Short-Term Prediction of Lagrangian Trajectories

Transcription

1 1398 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 Short-Term Prediction o Lgrngin Trjectories LEONID I. PITERBARG Center o Applied Mthemticl Studies, University o Southern Cliorni, Los Angeles, Cliorni (Mnuscript received August 000, in inl orm 5 Jnury 001) ABSTRACT Lgrngin prticles in cluster re divided in two groups: observble nd unobservble. The problem is to predict the unobservble prticle positions given their initil positions nd velocities bsed on observtions o the observble prticles. A Mrkov model or Lgrngin motion is ormulted. The model implies tht the positions nd velocities o ny number o prticles orm multiple diusion process. A prediction lgorithm is proposed bsed on this model nd Klmn ilter ides. The lgorithm perormnce is exmined by the Monte Crlo pproch in the cse o single predictnd. The prediction error is most sensitive to the rtio o the velocity correltion rdius nd the initil cluster rdius. For six predictors, i this prmeter equls 5, then the reltive error is less thn 0.1 or the 15-dy prediction, wheres or the rtio close to 1, the error is bout 0.9. The reltive error does not chnge signiicntly s the number o predictors increses rom 4 7 to Introduction The role o driters, current-ollowing devices, in ocen studies hs drsticlly incresed in recent decdes (Dvis 1991,b). Bsiclly, driter dt hve been used or the mpping o ocenic circultion (e.g., Swensson nd Niiler 1996; Buer et l. 1998), estimtion o mixing prmeters (e.g., Gri et l. 1995), nd ssimiltion in numericl models (e.g., Ishikw et l. 1996). Recently, new direction o reserch concerning the predictbility nd prediction o the Lgrngin motion in the upper ocen hs come orwrd (Ozgokmen et l. 000, 001; Cstellri et l. 001). In prticulr, this direction ws stimulted by need or serch-nd-rescue opertions in the se (Schneider 1998). Investigtion o the driter position predictbility is lso importnt rom the undmentl reserch viewpoint since it is prt o the generl Lgrngin predictbility problem in chotic systems. No doubt, the upper ocen cn nd should be treted s dynmicl stochstic system due to combined eects o deterministic men circultion nd eddy turbulence. For now, the best nd most common prediction tool in such systems is the extended Klmn ilter. Not surprising, just Klmn-type ilters were pplied to rel nd synthetic dt in the works cited bove. We notice tht the problem o predicting sole driter position ws considered with no comprehensive sensitivity nlysis. The gol o this pper is to extend the Klmn ilter Corresponding uthor ddress: Dr. Leonid I. Piterbrg, Center o Applied Mthemticl Sciences, University o Southern Cliorni, 104 W. 36th Plce, DRB 155, Los Angeles, CA E-mil: piter@mth.usc.edu pproch to predicting the positions o severl prticles nd studying the prediction error vi stochstic simultions. In prticulr, we set up to investigte the error dependence (sensitivity) on the velocity correltion rdius, the initil cluster rdius, the Lgrngin correltion time, nd the number o predictors. We concentrte on the short-term prediction, restricted by 15 dys, nd smll number o predictors (4 7). The cornerstone o our pproch is stochstic model o the Lgrngin turbulence in the upper ocen, irst reported in Piterbrg (000). The model implies tht the velocities nd positions o ny M prticles orm Mrkov process in 4M dimensions two velocity components nd two coordintes or ech prticle. In the simplest isotropic cse the model is completely deined by ew prmeters: the velocity vrince, Lgrngin correltion time, nd spce correltion rdius R. Typicl considered vlues re 0 km dy 1, 10 dys, R km. A prticulr version o the discussed lgorithm hs lredy been pplied or studying the Lgrngin motion predictbility in the tropicl Pciic using rel dt (Ozgokmen et l. 001). Unlike tht work, here the model is ormulted in the generl cse including three-dimensionl motion nd inhomogeneous environments. Yet, the prediction lgorithm is extended to predicting severl prticle positions. However, the error nlysis is ocused on the D isotropic cse nd one predictnd only. An importnt eture o the stochstic dierentil equtions describing cluster motion is tht the corresponding diusion mtrix depends on the stte vrible. More exctly, the velocity covrince mtrix de- 001 Americn Meteorologicl Society

