t ) and the left mst L s that f the frst tme nterval (frm t t t ) and matrx prduct s evaluated n the rght-t-left rder. 7

8 he gradent f the bservatnal term, J H H N = ( ( x) y ) R ( ( x) y ) 2 = s mre cmplcated becausex = M( x ( t)). If we ntrduce a perturbatn t the ntal state, thenδx = L( t, t) δx, s that ( H( x) y ) H M = = HL( t, t ) = H L( t, t ) x( t ) x x j j+ j= (6.8) H M As ndcated by (6.8), the matrces H, L are the lnearzed Jacbans, x x. herefre the gradent f the bservatn cst functn s gven by (remember J x ( t ) N = L HR x y = ( t, t ) ( H( ) ) J x y = x Ay) (6.9) Equatn (6.9) shws that every teratn f the 4D-Var mnmzatn requres the cmputatn f the gradent,.e., cmputng the ncrements ( H( x) y ) at the bservatn tmes t durng a frward ntegratn, multplyng them by HR and ntegratng these weghted ncrements back t the ntal tme usng the adjnt mdel. 8

9 Dente ( ( ) ) d = HR H x y = H R d whch we call the weghted bservatnal ncrement fr bservatns at tme t. Snce parts f the backward adjnt ntegratn are cmmn t several tme ntervals, the summatn n (6.9) can be arranged mre cnvenently: J x ( t ) = = N N L( t, t) d LL... L d = = = d + L d LL... L d + LL... L d 2 = d + L ( d + L ( d L ( d + L d ))) 2 2 Assume, fr example that the nterval f assmlatn s frm Z t 2Z, and that there are bservatns every 3 hurs (Fg. 5.6) d L d L d 2 L 2 d 3 L 3 d 4 Fg. 5.6: Schematc f the cmputatn f the gradent f the bservatnal cst functn fr a perd f 2 hurs, bservatns every 3 hurs and the adjnt mdel that ntegrates backwards wthn each nterval. 9

10 Prcedure f 4DVAR mnmzatn ) Frst ntegrate the (nnlnear) frward mdel and save the nnlnear trajectry (.e., mdel state at every tme step) n the assmlatn wndw. hs 4D state s needed fr defnng L and fr calculatng the bservatnal ncrements. 2) Cmpute the weghted negatve bservatn ncrements d = HR ( H( x ) y ) = H R d. 3) he adjnt mdel L ( t, t ) = L appled n a vectr advances t frm t tt. In ur case, ths vectr s d r the cmbnatn f d s at tmes later than t. hs vectr s the adjnt varable that the adjnt equatn advances. J 4) hen we can wrte (6.9) as = d + L( d + L( d2 + L2( d3 + Ld 4 4))) x (6.) 5) Frm (6.4) plus (6.9) r (6.) we btan the gradent f the cst functn, and the mnmzatn algrthm mdfes apprprately the cntrl varable x ( t ). 6) After ths change, a new frward ntegratn and new bservatnal ncrements are cmputed and the prcess s repeated untl cnvergence. herefre each mnmzatn teratn nvlves ne frward ntegratn f the nnlnear predctn mdel and ne backward ntegratn f the adjnt mdel. Because everythng n the adjnt mdel s reversed n rder, and the defntn f the adjnt peratr requres the value f the varables n the frward mdel (even wthn the ndvdual steps f a sngle tme step), many recalculatns are ften nvlved n rder t restre the values f these varables, the adjnt mdel s ften 2-3 tmes as expensve as the frward mdel. But stll, nly ne ntegratn s needed nstead f N f them as Eq. (6.9) mght suggest. hs s s because f the lnear nature f Eq. (6.9).

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12 Incremental Frm f 4DVAR 4D-Var can als be wrtten n an ncremental frm wth the cst functn defned by J( δx ) ( δx ) B ( δx ) HL( t, t ) δx d ) R HL( t, t ) δx d ) (6.) N = = f and the bservatnal ncrement defned as dj = y H( x ( t)) and dj = y H( x ( t)). Because H ( x ) H( x ) = H( x ) + δx = H( x ) + HL( t, t ) δx x( t ) pluggng the abve nt (6.) gves (6.). Wthn the ncremental frmulatn, t s pssble t chse a smplfcatn peratr that slves the prblem f mnmzatn n a lwer dmensnal space w than that f the rgnal mdel varables x : δw= Sδx S s meant t be rank defcent (as wuld be the case, fr example, f a lwer reslutn spectral truncatn r a lw-reslutn grd was used fr w than fr x ). 2

13 After the mnmum f the prblem s btaned fr J ( δ w ), x b = x g I δ + S w and a new uter teratn at the full mdel reslutn can be carred ut (Lrenc, 997). he teratn prcess can als be accelerated thrugh the use f precndtnng, a change f cntrl varables that makes the cst functn mre sphercal, and therefre each teratn can get clser t the center (mnmum) f the cst functn (e.g., Parrsh and Derber, 992, Lrenc, 997). 3

14 4DVAR versus 3D analyss methds (3DVAR, IO) When cmpared t a 3-D analyss algrthm n a sequental assmlatn system, 4D-Var has the fllwng characterstcs: t wrks nly under the assumptn that the mdel s perfect. Prblems can be expected f mdel errrs are large. t requres the mplementatn f the rather specal lt f wrk f the frecast mdel s cmplex. L peratrs, the s-called adjnt mdel. hs can be a n a real-tme system t requres the assmlatn t wat fr the bservatns ver the whle 4D-Var tme nterval t be avalable befre the analyss prcedure can begn, whereas sequental systems can prcess bservatns shrtly after they are avalable. hs can delay the avalablty f analyss. x a s used as the ntal state fr a frecast, then by cnstructn f 4D-Var ne s sure that the frecast wll be cmpletely cnsstent wth the mdel equatns and the fur-dmensnal dstrbutn f bservatns untl the end f the 4D-Var tme nterval (the cut ff tme). hs makes ntermttent 4D-Var a very sutable system fr numercal frecastng. 4D-Var s an ptmal and relatvely effcent (relatve t e.g., Kalman flter) assmlatn algrthm ver ts tme perd thanks t a therem that says that he evaluatn f the 4D-Var bservatn cst functn and ts gradent, J ( x) and J ( x ), requres ne drect mdel ntegratn frm tmes t t t n and ne sutably mdfed adjnt ntegratn made f transpses f the tangent lnear mdel tme-steppng peratrs M.. In fact t the applcatn f ths prcedure that made 4DVAR practcal. 4

15 4DVAR versus Kalman flter he mst mprtant advantage f 4D-Var s that f we assume that the mdel s perfect, and that the a prr errr cvarance at the ntal tme B s crrect, t can be shwn that the 4D-Var analyss at the fnal tme s dentcal t that f the extended Kalman Flter (Lrenc, 986, Daley, 99). hs means that mplctly 4D-Var s able t evlve the frecast errr cvarance frm B t the fnal tme. Unfrtunately, ths mplct cvarance s nt avalable at the end f the cycle, and nether s the new analyss errr cvarance. In ther wrds, 4D-Var s able t fnd the BLUE but nt ts errr cvarance. mtgate ths prblem, a smplfed Kalman Flter algrthm has been prpsed t estmate the evlutn f the analyss errrs n the subspace f the dynamcally mst unstable mdes (Fsher and Curter, 995, Chn and dlng, 996). 4DVAR s, hwever, much cheaper than the Kalman Flter. 5