Short-Term Prediction of Lagrangian Trajectories - AMS Journals

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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

VOLUME 18

Short-Term Prediction of Lagrangian Trajectories LEONID I. PITERBARG Center of Applied Mathematical Studies, University of Southern California, Los Angeles, California (Manuscript received 22 August 2000, in final form 5 January 2001) ABSTRACT Lagrangian particles in a cluster are divided in two groups: observable and unobservable. The problem is to predict the unobservable particle positions given their initial positions and velocities based on observations of the observable particles. A Markov model for Lagrangian motion is formulated. The model implies that the positions and velocities of any number of particles form a multiple diffusion process. A prediction algorithm is proposed based on this model and Kalman filter ideas. The algorithm performance is examined by the Monte Carlo approach in the case of a single predictand. The prediction error is most sensitive to the ratio of the velocity correlation radius and the initial cluster radius. For six predictors, if this parameter equals 5, then the relative error is less than 0.1 for the 15-day prediction, whereas for the ratio close to 1, the error is about 0.9. The relative error does not change significantly as the number of predictors increases from 4–7 to 20.

1. Introduction The role of drifters, current-following devices, in ocean studies has drastically increased in recent decades (Davis 1991a,b). Basically, drifter data have been used for the mapping of oceanic circulation (e.g., Swensson and Niiler 1996; Bauer et al. 1998), estimation of mixing parameters (e.g., Griffa et al. 1995), and assimilation in numerical models (e.g., Ishikawa et al. 1996). Recently, a new direction of research concerning the predictability and prediction of the Lagrangian motion in the upper ocean has come forward (Ozgokmen et al. 2000, 2001; Castellari et al. 2001). In particular, this direction was stimulated by a need for search-and-rescue operations in the sea (Schneider 1998). Investigation of the drifter position predictability is also important from the fundamental research viewpoint since it is a part of the general Lagrangian predictability problem in chaotic systems. No doubt, the upper ocean can and should be treated as a dynamical stochastic system due to combined effects of deterministic mean circulation and eddy turbulence. For now, the best and most common prediction tool in such systems is the extended Kalman filter. Not surprising, just Kalman-type filters were applied to real and synthetic data in the works cited above. We notice that the problem of predicting a sole drifter position was considered with no comprehensive sensitivity analysis. The goal of this paper is to extend the Kalman filter Corresponding author address: Dr. Leonid I. Piterbarg, Center of Applied Mathematical Sciences, University of Southern California, 1042 W. 36th Place, DRB 155, Los Angeles, CA 90089-1113. E-mail: [email protected]

q 2001 American Meteorological Society

approach to predicting the positions of several particles and studying the prediction error via stochastic simulations. In particular, we set up to investigate the error dependence (sensitivity) on the velocity correlation radius, the initial cluster radius, the Lagrangian correlation time, and the number of predictors. We concentrate on the short-term prediction, restricted by 15 days, and a small number of predictors (4–7). The cornerstone of our approach is a stochastic model of the Lagrangian turbulence in the upper ocean, first reported in Piterbarg (2000). The model implies that the velocities and positions of any M particles form a Markov process in 4M dimensions—two velocity components and two coordinates for each particle. In the simplest isotropic case the model is completely defined by few parameters: the velocity variance s 2 , Lagrangian correlation time t, and space correlation radius R. Typical considered values are s 5 20 km day 21 , t 5 2– 10 days, R 5 50–300 km. A particular version of the discussed algorithm has already been applied for studying the Lagrangian motion predictability in the tropical Pacific using real data (Ozgokmen et al. 2001). Unlike that work, here the model is formulated in the general case including three-dimensional motion and inhomogeneous environments. Yet, the prediction algorithm is extended to predicting several particle positions. However, the error analysis is focused on the 2D isotropic case and one predictand only. An important feature of the stochastic differential equations describing a cluster motion is that the corresponding diffusion matrix depends on the state variable. More exactly, the velocity covariance matrix de-

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pends on the particle positions at any time. This fact makes direct application of the Kalman filter impossible since it implies that the covariance matrix of the stochastic forcing does not depend on the state vector. However, the main idea of optimally combining the dynamics and observations can be realized. We just use the predicted and observed positions in the previous step for evaluating the stochastic forcing covariance. Of course, this approach does not yield the optimal estimator. Moreover, it is very difficult to determine analytically how close the proposed prediction is to the optimal one. In this situation there is nothing better than using stochastic simulations to evaluate the prediction skill. Using this approach we study the dependence of the prediction error on R, t, initial cluster radius R 0 , and the number of predictors. The paper is organized as follows. In section 2 the mathematical problem statement is given, in section 3 the model is formulated and explained, in section 4 the prediction algorithm is presented and an example is given where the exact prediction is reached, and results of simulations and their discussion are given in section 5. 2. Problem statement Mathematically, the Lagrangian prediction is a problem of filtering a multidimensional stochastic process driven by a system of stochastic differential equations. We explain that in a moment. For the sake of generality, the motion in the Euclidean space R d of any dimension d 5 1, 2, 3, . . . is considered. Let u(t, r) 5 U(t, r) 1 u9(t, r) be a decomposition of a d-dimensional velocity field into mean circulation and fluctuation. More exactly, U(t, r) is a deterministic velocity field, u9(t, r) is a random vector field with zero mean, and E{u9(t, r)} 5 0, E{ } denotes expectation. Consider M Lagrangian particles, that is, current-following fluid parcels, starting at the same time t 5 0 from different positions r10 , r 02 , . . ., r M0 . Their motion is covered by the following system of Md equations: drj 5 u(t, rj ), dt

rj (0) 5 r j0 ,

(1)

where j 5 1, . . . , M. We denote v j (t) 5 dr j (t)/dt the Lagrangian velocity of the jth particle, while u(t, r) is called the Eulerian velocity field. Both velocities are related in a simple way: v j (t) 5 u[t, r j (t)]. Since the right-hand side of (1) includes the velocity fluctuations as well as the mean flow we call the equations stochastic differential equations and call the system itself a dynamical stochastic system. Assume that trajectories of the first p , M particles r1 (t), r 2 (t), . . . , r p (t) are observed during time interval