2 AUGUST 001 PITERBARG 1399 pends on the prticle positions t ny time. This ct mkes direct ppliction o the Klmn ilter impossible since it implies tht the covrince mtrix o the stochstic orcing does not depend on the stte vector. However, the min ide o optimlly combining the dynmics nd observtions cn be relized. We just use the predicted nd observed positions in the previous step or evluting the stochstic orcing covrince. O course, this pproch does not yield the optiml estimtor. Moreover, it is very diicult to determine nlyticlly how close the proposed prediction is to the optiml one. In this sitution there is nothing better thn using stochstic simultions to evlute the prediction skill. Using this pproch we study the dependence o the prediction error on R,, initil cluster rdius R 0, nd the number o predictors. The pper is orgnized s ollows. In section the mthemticl problem sttement is given, in section 3 the model is ormulted nd explined, in section 4 the prediction lgorithm is presented nd n exmple is given where the exct prediction is reched, nd results o simultions nd their discussion re given in section 5.. Problem sttement Mthemticlly, the Lgrngin prediction is problem o iltering multidimensionl stochstic process driven by system o stochstic dierentil equtions. We explin tht in moment. For the ske o generlity, the motion in the Eucliden spce R d o ny dimension d 1,, 3,... is considered. Let u(t, r) U(t, r) u (t, r) be decomposition o d-dimensionl velocity ield into men circultion nd luctution. More exctly, U(t, r) is deterministic velocity ield, u (t, r) is rndom vector ield with zero men, nd E{u (t, r)} 0, E{ } denotes expecttion. Consider M Lgrngin prticles, tht is, current-ollowing luid prcels, strting t the sme time t rom dierent positions r1, r,..., rm. Their motion is covered by the ollowing system o Md equtions: dr j 0 dt u(t, r ), r (0) r, (1) j j j where j 1,..., M. We denote v j (t) dr j (t)/dt the Lgrngin velocity o the jth prticle, while u(t, r) is clled the Eulerin velocity ield. Both velocities re relted in simple wy: v j(t) u[t, r j(t)]. Since the right-hnd side o (1) includes the velocity luctutions s well s the men low we cll the equtions stochstic dierentil equtions nd cll the system itsel dynmicl stochstic system. Assume tht trjectories o the irst p M prticles r 1 (t), r (t),...,r p (t) re observed during time intervl (0, T), while the trjectories o the remining prticles r p 1 (t), r P (t),...,r M (t) re not observed. We cll the irst nd second prticles predictors nd predictnds, respectively. The problem is inding the best prediction, rˆ p 1 (T), rˆ p (T),...,rˆ M(T) o the positions o the unobserved prticles given the bove predictor observtions, men low U(t, r), the initil positions o the predictnds, nd, inlly, sttistics o the velocity luctutions tht will be speciied below. Henceorth, by the best prediction we men the prediction minimizing the men-squre devition, M j j p 1 rˆ (T) r j(t) min. It is well known (e.g., Liptser nd Shyryev 1978) tht the solution to this problem is given by the conditionl expecttion rˆ (T) E{r (T) r (t), r (t),...,r (t), 0 t T}, j j 1 p j p 1,...,M, () given or ll the observtions. Theoreticlly speking, these conditionl expecttions re expressed in terms o the joint probbility distribution o ll M trjectories, which is distribution in unctionl spce o Mdvector unctions. I the Lgrngin displcement is Mrkovin process, which is the cse or white noise velocity luctution, then the expecttions depend on r 1 (T), r (T),...,r p (T) only. Under some conditions stochstic dierentil eqution cn be derived or the optiml prediction in this cse (e.g., Liptser nd Shyryev 1978). Finlly, i the joint distribution o r 1 (T), r (T),..., r p (T) is Gussin, then the dependence is liner nd the corresponding weighing coeicients re determined by the men low nd sttistics o the second order, such s the Lgrngin velocity covrince tensor. Unortuntely, Mrkovin nd Gussin pproximtions or the prticle displcement re not stisctory or rel ocenic lows. As or the ormer, there is strong indiction (Gri et l. 1995; Gri 1996) tht the Lgrngin velocity hs signiicnt correltion time o 1 10 dys, nd its time evolution cn be well pproximted by sttionry Mrkovin process. Thus, the displcement cn be viewed s n integrl o Mrkovin process. Regrding Gussinity, notice tht even i the Eulerin velocity luctution is Gussin white noise in time, the joint distribution o prticle pir is not Gussin. Thus, one should rule out ny eort to get n exct expression or the optiml prediction. However, i the joint process o Lgrngin velocities [v 1 (t), v (t),..., v M (t)] is diusion, then so is the joint process o velocities nd positions [v 1 (t), r 1 (t), v (t), r (t),..., v M (t), r M (t)], nd one cn try to pply n extended Klmn ilter (EKF). Now we ormulte the model where this is the cse.

3 1400 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 FIG. 1. () Trjectories o Lgrngin prticles in stochstic isotropic low. Predictor trjectories (solid lines), predictnd trjectory (circles), nd Klmn ilter prediction (strs). Predictors strt rom the vertices o right hexgon (squres) nd predictnd strts rom its center. The observtion time T 15 dys, Lgrngin correltion time 3 dys, number o predictors p 6, hexgon rdius R 0 5 km, velocity spce correltion rdius R 00 km. (b) Sme s in () with R 100 km.

