The representer method, the ensemble Kalman filter and the ensemble Kalman smoother: A comparison study using a nonlinear reduced gravity ocean model

Transcription

1 Ocean Modelling () The representer ethod, the enseble Kalan filter and the enseble Kalan soother: A coparison study using a nonlinear reduced gravity ocean odel Hans E. Ngodock a, *, Gregg A. Jacobs b, Mingshi Chen a, a Departent of Marine Science, University of Southern Mississippi, Stennis Space Center, Mississippi, United States b Naval Research Laboratory, Code, Stennis Space Center, Mississippi, United States Received February ; received in revised for August ; accepted August Available online Septeber Abstract This paper copares contending advanced data assiilation algoriths using the sae dynaical odel and easureents. Assiilation experients use the enseble Kalan filter (EnKF), the enseble Kalan soother (EnKS) and the representer ethod involving a nonlinear odel and synthetic easureents of a esoscale eddy. Twin odel experients provide the truth and assiilated state. The difference between truth and assiilation state is a ispositioning of an eddy in the initial state affected by a teporal shift. The systes are constructed to represent the dynaics, error covariances and data density as siilarly as possible, though because of the differing assuptions in the syste derivations subtle differences do occur. The results reflect soe of these differences in the tangent linear assuption ade in the representer adjoint and the teporal covariance of the EnKF, which does not correct initial condition errors. These differences are assessed through the accuracy of each ethod as a function of easureent density. Results indicate that these ethods are coparably accurate for sufficiently dense easureent networks; and each is able to correct the position of a purposefully isplaced esoscale eddy. As easureent density is decreased, the EnKS and the representer ethod retain accuracy longer than the EnKF. While the representer ethod is ore accurate than the sequential ethods within the tie period covered by the observations (particularly during the first part of the assiilation tie), the representer ethod is less accurate during later ties and during the forecast tie period for sparse networks as the tangent linear assuption becoes less accurate. Furtherore, the representer ethod proves to be significantly ore costly ( ties) than the EnKS and EnKF even with only a few outer iterations of the iterated indirect representer ethod. Ó Elsevier Ltd. All rights reserved. Keywords: Data assiilation; Representer ethod; Enseble Kalan filter; Enseble Kalan soother * Corresponding author. E-ail address: Hans.Ngodock@nrlssc.navy.il (H.E. Ngodock). Present address: Departent of Matheatics, Colorado University, Denver, CO, USA. -/$ - see front atter Ó Elsevier Ltd. All rights reserved. doi:./j.oceod...

2 H.E. Ngodock et al. / Ocean Modelling () 9. Introduction There is an abundance of data assiilation algoriths, each falling into one of two categories: sequential or variational. The sequential ethods are built on the application of the Kalan filter (Kalan and Bucy, 9), and include the extended Kalan filter (EKF) (e.g. Miller et al., 99), and soe reduced-rank EKF such as the reduced-order extended Kalan filter (ROEK) of Cane et al. (99), the sequential evolutive extended Kalan filter (SEEK) of Pha et al. (99), the error subspace statistical estiation (ESSE) of Lerusiaux and Robinson (999), the sequential evolutive interpolated Kalan filter (SEIK) of Pha et al. (99), and the reduced-order inforation filter (ROIF) of Chin et al. (). Other ipleentations of the KF to nonlinear dynaics use Monte-Carlo techniques for the coputation of the forecast error covariance atrix. Exaples include the enseble Kalan filter or EnKF (Evensen, 99), and its any variants (see Tippett et al., ). The list of the sequential ethods above is not exhaustive. It ay be suppleented by the soother version or ipleentation (e.g. the Kalan soother (KS), the enseble Kalan soother (EnKS)... etc.) and by the optial interpolation (OI) based ethods. The variational ethods are derived fro the optial control theory (Le Diet and Talagrand, 9) and objective analysis (Sasaki, 9, 9). They have been ipleented in eteorology and oceanography as three- and four-diensional variational assiilation ethods, better known as D-Var (Sasaki, 9) and D-Var (Lewis and Derber, 9; Derber, 9) respectively. The D-Var algorith is coputationally expensive. Efforts have been ade to ake it affordable, e.g. the increental D-Var of Courtier et al. (99). The D-Var ay be ipleented in its dual forulation, where the iniization of the cost function is done in the data subspace, e.g. the representer algorith (Bennett, 99, ). Each of these sequential and variational ethods is being used (or being considered to be used) at any operational eteorological and oceanographic centers, acadeic and research institutions. Considering the diverse assiilation counity, a growing interest in the field and liited coputer resources, one fundaental question arises: which ethod is ore accurate at less cost and under which circustances? The answer to this question should be inferred fro a careful and rigorous study that copares the assiilation ethods within the sae context, which includes odel dynaics and associated external forcing, initial and boundary conditions, data, data error covariance, odel error covariance or its representation, and of course, the sae estiator. The study should be conducted with a odel of high coplexity as currently used in the atospheric and oceanographic counities. Also, the coparison should be carried out in various assiilation settings (e.g. different circulation events or dynaical cases, different ethods for generating or prescribing the odel error covariance, different datasets and data types) to assess the ipact of different input factors on the outcoe of the assiilation. A sensitivity study with respect to each of these control paraeters would iply an extensive ultidiensional proble. Such an extensive study presents ajor difficulties: () the large nuber of assiilation ethods cannot be considered for one coparison study; () there is not a ethod to reconcile the representation of the odel error covariance for all the sequential ethods; () it would require a considerable aount of tie to develop the adjoint of a coplex odel, and should the adjoint exist, the assiilation experients would be coputationally costly. Moreover, the use of a coplex odel ay haper the insights that we can gain fro a siple odel. All these factors place the coparison study beyond the scope of one study and would require a collaborative effort of any assiilation practitioners using the sae odel with different assiilation techniques. The alternative is to narrow the scope of an individual study, and soe progress has been ade in this direction. Soe recent studies have copared assiilation ethods. Brusdal et al. () copared the EnKF, SEEK and the EnKS using the Miai Isopycnal coordinate ocean odel. Their study was a deonstration of the potential of these ethods for operational ocean forecasting systes. That coparison was between sequential ethods only. Caya et al. () copared a strong constraint D-Var with the enseble square root filter (EnSRF) Tippett et al. (), using a cloud odel and siulated radar observations. The rs error of odel variables was coputed during the assiilation window, and for the forecast. The D-Var yielded the lowest rs error. However, with the assiilation window confined to the first fifth of the entire tie interval of the experient (assiilation and forecast), the sequential ethod was at disadvantage because of its inability to correct the initial condition.

