1. Introduction

One of the main goals of data assimilation is to improve the forecast performance of a dynamical model. Many data assimilation algorithms are based on Kalman filtering, where the state of a system is estimated by combining all the information that is available about the system in accord with their statistical uncertainty. The main computational issue in Kalman filter–based data assimilation is the propagation of the forecast error covariance matrix. A number of methods have been proposed to solve this problem, for example the ensemble Kalman filter (EnKF) proposed by Evensen (1994). In EnKF, the forecast error covariance matrix is approximated by using a Monte Carlo method. Another method, which can also be considered as an ensemble method, is the reduced rank square root (RRSQRT) Kalman filter proposed by Verlaan and Heemink (1997). The RRSQRT Kalman filter approximates the error covariance by a matrix of a lower rank. An overview about the ensemble assimilation development can be found for example in Evensen (2003).

A less computationally demanding assimilation algorithm is based on the steady-state solution of the Kalman filter as suggested by Heemink and Kloosterhuis (1990). This approach is built on the fact that for systems where the covariance of the forecast error is almost time invariant, it is possible to compute the Kalman gain offline until it has reached approximately a constant solution. Applying the constant Kalman gain in a data assimilation system reduces the computational demands to almost the same as the standard model execution without data assimilation. However, this approach still requires the solution of the Riccati equation used subsequently in the Kalman filter formulation or it requires a more elaborate scheme for the generation of the steady-state gain. Heemink and Kloosterhuis (1990) used simpler dynamics for propagating the covariance matrix and a Chandrasekhar-based algorithm to compute a steady-state Kalman gain. Another approximation can be introduced by defining the error at a coarser grid size (Fukumori and Malanotte-Rizzoli 1995). The usage of a simpler dynamical model propagator for the error propagation is also suggested by Dee (1991). Sørensen et al. (2004a) computed the steady gain as a long-term average of the Kalman gains estimated by an EnKF and demonstrated its performance in the North and Baltic Sea system. The EnKF, however, requires a large number of ensemble members to get the ensemble mean statistics.

In this paper, a new method for computing the steady gain is proposed, which is based only on two forecast realizations. If the covariance of forecast error is almost time invariant we can use the time average instead of ensemble mean. This has the advantages of being computationally very cheap and very easy to implement. The two forecast realizations may be generated with two differing but similarly skillful prediction models. Moreover, it is also possible to use the algorithm for the case where the two realizations are generated by running the dynamical model twice, driven by two different inputs. In oceanography, for example, it becomes increasingly popular to compute the forecast error statistics by using two different wind fields, produced by two different atmospheric models, as input to an oceanic model under study (Alves and Robert 2005; Leeuwenburgh 2007). In this case no explicit model error specification is required.

The proposed algorithm is based on the assumption that the error process of the system of interest is weakly stationary. That is, the mean and covariance of the forecast error of the dynamical system being studied is constant in time. However, the algorithm may also be applicable for a system where the error statistics vary slowly in time. The proposed algorithm makes use of two different forecasts performed for the same period. The difference between the two forecasts provides information on the forecast error statistics. To estimate the forecast error statistics from the time series of these differences, it is necessary to assume that the noise process is ergodic. In this case the covariance estimates can be computed by averaging the realization over time. In practice, ergodicity assumption may be taken for a stationary random process with a short correlation time. If we can select some samples of the random process at a time interval that is sufficiently longer than its correlation time, the samples are effectively uncorrelated. This allows us to consider the series as independent realizations of the same distribution. In this case, averaging over time is equivalent to averaging over ensembles. The covariance estimate is used to define a gain matrix for a sequential steady-state analysis system. The proposed algorithm also consists of an iterative procedure for improving the covariance estimates. The procedure requires a fixed observing system during the iteration.

The algorithm proposed in this paper may be applied to systems where a steady-state Kalman filter can be applied. A number of applications of the steady-state Kalman filter can be found for example in shelf sea modelings (e.g., Canizares et al. 2001; Verlaan et al. 2005; Sørensen et al. 2006) and coastal sea modelings (e.g., Heemink 1990; Oke et al. 2002; Serafy et al. 2005). The application of an approximate Kalman filter, which includes the use of stationary error covariance, can also be found in oceanography (e.g., Fukumori et al. 1993). The effectiveness of the stationary assumption was examined by Fu et al. (1993) where the resulting solution was found to be statistically indistinguishable from the one obtained using the full Kalman filtering. It is interesting to mention here that the utility of a stationary covariance matrix in the Kalman gain was demonstrated even in the presence of evolving properties of the observation system (Fukumori 1995; Fukumori and Malanotte-Rizzoli 1995; Hirose et al. 1999). The proposed algorithm is expected to be applicable for any system similar to those mentioned above.

