Representer-based variational data assimilation in a ...ingria.coas.oregonstate.edu/pdf/kurapov_JGR_2007JC004117.pdf · Representer-based variational data assimilation in a nonlinear

Representer-based variational data assimilation in a

nonlinear model of nearshore circulation

Alexander L. Kurapov,1 Gary D. Egbert,1 J. S. Allen,1 and Robert N. Miller1

Received 20 January 2007; revised 2 August 2007; accepted 17 August 2007; published 30 November 2007.

[1] A representer-based variational data assimilation (DA) method is implemented with ashallow-water model of circulation in the nearshore surf zone and tested with syntheticdata. The behavior of the DA system is evaluated over a 1-hour time interval that is largecompared to timescales characteristic of instability growth and eddy interactions. Truereference solutions, from which the synthetic data are sampled, correspond to fullydeveloped unsteady nonlinear flows driven by a steady spatially varying forcingrepresenting the effect of breaking waves. Forcing and initial conditions are adjusted to fitthe data. The convergence of the nonlinear optimization algorithm and the accuracyof the forcing and state estimates depend on the choice of the forcing error covariance C.In a weakly nonlinear (equilibrated waves) regime, using C that allows only a steadyforcing correction yields a convergent and accurate solution. In a more strongly nonlinearregime, the DA system cannot find sufficient degrees of freedom in the steadyforcing to control eddy variability. Implementing a bell-shaped temporal correlationfunction in C with the 1-min decorrelation scale yields a convergent linearized inversesolution that describes correctly the spatiotemporal variability in the eddy field. Thecorresponding estimate of forcing, however, is not satisfactory. Accurate estimates of boththe flow and the forcing can be achieved by implementing a composite C with a temporalcorrelation separated into an O(1) steady and small amplitude time-variable parts.

Citation: Kurapov, A. L., G. D. Egbert, J. S. Allen, and R. N. Miller (2007), Representer-based variational data assimilation in a

nonlinear model of nearshore circulation, J. Geophys. Res., 112, C11019, doi:10.1029/2007JC004117.

1. Introduction

[2] Variational data assimilation (DA) methods have beenimplemented in oceanography for synthesis of observationsand models with goals to quantify and to improve under-standing of ocean dynamical processes and forcing mech-anisms [e.g., Egbert et al., 1994; Bennett et al., 1998, 2000,2006; Egbert and Ray, 2000; Bennett, 2002; Yaremchuk andMaximenko, 2002; Kurapov et al., 2003; Foreman et al.,2004; Stammer et al., 2004; Stammer, 2005; Wunsch andHeimbach, 2007]. There are also advantages in utilizingvariational DA for operational applications, to provideimproved initial conditions for forecast models [e.g., Rabierand Courtier, 1992; Ehrendorfer, 1992; Rosmond and Xu,2006; Di Lorenzo et al., 2007]. Variational methods find anestimate of a system state that fits a dynamical model andavailable observations in a least squares sense. The optimalsolution minimizes a penalty functional that is a sum ofsquared norms of residuals (deviations from the priorestimates) in model inputs and data, integrated over thecomputational domain and specified time interval. Thepenalized model inputs may include initial and boundaryconditions, external forcing, model parameters, and interior

dynamical errors. Because of nonlinearities in computationalocean models, variational oceanographic problems are gen-erally solved iteratively, utilizing companion tangent linearand adjoint numerical codes.[3] The variational representer-based method [Chua and

Bennett, 2001; Bennett, 2002] finds a minimizer of thepenalty functional by solving a series of linearized optimi-zation problems. On each iteration, a correction to the priorsolution is sought in the subspace spanned by the repre-senter functions corresponding to the observations. Therepresenter method offers greater flexibility in the choiceof norms of the model penalty terms than the so calledadjoint method or 4DVAR [e.g., Wunsch and Heimbach,2007], providing means for making explicit and nontrivialstatistical hypotheses about prior errors in the model inputs.For instance, by choosing the norms, one can specifynontrivial spatial and temporal correlations among inputerrors, as well as additional dynamical constraints oncorrections at the initial time [Rosmond and Xu, 2006] orat the open boundaries [Kurapov et al., 2003]. Thesechoices may affect both the computational efficiency ofthe optimization algorithm and the inverse solution accuracy,as shown here.[4] The objective of the present study is to assess the

utility of the representer method implemented for a nonlin-ear model of forced dissipative oceanic flows, in whichspatiotemporal variability is dominated by instabilities andinterior eddy interactions. Such behavior can be found, for

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1College of Oceanic and Atmospheric Sciences, Oregon StateUniversity, Corvallis, Oregon, USA.

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example, in two-dimensional (shallow water) models ofcirculation in the nearshore surf zone [Allen et al., 1996;Slinn et al., 2000] and in high-resolution three-dimensionalmodels of stratified shelf flows [Durski and Allen, 2005].Previous studies on variational DA in the presence ofinstabilities have mostly been focused on applications inwhich the assimilation interval was comparable to thetimescale of linear instability growth and the initial con-ditions were a major source of error [e.g., Rabier andCourtier, 1992; Xu and Daley, 2002]. In our study, therepresenter method is applied over time intervals thatexceed substantially both the characteristic timescale oflinear instability growth or the timescales associated witheddy motions (which can be evaluated using the frequencyspectrum, see below). Assimilation in such long timewindows may be necessary to obtain an accurate estimateof forcing that varies on a slower timescale. If temporal flowvariability is a result of intrinsic nonlinear interactions ratherthan a direct effect of time-variable forcing, it is not obviousthat the variational method, trying to fit the data by adjust-ing the forcing, would result in a convergent or accuratesolution. In our study, we explore the impact of the forcingerror covariance, which must be specified prior to assimi-lation, on solution convergence and on the accuracy of theinverse estimates of both the flow and forcing.[5] To explore these issues, the representer method is

implemented with an idealized shallow-water model ofnearshore circulation over variable beach bathymetry andtested on synthetic data generated by the nonlinear model.The dynamical content is close in many aspects to that ofSlinn et al. [2000]. The alongshore current is forced bygradients in the radiation stresses resulting from breakingwaves. Over the time periods considered, the forcing isassumed to be steady. If the linear friction coefficient issufficiently small, the current is subject to shear instabilitiesthat evolve to a weakly nonlinear equilibrated shear waveregime or, as friction is reduced further, to a more stronglynonlinear irregular eddy regime. We consider DA experi-ments in both unsteady regimes.[6] Aside from the pioneering study by Feddersen et al.

[2004] who assimilated time-averaged observations in alinear one-dimensional (cross-shore coordinate) model,there has been little prior work on DA in the nearshore

ocean. However, advanced DA methods might be used tocombine existing extensive field observations with modelsproviding better understanding of some of the complex flowprocesses in the nearshore region. Although our focus hereis on fundamental aspects of DA in a nonlinear oceanicsystem, this study certainly represents a next step towardDA in the nearshore surf zone by utilizing a model thatwould be more directly comparable to the time-variableobserved fields than the one-dimensional model.