(0, T), while the trajectories of the remaining particles r p11 (t), r P12 (t), . . . , r M (t) are not observed. We call the first and second particles predictors and predictands, respectively. The problem is finding the best prediction, rˆ p11 (T), rˆ p12 (T), . . . , rˆ M (T) of the positions of the unobserved particles given the above predictor observations, mean flow U(t, r), the initial positions of the predictands, and, finally, statistics of the velocity fluctuations that will be specified below. Henceforth, by the best prediction we mean the prediction minimizing the mean-square deviation,

O |rˆ (T ) 2 r (T )| → min. M

2

j

j

j5p11

It is well known (e.g., Liptser and Shyryaev 1978) that the solution to this problem is given by the conditional expectation rˆj (T ) 5 E{rj (T ) | r1 (t), r2 (t), . . . , rp (t), 0 # t # T}, j 5 p 1 1, . . . , M,

(2)

given for all the observations. Theoretically speaking, these conditional expectations are expressed in terms of the joint probability distribution of all M trajectories, which is a distribution in a functional space of Mdvector functions. If the Lagrangian displacement is a Markovian process, which is the case for a white noise velocity fluctuation, then the expectations depend on r1 (T), r 2 (T), . . . , r p (T) only. Under some conditions a stochastic differential equation can be derived for the optimal prediction in this case (e.g., Liptser and Shyryaev 1978). Finally, if the joint distribution of r1 (T), r 2 (T), . . . , r p (T) is Gaussian, then the dependence is linear and the corresponding weighing coefficients are determined by the mean flow and statistics of the second order, such as the Lagrangian velocity covariance tensor. Unfortunately, Markovian and Gaussian approximations for the particle displacement are not satisfactory for real oceanic flows. As for the former, there is a strong indication (Griffa et al. 1995; Griffa 1996) that the Lagrangian velocity has a significant correlation time of 1–10 days, and its time evolution can be well approximated by a stationary Markovian process. Thus, the displacement can be viewed as an integral of a Markovian process. Regarding Gaussianity, notice that even if the Eulerian velocity fluctuation is a Gaussian white noise in time, the joint distribution of a particle pair is not Gaussian. Thus, one should rule out any effort to get an exact expression for the optimal prediction. However, if the joint process of Lagrangian velocities [v1 (t), v 2 (t), . . . , v M (t)] is a diffusion, then so is the joint process of velocities and positions [v1 (t), r1 (t), v 2 (t), r 2 (t), . . . , v M (t), r M (t)], and one can try to apply an extended Kalman filter (EKF). Now we formulate the model where this is the case.

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FIG. 1. (a) Trajectories of Lagrangian particles in stochastic isotropic flow. Predictor trajectories (solid lines), predictand trajectory (circles), and Kalman filter prediction (stars). Predictors start from the vertices of a right hexagon (squares) and predictand starts from its center. The observation time T 5 15 days, Lagrangian correlation time t 5 3 days, number of predictors p 5 6, hexagon radius R 0 5 25 km, velocity space correlation radius R 5 200 km. (b) Same as in (a) with R 5 100 km.

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FIG. 1. (Continued) (c) Same as in (a) with R 5 50 km.

3. Model Here we suggest a Markov diffusion model for the stochastic flow. In the framework of that model, the joint motion of M particles (M-particle motion) is described as a 2dM-diffusion process (velocity and position), whose drift is determined by the mean flow and the diffusion matrix is expressed in terms of some covariance matrix. Namely, we assume that the fields of the Lagrangian velocity v(t, v 0 , r 0 ) and position r(t, r 0 ) corresponding to any initial conditions v 0 and r 0 , satisfy the following Ito stochastic differential equations: dv 5 a(v, r)dt 1 dw(t, r),

dr 5 [U(r) 1 v]dt,

(3)

where a is a deterministic acceleration; w(t, r) is a Wiener processes (Brownian motion) in a Hilbert space, that is, Ew(t, r) 5 0, E{w(t1, r1 )w(t 2 , r2 )T} 5 min(t1, t 2 )B(r1, r2 ); B(r1 , r 2 ) is a given covariance tensor; and U(r) is a given deterministic vector field, which is not in general the mean flow. Equations (3) determine a Brownian stochastic flow in a Euclidean space R 2dM . In mathematics, a flow means a family {Fts } of continuous maps of R 2dM in itself defined for any time moments t # s, such that Ftt is an identical map and Fts+Fsu 5 Ftu for any t # s # u, where the circle means the successive application of the maps starting from the far left. If the maps are random, the flow is called stochastic. Finally, if Ftsz–