3 H.E. Ngodock et al. / Ocean Modelling () The data assiilation and odel evaluation experient (DAMEE) project consisted of a coparison of assiilation results fro different ocean odels using different assiilation schees on the sae region (Willes et al., 99). In such a study, it is difficult to draw consistent conclusions since any input paraeters were different across the experients. Kurapov et al. () also conducted a coparison study between the generalized inverse or GIM (the nae adopted for the representer ethod), the EnKF and the OI, using a linear baroclinic coastal ocean odel. They reported that both the representer and the EnKF ethods could fit the data satisfactorily, and that the representer ethod was able to copensate for poorly specified statistics for the odel and data errors. Another coparison between the representer ethod and the EnKF was carried out by Reichle et al. () with a land surface hydrology odel and synthetic icrowave radiobrightness data. They found that the EnKF could be less accurate than the representer ethod. However, the EnKF could be very inexpensive (in ters of coputational cost) when a reasonable analysis was achieved with a oderate enseble size. In all these studies, the nuber of assiilation ethods to be copared was reduced to two or three. The criteria for coparison were () the fit to the data, easured by the rs error of the assiilated solution, () the forecast error associated with each ethodõs estiate of the odel state at the final tie (e.g. Caya et al., ), and () the coputational cost of each ethod involved. The study here is different fro the DAMEE experient, and fro the study by Kurapov et al. () in the following sense: we use a nonlinear odel and ipleent both the EnKF and the representer ethod for the sae odel and datasets. Our approach is siilar to that of Reichle et al. () and Caya et al. (), coparing a sequential ethod with a variational, and not sequential with sequential, or variational with variational. We consider a weak-constraint D-Var, a change fro Caya et al. () who used a strong constraint D-Var. The strong constraint algorith ay becoe probleatic when the assiilation tie interval is longer than the predictability tie of the odel, i.e. the tie range of influence of the initial condition. The coparison within this work follows the work of Reichle et al. (), i.e. a coparison between the representer ethod and the EnKF. Here an ocean odel with esoscale activity is eployed. In addition, the EnKS solution is coputed. It allows not only an assessent of the iproveent in the EnKF solution by using the soother version of the sequential ethod, but also to copare a sequential soother (EnKS) to a variational soother (representer ethod). The variational ethod ay be forulated as either strong constraint (i.e. assuing a perfect dynaical odel), or weak constraint (i.e. allowing errors in the odel). Note that the errors in the dynaical odel can arise fro any sources: bad paraeterization, neglected and/or unresolved processes, erroneous external forcing... etc. In the weak constraint forulation, the search for the iniu of the cost function can be carried out in the state space, which is prohibitively costly for odels of high diensions, or in the data subspace, which is generally uch saller. The representer ethod is a weak constraint variational ethod that iniizes a cost function in the data space. It will be applied as the variational ethod in this study in the for of the iterated indirect representer ethod (Bennett, ; Chua and Bennett, ; Ngodock et al., ; Muccino and Bennett, ). A variational ethod is ipleented on the preise that the tangent linear odel (TLM) of the nonlinear odel is available and stable for the length of the assiilation window. The adjoint is based on the TLM. In this study, we used the full TLM of the ocean odel under consideration, and the adjoint thereof. The iterated indirect representer ethod is an attept to address the nonlinear properties of the syste within the context of the linear iniization. For the first iteration, a solution of the full nonlinear odel is used as the background state within the TLM. The representer ethod provides an optial estiate of the correction to the background state. The next iteration uses the iproved background state, and so on. The successive corrections within each iteration are expected to becoe saller, and the TLM approxiation ore accurate. The original Kalan filter (KF) algorith is designed for linear odels. For nonlinear odels, the extended Kalan filter (EKF) is used, where the propagation of the odel error covariance atrix is done with a linearized version of the odel. Within these fraeworks, the large diension error covariance atrix has to be coputed, and the linearization of the odel ight not work for strongly nonlinear odels, e.g. Evensen (99) and Miller et al. (99). This akes the KF and EKF unlikely candidates for sequential data assiilation using nonlinear odels with large diensions. The EnKF avoids the large diensional atrices of the KF and EKF and the oversiplifications of the reduced-order filters by generating the odel error covariance atrix based on enseble integrations with the nonlinear odel dynaics. Thus, the EnKF does