Some twin experiments are performed to evaluate the performance of the proposed algorithm. This allows us to evaluate the results by comparing them to the “truth.” The experiments are intended to demonstrate that for a number of applications, the results produced by the proposed algorithm converge to the optimal ones. For these experiments, the perturbations are drawn from a Gaussian distribution with a mean zero and a given covariance matrix. The results are compared to those produced by a classical Kalman filter algorithm.

The paper is organized as follows. Section 2 explains the algorithm along with its mathematical formulations. The first experiment, which uses a simple one-dimensional wave model, is discussed in section 3. The second experiment uses the operational model for storm surge prediction, the Dutch Continental Shelf Model, which is described and discussed in section 4. An experiment with the three-variable Lorenz model is given in section 5. The paper concludes in section 6 with a discussion of the results.

2. Algorithm

The algorithm proposed in this paper belongs to the class of iterative methods for finding steady-state Kalman gain. A number of such methods have been proposed, for example by Mehra (1970), Mehra (1972), and Rogers (1988), which are based on some algebraic manipulations on the Kalman filter equations. However, these methods still require propagating matrices that have the same dimension as the covariance matrix into the dynamical system of interest. Therefore, they are not applicable for earth systems problems, which usually have big dimension.

Our algorithm is based on estimating the forecast error covariance matrix using two forecast realizations, which are perturbed by statistically independent perturbations. This covariance matrix is used to define the Kalman gain matrix for a sequential steady-state analysis system. The algorithm consists of an open-loop step and a closed-loop step. In the open-loop step, the covariance is computed without making use of any observations of the process. The closed-loop step consists of an iterative procedure for improving the covariance estimate by assimilating available observations. The result of the open-loop step serves as the starting point for the closed-loop iteration.

a. Statistics from two independent realizations

Consider a discrete random process X^f(t_k) ∈ Rⁿ that can be decomposed into deterministic and stochastic components as follows:

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where x(t_k) ∈ Rⁿ is a deterministic process and X̃(t_k) ∈ Rⁿ a weakly stationary random process. Suppose we can draw two realizations of X^f(t_k), x^f₁(t_k) and x^f₂(t_k), obtained from two statistically independent realizations of X̃(t_k) added to the deterministic part. We define e_1,2(t_k) as the difference between these two series:

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It is easy to see that e_1,2(t_k) is a weakly stationary random process and has the following properties:

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where 𝗘[] is the expectation operator and 𝗣^f(t_k) ∈ R^n×n is the covariance matrix of the random process X^f(t_k) defined as 𝗣^f(t_k) = 𝗘[{X^f(t_k) − 𝗘[X^f(t_k)]}{X^f(t_k) − 𝗘[X^f(t_k)]}^T]. Since the random process is weakly stationary, its covariance matrix is constant, 𝗣^f(t_k) = 𝗣^f.

Given a sample e_1,2(t_k), k = 1, . . . , N, and assuming that the random process is ergodic, we can calculate the covariance by averaging the realization over time:

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Note that two samples of X^f(t_k) are required to eliminate the trend within each sample, so that we can use time averaging for computing the covariance.

b. Open-loop estimate

We first discuss the open-loop estimate, where the estimation of the covariance matrix of a random process is done without making use of any observations. Consider a random process governed by a dynamic model 𝗠 as the following:

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where X^f(t_k) ∈ Rⁿ denotes the state vector, u(t_k) ∈ R^p is the deterministic forcing term, 𝗕 ∈ R^n×p is the transition matrix between the forcing and the state, and W(t_k) ∈ Rⁿ is a white noise vector with covariance 𝗤 ∈ R^n×n. From the governing equation of X^f(t_k) we can derive the evolution equation of the covariance 𝗣^f:

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If 𝗠 is stable it will reach a steady-state condition where 𝗣^f(t_k+1) = 𝗣^f(t_k) = 𝗣^f according to 𝗣^f = 𝗠𝗣^f𝗠^T + 𝗤.