2. A Dynamical Model

[7] A Cartesian coordinate system (x, y) is utilized, withthe x axis aligned alongshore, the y axis positive in theoffshore direction, and the origin at the shore. The modeldomain is rectangular, 256 � 200 m, representing a beacharea with a straight coastline and along- and cross-shorevariable bathymetry including a sand bar parallel to thecoast (Figure 1). The depth is approximately 0.06 m atinterior grid points closest to the coast and 4.1 m near theoffshore boundary. No normal flow boundary conditions areapplied at the coast and offshore boundaries; periodicboundary conditions are utilized in the alongshore direction.The dynamics are described by the nonlinear shallow-waterequations

@z@t

þ @ Duð Þ@x

þ @ Dvð Þ@y

¼ 0; ð1Þ

@ Duð Þ@t

þ @ Duuð Þ@x

þ @ Dvuð Þ@y

¼ �gD@z@x

þ fX � ru� ar2 Hr2u� �

;

ð2Þ

@ Dvð Þ@t

þ @ Duvð Þ@x

þ @ Dvvð Þ@y

¼ �gD@z@y

þ fY � rv� ar2 Hr2v� �

;

ð3Þ

where z(x, t) is the free surface elevation, u(x, t) and v(x, t)are, respectively, the alongshore and cross-shore compo-nents of the depth-averaged velocity, x = (x, y), H(x) is thebathymetric depth, D = H + z is the total water depth, g isthe gravitational acceleration, f{X,Y}(x, t) are forcingcomponents, and r is the linear friction coefficient. Weakbiharmonic horizontal friction terms, with a = 1.25 m4 s�1

as in work by Slinn et al. [2000], are included in (2) and (3)to provide numerical dissipation at small length scales.[8] Details of the space and time discretization are

adopted from the Regional Ocean Modeling System(ROMS) [Haidvogel et al., 2000]. The equations are dis-cretized on a staggered rectangular C-grid, with a second-order centered difference scheme utilized for the advectionterm. The equations are integrated in time using a predictor-corrector (leapfrog-Adams-Bashforth) scheme. For compu-tations discussed here, the grid resolution is 2 m in eitherdirection and the time step is 0.15 s. At the coast andoffshore boundaries, in addition to the condition v = 0, weset @2v/@y2 = 0, @u/@y = 0, and @3u/@y3 = 0 when discretiz-ing the biharmonic terms [Allen et al., 1996].

Figure 1. Model domain and observational locations(circles). Bathymetric contours are every 0.2 m.

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[9] To obtain the true solution for our DA experiments,the nonlinear model forcing f{X,Y}(x) is steady. This approx-imates the effect of obliquely incident breaking waves andis computed using a wave model as described by Slinn et al.[2000, Appendix C], utilizing the Thornton and Guza[1983] formulation for wave energy dissipation. The forcingis computed in a larger domain extending 512 m in theoffshore direction, to deeper water. The surface wavesare assumed to arrive in the nearshore zone obliquely fromthe right. The angle between the direction of wave phasepropagation in deep water far offshore and the x axis is135�. At y = 512 m, the wave rms height is 0.7 m, andthe peak period of the narrow band wave energy spectrum is8 s. On average, the alongshore component fX (Figures 2aand 2b) forces an alongshore current in the negativedirection (from right to left Figures 2a and 2b). The cross-shore component fY (Figures 2c and 2d) supports theelevated level (set-up) of the water near the coast. Themagnitude of both fX and fY is larger over the offshore sideof the bar and over the slope near the coast, although it issharply reduced to 0 very close to the coast.[10] The nonlinear model is spun up from rest. After an

initial adjustment the flow regime depends on the magni-tude of the friction coefficient r. If r is large enough, theflow becomes steady and the dominant dynamical balance islinear, between the forcing and the bottom friction. Forlower values of r the alongshore current may develop shearinstabilities. Two cases in which the well-developed flow isunsteady are considered here, with r = 0.004 m s�1 and r =0.002 m s�1. After approximately two model hours, in thefirst case the area-averaged, depth-integrated kinetic energyasymptotes to a constant value, while in the second case itapproaches a state with irregular 10-min timescale fluctua-tions about a larger constant value (Figure 3). In the case r =0.004 m s�1, the model solution evolves into a weakly

nonlinear equilibrated shear wave regime. The instanta-neous velocity and vorticity (x = @v/@x � @u/@y) fields att = 5 hours corresponding to this regime are shown inFigures 4b and 4c. After the solution equilibrates, this flowstructure translates from right to left disturbed only by veryminor variations in the intensity of the eddies. The dominantshear wave period is near 6 min. The alongshore currentaveraged in time and in the alongshore direction (Figure 4a)has peaks near the coast (0.8 m s�1) and over the bar(0.4 m s�1). In the more strongly nonlinear case with r =0.002 m s�1 (Figures 4d, 4e, and 4f), the flow can bedescribed as irregular, with more intense eddies and a moreenergetic mean alongshore current; the peak values are now1.2 m s�1 near the coast and 0.5 m s�1 near the sand bar.[11] To provide a low-dimensional representation of the

two solutions that emphasizes the difference in their non-linear behavior, the kinetic energy of deviations from thetime mean is analyzed in the 4-hour interval beginning at t =

Figure 2. True steady forcing, (top) alongshore and (bottom) cross-shore components: (a, c) alongshoreaverages and (b, d) spatial maps. Negative values are shaded. The contour interval is 2.5 � 10�3 and 1 �10�2 m2 s�2 in Figures 1b and 1d, respectively. The dashed line is the 2-m bathymetric contour.

Figure 3. Area-averaged, depth-integrated kinetic energyin nonlinear runs.


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5 hours. In Figure 5, the depth-integrated and area-averagedkinetic energy of the velocity deviations, K, is plottedagainst dK/dt. The trajectory corresponding to the case r =0.004 m s�1 is nearly a point compared to the irregulartrajectory for the case r = 0.002 m s�1. In the first case, thedomain-averaged energy of the time-variable part of theflow is preserved, which is characteristic of the equilibratedshear waves, while in the second, more strongly nonlinearcase there are substantial irregular fluctuations in K anddK/dt.[12] The frequency-wave number (w � k) power spectra

of the vorticity x(x, t) corresponding to the 1-hour timeinterval between t = 5 and 6 hours are shown in Figure 6 foralongshore profiles y = 8 m (nearshore slope), 38 m (troughbetween the shore and the bar), 65 m (the bar), and 98 m(offshore from the bar). In the equilibrated wave case, thespectra are discrete (Figures 6a–6d), with most powerconcentrated near (w, k) = (0.003,0.01) (cycles s�1, m�1),corresponding to motions with a nearly 6-min period and100-m wave length. In the more irregular case the spectraare continuous (Figures 6e–6h), with substantial power inthe motions near w = 1/(60s) = 0.017 s�1 on the inshore sideof the bar (Figures 6e and 6f). The dominant phase speedw/k is 0.75 m s�1 on the inshore side and 0.5 m s�1 on theoffshore side of the bar.