4 not require any linearization of the odel. Miller et al. (999) and Evensen (99) show that the enseble approach resolves the ajor difficulties associated with the EKF. Many factors influence the assiilation solution accuracy: data density, representation of odel error covariance, properly assigning errors to the appropriate sources and the enseble generation and size are a few. This study does not pursue each of these factors individually. Rather it focuses on the data density since it can easily be ade coon to both the sequential and variational ethods. The odel error covariance can be reconciled between the ethods in the sense that the statistical properties (variance and decorrelation scales) used in the enseble generation are the sae for the odel error covariance prescribed in the representer ethod. The selected ethods are expected to perfor satisfactorily accurately with dense data coverage. As data density decreases, the solution accuracy of each ethod is expected to degrade. An indication of which ethod is ore capable is easured by retaining accuracy (in the analysis and forecast) as data density decreases. It should be entioned that at any final analysis tie, the EnKS and the EnKF should always provide the sae solution. Also, both the representer and the EnKS should yield the sae solution at the analysis ties, provided that: () the odel is linear, and () the EnKS uses an infinitely large enseble. Such consistency results are not expected when a nonlinear odel is used as is the case here, even if a large enseble were used, which would be ipractical. Other reasons why we should expect differences follow. The representer ethod solves for the odal trajectory, i.e. the ode of the conditional joint pdf, while the general filter (enseble or not) solves for the arginal conditional pdf. These will differ for nonlinear dynaics. This can be seen fro the fact that with no observations the representer ethod results in the central forecast as the estiator, while the EnKF results in the enseble ean (Evensen,, personal counication). Furtherore, the EnKF cannot correct the initial conditions. The ethod has to progressively adjust toward the data until it starts fitting the data. The enseble generates a good spatial covariance atrix that takes into account the nonlinear evolution of the odel errors. The EnKS uses the EnKF solution at the latest analysis tie as first guess. The ethod then propagates this inforation backwards through a tie space covariance atrix generated by an enseble of odel trajectories, and is able to update the solution at all previous analysis ties. This backward propagation is deterined by the tie correlation estiated fro the enseble of trajectories. Thus, the accuracy of the previous analyses depends on the accuracy of the EnKF analysis in the first place, and the tie correlation in the EnKS covariance atrix. Nonetheless, the EnKS has a better covariance atrix than the EnKF because the trajectories that ake up the enseble coprise analyzed states at all previous analysis ties. In the representer ethod, the covariance functions are prescribed. The potential of being wrong, i.e. prescribing covariance functions that do not reflect the ÔrealityÕ of the ÔstateÕ errors is greater. Specifying odel errors is equally difficult in either ethod. The state error variance is ore correctly represented in the enseble ethods because of the ability to take into account the nonlinearities. However, adding odel error covariances to the enseble is just as difficult as specifying the in the variational ethod. The TLM and the adjoint will iss the nonlinear interactions and propagation of the odel errors. This can be copensated by the ability of the ethod to propagate the inforation backward and forward fro any data to the entire odel space tie doain. However, the ethod is clearly liited by the accuracy of the TLM. This is especially crucial when strong nonlinearities are present. Owing to all these potential sources of disagreeent between the selected ethods, this study will not focus on such aspects like choosing the right enseble size or prescribing the right covariance functions for the representer ethod. Since a large enseble is ipractical and since the literature abounds with applications using liited-size ensebles, this paper focuses on what accuracy is achieved with the representers and a liited-size enseble for EnKF and EnKS as one solves the assiilation proble. Thus, this paper is not an attept to cross-validate the assiilation ethods, justify the choice of covariance functions of enseble size. In the next section, we describe the dynaical odel. Assiilation experient setup and evaluations are discussed in Section, and results of the experients are exained in detail in Section. In Section, we discuss the cost of ipleenting each of these ethods, and concluding rearks follow in Section.. The dynaical odel H.E. Ngodock et al. / Ocean Modelling () A nonlinear reduced gravity (priitive equation) odel is used to siulate an idealized eddy shedding off the loop current (hereafter LC) in the Gulf of Mexico (hereafter GOM). It is the sae as the layer