4 AUGUST 001 PITERBARG 1401 FIG. 1.(Continued) (c) Sme s in () with R 50 km. 3. Model Here we suggest Mrkov diusion model or the stochstic low. In the rmework o tht model, the joint motion o M prticles (M-prticle motion) is described s dm-diusion process (velocity nd position), whose drit is determined by the men low nd the diusion mtrix is expressed in terms o some covrince mtrix. Nmely, we ssume tht the ields o the Lgrngin velocity v(t, v 0, r 0 ) nd position r(t, r 0 ) corresponding to ny initil conditions v 0 nd r 0, stisy the ollowing Ito stochstic dierentil equtions: dv (v, r)dt dw(t, r), dr [U(r) v]dt, (3) where is deterministic ccelertion; w(t, r) iswiener processes (Brownin motion) in Hilbert spce, tht is, Ew(t, r) 0, T E{w(t 1, r 1)w(t, r )} min(t 1, t )B(r 1, r ); B(r 1, r ) is given covrince tensor; nd U(r) is given deterministic vector ield, which is not in generl the men low. Equtions (3) determine Brownin stochstic low in Eucliden spce R dm. In mthemtics, low mens mily { ts } o continuous mps o R dm in itsel deined or ny time moments t s, such tht tt is n identicl mp nd ts su tu or ny t s u, where the circle mens the successive ppliction o the mps strting rom the r let. I the mps re rndom, the low is clled stochstic. Finlly, i ts z su z nd su z re independent or t s u, z R dm, then the low is clled Brownin. We point out tht (3) deines Brownin low in the velocity-position phse spce, while its projection to the position phse spce is not Brownin. In prticulr, Eqs. (3) imply tht the motion o ny M prticles is diusion in dm dimensions (see Kunit 1990 or rigorous proo, or Piterbrg 1998 or explntions). Nmely, or M prticles introduce the stte nd drit vectors by v1 1 r 1 U(r 1) v 1 v z r, A(z) U(r ) v, vm M r U(r ) v M M M where m (v m, r m ). Then it ollows rom (3) tht z(t) isdm-dimensionl diusion process with the drit vector A(z) nd diusion mtrix D(z) given by with d d blocks D(z) [D (z)], ij ] [ B(r i, r j) 0 Dij. 0 0 More ccurtely,

5 140 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 FIG.. () Dependence o the prediction error on the prediction time nd the velocity spce correltion R nd its comprison with dispersion. Dispersion (dimonds), error or R 50 km ( ), R 100 km (circles), R 150 km (tringles), R 00 km (strs), R 50 km (squres), 3 dys, R 0 50 km. (b) Dependence o the reltive prediction error on the prediction time nd the initil cluster rdius R 0 nd its comprison with dispersion. Dispersion (dimonds), error or R 0 15 km ( ), R km (circles), R 0 75 km (tringles), R 0 50 km (strs), R 5 km (squres), 3 dys, R 50 km.

6 AUGUST 001 PITERBARG / dz A(z)dt D(z) dw, (4) where W(t) is stndrd Wiener process in dm dimensions, tht is, E{W(t 1 )W(t ) T } min(t 1, t )I, where I is the dm dm unit mtrix. In prticulr, (4) implies the common Lngevin eqution or the one-prticle motion 1/ dv (v, r)dt B(r, r) dw(t), dr [U(r) v]dt, where w(t) is stndrd d-dimension Wiener process. The two-prticle motion is covered by dv1 (v 1, r 1)dt 11(r 1, r )dw 1(t) 1(r 1, r )dw (t), dr [U(r ) v ]dt, dv (v, r )dt (r, r )dw (t) (r, r )dw (t), dr [U(r ) v ]dt, (5) where (w 1, w ) is stndrd d Wiener process nd d d mtrix ( ij ) stisies [ ] B(r 1, r 1) B(r 1, r ) *. B(r, r ) B(r, r ) 1 3 Further, we concentrte on the sttionry homogeneous cse deined by (v, r) v, U(r) U, B(r 1, r ) B(r1 r ), where nd U re positive constnt mtrix nd constnt vector, respectively. It cn be shown tht under these ssumptions nd specil initil conditions the Eulerin velocity is homogeneous-in-spce nd sttionry-in-time rndom ield. Detils on reltion o the discussed Lgrngin model nd Euler equtions cn be ound in PIT. Under the homogeneity conditions (4) becomes, 1/ dv vdt B(0) dw, dr (U v)dt. In the most importnt pplictions o the D cse, i mtrices nd B(0) re digonl, 1/ u 0 u/ u 1 0 1/ 0 /, B(0), then the velocity components re uncorrelted nd we hve or ech component (i 1, ), du (u/ )dt / dw, u u u 1 d ( / )dt / dw, dx (U u)dt, dy (V )dt, where u, nd u, re vrinces nd Lgrngin correltion times or the corresponding component (south north or est west). Thus, we hve the well-known rndom light model or the one-prticle motion studied in Thomson (1986) nd Gri (1996). The Coriolis term could be incorported into this model by ssuming, 1/ u, 1/ where is the Coriolis prmeter. Notice the ollowing our etures o the model, which re importnt rom the ppliction viewpoint. First, the one-prticle motion is described by rndom light model, which mthemticlly is nothing more thn the well-known Ornstein Uhlenbeck process or the Lgrngin velocity. Such model is very common nd well ounded in ocenogrphy nd meteorology (Thomson 1986; Gri et l. 1995; Gri 1996; Roden 1996; Reynolds 1998). Second, the model cn be deduced rom resonble stochstic eqution or the corresponding Eulerin velocity ield; hence, it hs solid physicl