3. Data Assimilation (DA) Problem

3.1. Penalty Functional

[13] In the following, it is convenient to denote the stateas a vector field, u(x, t) = [z u v]0 (the prime denotes matrix

transpose); the two forcing components are also combinedin a vector field f(x, t) = [fX fY]

0. In our experiments, thetrue model starts from initial conditions corresponding to thefully developed flow, sampled from the nonlinear solutions

Figure 4. Nonlinear solutions, (top) r = 0.004 and (bottom) 0.002 m s�1: (a, d) mean u, averaged over xand t (5–9 hours after spin-up from rest), (b, e) instantaneous velocity vectors at t = 5 hours (the half-tonecontour is H = 2 m), and (c, f) vorticity at the same time (positive values shaded, contours every 0.01 s�1).

Figure 5. K versus dK/dt trajectories, where K is the area-averaged and depth-integrated kinetic energy of deviationsfrom the time-mean current obtained from the nonlinearsolutions at t = 5–9 hours. To help the viewer see thetrajectories, line intensity changes from light (earlier times)to dark (later times).


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described above at t = 5 h (see Figure 4). To explore thenonlinear aspect of the optimization algorithm, we deliber-ately choose prior inputs that differ substantially from thetruth, namely, uprior(x, 0) = uprior

0 = 0 and f prior = 0 (theprior state is thus 0). The time series observations of z, u,and v are sampled from the true solution at 32 locationsshown in Figure 1 once every minute from t = 1 to 59 min;the total number of observations is K = 5,664. Randomnoise is added to the observations with standard deviationsd = 0.02 m for z and 0.02 m s�1 for u and v.[14] In symbolic form, the nonlinear (NL) model can be

written as

@u

@t¼ N uð Þ þ fprior þ e ð4Þ

u x; 0ð Þ ¼ uprior0 þ e0; ð5Þ

where u is the true state, N is the NL model operator, e(x, t)is the forcing error, and e0(x) is the initial condition error.The observations are written in the general form as

L uð Þ ¼ZS

dx1

Z T

0

dt1

g01 x1; t1; x; tð Þg02 g0K

0BB@

1CCAu x1; t1ð Þ ¼ dþ ed ; ð6Þ

where [0, T] is the assimilation time interval, S is a domainarea, dx = dx dy, d = {dk} is a vector of observations (ofsize K � 1), L is a linear operator matching the model stateu(x, t) and the observations, gk define the observationalfunctionals (sampling rules for each datum dk, i.e., impulsesat the observation locations xk and times tk in our example),and ed is the observation error.[15] In (4)–(6), errors e, e0, and ed can be interpreted as

random processes or vectors, satisfying the equations with

Figure 6. Frequency-wave number power spectra P of the vorticity x(x, t) at different y (top to bottom),using 1-hour time series. Cases are: (left) the true NL solution, r = 0.004 m s�1, (middle) the true NLsolution, r = 0.002 m s�1, and (right) the inverse NL solution (iteration 7, using the composite covarianceC), r = 0.002 m s�1.


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the true state u. The optimal, inverse, solution u minimizesthe following penalty functional:

J uð Þ ¼k e k2 þ k e0 k20 þ k ed k2d ; ð7Þ

where e, e0, and ed are residuals, satisfying (4)–(6) with u.The norms in (7) are defined using inverse prior errorcovariances in the corresponding inputs, C�1, C�1

0 , and C�1d,

that must be specified prior to assimilation,

k e k2¼Z T

0

dt1

ZS

dx1

Z T

0

dt2

ZS

dx2 e0 x1; t1ð ÞC�1 x1; t1; x2; t2ð Þ

e x2; t2ð Þ; ð8Þ

k e0 k20¼ZS

dx1

ZS

dx2 e00 x1ð ÞC�1

0 x1; x2ð Þe0 x2ð Þ; ð9Þ

k ed k2d¼ e0dC�1d ed : ð10Þ

The inverse C�1 is defined such thatZ T

0

dt

ZS

dxC x1; t1; x; tð ÞC�1 x; t; x2; t2ð Þ ¼ d x1 � x2ð Þd t1 � t2ð Þ;

ð11Þ

where C(x1, t1, x, t) = he(x1, t1) e0(x, t)i, and hi denotes theensemble mean [Bennett, 2002]; C�1

0 is defined similarly.[16] In our study, the data error covariance Cd is diagonal

with the variances s2d on the main diagonal corresponding tothe noise level in the data. The choice of the initial andforcing error covariances is less trivial. In models with finespatial resolution, the assumption that errors are uncorrelatedin space and time (e.g., C = d(x1 � x2)d(t1 � t2) and C0 =d(x1 � x2)) could result in an irregular solution that fits thedata only in the vicinity of the observational locations[Egbert and Bennett, 1996; Bennett, 2002]. Implementationof the covariances with nonzero spatial and temporaldecorrelation scales helps to obtain smooth solutions.[17] We assume that the errors in fX and fY are statistically

independent from each other. For each forcing component,the error covariances are considered in the following generalform, allowing nonzero correlation in each spatial dimen-sion and in time, and also cross-shore dependence of thevariance:

C x1; t1; x2; t2ð Þ ¼ g2s y1ð Þs y2ð ÞCX x1 � x2ð Þ CY y1 � y2ð ÞCT t1 � t2ð Þ; ð12Þ

where

CX xð Þ ¼ exp � 1� cos qxð Þqlxð Þ2

!; ð13Þ

CY yð Þ ¼ exp � y2

2ly� �2

!; ð14Þ

CT tð Þ ¼ exp � t2

2ltð Þ2

!: ð15Þ

[18] Expression (13), in which q = 2p/Lx and Lx is theperiodic channel length, provides a functional form that isclose to bell-shaped while accounting for periodicity in thealongshore direction [Menard, 2005]. Numerical values forthe scaling factor g and decorrelation scales lx, ly, and ltutilized in our studies are specified below.[19] The covariance of (12) is standard for the representer

method [Chua and Bennett, 2001]. However, since the trueand prior forcings are both steady in our case, it is natural tofirst consider a special case with lt ! 1 (C = C1). Such acovariance does not allow temporal variation in the forcingcorrection. However, as we shall show, such a severerestriction on the available degrees of freedom in the forcingcorrection may impede convergence to the true time-variable solution. Using the standard covariance (12) witha finite lt that is smaller than the characteristic eddyvariability timescale (C = Clt<1) may result in betterconvergence, but if rapid variability is allowed in theinverse forcing, the question naturally arises of whether itstime-mean will provide an accurate estimate of the constantforcing. In practice, this question is important if one wantsto analyze the forcing estimates to obtain inferences aboutdominant driving mechanisms. In our study, we have beenled to consider a composite covariance,