Fsuz and Fsuz are independent for t , s , u, z ∈ R 2dM , then the flow is called Brownian. We point out that (3) defines a Brownian flow in the velocity-position phase space, while its projection to the position phase space is not Brownian. In particular, Eqs. (3) imply that the motion of any M particles is a diffusion in 2dM dimensions (see Kunita 1990 for a rigorous proof, or Piterbarg 1998 for explanations). Namely, for M particles introduce the state and drift vectors by 

v1  r1 v2 z 5  r2  , ··· vM  rM 



a1  U(r1 ) 1 v1 a2 A(z) 5  U(r2 ) 1 v2  , ··· aM U(rM ) 1 vM

 

 

 

where a m 5 a(v m , r m ). Then it follows from (3) that z(t) is a 2dM-dimensional diffusion process with the drift vector A(z) and diffusion matrix D(z) given by D(z) 5 [D ij (z)],

with 2d 3 2d blocks D ij 5

More accurately,

[

]

B(ri , rj ) 0 . 0 0

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FIG. 2. (a) Dependence of the prediction error on the prediction time and the velocity space correlation R and its comparison with dispersion. Dispersion (diamonds), error for R 5 50 km (1), R 5 100 km (circles), R 5 150 km (triangles), R 5 200 km (stars), R 5 250 km (squares), t 5 3 days, R 0 5 50 km. (b) Dependence of the relative prediction error on the prediction time and the initial cluster radius R 0 and its comparison with dispersion. Dispersion (diamonds), error for R 0 5 125 km (3), R 0 5 100 km (circles), R 0 5 75 km (triangles), R 0 5 50 km (stars), R 5 25 km (squares), t 5 3 days, R 5 250 km.

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dz 5 A(z)dt 1 D(z)1/2 dW,

(4)

where W(t) is a standard Wiener process in 2dM dimensions, that is, E{W(t1 )W(t 2 )T } 5 min(t1 , t 2 )I, where I is the 2dM 3 2dM unit matrix. In particular, (4) implies the common Langevin equation for the one-particle motion dv 5 a(v, r)dt 1 B(r, r)1/2 dw(t),

dr 5 [U(r) 1 v]dt,

where w(t) is a standard d-dimension Wiener process. The two-particle motion is covered by dv1 5 a(v1, r1 )dt 1 s11 (r1, r2 )dw1 (t) 1 s12 (r1, r2 )dw2 (t), dr1 5 [U(r1 ) 1 v1 ]dt, dv2 5 a(v2 , r2 )dt 1 s21 (r1, r2 )dw1 (t) 1 s21 (r1, r2 )dw2 (t), dr2 5 [U(r2 ) 1 v2 ]dt,

(5)

where (w1 , w 2 ) is a standard 2d Wiener process and 2d 3 2d matrix s 5 (s ij ) satisfies

ss * 5

[

]

B(r1, r1 ) B(r1, r2 ) . B(r2 , r1 ) B(r2 , r3 )

Further, we concentrate on the stationary homogeneous case defined by a(v, r) 5 2Lv,

U(r) [ U,

B(r1, r2 ) 5 B(r1 2 r2 ), where L and U are a positive constant matrix and constant vector, respectively. It can be shown that under these assumptions and special initial conditions the Eulerian velocity is a homogeneous-in-space and stationary-in-time random field. Details on relation of the discussed Lagrangian model and Euler equations can be found in PIT. Under the homogeneity conditions (4) becomes, dv 5 2Lvdt 1 B(0)1/2 dw,

dr 5 (U 1 v)dt.

In the most important applications of the 2D case, if matrices L and B(0) are diagonal, L5

1

1/t u 0

2

0 , 1/t y

B(0) 5

1

2su2 /t u 0

2

1 , 2sy2 /t y

then the velocity components are uncorrelated and we have for each component (i 5 1, 2), du 5 2(u/t u )dt 1 suÏ2/t u dw1, dy 5 2(y /t y )dt 1 sy Ï2/t y dw2 , dx 5 (U 1 u)dt,

dy 5 (V 1 y )dt,

where s and t u,y are variances and Lagrangian correlation times for the corresponding component (south– north or east–west). Thus, we have the well-known ‘‘random flight’’ model for the one-particle motion studied in Thomson (1986) and Griffa (1996). The Coriolis term could be incorporated into this model by assuming, 2 u,y

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L5

12 f

1/t u

2

f , 1/t y

where f is the Coriolis parameter. Notice the following four features of the model, which are important from the application viewpoint. First, the one-particle motion is described by a ‘‘random flight’’ model, which mathematically is nothing more than the well-known Ornstein–Uhlenbeck process for the Lagrangian velocity. Such a model is very common and well founded in oceanography and meteorology (Thomson 1986; Griffa et al. 1995; Griffa 1996; Rodean 1996; Reynolds 1998). Second, the model can be deduced from a reasonable stochastic equation for the corresponding Eulerian velocity field; hence, it has a solid physical background. Third, it contains a few well-estimated or well-determined parameters. Finally, it yields a mathematically consistent description of the Lagrangian motion of any number of particles. Notice that even the two-particle motion modeling is a nontrivial problem. Some physically reasonable models have been formulated for this purpose (e.g., Thomson 1990; Sawford 1993; Borgas and Sawford 1994). These models are three-dimensional and also based on diffusion equations like (5) with a general 6D drift and diagonal covariance matrix of the stochastic forcing; that is, the forcing is assumed to be a white noise in space as well as in time. The citations above address finding a drift such that the two-point Eulerian velocity distribution is Gaussian. It can be easily seen that there is no such drift making both one- and two-point Eulerian velocity distributions Gaussian in the case of general space covariance. Another approach for constructing the two-particle motion is addressed in Pedrizzetzi and Novikov (1994). In the same Markov framework the drift is found from physical considerations and then the corresponding two-point Eulerian velocity distribution is computed from the corresponding Fokker–Ulank equation. The obtained solution has nothing to do with the Gaussian distribution even though it was used as an initial condition. Similar results have been claimed by Pope (1987). It is important to point out that the ‘‘well-mixed condition,’’ being a milestone in the cited papers, is not an independent assumption, but rather an exact consequence of the Markov property in the case of the divergence-free Eulerian velocity field (Pedrizzetti and Novikov 1994). The focus of this study is quite different. We are not interested in determining the probabilistic distribution of the Eulerian velocity field corresponding to the above Lagrangian model. Instead, we concentrate on the above prediction problem. In particular, we study the dependence of the prediction skill on the ratio of the space correlation radius R and the initial cluster diameter R 0 . 4. Prediction algorithm In practice, we encounter measurements separated by a finite timescale, Dt, which can be the same order as the Lagrangian correlation time. Let us set