bckground. Third, it contins ew well-estimted or well-determined prmeters. Finlly, it yields mthemticlly consistent description o the Lgrngin motion o ny number o prticles. Notice tht even the two-prticle motion modeling is nontrivil problem. Some physiclly resonble models hve been ormulted or this purpose (e.g., Thomson 1990; Sword 1993; Borgs nd Sword 1994). These models re three-dimensionl nd lso bsed on diusion equtions like (5) with generl 6D drit nd digonl covrince mtrix o the stochstic orcing; tht is, the orcing is ssumed to be white noise in spce s well s in time. The cittions bove ddress inding drit such tht the two-point Eulerin velocity distribution is Gussin. It cn be esily seen tht there is no such drit mking both one- nd two-point Eulerin velocity distributions Gussin in the cse o generl spce covrince. Another pproch or constructing the two-prticle motion is ddressed in Pedrizzetzi nd Novikov (1994). In the sme Mrkov rmework the drit is ound rom physicl considertions nd then the corresponding two-point Eulerin velocity distribution is computed rom the corresponding Fokker Ulnk eqution. The obtined solution hs nothing to do with the Gussin distribution even though it ws used s n initil condition. Similr results hve been climed by Pope (1987). It is importnt to point out tht the well-mixed condition, being milestone in the cited ppers, is not n independent ssumption, but rther n exct consequence o the Mrkov property in the cse o the divergence-ree Eulerin velocity ield (Pedrizzetti nd Novikov 1994). The ocus o this study is quite dierent. We re not interested in determining the probbilistic distribution o the Eulerin velocity ield corresponding to the bove Lgrngin model. Insted, we concentrte on the bove prediction problem. In prticulr, we study the dependence o the prediction skill on the rtio o the spce correltion rdius R nd the initil cluster dimeter R Prediction lgorithm In prctice, we encounter mesurements seprted by inite timescle, t, which cn be the sme order s the Lgrngin correltion time. Let us set

7 1404 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 z(n) z(n t). It is shown in Ozgokmen et l. (001, their ppendix A) tht the pproximtion o (4) with (v, r) v o order ( t) 3/ results in the ollowing irst-order discrete Mrkov model: z(n) [z(n 1)] q[n, r(n 1)], (6) where (I t )v1 r1 t[u(r 1) v ] r1 (I t )v r r, (z) r t[u(r ) v ], (7) r M (I t )v M r t[u(r ) v ] M M M I is the unit mtrix, nd the covrince mtrix o the stochstic prt is expressed in the orm, E{q(n, r)q(n, r)*} Q(r) (Q ij), where 4 4 blocks Q ij re given by 1 6 tb(r i, r j) 3( t) B(r i, r j) Qi,j. (8) 6 3 3( t) B(r, r ) ( t) B(r, r ) i j i j Note tht the strightorwrd irst-dierence pproximtion o (3), 1/ v(n) (I t )v(n 1) ( t) [n, r(n 1)], r(n) r(n 1) {U[r(n 1)] v(n 1)} t, (9) where (n, r) is sequence o independent rndom ields, leds to system similr to (6), but with degenerte covrince mtrix o the stochstic prt. This ct would prevent n implementtion o the EKF pproch. The second drwbck o (9) is tht it does not give ny explicit reltion between the covrince mtrix o (n, r) nd B(r i, r j ). Our pproch to the discretiztion is dierent. We irst linerize the originl timecontinuous system t the neighborhood o the current stte. Then solve the equtions exctly t n intervl o durtion t, nd, inlly, truncte the terms o order higher thn ( t) 3/ (see Ozgokmen et l. 001, their ppendix A, or detils). Let us supply the stte Eq. (6) with the observtion eqution, y(n) Hz(n), where the observtionl pd Md mtrix is H (I 0), p nd I p is the pd pd unit mtrix. The distinguished eture o the system (6) is tht the noise depends on the stte. I this ws not the cse, tht is, q(n, r) q(n), then one could use the common EKF or the system (6). Als, the stndrd pproch is not pplicble or hndling the considered cse. Moreover, systems with noise depending on the stte re poorly studied in mthemtics. The diiculty here is tht even though the sequence {q(n, r) n 1,,... } itsel is sequence o independent Gussin rndom ields, it loses this property ter substituting r(n 1) insted o r. More exctly, the sequence {q[n, z(n 1)] n 1,,... }is not sequence o independent rndom vectors. For the bove reson we suggest prediction lgorithm tht inherits only the ide o the Klmn ilter nd is not optiml. However, we will show tht in one simple cse it works s optiml. The suggested lgorithm cn be viewed s n dptive Klmn ilter since its weighing coeicients depend on the previous observtions. Here we give the computtionl ormuls only or the cse o smll t ( t A t G K 1). The generl cse, s well s the derivtion, cn be ound in Osgokmen et l. (001, their ppendix B). Nmely, the predicted vlues o the velocities nd positions or the unobservble prticles corresponding to subs i p 1,...,M re computed by p i i ij j j j 1 p i i ij j j j 1 v (n) v (n) K [v (n) v (n)], r (n) r (n) K [r (n) r (n)]. (10) The superscripts nd stnd or nlyzed nd orecsted, respectively, in ccordnce with the common Klmn ilter nottion. Notice tht or the observed prticles, the nlyzed vribles coincide with observtionl ones: vi(n) v i (n), ri(n) r i (n), nd i 1,...,p. The ore- csted vlues re computed by v k(n) (I t )v k(n 1), r k(n) r k(n 1) {U[r k(n 1)] v k(n 1)} t, k 1,...,M, nd p K B[r (n 1), r (n 1)]B 1 ij i k kj (n 1), k 1 i p 1,...,M, j 1,...,p, (11) 1 where B (n) re entries o {B[r k (n), r j (n)]} 1 kj. In contrst to (10), the nive pproximtion (9) o order t gives p v i (n) v i (n) K ij[v j(n) v j(n)], j 1 r i (n) r i(n), (1) tht is, diers rom (10) by bsence o the correcting term in the position prediction. For simultions, we use

8 AUGUST 001 PITERBARG 1405 [ ] (1 y /R ) exp( r /R ) (xy/r ) exp( r /R ) 1/ 0 B(r),, (13) (xy/r ) exp( r /R ) (1 x /R ) exp( r /R ) 0 1/ where R is the spce correltion scle nd is the Lgrngin correltion time common or both components. The bove covrince mtrix corresponds to the orcing stremunction with the covrince proportionl to exp( r /R ). It is worthy to give the predictions ormuls or only one predictor in the coordinte-wise orm. Nmely, let (x 1, y 1 ) nd (x, y ) be the coordintes o the predictor nd predictnd, respectively, nd (u 1, 1 ) nd (u, ) be their velocities. Set x x x 1, y y x 1, nd x y. Then (10) becomes u (n) u (n 1) [u (n) u (n 1)] n c n 1[ 1(n) 1(n 1)], (n) (n 1) c [u (n) u (n 1)] n b n 1[ 1(n) 1(n 1)], x (n) x (n 1) u (n 1) t [x (n) x (n 1) u (n 1) t] n c n 1[y 1(n) y 1(n 1) 1(n 1) t], y (n) y (n 1) (n 1) t where c b [x (n) x (n 1) u (n 1) t] n [y (n) y (n 1) (n 1) t], n n [1 y(n) /R ] exp[ (n) /R ], bn [1 x(n) /R ] exp[ (n) /R ], cn [ x(n)y(n)/r ] exp[ (n) /R ], (14) 1 t. (15) Now n exmple is given in the generl sitution where the prediction ormuls in (10) result in n exct prediction, while (1) cn give n error growing exponentilly in time. Becuse o this, justiiction or keeping the terms o order o ( t) 3/ is provided. Nmely, we strt with the equtions v v (t), ṙ Gr v, (16) where the turbulent velocity does not depend on the position t ll, G is constnt mtrix chrcterizing the sher o the men low, E{ (t)} 0, E{ (t ) (t )*} (t t )I, 1 1 where 1 nd re sclr prmeters. The model (16) oversimpliies the relity; however, it ws used by severl uthors or theoreticl considertions o the turbulent diusion in the ocen (e.g., Zmbinchi nd Gri 1994). Since in (16) the luctution velocity is the sme or ll prticles, it is enough to consider only one predictor tht gives complete inormtion on the luctution velocity in ll spce. Thereore, the cse o only two prticles is considered when one o them is predictor nd nother is predictnd. The linerity o (16) llows the exct discretiztion given by (6) nd (7): v(n) v(n 1) (n), r(n) Fr(n 1) Sv(n 1) (n), where exp( t ), F exp( tg), 1 S (exp( tg) exp( t )I)(G I), E{ (n) (n)*} t I, 3 ( t) E{ (n) (n)*} I, 3 ( t) E{ (n) (n)*} I. Thus, we hve or the predictnd v M(n) v M(n 1) (n), r (n) Fr (n 1) Sv (n 1) (n), (17) M M M nd the sme equtions or the predictor (with dierent initil conditions): v p(n) v p(n 1) (n), r (n) Fr (n 1) Sv (n 1) (n). (18) p p p From (18) we cn ind (n) nd (n) exctly nd substitute them to (17). The result is v M(n) v M(n 1) v p(n) v p(n 1), r (n) Fr (n 1) Sv (n 1) r (n) M M M p Fr p(n 1) Sv p(n 1). (19) Now pply the prediction ormul (10). Since the correltion between the predictor nd predictnd velocities is 1, we should set K 1 1. This results in v M(n) v M(n 1) v p(n) v p(n 1), r M(n) Fr M(n 1) Sv M(n 1) r p(n) Fr p(n 1) Sv p(n 1). (0) Thus, the predicted velocities nd positions re computed by the sme ormuls (0) s the rel ones (19)

9 1406 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 with the sme initil conditions. Hence, the prediction is bsolutely exct. Now we shll ind the prediction error in the cse o the simpliied lgorithm (1). By subtrcting (17) rom (1), we get r (n) r (n) F[r (n 1) r (n 1)]. M M M M Since (n) is independent o r M (n 1) r M (n 1), we obtin or the error s(n) E{[r M(n) r M(n)]*[r M(n) r M(n)]} n 1 Sp{[(FF*) I](FF* I) q}, where q E (n)* (n). By substituting the expression or F we get s(n) Sp {exp[n t(g G*)] I} {exp[n t(g G*)] I} q. (1) Note tht q does not depend on the observtion time T n t. From (1) it ollows tht i the symmetrized grdient mtrix G G G* hs positive eigenvlue, then s(n) grows exponentilly with T. More precisely, i t is ixed nd T, then s(n) 3 ( t) d 3 exp( T), m where m is the mximum eigenvlue o G. 5. Simultions nd discussion The experiment conditions re s ollows. Initilly, the observed prticles (predictors) re locted in the vertices o right polygon with the rdius R 0, while the unobserved prticle is relesed t the center o the polygon t the sme time. The initil velocities or ll the prticles