Now let x^f₁(t_k) and x^f₂(t_k), k = 1, . . . , N, be two independent samples from this process. If the process is ergodic, we can estimate 𝗣^f by using Eq. (5).

c. Closed-loop estimate

Suppose now the observation y^o(t_k) ∈ R^m of the dynamical system [Eq. (6)] is available and we assume that the observation is related to the unknown true state x^t(t_k) through

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where 𝗛 ∈ R^m×n is a linear observation operator and V(t_k) ∈ R^m is a white noise vector with a constant covariance 𝗥 ∈ R^m×m representing observation uncertainty. Combining the observations and the dynamical system, we write the governing equation for X^f(t_k) as follows:

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It is easy to show that the corresponding covariance matrices evolve in time according to

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For a stable and time-invariant system, a steady-state condition will occur, where 𝗣^f(t_k+1) = 𝗣^f(t_k) = 𝗣^f and 𝗣^a(t_k+1) = 𝗣^a(t_k) = 𝗣^a.

Let x^f₁(t_k) and x^f₂(t_k), k = 1, . . . , N, be two independent samples from this process. Then if we assume that the process is ergodic, we can estimate 𝗣^f by using Eq. (5).

The constant gain matrix 𝗞 ∈ R^n×m is chosen such that it minimizes the covariance 𝗣^a. It can be shown that such 𝗞 is given by

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Note that the closed-loop system of Eqs. (9)–(10) is actually a forecast-analysis system.

d. Proposed algorithm

The algorithm is initialized by generating two independent realizations of the open-loop process in Eq. (6). This is done by running the open-loop system twice for the same period and by perturbing both runs using statistically independent perturbations drawn from a given distribution. Having two independent realizations, the covariance is computed using Eq. (5) and subsequently the open-loop gain matrix by using Eq. (13). The next step is the closed-loop iteration. In the first iteration, the gain matrix computed in the open-loop step is inserted into Eq. (10) and the closed-loop system [i.e., Eqs. (9)–(10)] is run twice for the same period. The forecasts in both runs are perturbed using statistically independent perturbations drawn from the model error distribution. Moreover, independent perturbations representing observation noise are also generated from a given distribution. The two independent realizations of the closed-loop forecasts in Eq. (9) are used to compute the forecast error covariance and the corresponding gain matrix by using Eqs. (5) and (13), respectively. This gain matrix is subsequently inserted into the closed-loop system in Eq. (10) for the next iteration. The closed-loop estimation is repeated until the covariance does not change anymore. At each iteration, the gain matrix computed in the previous iteration is inserted into the closed-loop system in Eq. (10).

To summarize, the proposed algorithm consists of the following steps:

• Open-loop step:

  • generate two realizations of open-loop process, Eq. (6): x^f₁(t_k) and x^f₂(t_k), k = 1, . . . , N;

  • estimate 𝗣^f using Eq. (5);

  • stimate 𝗞 using Eq. (13).

• Closed-loop step:

  • insert 𝗞 into Eq. (10);

  • generate two realizations of closed-loop process, Eq. (9): x^f₁(t_k) and x^f₂(t_k), k = 1, . . . , N;

  • estimate 𝗣^f using Eq. (5);

  • estimate 𝗞 using Eq. (13); and

  • repeat closed-loop step until 𝗣^f, and therefore also 𝗞, have converged.

From the previous section, since 𝗞 is chosen to minimize the covariance 𝗣^a, the covariance 𝗣^f at one iteration step will be smaller than the one at the previous iteration. Hence, it can be seen that if the iteration process has converged, then 𝗞 is the optimal gain. Note that for linear, time-invariant systems, the corresponding steady-state Kalman gain is a solution to the iteration.

This algorithm has much in common with the analysis-ensemble method (Fisher 2003). The novel aspects of this algorithm are the use of the open-loop estimate as an initial guess for the closed-loop iterations and the iterative improvement of the estimated covariances. Our algorithm is also similar as the ensemble Kalman filter (Evensen 2003). The difference is that instead of averaging over ensembles, the forecast error covariance is estimated by averaging over time.

Note that the equations of the closed-loop system are the same as in the Kalman filter formulation. The difference is that in our algorithm 𝗞 is updated every iteration step, while in the Kalman filter it is updated every time step. It is a well-known result that for a stable and time-invariant system the Kalman gain will converge. For the same system the gain computed by the proposed algorithm is expected to converge as well.