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z(n) [ z(nDt). It is shown in Ozgokmen et al. (2001, their appendix A) that the approximation of (4) with a(v, r) 5 2Lv of order (Dt) 3/2 results in the following first-order discrete Markov model: z(n) 5 F[z(n 2 1)] 1 q[n, r(n 2 1)],

(6)

where (I 2 DtL)v1



 



r1  r  r 5  2 , · · ·  rM 



r1 1 Dt[U(r1 ) 1 v2 ] (I 2 DtL)v2

 

F(z) 5  r2 1 Dt[U(r2 ) 1 v2 ]  ,

rM

··· (I 2 DtL)vM

(7)

1 Dt[U(rM ) 1 vM ]

I is the 2 3 2 unit matrix, and the covariance matrix of the stochastic part is expressed in the form,

E{q(n, r)q(n, r)*} [ Q(r) 5 (Q ij ),

1

is, q(n, r) [ q(n), then one could use the common EKF for the system (6). Alas, the standard approach is not applicable for handling the considered case. Moreover, systems with noise depending on the state are poorly studied in mathematics. The difficulty here is that even though the sequence {q(n, r) | n51,2, . . . } itself is a sequence of independent Gaussian random fields, it loses this property after substituting r(n 2 1) instead of r. More exactly, the sequence {q[n, z(n 2 1)] | n51,2, . . . } is not a sequence of independent random vectors. For the above reason we suggest a prediction algorithm that inherits only the idea of the Kalman filter and is not optimal. However, we will show that in one simple case it works as optimal. The suggested algorithm can be viewed as an adaptive Kalman filter since its weighing coefficients depend on the previous observations. Here we give the computational formulas only for the case of small Dt (Dt\ A \ 1 Dt\ G \ K 1). The general case, as well as the derivation, can be found in Osgokmen et al. (2001, their appendix B). Namely, the predicted values of the velocities and positions for the unobservable particles corresponding to subs i 5 p 1 1, . . . , M are computed by

O r (n) 5 r (n) 1 O p

where 4 3 4 blocks Q ij are given by Q i, j 5

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via (n) 5 vif (n) 1

2

3(Dt) 2 B(ri , rj ) 1 6DtB(ri , rj ) . 6 3(Dt) 2 B(ri , rj ) 2(Dt) 3 B(ri , rj )

K ij [vj (n) 2 v jf (n)],

j51

(8)

p

a i

f i

K ij [rj (n) 2 r jf (n)].

(10)

j51

Note that the straightforward first-difference approximation of (3), v(n) 5 (I 2 DtL)v(n 2 1) 1 (Dt)1/2«[n, r(n 2 1)], r(n) 5 r(n 2 1) 1 {U[r(n 2 1)] 1 v(n 2 1)}Dt,

(9)

where «(n, r) is a sequence of independent random fields, leads to a system similar to (6), but with a degenerate covariance matrix of the stochastic part. This fact would prevent an implementation of the EKF approach. The second drawback of (9) is that it does not give any explicit relation between the covariance matrix of «(n, r) and B(r i , r j ). Our approach to the discretization is different. We first linearize the original timecontinuous system at the neighborhood of the current state. Then solve the equations exactly at an interval of duration Dt, and, finally, truncate the terms of order higher than (Dt) 3/2 (see Ozgokmen et al. 2001, their appendix A, for details). Let us supply the state Eq. (6) with the observation equation, y(n) 5 Hz(n), where the observational 2pd 3 2Md matrix is H 5 (I p 0),

and I p is the 2pd 3 2pd unit matrix. The distinguished feature of the system (6) is that the noise depends on the state. If this was not the case, that

The superscripts a and f stand for ‘‘analyzed’’ and ‘‘forecasted,’’ respectively, in accordance with the common Kalman filter notation. Notice that for the observed particles, the analyzed variables coincide with observational ones: vai (n) 5 vi (n), rai (n) 5 ri (n), and i 5 1, . . . , p. The forecasted values are computed by vkf (n) 5 (I 2 DtL)vka (n 2 1), r kf (n) 5 r ka (n 2 1) 1 {U[r ka (n 2 1)] 1 vka (n 2 1)}Dt, k 5 1, . . . , M,

O

and

p

K ij 5

B[r ia (n 2 1), rk (n 2 1)]B21 kj (n 2 1),

k51

i 5 p 1 1, . . . , M, j 5 1, . . . , p, 21 kj

(11) 21

where B (n) are entries of {B[r k (n), r j (n)]} . In contrast to (10), the ‘‘naive’’ approximation (9) of order Dt gives