re Gussin independent vribles. The simulted motion is covered by the ollowing inite dierence equtions: v(n) exp( dt/ )v(n 1) 1/ D[r(n 1)] dt/ n, r(n) r(n 1) v(n)dt, n 1,,...,N, () where v (v 1,..., v M ), r (r 1,..., r M ), D(r) [B(r i, r j )] with B(r) given in (13), nd n re independent stndrd Gussin M-vectors. Thus, we ocus on the isotropic cse with no men low t ll, since the gol o this pper is to study the eect o stochstic currents. The ollowing prmeters re ixed during ll the experiments: simultion step, dt 1 h; ssimiltion step, t 1 h; men-squre velocity, 0 km dy 1 ; nd the mximum prediction time T m Ndt 15 dys; while the rest o prmeters typiclly vry in the ollowing rnges: Lgrngin correltion time, 1 10 dys; velocity spce scle, R 0 50 km; initil cluster rdius, R km; number o predictors, p M 1 1 6; nd prediction time, T 1 15 dys. Figures 1 c demonstrte trjectories o the observed prticles (solid, unmrked lines), the predictnd (circles), nd the Klmn ilter (KF) prediction (strs) or smll, medium, nd lrge rtio R/R 0, respectively. In the irst cse, R 00 km, while R 0 5 km is ixed or ll three experiments. The prediction in this cse is very good due to the ct tht three out o six predictors re closely ollowing the predictnd becuse o high correltion between their velocities. In ct, or ll the moments the error o prediction is less thn 15 0 km. The interesting thing is tht the predictions on the 13th nd 14th dys re better thn in the intermedite term s 5 10 dys. Thus, the error is not ccumulted. When R 100 km the prediction is still very good. In ct, it is not worse thn in the previous cse, even though only single predictor out o six is stying close to the predictnd during 15 dys, while three prticles got wy rom the re nd two prticles stuck ner the initil polygon. So, the lgorithm works in wy such tht good prediction is ensured by hving single lucky predictor. It would be nive to suppose tht the unobservble prticle should ollow the mjority o the observble ones. Under such guess, one would look or the predictnd somewhere in the northwest corner where two predictors heded. In ct, the prediction skill is completely determined by whether one or more predictors re stying close to the predictnd during the observtion time. Let us cll such predictors the signiicnt predictors. The lgorithm utomticlly picks up signiicnt predictors by djusting weights K ij in (10), which decy rpidly with the distnce between predictnd nd the corresponding predictor. In prticulr, in the cse o sole signiicnt predictor the prediction ormul turns to (14). From this ormul nd explicit expressions (15) or the coeicients, one cn see tht i the distnce between two prticles is smll, then nd b re close to 1, while c is bout 0. Hence, the velocity nd position o the unobserved prticle pproch the corresponding prmeters or the predictor. Finlly, in the cse R 50 km, the prediction completely ils. The prticles re becoming uncorrelted rom the very beginning. All the predictors re running wy one rom nother s well s rom the predictnd. The result is unortunte: the error is bout 50 km in 15 dys, tht is, close to the dispersion. Thus, the rtio R/R 0 is crucil or the prediction skill. Now we look t this dependence more creully. Introduce the men-squre-error prediction error nd dispersion [ ] 1/ L 1 M,k M,k L k 1 s(n) r (n) r (n), [ ] 1/ L 1 M,k M,k L k 1 r(n) r (0) r (n), where L is the number o runs, typiclly ssumed 100; nd r M,k (n) nd (n) re the position nd its prediction r M,k

10 AUGUST 001 PITERBARG 1407 FIG. 3. () Dependence o the reltive prediction error on the Lgrngin correltion time nd the velocity spce correltion rdius; R 0 50 km. (b) Dependence o the reltive prediction error on the velocity spce correltion rdius; R 0 50 km, 3 dys. o the predictnd t time n in the kth run. The dependence o s(n) on the prediction time or ixed R 0 50 km nd dierent R 50, 100, nd 00 km is shown in Fig.. The dimond line shows the dispersion r(n). In greement with the clssicl theory, the dispersion is liner t the initil stge nd proportionl to t, where t ndt, or lrge t. It is bout 90 km in T 15 dys. Notice tht the displcement cn be theoreticlly estimted s T 68 km using the diusivity or- mul D or single component (Gri 1996). The curves line up with the correltion rdius: the lrger R, the less error s(n) or ny time moment n. The explntion is simple in the bsence o the men low, or the lrge correltion scle the probbility is high tht one o the predictors is close enough to the predictnd, ensuring good prediction conditions. The error o the