5 H.E. Ngodock et al. / Ocean Modelling () version of the reduced gravity odel introduced by Hurlburt and Thopson (9). The dynaical equations are: ohu fhv þ g h oh ot ox ¼ A o hu M ox þ o hu þ s x drag oy x ; ohv ot þ fhu þ g h oh oy ¼ A o hv M ox þ o hv þ s y drag oy y ; ðþ oh ot þ ohu ox þ ohv oy ¼ ; where u and v are the zonal and eridional coponents of velocity, h is the layer thickness, f is the Coriolis paraeter (here a b-plane is adopted), g is the acceleration due to gravity, g is the reduced gravity, A M is the horizontal eddy diffusivity, coputed based on the prescribed Reynolds nuber Re, the axiu inflow velocity and half the width of the inflow port. The odel paraeters are listed in Table. Hurlburt and Thopson (9) showed that it is possible to siulate the eddy shedding by specifying tieinvariant transport at the inflow and outflow open boundaries (see the odel doain in Fig. ). In this case the wind stress and the botto drag are neglected. With a transport of Sv at inflow and outflow ports, we can siulate an eddy shedding with a period of about onths. The coputations are carried out on an Arakawa C-grid (Mesinger and Arakawa, 9), with unifor space steps Dx = k and Dy =. k and a tie step of in. The equations are solved for the transports hu and hv and for the layer thickness h, and the velocities are coputed by dividing the transports by the instantaneous layer thickness. Thus, the nonlinearities are present only in the pressure gradient ters and in retrieving the velocities fro the transport, aking the tangent linearization easier. The doain is k 9 k, with the inflow port k wide, centered k fro the eastern boundary, and the outflow port k wide, centered Table Table of odel paraeters b f g g Re s s 9. /s. /s. Fig.. The odel doain is an idealized Gulf of Mexico representation with inflow and outflow ports. The selected diagnostic locations of the assiilation solution are arked with bullets.

6 H.E. Ngodock et al. / Ocean Modelling () k fro the southern boundary. A scheatic representation of the GOM is a rectangular doain with inflow and outflow ports, and the odel grid is fro direct alignent with east west and north south (see Fig. ). In the reduced gravity odel the initial upper-layer thickness is with the flow at rest. Fro the reduced gravity layer thickness, the corresponding sea surface height (SSH) is given as: SSH ¼ðh Þ=ð þ g=g Þ. ðþ This linear relationship yields a linear easureent functional when the SSH data are assiilated into the odel. The reduced gravity is g = gdq/q, where q is the reference density, Dq is the density difference between the active layer and the otionless botto layer, and g is the acceleration due to gravity.. Experient setup and evaluation etrics.. Experient initialization, generating initial conditions and enseble ebers The assiilation experients are set up to test the ability to correct the position of the esoscale eddy field. This is accoplished through a twin odel experient in which one odel run provides an initial condition for the assiilation as well as the data for the assiilation. In order to provide an eddy position error, the observations are offset in tie. The dynaical odel of Section is spun up fro rest for years, which is sufficient tie to reach a statistical equilibriu (Hurlburt and Thopson, 9), and provides the initial condition for the subsequent setup. The reference or truth solution is constructed by integrating the nonlinear odel fro the initial condition for an additional onths and storing only the last onths fro which the observations are sapled. Thus the assiilation tie window is onths. The background solution, which the assiilation experients are supposed to correct, differs fro the reference solution by being lagged in tie. The SSH easureents are obtained fro the layer thickness h through Eq. (). We have assigned a standard deviation of c to the SSH and c/s to the velocity easureents in all the assiilation experients. For all assiilation experients, the data error covariance atrix is assued diagonal, the variance being the square of the respective easureent standard deviation. To generate enseble ebers, the odel is integrated fro the initial condition for onths and forced by a rando wind stress. The rando wind stress is coputed as a standard deviation ultiplying a rando field with unity variance. Each realization of the rando field is generated fro the inverse Fourier transfor of the prescribed wave nuber spectru with a rando phase at each wave nuber (Evensen, 99). The wave nuber spectru is a Gaussian function with e-folding length of k in both x and y directions. The standard deviation ( /s ) of the wind stress field represents a typical stress of. N/ plus uncertainty in the dynaical equations. The perturbation wind stress applied to each enseble is recoputed every observation interval, which is days or days depending on the sapling network. This wind stress perturbation represents not only errors in the external oentu flux, but also errors in the syste dynaics (Eq. ()). A different rando wind stress is provided to each enseble eber to generate as any ebers as are needed in the enseble. The -onth tie period in the enseble spin-up ensures two things: first, each enseble covers a large portion of physically realizable solutions. That is, the enseble properly represents the odel pdf. Second, the -onth offset provides a significant deviation fro the observations. This constructs the initial enseble (i.e. after the -onth spin-up with rando wind stress). The odel error covariance functions used in the representer ethod are given by! Qðx; t; x ; t Þ¼Vðx; x Þ exp ðx x Þ exp jt t j ðþ L s where L = k, i.e. the sae decorrelation length as in the rando fields used in the wind stress perturbations, and s = day is the tie decorrelation scale. The variance V(x,x ) is constant in space and tie and is equal to the square of the standard deviation of the wind stress perturbations. Finally, the dynaical errors in the representer ethod are applied only to the oentu equations (Jacobs and Ngodock, ), and there is