C ¼ C1 þ aClt<1; ð16Þ

where a is a small parameter. With (16), deviations of theinverse forcing from its time-mean are allowed, so there areenough degrees of freedom to allow control of the time-dependent flow, but these deviations are kept small enoughto allow accurate estimation of the time-mean forcing.[20] To define C0, we assume that initial errors in z, u,

and v are statistically independent, with the covariance ofeach component in the following form:

C0 x1; x2ð Þ ¼ g20s0 y1ð Þs0 y2ð ÞCX x1 � x2ð ÞCY y1 � y2ð Þ: ð17Þ

[21] In our DA experiments, lx = 40 m, ly = 20 m, g0 =0.05 m, 0.5 m s�1, and 0.2 m s�1 for initial z, u, and vrespectively. To reduce the error variance in the offshoredirection, s0(y) = exp(�y/Y), with Y = 70 m. For the forcingerror covariance, g = 0.01 m2 s�2; lx = 70 m, and ly = 20 m.The scaling function s(y) is similar to s0, utilized for theinitial conditions, except very close to the coast (y < 10 m)where it is linearly reduced to zero. This last modificationhelps to reduce forcing variations in very shallow water thatotherwise may cause numerical instabilities associated withthe beach drying (z < �H).

3.2. Minimization Method

[22] The representer-based minimization algorithm [Chuaand Bennett, 2001] attempts to find a solution to thenonlinear Euler-Lagrange equations, providing necessaryconditions for the minimum of (7), iteratively via a seriesof linearized optimization problems (so called ‘‘outer loop’’iterations). For details of derivation of the Euler-Lagrange


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equations specific to the general form of the model and datadefined in (4)–(6) see the Appendix. On iteration n + 1, thenonlinear equations are linearized with respect to theprevious iteration tangent linear inverse solution un. Onthe first outer loop iteration, the system is linearized withrespect to u0 = uprior(= 0 here).[23] The linearized inverse solution is the sum of the

linearized prior (or ‘‘forward’’) solution unF and the correc-tion yn,

un ¼ unF þ yn: ð18Þ

[24] Solution unF satisfies the full state tangent linear (TL)equations and prior initial conditions,

@unF@t

¼ N un�1� �

þ A un�1�

unF � un�1� �

þ fprior; ð19Þ

unF x; 0ð Þ ¼ uprior0 ; ð20Þ

where

A un�1�

¼ @N

@u

��u¼un�1

ð21Þ

is the TL operator (in our notation, a 3 � 3 matrix operatorwith elements depending on un�1). Note that in (19),N(un�1) � A[un�1]un�1 provides additional forcing.[25] The correction yn satisfies the perturbation TL

equations,

@yn

@t¼ A un�1

� yn þ

ZS

dx1

Z T

0

dt1 C x; t; x1; t1ð Þl x1; t1ð Þ; ð22Þ

yn x; 0ð Þ ¼ZS

dx1 C0 x; x1ð Þl x1; 0ð Þ: ð23Þ

[26] The evolution of the sensitivity to inputs l(x, t)appearing in (22) and (23) is described by the adjoint(AD) system,

� @l@t

� A0 un�1�

l ¼ g1j . . . jgK½ �b ¼X

kgkbk ; ð24Þ

l x;Tð Þ ¼ 0; ð25Þ

where

b ¼ C�1d d� L unð Þ½ �: ð26Þ

[27] Since the inverse solution appears in (26), the TL andAD systems (22)–(25) are coupled. To decouple them, thecorrection can be written as a linear combination of repre-senter functions rk, k = 1, . . ., K. Each representer functiondescribes the spatiotemporal evolution of a correction to unFthat would result from assimilation of a single observation.The representer function rk is obtained by solving (22)–(25)

with the r.h.s of (24) replaced by gk (as a result, rk = yn).Then, b is obtained by solving a K � K linear system,

Pb ¼ d� L unF� �

; ð27Þ

where P = R + Cd and R = L([r1jr2j . . . jrk]) is therepresenter matrix.[28] When K is large, computation and storage of each

representer is impractical and unnecessary. Instead, theindirect representer approach is utilized [Egbert et al.,1994; Chua and Bennett, 2001]. System (27) is solvediteratively using the preconditioned conjugate gradientmethod [Golub and Van Loan, 1989], using a series of‘‘inner loop iterations.’’ The algorithm requires a rule, bywhich the matrix R multiplies an arbitrary vector b. Thismultiplication is achieved by solving (22)–(25) with b = band sampling the resulting tangent linear solution, to obtainL (yn). In our implementation, system (27) is solved untile0e/h0h < 10�3, where h is the rhs of (27) and e the misfit inthis equation.[29] In most cases in our study, the linear system (27) is

so poorly conditioned that the inner loop iterations fail toconverge without preconditioning. Instead of a poorlyconditioned system Pb = q, a better conditioned system~P~b = ~q, where ~P = Pp

�1/2PPp�1/2, ~b = Pp

1/2 b, and ~q = Pp�1/2q,

is solved. Matrix Pp should be easy to invert since thepreconditioned conjugate gradient algorithm requires solv-ing Pp z = w, where z and w are arbitrary vectors, directly[see Golub and Van Loan, 1989]. The method suggested byEgbert [1997] is found to be very effective. It is based oncomputation of a small subset of representers directly andusing those to form Pp. Obtaining inversion in a 1-hourinterval with K = 5,664 observations, we use 624 repre-senters to build the preconditioner. On the first outer loopiteration (utilizing the zero solution as the background forlinearization), 150–300 inner loop iterations are then re-quired. The second and higher outer loop iterations (utiliz-ing a better background solution) involve 20–50 inner loopiterations. We find that the number of the inner loopiterations depends on the number of degrees of freedom inthe corrected inputs. Generally, fewer inner loop iterationsare needed if inversion is obtained in a smaller assimilationtime interval. Also, in a given time interval, fewer innerloop iterations are needed in a case with C = C1 than withC = Clt<1, i.e., when fewer effective degrees of freedom areallowed in the control space.