The computational cost of the proposed algorithm depends on the length of the two samples as well as on the number of observations used for assimilation. It should be noted here that the computation of the gain matrix 𝗞 requires only the computation of the two terms 𝗣^f𝗛^T and 𝗛𝗣^f𝗛^T, which we compute as

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and

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For a limited number of observations per analysis step this is a limited amount of work. Moreover, since the statistical computations are performed offline, the computational cost is not an issue as long as it is feasible for the system of interest. Once the gain matrix is computed and implemented in a data assimilation system, the computational cost will be the same as an assimilation system using a steady-state Kalman filter or optimal interpolation. Furthermore, the implementation is simple since only the analysis equation [i.e., Eq. (10)] is added to the code of the model.

From the description above, we can see that the algorithm is applicable for an assimilation system with a fixed observation network. Moreover, it also assumes the error process of the system under study to be weakly stationary. The stationarity assumption impedes the algorithm from having the ability to include a flow dependence in the forecast error statistics. However, in a number of real-time forecasting systems, it has been shown that it is sufficient to use a stationary covariance for data assimilation (e.g., Heemink and Kloosterhuis 1990; Sørensen et al. 2004b). In these applications, a steady-state Kalman filter is found to be sufficiently accurate. In particular, Canizares et al. (2001) used an ensemble Kalman filter for a shelf sea model and showed that the forecast error covariance matrix in this specific case tends to a quasi–steady state after a few days of assimilation. Moreover, Sørensen et al. (2006) studied and compared the properties of EnKF, the RRSQRT Kalman filter, and the steady-state Kalman filter using an idealized bay and demonstrated the effectiveness of the steady-state Kalman filter over the other methods for the given system. Hence, the steady-state Kalman filter is recommended because of its low computational cost, especially if an operational setting is considered. A constant forecast error covariance matrix is also used in other data assimilation systems, which work with optimal interpolation (e.g., Demirov et al. 2003; Borovikov et al. 2005; Sun et al. 2007) and three-dimensional variational data assimilation (3DVAR; e.g., Daley and Barker 2001; Devenyi et al. 2004).

Although in this paper the algorithm is developed for using two realizations, it can be extended for a larger number of realizations. With a larger number of realizations, it is possible to use samples over a shorter time. This may relax the stationarity assumption. Note that for a sufficiently large number of realizations, it is possible to compute the covariance estimate by averaging over ensembles over one simulation time step. In this case the algorithm reduces to the ensemble Kalman filter. The choice of using only two realizations is set by a practical consideration. The algorithm is intended to be used for a future study using an operational storm surge prediction model, where two wind fields produced by two different meteorological models are available. Assuming that the two wind fields are generated from the same atmospheric random process, the two storm-surge forecasts are produced by running the storm-surge model twice, each run is driven by one of the two wind fields. In this case, no model error statistics have to be specified explicitly.

3. Experiment using a simple wave model

To check the validity of the algorithm proposed in this paper we performed an experiment using a simple wave model. This model is a simplified version of the De St. Venant equation, which can be applied for modeling tidal flow in a long and narrow estuary. The governing equations of this model are

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where ξ(x, t) is the water level above a reference plane, u(x, t) is the average current velocity over a cross section, D is the water depth, g is the acceleration of gravity, and c is the friction constant. This system of equations is solved numerically by using a finite-difference method. A Crank–Nicholson scheme with the implicity factor θ = 0.4 is used and the discretization in space is based on the use of a staggered grid.

The uncertainty of the model is assumed to be caused by an inaccurate sea–estuary boundary. A colored noise process, denoted by w_b(t_k), modeled as an AR(1) process is used to model this uncertainty:

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where α_b is a correlation parameter defined by α_b = e^−Δt/T_c, Δt is the time step, T_c is the correlation time parameter, and ϵ is a Gaussian white noise process. The correlation time is set to 2 h and the standard deviation of ϵ is chosen to be 20 cm for this study. Moreover, a Gaussian white noise process with standard deviation of 2 cm is used to represent the observation error.