7 9 9 9 H.E. Ngodock et al. / Ocean Modelling () no cross-correlation between the odel error coponents. The continuity equation is considered strong constraint. The initial enseble ean is also taken as the initial condition for the representer experients. The background used in the variational solution for the tangent linearization of the dynaical equations (for the first iteration of the indirect representer ethod) is obtained fro integrating the nonlinear odel for onths fro the initial enseble ean. Fig. a shows the difference between the reference and the background initial conditions. The reference initial condition has a loop current eddy (hereafter LCE) in the center of the odel doain, and the renant of a LCE at the northwestern corner of the odel doain. In the next onths, the LCE in the iddle of the doain will quasi-linearly propagate westward, crash against the western boundary and slowly ove northward while dissipating. In the eantie, the loop current intrudes further into the doain and by the wake of the fourth onth another LCE is about to shed fro the LC. In the background initial condition, the LCE is just about to shed fro the LC, and a previously shed LCE has reached the western coast of the GOM. Fig. b shows that there is a substantial spread in the enseble. Although there is ore spatial structure in the spread with ebers, the agnitude of the spread did not increase with ebers. The values in Fig. b indicate that for SSH, the difference between ebers varies fro c to c, where a change of c in SSH corresponds to a change of in the layer thickness according to Eq. (). A standard set of diagnostic stations is used throughout the evaluations in this study. The station locations are shown in Fig.. They are selected in such a way that they are coon to all the sapling networks; locations are distributed along the path of the LCE, location is in the region where the LCE detaches, and 9. /sec 9 K K 9 9 a K. /sec K K b 9 9.e.e.e.e /sec 9 9.e K.e.e.e K 9 9.e.e.e.e. /sec K 9 9.e.e.e.e e Fig.. (a) Initial condition of the reference solution (left) fro which observations are taken and the initial enseble ean (right) which is also the initial condition of the background for the variation assiilation. The contour lines are for SSH, and the contour interval is c. The correction that the assiilation systes ust affect is to change the position of the loop current eddy. (b) The square root of the enseble spread around the initial enseble ean using ebers (left) and ebers (right).

8 H.E. Ngodock et al. / Ocean Modelling () location is north of the LCE detachent region. The background displays a phase lag of about onths fro the ÔtruthÕ solution as shown on saple tie series in Fig.. The rs errors between the background and the ÔtruthÕ solution, coputed at the diagnostic stations over the -onth and the last onths of the assiilation are shown in Table. The coputation of the rs is restricted to the diagnostic stations for the following reason: the variability in the odel doain is due to the LC intrusion and the LCE shedding and westward propagation. Elsewhere there is little variability, if at all. That is the reason we have selected diagnostic stations along the LCE path and in the region of the LCE s s s s s s s s s s Fig.. Tie series of the reference solution fro which data is taken (solid line) and the background that ust be corrected (dashed line) at diagnostic locations (top to botto rows respectively). The coluns fro left to right are for SSH, u velocity and v velocity respectively, and the x-axis is the nuber of days. Table Prior rs errors (between the reference and the background solutions) at the selected diagnostic locations coputed for the -onth and the last -onth of the assiilation tie window -onth Last -onth SSH u v SSH u v Location Location Location Location Location The units are eters () for SSH and eters per second (/s) for both coponents of velocity.

9 H.E. Ngodock et al. / Ocean Modelling () shedding. Often, the rs isfits are coputed based on a spatial average. However, that would lower the rs (by the contribution of the regions with no variability coon to the odel and the data) and cloud the ipact of the assiilation. The assiilation proble at hand can arguably be presented as an initialization proble. However, it is known that the sequential filter cannot estiate the initial condition fro future observations. While correcting an error in the initial condition (a isplaced eddy) is iportant, it is equally iportant to consider the fit to the future observations and accuracy of a subsequent forecast beyond the observation interval... Nonlinearities Nonlinearities in the odel are represented by strong advection when the LCE sheds fro the LC. Also, as the LCE propagates westward, nonlinear interactions between advection, latitudinal variations in Coriolis force and the pressure gradient ters are iportant. Nonlinearities influence the perforance of the EnKF, in the sense that the probability distribution function of the odel error ay not be Gaussian, when propagated by nonlinear dynaics. On the other hand, a nonlinear odel ust be preferably tangent linearized, with a stable TLM and the adjoint thereof, prior to using the representer ethod. Thus, the significant difference between the background and the reference solution (and thus the data, see Table and Figs. and ), and the nonlinearity of the odel provide a challenging fraework for all ethods, and hence the coparison. In these experients, the TLM is stable for the assiilation window, and we have ipleented the adjoint of the full TLM. The stability of the TLM was tested by the growth of sall perturbations using the first-order TaylorÕs expansion. For several easureent types (u, v, and ssh) and sites in the space tie doain the representer atrix is syetric, which insures that the adjoint covariance TLM syste is consistent... Enseble size The EnKF (and EnKS) solution is coputed with increasing enseble size, using,,, and ebers respectively. Results not reported here show that there is a substantial iproveent to the solution of both the EnKF and EnKS as the enseble size is increased fro to to. Beyond ebers, there is little change in the solutions. Therefore, for the coparison experients below, we will consider the EnKF and EnKS solutions that used the enseble of ebers... Observation networks and evaluation etrics It is expected all assiilation ethods perfor well with an accurate observation network that covers ost of the state space. In fact, if all the state space is observed with satisfactory sall errors, we do not need to assiilate at all. However, the observations seldo cover all the state space (all odel state variables observed throughout space and tie), and the accuracy of the assiilation is affected by the observation density. Therefore, the experients here have eight data networks ranging fro sparse (network ) to dense (network ) to investigate the accuracy of the assiilation ethods. Table describes the networks, the total nuber of easureents fro each network. The accuracy of each ethod is easured by the rs error (data inus analyzed solution) of each odel state variable coputed at the easureent sites. This rs is square root of the ean of squared errors over of the assiilation tie. A successful assiilation experient should fit the observations and the dynaics to within respective errors. This iplies that the rs should be less than the easureent standard deviation. This is the definition of ÔaccuracyÕ we will be using in the assiilation experients. It is known that the EnKF, being a sequential filter, does not correct the initial condition. Since the specified initial condition is lagged in tie to represent a isplaced eddy, soe tie is required for the EnKF to adjust the odel state to atch the data. Thus, it should not be expected that the EnKF typically perfor well early in the assiilation tie window. The EnKS is ipleented sequentially following Evensen and van Leeuwen (). However, being a soother, it uses all the data within the tie interval to adjust the odel trajectory, which can include the earliest observation tie. The representer ethod also has the ability to