3.3. Numerical Implementation Details

[30] The TL and AD numerical codes have been devel-oped manually following recipes of adjoint generation[Giering and Kaminski, 1998; Moore et al., 2004]. Thebackground state in both the TL and AD models is providedas a series of snapshots every Dt; instantaneous backgroundfields are obtained by linear interpolation. In cases with r =0.004 m s�1 (the equilibrated wave regime), we chooseDt =1 min. With r = 0.002 m s�1 (a more strongly nonlinearregime), Dt = 15 s to better represent higher-frequencyvariability. In both the NL and TL models, the discretespace of inputs F includes the initial conditions and thesnapshots of forcing fields provided at selected times;instantaneous forcing at every time step is obtained bylinear interpolation. In our implementations, if C = C1,


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the forcing can be represented by two identical snapshots, att = 0 and T. In cases allowing time variability in the forcing,forcing snapshots are specified every Dt. The discreteoutput space Y includes the snapshots of the fields at timest = 0,Dt, 2Dt, . . .,T. Copies of the interior fields at the edgesof the periodic boundaries are excluded from F and Y.[31] Subject to these definitions of the input and output

spaces, adjoint symmetry of the TL and AD codes has beenmaintained to within computer accuracy. Tests of thesymmetry involved generating a set of random input vectorsQ = [f1j . . .jfn] and verifying that the n � n matrix Q0 [AD][TL] Q, where [TL] and [AD] denote the implementation ofthe TL and AD codes, is symmetric and positive definite.[32] In the strongly nonlinear regime, small-scale tran-

sient flow convergence zones appearing near the coast arealiased in the time series sampled every 15 s. To avoidnumerical problems, temporal filtering (using a seven-pointpolynomial filter) has been applied to the inverse TLsolutions before using them as the background state onthe next outer loop iteration.

4. Integration Limit Set by Dataless Iterations

[33] Dataless (or ‘‘Picard’’) iterations are a standard testof convergence of the full state TL model (19) to the NLmodel [e.g., see Chua and Bennett, 2001; Muccino and Luo,2005]. On each iteration, (19) is integrated with the true(instead of prior) initial conditions and forcing. On the firstiteration, u0 = 0. Figure 7 shows time series plots of thearea-averaged root mean square (RMS) error, i.e., the RMSdifference between the true NL solution and TL solutionsun, separately for z, u, and v. The series of the TL solutionsconverge to the true NL solution only for a limited periodof time, tlim � 15 min in the case with r = 0.004 m s�1

(Figures 7a–7c) and tlim � 7 min with r = 0.002 m s�1

(Figures 7d–7f). Analysis of the flow fields shows thatiteration 1 yields a solution that equilibrates toward a steadysheared current. On iteration 2, shear instabilities appear andgrow exponentially on this steady background. Furtheriterations, utilizing growing solutions as the backgroundin (19), result in the abrupt growth (finite-time blow-up) asthe time approaches tlim. However, the TL solutions appar-ently converge to the true NL solution for t < tlim.[34] Through a series of additional experiments on an

alongshore uniform bathymetry (not shown), we verifiedthat tlim is proportional to the dominant timescale for growthof the linear shear instabilities on the unstable currentprofile u(y). Although tlim may indicate a characteristictimescale for the validity of the tangent linearization, itshould not stop us from attempting DA over larger timeintervals, as discussed next.

5. Data Assimilation Results

[35] The representer method is implemented in a timewindow of length T = 1 h, which is substantially longer thanboth the limit tlim suggested by the dataless iterations, andthe dominant timescale of cross-shore variations (severalminutes). Our study progresses from the equilibrated wavecase (section 5.1) to the more strongly nonlinear case(section 5.2). The quality of DA results are evaluated interms of several criteria:[36] 1. Do the series of inverse TL solutions converge to a

solution that is close to the true nonlinear state? If so, theinverse TL solution provides an accurate dynamically con-strained space- and time-continuous interpolation of asparse data set.[37] 2. Is the inverse estimate of the forcing close to the

true forcing? If so, the DA system can qualify as a tool forscientific analysis of dynamical driving mechanisms. We

Figure 7. Time series of the area-averaged RMS error, TL solutions from dataless iterations, (left) r =0.004 m s�1 and (right) r = 0.002 m s�1, (top) z, (middle) u, and (bottom) v.


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shall see that success in terms of criterion 1 does not alwaysimply that criterion 2 is met. In fact, the performance willdepend on an appropriate choice of C.[38] 3. Do inverse NL solutions, obtained by running the

NL model with the inverse initial conditions and forcing oneach outer loop iteration, converge to a solution close to thetrue state? In practice, it may be desirable to obtain anonlinear solution that fits the data, for example, to facilitateterm balance analysis. We shall see that convergencesatisfying criterion 1 does not guarantee success in termsof criterion 3.[39] Approaching these goals, we gain understanding of

the following fundamental questions. Is tangent lineargrowth an obstacle to variational DA over a large timeinterval, for example, extending far beyond tlim? Canvariability in the eddy field, resulting from interior nonlin-ear flow interactions, be captured in a DA system? What isthe impact of the prior forcing error covariance, C, both onconvergence of the representer method and on estimates offorcing?

5.1. Equilibrated Wave Regime: r = 4 � 10�3 m s�1

[40] With C = C1, the series of inverse TL solutionsconverges to a solution that fits the data and is uniformlyclose to the true solution in the full 1-hour time interval, ascan be seen in the time series plots of the area-averagedRMS error (Figure 8). Convergence is also apparent inFigure 9 (solid black line), in which the time- and area-averaged u- and v-RMS errors in un are plotted as functionsof the outer loop iteration number. The series of inverse NLsolutions (half-tone line in Figure 9, iterations 3–5 only)also converges to a solution of the same accuracy as u5.Over the entire assimilation time interval, instantaneousvorticity fields from iteration 5 TL and NL solutions arequalitatively very close to each other, and to the true state.In the equilibrated wave regime, the spatiotemporal evolu-tion of the eddy field can be described deterministically byboth the TL and NL inverse solutions.[41] On iteration 2 in the case with C = C1, a sizable, but

erroneous, correction is assigned to the alongshore compo-

nent fX in a narrow area offshore of the observational array.This correction is positive on average, forcing a stronglysheared alongshore current at y > 120 m (Figure 10a), wherethere are no observations (see Figure 1). Apparently, the DAsystem attempts to associate small-scale variability in theassimilated fields with the small-scale changes in the time-mean forcing. Additional iterations are necessary to reducethis erroneous effect. In some runs, in which we experi-mented with the spatial details of C = C1, the positivecurrent offshore of the observational array in early iterationsolutions was prone to shear instabilities that could not beconstrained by DA.[42] The case with the assumed standard bell-shaped

temporal correlation in the forcing error, C = Clt<1, lt =300 s (dashed line in Figure 9), converges more rapidly than

Figure 8. Time series of area-averaged RMS error, inverseTL solutions, r = 0.004 m s�1, C = C1: (a) z, (b) u, and (c) v.

Figure 9. Time- and area-averaged RMS error as afunction of the outer loop iteration number, inverse TL,and NL solutions, r = 0.004 m s�1: (top) u and (bottom) v.

Figure 10. Time- and alongshore-averaged u(y), the trueand inverse TL solutions, r = 0.004 m s�1: (a) C1 and(b) Clt<1.