11 1408 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 best prediction corresponding to the lrgest R 50 km is just bout 30 km, tht is, 10% o the dispersion or the prediction time o 15 dys. The dependence o the prediction error on the prediction time or ixed R 00 km nd dierent R 0 5, 50, 75, 100, nd 15 km is shown in Fig. b. First, notice tht the dispersion in this study is bout 75 km nd hence grees with the theory nd the previous experiment. The ltter indictes tht 100 runs is enough or resonble sttisticl inerence. Recll tht the sttistics o n individul prticle is determined by, T, only, while the two-prticle sttistics is essentilly dependent on R nd R 0. The experiments resulting in Figs.,b re in ull greement or vlues R 50 km, R 0 50 km (squres in the irst cse nd strs in the second cse). The inl error is bout 30 km in the irst cse nd close to 5 km in the second one. Also, the picture in Fig. b is similr to tht o Fig. in the sense tht the error lines up with the vlue o R 0 : the less R 0, the better the prediction. However, there is n essentil dierence. Chnging R 0 in wide enough rnge, km, does not cuse drstic chnges in the prediction skill. The error grows rom 5 to 50 km. There is lmost no dierence in the error or initil vlues o the cluster rdius 5 nd 50 km. The simultion results show tht the error is not determined by the rtio R/R 0, but rther is complex unction o R nd R 0. Introduce the reltive error s(n) s r(n). r(n) Figure 3 illustrtes the dependence o s r (T)onRnd or ixed R 0 50 km nd M 7. As we observed erlier, the error decys st with incresing R or ll vlues o in the rnge 1 15 dys. To stress this once gin we give rgment o this dependence or dys in Fig. 3b. However, the error lmost does not chnge with chnging in the brod rnge, 4 dys. At the irst glnce it seems strnge becuse the incresing correltion time should increse the inerti nd hence improve the predictbility. Remember, however, tht we discuss the reltive error. In ct, incresing the correltion timescle implies incresing the dispersion r(t) t. Thereore, our simultions indicte tht the bsolute error s should be inverse proportionl to. The ct o wek dependence o the error on the Lgrngin time is lso supported by the grph in Fig. 4, where the dependence o the 15-dy reltive prediction error on is shown or much bigger rnge o. The conclusion is the sme: the reltive error lmost does not depend on the Lgrngin correltion time ter 3 dys. It is worthwhile to compre this conclusion with study o the Lypunov exponent L in PIT, or the sme stochstic model. It ws shown tht L, s unction o two time prmeters nd 0 R/, only slightly vries long with, while showing strong FIG. 4. () Long-term dependence o the reltive prediction error on the Lgrngin correltion time; R 0 50 km, R 50 km. (b) Dependence o the reltive prediction error on the Lgrngin correltion time or dierent number o driters: M 3 (squres), M 5 (dimonds), M 7 (circles). R 0 50 km, R 50 km. dependence on 0. Since the quntity 1/L chrcterizes the predictbility limit, one cn conclude tht the presented study is in greement with the Lypunov exponent considertion. Another interesting nd unexpected inding is tht the error shrply increses ter 3 dys. The sme eture we cn observe in Fig. 4b, where the dependence on is shown or dierent number o predictors. Notice tht there is moderte improvement in the prediction when the number o predictors grows. For 3 dys the errors or p, 4, nd 6 re 0.05, 0.09, nd 0.17, respectively; or 10 dys the igures re 0.15, 0., nd 0.6, respectively. In ll cses the error increses in the 1 5-dy rnge nd then becomes lt. Finlly, the prediction error dependence o the num-

12 AUGUST 001 PITERBARG 1409 FIG. 5. () Dependence o the reltive prediction error on the number o driters or dierent spce correltion scles: R 100 km (dimonds), R 00 km (circles), 3 dys, R 0 50 km. Initilly the predictors re plced t the vortices o the right polygon nd the predictnd is plced t its center. (b) Sme s in () with the predictors initilly plced equidistntly long stright line. The distnce rom the initil predictnd position to the line is 5 km nd the distnce between the end predictors is 100 km. The end predictors hve the sme distnce rom the predictnd. (c) Sme s in () with the predictors initilly plced rndomly over the squre o 100 km 100 km, centered t the predictnd position. The predictors re distributed uniormly. ber o predictors, p M 1, ws studied. For p 7, the p p mtrix D [B(r i, r j )], which is used in clcultion o the weights (11), sometimes becomes illconditioned, resulting in signiicnt computtionl error. For this reson the lgorithm ws slightly modiied by chnging D to D I whenever det(d) 10 10, where is smll number nd I is the p p unit mtrix. This regulriztion procedure hs been used in Ozgokmen et l. (000) or rel dt; however, its ccurcy ws not studied. Figure 5 demonstrtes the prediction skill or R 100 km nd R 00 km s unction o the predictor number. In the irst cse, the error decreses only rom 0.4 to 0.36 s p increses rom 7 to 0. In the second cse, the error even increses rom 0.14 to 0.8 or the sme vlues o p. Such trend is typicl or other experiments not included in this pper, which covered wider rnge o R nd dierent vlues o. The sme conclusion ollows rom our considertion o initil predictor conigurtions dierent rom the right polygon conigurtion. First, the predictors were locted long the stright line distnced rom the predictnd. Detils re given in the cption to Fig. 5b. Then, we tried rndom distribution o the predictors round the predictnd. In both cses (Figs. 5b,c) the dependence o the error on the number o predictors remins bsiclly the sme. Roughly, or p 7 the error is bout i R 00 km, nd bout i R 100 km. The initil right polygon conigurtion