10 H.E. Ngodock et al. / Ocean Modelling () Table The nuber of observations provided by each sapling network Network Assiilation tie covered k days k days k days k days k days k days k days k days Experient onth,,9 9,9 Experient onth Experient onth Only SSH The observations are uniforly distributed in the space and tie doain covered by each experient. The second colun provides soe detail on the differences between the experients. The abbreviations on the first row provide the spatial and teporal resolution of the observations. For exaple, k and days stand for a data network sapling every k (in both the x and y directions) and every days respectively. adjust the odel trajectory. As a consequence, these ethods will be copared using three etrics: first, we copare the rs error of the three solutions (i.e. representer, EnKS and EnKF) over the last onths of assiilation, so that the EnKF is not directly put at disadvantage. Secondly, the study copares the rs of the representer and the EnKS solution errors over the entire tie interval. Thirdly, we copare the forecast rs error fro the three ethods.. Experients.. Analyzed errors and relative assiilation syste accuracies The first set of experients consists of assiilating the data fro each of the eight networks individually using the representer, the EnKS and the EnKF ethods. The data consist of SSH, u and v observations. The rs error at the five diagnostic stations for each network is coputed over the last onths of the assiilation window, and plotted against the network nuber (Fig. ) rather than the total nuber of easureents fro the network, in order to aintain an equidistant x-axis in the plots. The correspondence between the network nuber and the nuber of easureents fro the network is given in Table. We will use the sae legend in Figs., where the dotted line will represent the observation standard deviation and the assiilated solution and forecast rs will be plotted shown for the representer (solid line), the EnKS (dashed lined) and the EnKF (dashed-dotted line). For the EnKF, there are only a few locations and state variables for which the rs is below or close to the data standard deviation for networks (the denser networks, Fig. ). The rs nonuniforly decreases as the network nuber increases. The EnKF errors have large peaks for networks and (using the longer days sapling), particularly at diagnostic locations, and, which are in the vicinity of the LCE shedding area. These networks saple data every days, indicating that the ethod is ore accurate with data sapled ore frequently in tie (every days in this case), even though the spatial sapling ay be ore sparse. The rs errors associated with the EnKF solution show that the ethod is the least capable of driving the solution toward the reference. Nevertheless, considering the prior rs errors in Table, Fig. shows that the EnKF is able to reduce the prior rs by as uch as 9% (e.g. network and location for SSH and v, also location for SSH), albeit for prior rs reductions lower than % at any locations: SSH and u at location and network ; u at location and networks, location and networks and. For the EnKS, the solution is significantly ore accurate than that of the EnKF. The rs errors associated with the EnKS are generally lower that those of the EnKF with only one exception: u at location and networks. This unexpected behavior ay be solely due to the prior rs being already lower than the easureent standard deviation. Other than this one anoaly, the EnKS displays iproveents to the EnKF that exceed one standard deviation of the observation noise level (even for sparse networks), which translate into a higher reduction of the prior rs errors. The representer rs errors are generally lower than those of the EnKF and EnKS. There are only a few locations where the representer is less accurate that the other two ethods: SSH at location and for networks, v at locations and for networks.

11 H.E. Ngodock et al. / Ocean Modelling () s s s s s s s s s s Data std Rep EnKS EnKF Fig.. The rs of all state variables: SSH (left colun), u (center colun) and v (right colun) at all five diagnostic locations (fro location on top row to location on botto row). The x-axis is the network nuber. The association of the network nuber and the total nuber of data is given in Table. The rs averaging covers the last onths of the onth assiilation tie interval. Solid lines are the representer rs errors, dashed lines are the EnKS errors, dashed-dotted lines are the EnKF errors, and dotted lines are the observation noise level... EnKS and representer coparison Because the EnKF does not correct the initial condition, coparisons in Fig. are liited to the final onths of the assiilation window. However, both the EnKS and representer ethods correct the initial condition, and a fair coparison ay be conducted over the entire assiilation window. In this second coparison, the rs error of the EnKS and the representer solutions is coputed over the entire assiilation window. Results are plotted in Fig., and show that the representer is ore accurate than the EnKS as it retains its accuracy with ore sparse networks. The larger differences between the two ethods in Fig., as copared to Fig., indicate that representer ethod fits the data uch better in the first half of the assiilation window than its does in the second half. The converse sees to be true for the EnKS, although not uniforly across the networks and locations, and that ay be due to initial condition inaccuracy in the EnKS solution... Initial condition effects The EnKF analysis tie series of the assiilated solution (not shown), requires between and days (depending on odel variables) to begin to atch the data. This is a direct effect of the purposeful wrong initial condition. Eventually, the influence of the initial condition will decay over tie, resulting in a better