For this experiment, the values of the other parameters are D = 10 m, c = 0.0002 s⁻¹, L = 60 km, time step Δt = l min, and number of grid points for spatial discretization n = 80, hence, Δx = L/(n − 1). Moreover, at the sea–estuary boundary, x = 0, a periodic water level is generated according to ξ_b(t_k) = 0.5sin(2πt_k/T), where T = 3 h.

Using this model, the open- and closed-loop processes described in section 2 are carried out. At each iteration, the model is run twice, each with a different noise realization, for the same simulation period of 600 h. An artificial measurement generated at location x = 0.3L is inserted into the model at each time step in the closed-loop runs. The gain matrix computed at one iteration is inserted into the model used in the next iteration. The procedure is repeated until no significant change in the gain matrix is apparent. The results in term of the water level gain matrices are presented in Figs. 1a–d. We observe that there is no significant difference between the gain matrix estimated after the second closed-loop iteration from the one estimated after the third iteration. This indicates that the algorithm has converged.

To evaluate the performance of the algorithm, we compare the gain matrix estimated by using our algorithm to the one obtained by using a RRSQRT Kalman filter (Verlaan and Heemink 1997). The result, shown in Fig. 1e, has been obtained by running the RRSQRT Kalman filter until the Kalman gain matrix has become constant. The number of modes used to produce this result is 50, which turned out to be sufficient to estimate the Kalman gain accurately. Comparing the gain matrix estimated at the third closed-loop iteration with the one obtained using the RRSQRT Kalman filter, we see that they are very similar with each other. This indicates that for this model the algorithm proposed in this paper can reproduce the steady-state Kalman filter gain.

4. Experiment using the Dutch Continental Shelf Model

The second experiment is carried out by applying the proposed algorithm to the Dutch Continental Shelf Model (DCSM). This is an operational model used in the Netherlands for real-time storm surge prediction. In the operational setup, a steady-state Kalman filter based on the work of Heemink and Kloosterhuis (1990) has been implemented to improve the forecast’s accuracy of the model. Hence, in this study, it is possible to compare the gain matrix computed by using our proposed algorithm to the one computed using this steady-state Kalman filter.

The governing equations used in the DCSM are the nonlinear 2D shallow-water equations. However, for the propagation of the forecast error covariance matrix a linearized system of equations is used instead (see e.g., Verlaan 1998; ten Brummelhuis and Heemink 1990; Heemink and Kloosterhuis 1990):

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where u and υ are depth-averaged currents in the x and y directions, respectively; ξ is the water level above the reference plane; D is the water depth below the reference plane; g is the acceleration of gravity; f is the coefficient for the Corriolis force; c_f is the coefficient for the linear friction; and τ_x and τ_y are the wind stresses in the x and y direction, respectively.

These equations are discretized using an alternating directions implicit (ADI) method and a staggered grid that is based on the method by Leendertse (1967) and Stelling (1984). In the implementation, the spherical grid is used instead of the rectangular (e.g., Verboom et al. 1992).

Two sources of uncertainty are usually considered: uncertainty at the open boundary and uncertainty of the meteorological forcing. However in this experiment we assume that the source of uncertainty is only due to erroneous wind forcing. To account for this uncertainty noise processes w_u and w_υ are added directly to the u and υ velocities instead of to the depth average momentum equations. It is generally believed that the noise process is correlated in time, hence an AR(1) process is chosen to represent the uncertainty in each direction. Denoting w_u(x, y, t_k) and w_υ(x, y, t_k) as the zonal and meridional directions, respectively, the noise processes are written as

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where α_w is a correlation parameter and ϵ_u and ϵ_υ are white noise processes. The spatial characteristic is specified by using a spatial correlation given by

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where

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and d_w is a characteristic distance for the spatial correlation. The noise term is introduced on a coarser grid to reduce the number of noise inputs. The noise terms at other grid points are interpolated subsequently by using bilinear interpolation.

The DCSM covers an area in the northeast European continental shelf (i.e., 48°–62°N, 12°W–13°E) as shown in Fig. 2. The resolution of the spherical grid is 1/8° × 1/12°, which is approximately 8 × 8 km. With this configuration there are 201 × 173 grids with 19 809 computational grid points. The time step is Δt = 10 min and the friction coefficient c_f = 0.0024. For the wind input noise parameters the following values are used: σ_ε = 0.003 m s⁻¹ (time step)⁻¹, α_w = 0.9 (time step)⁻¹, which corresponds to a correlation time of 1.6 h, and d_w = 19 grid cells. The noise is defined at a coarser grid, where each coarser grid consists of 28 × 28 computational grid cells. Moreover, a white noise process with standard deviation σ_m = 0.10 m is used to represent the uncertainty of the observations. There are eight observation stations that are included in the assimilation, five of which are located along the east coast of the United Kingdom and the others are along the Dutch coast (see Fig. 2).