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the one with C = C1, even though the assumption C = C1

is actually consistent with the steady forcing used in theseexperiments. In the case with C = Clt<1, the optimizationalgorithm associates small-scale time-varying variability inobservations with the allowed temporal deviations in theforcing, rather than its time mean, resulting in a better (thanwith C = C1) solution on iteration 2 (Figure 10b) and fasterconvergence overall.

[43] A case with the composite covariance (16) was runwith a = 0.05 and lt = 300 s. The time- and area-averagedRMS error (dotted line in Figure 9) is close to that with C =C1 for u, and is closer to that with C = Clt<1 for v.[44] For all three covariances considered, the main fea-

tures of the true forcing (see Figure 2) are reproduced in theinverse estimates after five outer loop iterations (Figures 11,12, and 13). However, in the alongshore- and time-averaged

Figure 11. Similar to Figure 2, but showing the inverse forcing, r = 0.004 m s�1, C = C1, iteration 5solution.

Figure 12. Inverse forcing, r = 0.004 m s�1, C = Clt<1, iteration 5 solution, (top) alongshore and(bottom) cross-shore components: (a, c) the alongshore and time averages (lines) ± the standard deviationfrom the time average (shaded) and (b, d) the time-average fields. In the maps, negative values areshaded. The contour interval is 2.5 � 10�3 and 1 � 10�2 m2 s�2 in Figures 12b and 12d, respectively.The dashed line is the 2-m bathymetric contour.


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estimates of fX in the case with C = C1 (Figure 11a), theerroneous variability with positive sign in the across-shoredirection is still present for y > 120 m. The standarddeviation of fX in the case C = Clt<1 is larger than thetime-mean in the trough between the coast and the sand bar(see Figure 12a). Utilizing the composite covariance pro-vides additional degrees of freedom compared to the casewith the steady covariance and thus helps to avoid thewiggles in the average profile fX(y) (see Figure 13a). Atthe same time, the magnitude of temporal deviations of theforcing is sensibly smaller than in the case using thestandard bell-shaped time correlation.[45] In the equilibrated wave regime, the 1-hour time

series solution with C = C1 can be obtained in a way that ismore computationally economical than direct minimizationin the full 1-hour interval. First, minimization can beperformed in a short 5-min time interval beginning on timet = 0. This converges in three outer loop iterations providingan estimate of the initial conditions and steady forcing thatare then used to run the NL model for a longer period. Thetime series RMS error of that inverse NL solution is shownin Figure 14 as the thick dashed line. Next, optimization is

approached in the 15-min interval (also beginning at t = 0),using the above-mentioned NL solution as u0. Convergenceis now obtained in two outer loop iterations. The improvedestimates of the inputs from the 15-min inversion are now

Figure 13. Similar to Figure 12, but for the case with the composite covariance (16).

Figure 14. Area-averaged v RMS error of the prior andnonlinear inverse solutions obtained solving optimizationproblems sequentially in time windows of increasing length(5, 15, and then 60 min); r = 0.004 m s�1, C = C1.

Figure 15. (left) Alongshore and (right) cross-shorecomponents of inverse forcing obtained solving optimiza-tion problems sequentially in time windows of increasinglength T: (a, b) 5 min, (c, d) 15 min, and (e, f) 60 min; r =0.004 m s�1, C = C1. Shades and contours are plottedsimilar to Figure 2.


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used to run the NL model for 1 hour (thin solid line inFigure 14). Finally, this NL solution is used as u0 duringinversion in the 1-hour interval. Only one iteration isneeded, and the RMS error of the resulting inverse NLsolution (thin dashed line in Figure 14) is close to that of thesolution obtained directly in the 1-hour interval.[46] Comparison of the inverse forcing fields obtained in

the 5-, 15-, and 60-min time intervals (Figure 15) to theirtrue values (see Figure 2) suggests that assimilation in thelongest interval is essential to obtain an accurate forcingestimate. For instance, in the forcing estimate obtainedafter 5 min there is no local maximum of fY over the bar(Figure 15b). This feature remains underestimated in the15-min solution (Figure 15d), but is reproduced accuratelyin the 1-hour estimate (Figure 15f).

5.2. Irregular Flow Regime: r = 0.002 m s�1

[47] The economical way of obtaining the inverse solu-tion (with C = C1) described at the end of the previoussection does not work in the irregular flow regime. Inver-sion in a 5-min interval converges to a solution that is betterthan the prior. However, the inverse NL solution resultingfrom the 15-min inversion does not provide further im-provement over the background state.[48] An attempt to assimilate the data directly in the 1-hour

interval with C = C1, beginning with u0 = 0, also fails(Figure 16). Shear instabilities grow exponentially in theiteration 3 TL solution inshore of the bar and the solver failson iteration 4 owing to numerical instabilities. A similarattempt utilizing C = Clt<1 with lt = 300 s again results in noconvergence. However, the case with lt = 60 s yields a seriesof TL inverse solutions converging to a state with a lowRMS error. So, allowing degrees of freedom in the forcingcorrection with temporal scales shorter than 5 min isimportant. Analysis of the frequency/wave number spec-trum of the true solution (see Figures 6e–6h) suggests thatmotions with frequencies near 1/60 = 0.017 cycles s�1 areenergetic only inshore of the bar. Adjusting the forcing inthe high-frequency part of the spectrum helps to controleddy behavior in this area, where an energetic dipole-shapedvorticity structure with a sharp front travels along the coast(see Figure 4f).[49] Despite the fact that the inverse TL solution is

accurate in the case lt = 60 s, the corresponding estimateof the forcing is not quite satisfactory (Figure 17). Themagnitude of time variations of the alongshore componentfX is comparable to its time-mean at all y (Figure 17a). Themaximum in fX over the bar is shifted to the left (Figure17b) compared to the true forcing (see Figure 2b). The firstmaximum of the cross-shore component fY (near the coast)is substantially underestimated and the second maximum(over the bar) is absent (Figures 17c and 17d). In an

Figure 16. Time series of area-averaged RMS error, r =0.002 m s�1, C = C1: (a) z, (b) u, and (c) v.

Figure 17. Similar to Figure 12, but for the case with r = 0.002 m s�1, C = Clt<1.