O p

via (n) 5 vif (n) 1

K ij [vj (n) 2 v jf (n)],

j51

r ia (n) 5 r if (n),

(12)

that is, differs from (10) by absence of the correcting term in the position prediction. For simulations, we use

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B(r) 5

[

]

s 2 (1 2 y 2 /R 2 ) exp(2r 2 /R 2 ) 2(xy/R 2 ) exp(2r 2 /R 2 ) , 2t 2(xy/R 2 ) exp(2r 2 /R 2 ) (1 2 x 2 /R 2 ) exp(2r 2 /R 2 )

where R is the space correlation scale and t is the Lagrangian correlation time common for both components. The above covariance matrix corresponds to the forcing streamfunction with the covariance proportional to exp(2r 2 /R 2 ). It is worthy to give the predictions formulas for only one predictor in the coordinate-wise form. Namely, let (x1 , y1 ) and (x 2 , y 2 ) be the coordinates of the predictor and predictand, respectively, and (u1 , y 1 ) and (u 2 , y 2 ) be their velocities. Set x 5 x a2 2 x1 , y 5 y a2 2 x1 , and r 2 5 x 2 1 y 2 . Then (10) becomes

L5

10

1/t

2

0 , 1/t

erations of the turbulent diffusion in the ocean (e.g., Zambianchi and Griffa 1994). Since in (16) the fluctuation velocity is the same for all particles, it is enough to consider only one predictor that gives complete information on the fluctuation velocity in all space. Therefore, the case of only two particles is considered when one of them is a predictor and another is a predictand. The linearity of (16) allows the exact discretization given by (6) and (7): v(n) 5 av(n 2 1) 1 j (n),

u 2a (n) 5 au 2a (n 2 1) 1 a n21 [u1 (n) 2 au1 (n 2 1)]

r(n) 5 Fr(n 2 1) 1 Sv(n 2 1) 1 h (n),

1 cn21 [y 1 (n) 2 ay 1 (n 2 1)],

where

y 2a (n) 5 ay 2a (n 2 1) 1 cn21 [u1 (n) 2 au1 (n 2 1)]

a 5 exp(2Dtl),

1 bn21 [y 1 (n) 2 ay 1 (n 2 1)],

F 5 exp(DtG),

S 5 (exp(DtG) 2 exp(2Dtl)I)(G 1 lI)21,

E{j (n)j (n)*} ; Dts 2 I,

x 2a (n) 5 x 2a (n 2 1) 1 u 2a (n 2 1)Dt 1 a n21 [x1 (n) 2 x1 (n 2 1) 2 u1 (n 2 1)Dt]

E{h (n)h (n)*} 5

(Dt) 3s 2 I, 3

E{j (n)h (n)*} 5

(Dt) 2s 2 I. 2

1 cn21 [y1 (n) 2 y1 (n 2 1) 2 y 1 (n 2 1)Dt], y (n) 5 y2a (n 2 1) 1 y 2a (n 2 1)Dt a 2

1 cn21 [x1 (n) 2 x1 (n 2 1) 2 u1 (n 2 1)Dt] 1 bn21 [y1 (n) 2 y1 (n 2 1) 2 y 1 (n 2 1)Dt], (14) where

Thus, we have for the predictand vM (n) 5 avM (n 2 1) 1 j (n), rM (n) 5 FrM (n 2 1) 1 SvM (n 2 1) 1 h (n),

bn 5 [1 2 x(n) 2 /R 2 ] exp[2r (n) 2 /R 2 ],

vp (n) 5 avp (n 2 1) 1 j (n),

cn 5 [2x(n)y(n)/R 2 ] exp[2r (n) 2 /R 2 ],

rp (n) 5 Frp (n 2 1) 1 Svp (n 2 1) 1 h (n). (15)

Now an example is given in the general situation where the prediction formulas in (10) result in an exact prediction, while (12) can give an error growing exponentially in time. Because of this, a justification for keeping the terms of order of (Dt) 3/2 is provided. Namely, we start with the equations v˙ 5 2lv 1 j (t),

r˙ 5 Gr 1 v,

(16)

where the turbulent velocity does not depend on the position at all, G is a constant matrix characterizing the shear of the mean flow, E{j (t)} 5 0,

(17)

and the same equations for the predictor (with different initial conditions):

a n 5 [1 2 y(n) 2 /R 2 ] exp[2r (n) 2 /R 2 ],

a 5 1 2 lDt.

(13)

E{j (t1 )j (t 2 )*} 5 s 2 d(t1 2 t 2 )I,

where l 5 t 21 and s 2 are scalar parameters. The model (16) oversimplifies the reality; however, it was used by several authors for theoretical consid-

(18)

From (18) we can find j (n) and h(n) exactly and substitute them to (17). The result is vM (n) 5 avM (n 2 1) 1 vp (n) 2 avp (n 2 1), rM (n) 5 FrM (n 2 1) 1 SvM (n 2 1) 1 rp (n) 2 Frp (n 2 1) 2 Svp (n 2 1).

(19)

Now apply the prediction formula (10). Since the correlation between the predictor and predictand velocities is 1, we should set K12 5 1. This results in vMa (n) 5 avMa (n 2 1) 1 vp (n) 2 avp (n 2 1), rMa (n) 5 FrMa (n 2 1) 1 SvMa (n 2 1) 1 rp (n) 2 Frp (n 2 1) 2 Svp (n 2 1).