12 H.E. Ngodock et al. / Ocean Modelling () s s s s s s s s s s Fig.. This analysis is the sae as Fig. except the rs errors are coputed over the entire assiilation window. Only the representer (solid line) and EnKS (dashed line) results are shown. The EnKF does not correct the initial condition, and thus is not coparable to the EnKS and representer solutions over the entire assiilation window. fit to the data by the EnKF at ties that are distant enough fro the initial tie. Therefore, a second set of experients (Experient in Table ) is designed where the assiilation period is extended to onths and the rs coputation covers the last onths. The total nuber of data involved in these experients is listed in Table. The assiilation experients are conducted using data only for networks, and. These are the (sparse) networks for which the EnKF and EnKS accuracy sharply degrades after satisfactory perforance for the denser networks (see Figs. and ). In coparison with Fig., the results in Fig. indicate that the sequential ethods have a uch lower rs (across the networks and for alost all state variables) than in the previous experient. This suggests that sequential ethods benefit fro a longer adjustent tie, and therefore are ore accurate in the second half of the assiilation window. The representer ethod on the other hand yields ixed results. In ost cases (e.g. zonal velocity at all locations, eridional velocity at locations, SSH at locations ) the rs slightly increased, especially using network, which saples every days. Soe iproveents in the representer solution rs are noted for the eridional velocity at locations for all networks involved and the SSH at locations for network, and location for network. The ain point here is that in going fro to onths the rs did iprove for the sequential ethods in the last onths. The iplication is that when the sequential ethods are given enough tie to adjust, the analysis accuracy iproves. However, at ost locations and networks, the representer is ore accurate than the EnKS and EnKF, with soe exceptions. The representer and the EnKS rs errors are also coputed over the entire -onth assiilation window. Results (Fig. ) show that the representer outperfors the EnKS. Note here that the EnKS rs increases while the representer rs is slightly lower than that of the second half window (Fig. ). This again indicates that while the representer is fitting the data slightly better in the first half of the assiilation window than it does

13 9 H.E. Ngodock et al. / Ocean Modelling () s s s s s s s s s s Fig.. The results here are siilar to those in Fig. except the assiilation window is onths, and the averaging of the rs covers the last onths, and the experients are carried out only for networks, and. in the second, the EnKS is fitting the data better in the second half than it does in the first. This finding becoes critical in the forecast coparison... SSH only The assiilation of only SSH data is otivated by the potential use of satellite altieter data. Experient covers onths as in Experient, but only SSH is assiilated. The assiilation is carried out for networks. The corresponding nuber of data is listed in Table. The rs errors are coputed for the last onths. The results (Fig. ) indicate that the accuracy in this experient follows the pattern of Experient. The representer ethod is accurate for SSH at locations, and for all networks, location for networks, and. At location the accuracy decreases for networks. However, contrary to the other ethods, the representer SSH rs error never exceeds standard deviations of the data error. For the zonal velocity the accuracy is achieved at location networks, location for all networks, location for networks and, and location for networks. Elsewhere, the solution accuracy decreases with an rs that soeties exceeds standard deviations, e.g. location for networks -, location for network. In the eridional velocity, accuracy is achieved only at locations and for all networks. For locations, the rs rapidly increases fro below standard deviations (networks and ) to ore than, as the data density decreases to network. The challenging stations for the v-coponent are located around the eddy shedding area, where stronger advective nonlinearities occur. This ay indicate inadequacies in the odel error covariance functions: the

14 H.E. Ngodock et al. / Ocean Modelling () s s s s s s s s s s Fig.. Coparison of the representer and EnKS for the -onth assiilation experient. The rs is coputed over the entire assiilation window. covariance prescribed for the representer lacks cross-correlation between variables, while this cross-correlation is inherently generated in the covariance coputed fro the enseble. On the other hand, the covariance in the variational ethod is not designed to represent the nonlinear interaction of the odel errors. The EnKF solution SSH is accurate only at location for networks, having an rs error less or equal to the observation standard deviation. At all other locations, the SSH rs is always above a standard deviation. As the data density decreases, the rs gradually increases to ore than standard deviations, e.g. network at locations and. For the zonal velocity, accuracy is achieved at location for all networks, location for networks, and location for networks. Elsewhere the solution rs error is greater than the observation standard deviation. The rs for the eridional velocity is alost always above the observation standard deviation, except at locations for networks. In general, except for a few locations and variables, the EnKF rs error is always greater than that of the representer. As we have observed in previous experients, the EnKS rs error is significantly lower than the EnKF. There are several locations and networks where the EnKS rs is lower than the observation standard deviation whereas the EnKF rs is well above. Also, there are locations (and networks) where the EnKS rs is lower than the representer rs, especially for the velocity coponents. The rs errors coputed over the -onth assiilation window of Experient are copared for representer and the EnKS. Results in Fig. 9 show that the rs error slightly increases for the representer ethod, whereas the increase is quite significant for the EnKS, particularly for the velocity. This increase in the rs indicates that, contrary to the previous experients, both ethods are fitting the data with a lower accuracy in the first half of the assiilation window than in the second. The EnKS rs error is lower or equal to the