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application to real data, errors of such magnitude couldresult in incorrect inferences about the forcing.[50] To obtain both an accurate estimate of the flow and

forcing, the case with the composite covariance, lt = 60 sand a = 0.2 is run. The parameter a is chosen to be largerthan in the weakly nonlinear case, to allow control of moreenergetic eddy variability. Note that the selected value of ais close to the ratio of the magnitudes of the time-dependentcross-shore velocity fluctuations and the mean alongshorecurrent. The composite covariance still allows temporalvariations on a 1-min scale, but now their magnitude isconstrained. Outer loop iterations converge to a TL solutionwith low RMS error (Figure 18), and the estimate of theforcing (Figure 19) is more accurate than in the previous

case, with smaller deviations from the time-mean, betterspatial structure in fX, and two maxima in the cross-shoredirection in fY.[51] We also find that the time variable component of

the forcing estimate helps to control eddy variability ininverse TL solutions without contributing to the averageenergy balance. In particular, comparing the time-averagedwork by the mean forcing and by its fluctuations, maxju1fX ;1 þ v1fY ;1j � 0.01 max jufX þ vfY j, where the overbardenotes time-averaging and index 1 the deviation from thetime mean. Chua and Bennett [2001] suggest that additionaldissipation terms can be introduced in the system to increasethe validity limit of the TL model and thus to improveconvergence. In our case, the convergence is achieved in theinterval substantially longer than tlim without introducingadditional energy sinks in the system.[52] We have demonstrated that the solution obtained

with the composite covariance meets quality criteria 1 and2 stated at the beginning of section 5. However, unlike inthe equilibrated wave regime, it is more difficult to satisfycriterion 3 in this more irregular regime. The RMS error ofthe iteration 7 inverse NL solution (Figure 20, half-toneline) is sensibly better than the prior (thick black solid line)for z and u (Figures 20a and 20b, respectively) since thetime-mean surface elevation set-up z and alongshore currentu are well predicted by the DA model in this turbulentregime. However, the area-averaged RMS error for thecross-shore velocity component v (Figure 20c), whichwould seem to be a good metric of eddy predictabilitysince v � 0, is only marginally better than the prior error.Closer examination of the flow fields shows that in fact theinverse NL solution is qualitatively quite similar to the truth.Figure 21 shows vorticity snapshots every 7.5 min, the truesolution on the left and the inverse NL (iteration 7) solutionon the right. The two major flow features, the dipole nearthe coast and a larger-scale eddy farther offshore, are

Figure 18. Time series of area-averaged RMS error,inverse TL solutions, r = 0.002 m s�1, composite C: (a) z,(b) u, and (c) v.

Figure 19. Similar to Figure 12, but for the case with r = 0.002 m s�1 and the composite C.


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reproduced in the inverse NL solution. The location of thelatter is well predicted, while the former may be misplacedin the inverse NL solution. The RMS error in v is in factvery sensitive to the location of this dipole vortex structuretraveling along the coast.[53] Note that the corrected initial conditions at t = 0

(Figure 21, top left) are overly smooth compared to the truenonlinear solution. Since the time-dependent correction isapplied to the forcing, not directly to the ocean state, sharpfrontal features are developed in the inverse solutions (bothTL and NL) at later times. Preserving sharp fronts would bepotentially more difficult if a sequential DA algorithm (e.g.,optimal interpolation [Kurapov et al., 2005] or ensembleKalman filter [Evensen, 1994]) were applied to this prob-lem. Those methods provide corrections to the ocean statesequentially when data are available. Every time the fore-cast state is corrected, the fronts would be smeared.

Figure 20. Time series of area-averaged RMS error,inverse NL solutions, r = 0.002 m s�1, composite C: (a)z, (b) u, and (c) v.

Figure 21. Vorticity snapshots, the true and inverse NL solutions, r = 0.002 m s�1, composite C,iteration 7. Positive values are shaded, and contours are every 0.01 s�1.


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[54] Spatial maps of the time-averaged RMS velocityerror are shown in Figures 22a–22d. In the inverse NLsolution, the RMS error of u is improved throughout thedomain compared to the prior RMS error, mostly since thetime-mean is better (see Figures 22a and 22c). The RMSerror of v is improved on the offshore side of the bar(Figures 22b and 22d), where the NL model, run with theinverse input estimates, deterministically reproduces spatio-temporal variability in the larger vortex structure. Likewise,the correlation coefficient between the true and inverse NLsolutions is above the confidence limit (0.5) offshore of thebar (Figures 22e and 22f). The frequency-wave numberpower spectra of vorticity in the true and inverse NLsolutions (Figure 6, middle and right, respectively), are verysimilar both inshore and offshore of the bar. So, thevariability in the inverse NL solution on eddy timescalesappears to be statistically correct throughout the domain.

6. Summary

[55] A key result of our study is that accurate time-continuous reconstruction of the nonlinear flow is possibleutilizing variational DA for periods that significantly exceed‘‘the linearity time interval’’ (as can be defined, for instance,by convergence of Picard iterations). As the DA algorithmconverges, the deviation of the inverse solution from thetruth may become small enough for the linearization to be

valid over large time intervals (1 hour in our nearshoreexample).[56] In dynamical regimes where instabilities and eddy

interactions drive much of the variability DA is a challenge,particularly if the objective is to improve understanding offorcing mechanisms. Physical instabilities in the dynamicscan result in unstable behavior in the inversion procedure.Relaxing tight constraints on the forcing, even though theseconstraints might be strictly correct, can result in stableinversion and better ‘‘control’’ of unsteady flows in whichinstabilities and eddy interactions dominate. However, bymodifying the covariance to allow fictitious variability inthe forcing, the accuracy of the forcing estimate may becompromised. In our example, we utilize the compositecovariance (16) to relax the constraint on the forcing (here,steady forcing) only a little, allowing control of the flowwith minimal impact on the steady forcing estimates.[57] In our strongly nonlinear case, variational DA, based

on a series of linearized optimization problems, is success-ful in terms of criteria 1 (convergence of the linearizedsolutions to the true NL solution) and 2 (improved forcingestimate), defined in the beginning of section 5. In terms ofcriterion 3 (convergence of NL solutions obtained usingoptimized initial conditions and forcing), the resulting NLsolutions yield an accurate estimate of the mean alongshorecurrent and an eddy field that reproduces qualitativelyprincipal frontal features as well as statistics of the time-

Figure 22. Time-averaged statistics shown as a function of spatial coordinates, for the zero prior andinverse NL solutions, r = 0.002 m s�1, composite C, for (left) u and (right) v fields, (top) RMS error ofthe prior, (middle) RMS error of the inverse, and (bottom) correlation coefficient between the true andinverse solutions (only values �0.5 are contoured). The dashed contour is H = 2 m.