(20)

Thus, the predicted velocities and positions are computed by the same formulas (20) as the real ones (19)

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with the same initial conditions. Hence, the prediction is absolutely exact. Now we shall find the prediction error in the case of the simplified algorithm (12). By subtracting (17) from (12), we get r M (n) 2 r Ma (n) 5 F[r M (n 2 1) 2 r Ma (n 2 1)]. Since h(n) is independent of r M (n 2 1) 2 r aM(n 2 1), we obtain for the error s(n) [ E{[rM (n) 2 rMa (n)]*[rM (n) 2 rMa (n)]} 5 Sp{[(FF*) n 2 I](FF* 2 I)21 q}, where q 5 Eh(n)*h(n). By substituting the expression for F we get s(n) [ Sp^{exp[nDt(G 1 G*)] 2 I} 3 {exp[nDt(G 1 G*)] 2 I}&q.

(21)

Note that q does not depend on the observation time T 5 nDt. From (21) it follows that if the symmetrized gradient matrix G [ G 1 G* has a positive eigenvalue, then s(n) grows exponentially with T. More precisely, if Dt is fixed and T → `, then s(n) ;

(Dt) 3s 2 d exp(l m T ), 3

where l m is the maximum eigenvalue of G . 5. Simulations and discussion The experiment conditions are as follows. Initially, the observed particles (predictors) are located in the vertices of a right polygon with the radius R 0 , while the unobserved particle is released at the center of the polygon at the same time. The initial velocities for all the particles are Gaussian independent variables. The simulated motion is covered by the following finite difference equations: v(n) 5 exp(2dt/t )v(n 2 1) 1 s D[r(n 2 1)]1/2Ï2dt/t «n , r(n) 5 r(n 2 1) 1 v(n)dt,

n 5 1, 2, . . . , N,

(22)

where v 5 (v1 , . . . , v M ), r 5 (r1 , . . . , r M ), D(r) 5 [B(r i , r j )] with B(r) given in (13), and « n are independent standard Gaussian 2M-vectors. Thus, we focus on the isotropic case with no mean flow at all, since the goal of this paper is to study the effect of stochastic currents. The following parameters are fixed during all the experiments: simulation step, dt 5 1 h; assimilation step, Dt 5 12 h; mean-square velocity, s 5 20 km day 21 ; and the maximum prediction time T m 5 Ndt 5 15 days; while the rest of parameters typically vary in the following ranges: Lagrangian correlation time, t 5 1 4 10 days; velocity space scale, R 5 20 4 250 km; initial cluster radius, R 0 5 20 4 100 km; number of predictors, p 5 M 2 1 5 1 4 6; and prediction time, T 5 1 4 15 days.

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Figures 1a–c demonstrate trajectories of the observed particles (solid, unmarked lines), the predictand (circles), and the Kalman filter (KF) prediction (stars) for small, medium, and large ratio R/R 0 , respectively. In the first case, R 5 200 km, while R 0 5 25 km is fixed for all three experiments. The prediction in this case is very good due to the fact that three out of six predictors are closely following the predictand because of high correlation between their velocities. In fact, for all the moments the error of prediction is less than 15–20 km. The interesting thing is that the predictions on the 13th and 14th days are better than in the intermediate term’s 5– 10 days. Thus, the error is not accumulated. When R 5 100 km the prediction is still very good. In fact, it is not worse than in the previous case, even though only a single predictor out of six is staying close to the predictand during 15 days, while three particles got away from the area and two particles stuck near the initial polygon. So, the algorithm works in a way such that a good prediction is ensured by having a single lucky predictor. It would be naive to suppose that the unobservable particle should follow the majority of the observable ones. Under such a guess, one would look for the predictand somewhere in the northwest corner where two predictors headed. In fact, the prediction skill is completely determined by whether one or more predictors are staying close to the predictand during the observation time. Let us call such predictors the significant predictors. The algorithm automatically picks up significant predictors by adjusting weights K ij in (10), which decay rapidly with the distance between predictand and the corresponding predictor. In particular, in the case of a sole significant predictor the prediction formula turns to (14). From this formula and explicit expressions (15) for the coefficients, one can see that if the distance between two particles is small, then a and b are close to 1, while c is about 0. Hence, the velocity and position of the unobserved particle approach the corresponding parameters for the predictor. Finally, in the case R 5 50 km, the prediction completely fails. The particles are becoming uncorrelated from the very beginning. All the predictors are running away one from another as well as from the predictand. The result is unfortunate: the error is about 250 km in 15 days, that is, close to the dispersion. Thus, the ratio R/R 0 is crucial for the prediction skill. Now we look at this dependence more carefully. Introduce the mean-square-error prediction error

[O [O

1 s(n) 5 L and dispersion r(n) 5

1 L

k51

L

k51

] ]

1/2

L

|r (n) 2 rM,k (n)| a M,k

2

,

1/2

a |rM,k (0) 2 rM,k (n)| 2

,

where L is the number of runs, typically assumed 100; and r M,k (n) and r aM,k(n) are the position and its prediction

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FIG. 3. (a) Dependence of the relative prediction error on the Lagrangian correlation time and the velocity space correlation radius; R 0 5 50 km. (b) Dependence of the relative prediction error on the velocity space correlation radius; R 0 5 50 km, t 5 3 days.