Using the specifications described above, three months of artificial observation data are generated at these assimilation stations. The generated observations are used in the closed-loop iteration to improve the gain matrix estimates. Figure 3 shows the gains in term of water level computed by using our algorithm as well as by using the operational steady-state Kalman filter. Here we show only the gains for the assimilation station Wick. By visual inspection we see that the iteration is convergent. Moreover, the results computed at the fourth closed-loop iteration shows similar structure and magnitude as the one computed by using the steady-state Kalman filter (Heemink and Kloosterhuis 1990). Some minor differences are apparent at points farther away from the assimilation station. This may be due to sampling error, which is a common difficulty in ensemble-based approaches. The limited number of samples may produce spuriously large magnitude of covariance estimates between greatly separated grid points where the covariance is actually small. However, despite these differences it is clear from the pictures that in general the algorithm that we propose in this paper can reproduce the gain estimated by using the steady-state Kalman filter.

Besides performing visual checks on the estimated gain, another evaluation was also carried out. This evaluation is done by implementing the gain matrices computed using our proposed algorithm in a data assimilation system similar to the operational one, where the nonlinear version of the system of Eqs. (28)–(30) has been used (see Stelling 1984). For this evaluation, the real water level data observed within the period of 1 November 2004–1 February 2005 have been assimilated. Figure 4 presents three water level time series at Wick for the period of 0000–2400 UTC 12 January 2005. These time series refer to water levels obtained from observations, forecasts using deterministic model without data assimilation, and forecasts with data assimilation using the gain matrix obtained in the open-loop step. This figure demonstrates that the differences between the forecasts with data assimilation and the observations are always smaller than the differences between the forecasts without data assimilation and the observations. To acquire a better idea about the filter performance, the root-mean-square (RMS) of the water level innovation is calculated over the whole simulation period:

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where ξ^o is the observed water level and ξ^a is the analyzed water level. To check whether the data assimilation works also at other locations, the RMS water level innovation is also computed in some validation stations, where the observation data has not been used for the assimilation. In this experiment we used eight assimilation stations and eight validation stations. Figures 5 and 6 show the RMS water level innovation at the assimilation and validation stations, respectively. In these figures we also present the RMS innovation of the water level obtained by using the steady-state Kalman filter. It is clear from these pictures that using the gain matrices computed at both the open- and closed-loop steps, the filter reduces the RMS values of the water level innovations at both the assimilation and validation stations. At most assimilation stations, the largest reduction of the RMS water level innovation is obtained by the assimilation setup with the gain matrix computed at the open-loop step. This should be expected as the open-loop gain is bigger in the absolute sense than the closed-loop ones. As a result the assimilation system with the open-loop gain forces the states to be closer to the observations. The closed-loop gains however are spatially smoother than the open-loop one. Moreover, we can also see that the RMS water level innovation converges to the one obtained using the steady-state Kalman filter.

5. Experiment using the three-variable Lorenz model

To verify whether the algorithm proposed in this paper can also be applied for an unstable dynamical system, we also performed experiments using a three-variable Lorenz model. The Lorenz model is a simplified system of the flow equations governing thermal convection. It was introduced by Lorenz (1963) and has been the subject of extensive studies motivated by its chaotic and strongly nonlinear nature, yet small in dimension. Because of these properties, it is often used as a test bed for examining various data assimilation methods for systems with strongly nonlinear dynamics (e.g., Miller et al. 1994; Evensen 1997).