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variable flow. It may be argued that the above-mentionedcriteria as measures of success are specific to our variationalmethod. With regard to criterion 1, a criticism might beraised that our best solutions satisfy the linearized, ratherthan nonlinear, dynamical equations and that some otherDA methods that do not involve linearized dynamics, forexample, ensemble or particle filters, would be a betterapproach. However, let us note that estimates, or ‘‘analy-ses,’’ obtained by the filters do not satisfy the nonlineardynamics either. The sequential fitting of the instantaneousocean fields to data is analogous to adding a correction termin the dynamical equations [e.g., see Kurapov et al., 2005].Besides, since the filtering methods do not estimate theforcing directly, their success cannot be evaluated in termsof criteria 2 or 3, used here for the variational approach.[58] With the steady forcing error covariance C = C1 the

DA method converges in the equilibrated waves regime,yielding accurate forcing and flow estimates. In this case, anaccurate estimate of the steady forcing is necessary toprovide correct steady energy input to the system. Giventhis, the flow meanders over the entire 1-hour interval arelargely determined by initial conditions, which define theinitial phase of the shear wave. So, the initial conditions(and steady forcing) provide an effective means of controlof the time-variable flow. In the irregular case, the flow haslimited memory of initial conditions (about 5 min), so theinitial conditions are no longer effective means of control-ling temporal variability over the 1-hour interval. Therepresenter algorithm with C = C1 is not stable in theirregular regime.[59] Our synthetic data set captures energetic eddy vari-

ability in the computational domain. The spatial distributionand areal coverage in our case are comparable, for instance,to an array of moored instruments in the SandyDuck-97field experiment [Noyes et al., 2004]. Driving the model toclose proximity of the true trajectory may be more difficultusing sparser data sets. The sensitivity of results to the dataarray, and more generally studies of convergence andaccuracy of variational DA in cases when eddy variabilityis not adequately sampled, are topics for future research.[60] In systems with feedback mechanisms between flows

and forcings an assumption of time-varying forcing errorswould be physically reasonable. For instance, in the near-shore ocean, currents affect wave characteristics modifyingthe forcing of the circulation [Ozkan-Haller and Li, 2003;Yu and Slinn, 2003]. In implementations with real data,analysis of the model residuals could provide insights intothese flow-forcing interactions. Conversely, physical under-standing of such feedbacks could provide a basis forrefining the covariance.[61] Variational DA in high-resolution models promises

to be a useful synthesis tool in the context of oceanographicfield programs and emerging ocean observing systems,providing both dynamically constrained interpolation ofsparse and diverse data sets, and improved estimates ofthe forcing and parameters. Our example demonstrates that,for accurate estimation of forcing that varies on timescalesmuch larger than the dominant scales of variability ofnonlinear flows, it may be necessary to approach datainversion in time intervals that are substantially longer thanthe scales associated with eddies (e.g., see discussion at theend of section 5.1). The accuracy of the forcing estimate,

and possibly inferences about driving mechanisms, willdepend on the choice of the prior forcing error statistics.[62] The experiment in the alongshore periodic channel is

analogous to consideration of the flow in a domain of largealongshore extent. If the data were assimilated in a limited-area model, correcting open boundary inputs could provideadditional means of control of the unsteady inverse solution.Effectiveness of such control in the context of unstablenearshore flows remains to be investigated. The complexityof boundary conditions that ensure well-posed mathematicalformulation for the shallow-water equations, for instance,nonsmooth dependence on the state [Blayo and Debreu,2005], present a potential difficulty for the variationalmethod. Resolution of these difficulties in the shallow-watermodel would be a step toward understanding the even morechallenging problems of open boundary control in models ofthree-dimensional stratified flows [Oliger and Sundstrom,1978].[63] Shear waves, instabilities, and eddies have been

observed and modeled in the nearshore surf zone [Noyeset al., 2004, 2005; Long and Ozkan-Haller, 2005]. Thesuccess of our idealized experiment may open new possi-bilities for DA in the nearshore ocean. This area can bewell-sampled using moored instruments as well as remotesensing [Holman et al., 1993; Chickadel et al., 2003]. DAmight be applied to resolve deficiencies of today’s near-shore circulation models, providing improved estimates ofthe forcing, parameterizations, and possibly bathymetry, aswell as improving scientific understanding of the circula-tion, wave, and morphological processes. Work in thatdirection will ultimately help develop forecasting capabili-ties in the nearshore ocean. At the same time, our resultshave general applicability beyond nearshore applications,for example, to other time-dependent oceanographic flowproblems in which the assimilation time interval is longcompared to eddy timescales.

Appendix A: Derivation of the Euler-LagrangeEquations

[64] Chua and Bennett [2001] explain the essence of theindirect representer method, including derivation of the TLand AD equations, using a simple one-dimensional advec-tion equation. Following their approach, here we providedetails of how to obtain the linearized equations (18)–(27)for the general form of the model and data (4)–(6) and thepenalty functional defined by (7)–(10). The variation of thefunctional near the optimal solution is dJ(u) = J(u + du) �J(u), where du(x, t) is an arbitrary small variation of thestate. The ‘‘adjoint’’ variable is defined as the weightedoptimum dynamical error estimate,

l x; tð Þ ¼Z T

0

dt1

ZS

dx1 C�1 x; t; x1; t1ð Þe x1; t1ð Þ: ðA1Þ

[65] Using (4)–(6) and N(u + du) � N(u) + A[u]du, dJ(u)is expanded, terms quadratic in du are dropped and inte-gration by parts is utilized to eliminate @(du)/@t. Appropri-ate terms are combined to yield dJ(u) =

Rt

RSdu

0q1 +RSdu

0(x, 0)q2 +RSdu

0(x, T)q3. Now, the necessary conditionof a minimum of J is dJ(u) = 0 for arbitrary du. Then,


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q{1,2,3} must be 0, yielding the following Euler-Lagrangeequations:

q1 ¼ � @l@t

� A0 u½ �l� g1j . . . jgK½ �C�1d d� L uð Þð Þ ¼ 0; ðA2Þ

q2 ¼ l x; Tð Þ ¼ 0; ðA3Þ

q3 ¼ZS

C�10 x; x1ð Þ u x1; 0ð Þ � u

prior0

� �dx1 � l x; 0ð Þ ¼ 0: ðA4Þ

[66] Equations (A2) and (A3) form the ‘‘backward’’ sys-tem. The ‘‘forward’’ system is obtained convolving (A1)with C and (A4) with C0 and using (11),

@u

@t¼ N uð Þ þ fprior þ

ZS

ds1

Z T

0

dt1 C x; t; x1; t1ð Þl x1; t1ð Þ;

ðA5Þ

u x; 0ð Þ ¼ uprior0 þ

ZS

C0 x; x1ð Þl x1; 0ð Þdx1: ðA6Þ

[67] Both the optimum state u and the adjoint variable lare unknown such that the backward and forward systemsare coupled and nonlinear. They are solved via a series ofthe outer loop iterations, as described in section 3.2.Linearization is provided with respect to un�1. Then,decoupling is done following the representer methodology[see Chua and Bennett, 2001; Bennett, 2002].

[68] Acknowledgments. The research was supported by the Office ofNaval Research (ONR) Physical Oceanography Program under grantN00014-06-1-0257.

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��J. S. Allen, G. D. Egbert, A. L. Kurapov, and R. N. Miller, College of

Oceanic and Atmospheric Sciences, Oregon State University, 104 COASAdministration Building, Corvallis, OR 97331-5503, USA. ([email protected])


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