15 9 H.E. Ngodock et al. / Ocean Modelling () s s s s s s s s s s Fig.. Sae as in Fig., except only SSH easureents are assiilated. observation standard deviation only for SSH at location (all networks), location (networks ), for the zonal velocity at location (networks and ), location (network ) and location (network ). The rs error for the eridional velocity is always greater than the standard deviation. The representer ethod on the other hand is ore accurate. The SSH rs error is less than the standard deviation at all locations and networks, except at location. Also for both the velocity coponents, the ethod is accurate at locations for networks... Forecast accuracy The coparison experients above were all carried out for the hindcast proble. Since these ethods are potential candidates for analysis-forecast cycling systes, they ust be copared also on perforance in the forecast after the analysis. It should be noted that theoretically, the EnKF and the EnKS produce the sae solution at the final analysis (Evensen and van Leeuwen, ), and thus the sae forecast. This is confired in our experients. Therefore, we will copare the forecast fro the representer and the EnKS solutions. The forecast is coputed as the nonlinear odel integrated forward fro the analysis at the final tie of the assiilation. It is carried out for onths, and the coparison is based on the rs error between the reference forecast and the forecast fro the assiilation experients. These forecast rs errors are coputed for Experient (all data assiilated at each network) and Experient (only SSH data assiilated). The first forecast coparison results in Fig. show an alost identical scenario across the diagnostic locations and variables: the representer rs error is lower than that of the EnKS/F for the dense networks. As the data density decreases, the representer rs increases alost uniforly and becoes greater than the EnKS/F

16 H.E. Ngodock et al. / Ocean Modelling () s s s s s s s s s s Fig. 9. Sae as in Fig., except only SSH easureents are assiilated. rs. The forecast rs error depends solely on the accuracy attained in the estiation of the final state of the solution. Thus, Fig. illustrates that the representer estiate of the final odel state is better than that of the EnKS/F for the denser networks, and the converse is true for the sparser networks. This result sees to contradict the better accuracy achieved by the representer ethod in the hindcast proble. One should reeber that the hindcast rs depends on the accuracy of the assiilated solution at all the analysis ties, whereas the forecast rs depends only on the accuracy of the assiilated solution at the final analysis. Also, we have seen in the hindcast experients that the representer ethod is ore accurate in the first half of the assiilation window than in the second. Therefore, the lesser accuracy in the estiate of the final condition (for the sparse networks) is not a contradiction or surprise. The perforance of the representer forecast rs degrades when only SSH data are assiilated (Experient ) as shown in Fig.. The rs error grows rapidly with decreasing data density, except for a few locations where it appears steady. The EnKS/F forecast rs, on the other hand, is ostly level and substantially lower than that of the representer, with a very few exceptions. The EnKF better accuracy at the final tie can be explained by its representation of the covariance atrix: the coputation of the covariance atrix uses the full nonlinear odel, and the analysis ay be iproving at each observation tie, i.e. getting closer to the data. In the EnKF, the covariance atrix P f at any tie is approxiated by enseble integrations using the full nonlinear dynaics. The counterpart of P f in the representer ethod is LCL T, where L is the TLM, L T is the adjoint and C is the prescribed odel error covariance. It is clear that unlike P f, LCL T isses the nonlinear interactions of the dynaics in its representation of the covariance (this is especially true in the presence on strong nonlinearities as during the eddy shedding), and C ay not represent the nonlinear dynaical evolution of the odel errors. These covariance flaws in the

17 9 H.E. Ngodock et al. / Ocean Modelling () s s s s s s s s s s Fig.. The forecast rs error for the representer (solid line) and the EnKS/EnKF (dashed line), with all data assiilated. The dotted line is the easureent noise level, and the x-axis is the network nuber. representer ethod ay be copensated either by the aount of data for dense observation networks, or, in the case of sparse networks, by the contribution of past and future observation for observation ties well within the assiilation window. As entioned above, the dynaics represented by the data are characterized by a strong nonlinear event in the last onth of the assiilation window, naely an eddy shedding. In the first three onths of the data, the dynaics are characterized by a westward propagation of an LCE while the LC is intruding into the doain. In the wake of the fourth onth, the LC further intrudes and a new LCE starts shedding, a result of stronger advective nonlinearities. The representer ethod is expected to be less accurate in fitting the data in this last segent because the TLM is no longer as accurate as in the first onths, and the covariance functions have becoe inadequate: they are issing the nonlinear interactions and cross-correlation of the odel errors. The enseble-based ethods on the other hand generate a covariance atrix based on these nonlinearities, and are therefore ore able to fit the data. To elicit the role of these strong nonlinearities and inaccuracy of the TLM toward the end of the assiilation window, we ade a -onth forecast fro the representer solution onth ( days) before the end of the assiilation, and a -onth forecast fro the representer solution onths ( days) before the final tie. These forecasts are copared to the EnKF/EnKS forecast and the representer forecast fro the end tie for onths, for network. The results in Fig. show that the representer forecasts and days prior to the end tie are ore accurate than that fro the end tie and the one fro EnKF. Thus the strong nonlinear event during the last onth has a big ipact on the representer solution. However, this is not to say that the representer ethod is unable to assiilate data accurately in the presence of strong nonlinearities. The assiilation experient