13 1410 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 gives slightly better results thn the liner nd rndom ones. There is no indiction tht incresing the number o predictors beyond 4 7 improves the prediction skill. It is quite possible tht the bove simplest regulriztion procedure is to blme or tht. This procedure helps to void drstic computtionl errors; however, it results in systemtic bis nd or this reson is not quite stisctory. Other regulriztion procedures should be studied. In summry, we cn conclude tht the prediction skill improves rtio R/R 0 increses, where, in generl, R 0 is chrcteristic rdius o the cloud o predictors. The lest prediction error is due to smll Lgrngin correltion time, 1 3 dys, or the prediction time T 15 dys. Under the simplest regulriztion procedure, the error slowly decreses s the number o predictors grows or rtio R/R 0 round 1, while the prediction skill slightly worsens or does not chnge or p lrger thn 4 7 in the cse R/R 0 4. Acknowledgments. The work ws supported by ONR Grnt N The uthor thnks Tmy Ozgokmen nd the nonymous reviewer or remrks tht helped improve the presenttion. REFERENCES Buer, S., M. S. Swenson, A. Gri, A. J. Mrino, nd K. Owens, 1998: Eddy-men low decomposition nd eddy-diusivity estimtes in the tropicl Pciic Ocen. Prt I: Methodology. J. Geophys. Res., 103, Borgs, M. S., nd B. L. Sword, 1994: A mily o stochstic models or two-prticle dispersion in isotropic homogeneous sttionry turbulence. J. Fluid Mech., 79, Cstellri, S., A. Gri, T. M. Özgökmen, nd P. -M. Poulin, 001: Prediction o prticle trjectories in the Adritic Se using Lgrngin dt ssimiltion. J. Mr. Syst., 9, Dvis, R. E., 1991: Lgrngin ocen studies. Ann. Rev. Fluid Mech., 3, , 1991b: Observing the generl circultion with lots. Deep-Se Res., 38, Gri, A., 1996: Applictions o stochstic prticle models to ocenogrphic problems. Stochstc Modelling in Physicl Ocenogrphy, R. Adler et l., Eds., Birkhuser, , K. Owens, L. Piterbrg, nd B. Rozovskii, 1995: Estimtes o turbulence prmeters rom Lgrngin dt using stochstic prticle model. J. Mr. Res., 53, Ishikw, Y. I., T. Awji, nd K. Akimoto, 1996: Successive correction o the men se surce height by the simultneous ssimiltion o driting buoy nd ltimetric dt. J. Phys. Ocenogr., 6, Kunit, H., 1990: Stochstic Flows nd Stochstic Dierentil Equtions. Cmbridge University Press, 347 pp. Liptser, R. S., nd A. N. Shiryev, 1978: Sttistics o Rndom Processes. Springer-Verlg, 44 pp. Ozgokmen, T., A. Gri, L. Piterbrg, nd A. Mrino, 000: On the predictbility o Lgrngin trjectories in the ocen. J. Atmos. Ocenic Technol., 17, , I. Piterbrg, A. J. Mrino, nd E. H. Ryn, 001: Predictbility o driter trjectories in the tropicl Pciic Ocen. J. Phys. Ocenogr., 31, Pedrizzetti, G., nd E. A. Novikov, 1994: On Mrkov modeling o turbulence. J. Fluid Mech., 80, Piterbrg, L. I., 1998: Drit estimtion or Brownin lows. Stochstic Processes Appl., 79, , 000: The top Lypunov exponent or stochstic low modeling the upper ocen turbulence. SIAM J. Appl. Mth., in press. Pope, S. B., 1987: Consistency conditions or rndom wlk models o turbulent dispersion. Phys. Fluids, 30, Reynolds, A. M., 1998: On the ormultion o Lgrngin stochstic models o sclr dispersion within plnt cnopies. Bound.-Lyer Meteor., 86, Roden, H. C., 1996: Stochstic Lgrngin Models o Turbulent Diusion. Meteor. Monogr., No. 48, Amer. Meteor. Soc., 84 pp. Sword, B. L., 1993: Recent developments in the Lgrngin stochstic theory o turbulent dispersion. Bound.-Lyer Meteor., 6, Schneider, T., 1998: Lgrngin driter models s serch nd rescue tools. M.S. thesis, Dept. o Meteorology nd Physicl Ocenogrphy, University o Mimi, 96 pp. Swensson, M. S., nd P. P. Niiler, 1996: Sttisticl nlysis o the surce circultion o the Cliorni current. J. Geophys. Res., 101, Thomson, D. J., 1986: A rndom wlk model o dispersion in turbulent lows nd its ppliction to dispersion in vlley. Qurt. J. Roy. Meteor. Soc., 11, , 1990: A stochstic model or the motion o prticle pirs in isotropic high-reynolds-number turbulence, nd its ppliction to the problem o concentrtion vrince. J. Fluid Mech., 10, Zmbinchi, E., nd A. Gri, 1994: Eects o inite scles o turbulence on dispersion estimtes. J. Mr. Res., 5,