of the predictand at time n in the kth run. The dependence of s(n) on the prediction time for fixed R 0 5 50 km and different R 5 50, 100, and 200 km is shown in Fig. 2a. The diamond line shows the dispersion r(n). In agreement with the classical theory, the dispersion is linear at the initial stage and proportional to Ït, where t 5 ndt, for large t. It is about 290 km in T 5 15 days. Notice that the displacement can be theoretically esti-

mated as 2sÏTt ø 268 km using the diffusivity formula D 5 2s 2y t for a single component (Griffa 1996). The curves line up with the correlation radius: the larger R, the less error s(n) for any time moment n. The explanation is simple—in the absence of the mean flow, for the large correlation scale the probability is high that one of the predictors is close enough to the predictand, ensuring good prediction conditions. The error of the

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best prediction corresponding to the largest R 5 250 km is just about 30 km, that is, 10% of the dispersion for the prediction time of 15 days. The dependence of the prediction error on the prediction time for fixed R 5 200 km and different R 0 5 25, 50, 75, 100, and 125 km is shown in Fig. 2b. First, notice that the dispersion in this study is about 275 km and hence agrees with the theory and the previous experiment. The latter indicates that 100 runs is enough for reasonable statistical inference. Recall that the statistics of an individual particle is determined by s, T, t only, while the two-particle statistics is essentially dependent on R and R 0 . The experiments resulting in Figs. 2a,b are in full agreement for values R 5 250 km, R 0 5 50 km (squares in the first case and stars in the second case). The final error is about 30 km in the first case and close to 25 km in the second one. Also, the picture in Fig. 2b is similar to that of Fig. 2a in the sense that the error lines up with the value of R 0 : the less R 0 , the better the prediction. However, there is an essential difference. Changing R 0 in a wide enough range, 25–100 km, does not cause drastic changes in the prediction skill. The error grows from 25 to 50 km. There is almost no difference in the error for initial values of the cluster radius 25 and 50 km. The simulation results show that the error is not determined by the ratio R/R 0 , but rather is a complex function of R and R 0 . Introduce the relative error sr (n) 5

s(n) . r(n)

Figure 3a illustrates the dependence of s r (T) on R and t for fixed R 0 5 50 km and M 5 7. As we observed earlier, the error decays fast with increasing R for all values of t in the range 1 4 15 days. To stress this once again we give a fragment of this dependence for t 5 2 days in Fig. 3b. However, the error almost does not change with t changing in the broad range, t . 4 days. At the first glance it seems strange because the increasing correlation time should increase the inertia and hence improve the predictability. Remember, however, that we discuss the relative error. In fact, increasing the correlation timescale implies increasing the dispersion r(t) ; 2s y Ïtt. Therefore, our simulations indicate that the absolute error s should be inverse proportional to Ït. The fact of weak dependence of the error on the Lagrangian time is also supported by the graph in Fig. 4a, where the dependence of the 15-day relative prediction error on t is shown for much bigger range of t. The conclusion is the same: the relative error almost does not depend on the Lagrangian correlation time after t 5 3 days. It is worthwhile to compare this conclusion with a study of the Lyapunov exponent L in PIT, for the same stochastic model. It was shown that L, as a function of two time parameters t and t 0 5 R/s, only slightly varies along with t, while showing a strong

FIG. 4. (a) Long-term dependence of the relative prediction error on the Lagrangian correlation time; R 0 5 50 km, R 5 250 km. (b) Dependence of the relative prediction error on the Lagrangian correlation time for different number of drifters: M 5 3 (squares), M 5 5 (diamonds), M 5 7 (circles). R 0 5 50 km, R 5 250 km.

dependence on t 0 . Since the quantity 1/L characterizes the predictability limit, one can conclude that the presented study is in agreement with the Lyapunov exponent consideration. Another interesting and unexpected finding is that the error sharply increases after t 5 3 days. The same feature we can observe in Fig. 4b, where the dependence on t is shown for different number of predictors. Notice that there is a moderate improvement in the prediction when the number of predictors grows. For t 5 3 days the errors for p 5 2, 4, and 6 are 0.05, 0.09, and 0.17, respectively; for t 5 10 days the figures are 0.15, 0.22, and 0.26, respectively. In all cases the error increases in the 1–5-day range and then becomes flat. Finally, the prediction error dependence of the num-

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FIG. 5. (a) Dependence of the relative prediction error on the number of drifters for different space correlation scales: R 5 100 km (diamonds), R 5 200 km (circles), t 5 3 days, R 0 5 50 km. Initially the predictors are placed at the vortices of the right polygon and the predictand is placed at its center. (b) Same as in (a) with the predictors initially placed equidistantly along a straight line. The distance from the initial predictand position to the line is 25 km and the distance between the end predictors is 100 km. The end predictors have the same distance from the predictand. (c) Same as in (a) with the predictors initially placed randomly over the square of 100 km 3 100 km, centered at the predictand position. The predictors are distributed uniformly.

ber of predictors, p 5 M 2 1, was studied. For p . 7, the 2p 3 2p matrix D 5 [B(r i , r j )], which is used in calculation of the weights (11), sometimes becomes illconditioned, resulting in a significant computational error. For this reason the algorithm was slightly modified by changing D to D 1 «I whenever det(D) , 10 210 , where « is a small number and I is the 2p 3 2p unit matrix. This regularization procedure has been used in Ozgokmen et al. (2000) for real data; however, its accuracy was not studied. Figure 5a demonstrates the prediction skill for R 5 100 km and R 5 200 km as a function of the predictor number. In the first case, the error decreases only from 0.42 to 0.36 as p increases from 7 to 20. In the second case, the error even increases

from 0.14 to 0.28 for the same values of p. Such a trend is typical for other experiments not included in this paper, which covered a wider range of R and different values of «. The same conclusion follows from our consideration of initial predictor configurations different from the right polygon configuration. First, the predictors were located along the straight line distanced from the predictand. Details are given in the caption to Fig. 5b. Then, we tried a random distribution of the predictors around the predictand. In both cases (Figs. 5b,c) the dependence of the error on the number of predictors remains basically the same. Roughly, for p . 7 the error is about 0.2–0.3 if R 5 200 km, and about 0.35–0.45 if R 5 100 km. The initial right polygon configuration

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gives slightly better results than the linear and random ones. There is no indication that increasing the number of predictors beyond 4–7 improves the prediction skill. It is quite possible that the above simplest regularization procedure is to blame for that. This procedure helps to avoid drastic computational errors; however, it results in a systematic bias and for this reason is not quite satisfactory. Other regularization procedures should be studied. In summary, we can conclude that the prediction skill improves a ratio R/R 0 increases, where, in general, R 0 is a characteristic radius of the cloud of predictors. The least prediction error is due to a small Lagrangian correlation time, t 5 1–3 days, for the prediction time T 5 15 days. Under the simplest regularization procedure, the error slowly decreases as the number of predictors grows for ratio R/R 0 around 1–2, while the prediction skill slightly worsens or does not change for p larger than 4–7 in the case R/R 0 . 4. Acknowledgments. The work was supported by ONR Grant N00014-99-0042. The author thanks Tamay Ozgokmen and the anonymous reviewer for remarks that helped improve the presentation. REFERENCES Bauer, S., M. S. Swenson, A. Griffa, A. J. Mariano, and K. Owens, 1998: Eddy-mean flow decomposition and eddy-diffusivity estimates in the tropical Pacific Ocean. Part I: Methodology. J. Geophys. Res., 103, 30 855–30 871. Borgas, M. S., and B. L. Sawford, 1994: A family of stochastic models for two-particle dispersion in isotropic homogeneous stationary turbulence. J. Fluid Mech., 279, 69–99. ¨ zgo¨kmen, and P. -M. Poulain, 2001: Castellari, S., A. Griffa, T. M. O Prediction of particle trajectories in the Adriatic Sea using Lagrangian data assimilation. J. Mar. Syst., 29, 33–50. Davis, R. E., 1991a: Lagrangian ocean studies. Ann. Rev. Fluid Mech., 23, 43–64. ——, 1991b: Observing the general circulation with floats. Deep-Sea Res., 38, 5531–5571. Griffa, A., 1996: Applications of stochastic particle models to ocean-

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ographic problems. Stochastc Modelling in Physical Oceanography, R. Adler et al., Eds., Birkhauser, 113–128. ——, K. Owens, L. Piterbarg, and B. Rozovskii, 1995: Estimates of turbulence parameters from Lagrangian data using a stochastic particle model. J. Mar. Res., 53, 212–234. Ishikawa, Y. I., T. Awaji, and K. Akimoto, 1996: Successive correction of the mean sea surface height by the simultaneous assimilation of drifting buoy and altimetric data. J. Phys. Oceanogr., 26, 2381–2397. Kunita, H., 1990: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, 347 pp. Liptser, R. S., and A. N. Shiryaev, 1978: Statistics of Random Processes. Springer-Verlag, 442 pp. Ozgokmen, T., A. Griffa, L. Piterbarg, and A. Mariano, 2000: On the predictability of Lagrangian trajectories in the ocean. J. Atmos. Oceanic Technol., 17, 366–383. ——, I. Piterbarg, A. J. Mariano, and E. H. Ryan, 2001: Predictability of drifter trajectories in the tropical Pacific Ocean. J. Phys. Oceanogr., 31, 2691–2720. Pedrizzetti, G., and E. A. Novikov, 1994: On Markov modeling of turbulence. J. Fluid Mech., 280, 69–93. Piterbarg, L. I., 1998: Drift estimation for Brownian flows. Stochastic Processes Appl., 79, 132–149. ——, 2000: The top Lyapunov exponent for a stochastic flow modeling the upper ocean turbulence. SIAM J. Appl. Math., in press. Pope, S. B., 1987: Consistency conditions for random walk models of turbulent dispersion. Phys. Fluids, 30, 2374–2379. Reynolds, A. M., 1998: On the formulation of Lagrangian stochastic models of scalar dispersion within plant canopies. Bound.-Layer Meteor., 86, 333–344. Rodean, H. C., 1996: Stochastic Lagrangian Models of Turbulent Diffusion. Meteor. Monogr., No. 48, Amer. Meteor. Soc., 84 pp. Sawford, B. L., 1993: Recent developments in the Lagrangian stochastic theory of turbulent dispersion. Bound.-Layer Meteor., 62, 197–215. Schneider, T., 1998: Lagrangian drifter models as search and rescue tools. M.S. thesis, Dept. of Meteorology and Physical Oceanography, University of Miami, 96 pp. Swensson, M. S., and P. P. Niiler, 1996: Statistical analysis of the surface circulation of the California current. J. Geophys. Res., 101, 22 631–22 645. Thomson, D. J., 1986: A random walk model of dispersion in turbulent flows and its application to dispersion in a valley. Quart. J. Roy. Meteor. Soc., 112, 511–529. ——, 1990: A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. J. Fluid Mech., 210, 113–153. Zambianchi, E., and A. Griffa, 1994: Effects of finite scales of turbulence on dispersion estimates. J. Mar. Res., 52, 129–148.