The governing equations of the Lorenz model are

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where x, y, and z are the dependent variables, varying over dimensionless time t. For this experiment, we applied the parameter values, as first used by Lorenz to obtain chaotic solutions: σ = 10, ρ = 28, and β = 8/3. The model was integrated with a second-order Euler numerical scheme with a time step Δt = 1/60 time units. To account for model uncertainty, three independent white noise processes are added to each variable at each time step. The white noise processes are drawn from a Gaussian distribution with zero mean and variance equal to 0.5Δt. Using this system, we computed a true reference solution for a duration of 40 time units and with initial values (x_o, y_o, z_o) = (1.508870, −1.531 271, 25.46 091). A sequence of observations is generated at intervals of 0.25 time units by adding to the reference solution a generated random number representing the observation error. For this study, the observations are generated for all variables x, y, and z and the observation noise is generated from an unbiased Gaussian distribution with variance equal to 2.0 independently for each variable.

We implemented the proposed algorithm with the Lorenz model described above to estimate the corresponding steady-state gain matrix. The two forecasts are generated by running the system twice for the same period, where each run is perturbed independently by noise, drawn from a distribution with the same statistics as has been used to generate the true solution. Since the system is unstable, the two realizations of the open-loop system follow completely different trajectories. Therefore, the open-loop estimate cannot be computed since the difference between the two realizations is not stationary. To solve this problem, the open-loop step is skipped and a reasonable gain matrix should be used as the starting point for the closed-loop iteration. This will induce a feedback mechanism, which makes the overall system stable. For the experiment presented in this paper we used the gain matrix 𝗞 = 0.5𝗜 as the starting point for the closed-loop iteration, where 𝗜 is a 3 × 3 identity matrix. The closed-loop estimation was iterated 15 times and the results are presented in Fig. 7. This figure shows the elements of the 3 × 3 gain matrix 𝗞, where the bars grouped per element show the estimates of the respective element produced from consecutive iterations. It can be seen from the figure that the estimates of the gain matrix converge after seven iterations. The method to compute the forecast error covariance matrix in this experiment is similar to the method used in Yang et al. (2006) to compute the estimate of the background covariance for a 3DVAR system. The difference is that they used the time average of the difference between the estimate and the true solutions. In reality, however, the true solution is never known.

The next step is to use the gain matrix estimated above in a data assimilation experiment to examine its performance. Figure 8a shows both the true reference as well as the model solution without any assimilation for variable x, while Fig. 8b presents the results when the constant gain matrix obtained from the 15th closed-loop iteration is used for assimilation. The model solution without any data assimilation follows completely different trajectory from the true one. On the other hand, Fig. 8b demonstrates that in this experiment the assimilation system is able to track the true solution. It is generally able to reproduce the correct amplitudes in the peaks of the solution. There are a few points where the solution deviates rather significantly from the true solution, for example at time t = 17, t = 24, t = 31, and t = 37. However, the solution is quickly recovered and it starts to trace the true solution again. For comparison we also conducted a data assimilation experiment using the ensemble Kalman filter (Evensen 2003). Here the ensemble consists of 1000 members and the results are presented in Fig. 8c. Qualitative examinations on these plots indicates that the EnKF performs better in tracking the true solution, especially at the times where the solution obtained with assimilation using the constant gain matrix deviates rather significantly from the true one. To acquire a better insight about the performance, the RMS error of all variables with respect to the true solution is shown in Fig. 9. It is clear from this figure that the solution obtained using EnKF is more accurate than the results using the algorithm proposed in this paper. It seems that for a strongly nonlinear system like the Lorenz model, the assimilation system should include a flow-dependent forecast error covariance estimate for accurate results. However, the computational cost of the proposed algorithm is much cheaper than the EnKF. In practice, a decision has to be made involving the trade-off between the desired accuracy and the computational cost required to achieve it.

6. Conclusions

In this paper a new filter algorithm is proposed. The algorithm is based on estimating the error covariance by using two forecast samples. This covariance is used to define a gain matrix for a steady-state data assimilation system. Intended for a steady-state operation, it is applicable for models where approximately a statistically stationary condition will occur. Its applicability also requires the models to have short memory to allow for the ergodicity assumption. The algorithm consists of an iterative procedure for improving the estimate of covariance.

Twin experiments have been performed to evaluate the performance of the algorithm. In the first experiment a simple one-dimensional wave model is used, while for the second one, we used the operational storm surge model: the Dutch Continental Shelf Model (DCSM). The evaluation is done by comparing the gain matrix computed using the proposed algorithm to the one computed by using the exact steady-state Kalman filter. Both experiments show that the results using the proposed algorithm converge to the ones produced by using the classical Kalman filter.

An experiment using the three-variable Lorenz model has also been conducted to see the possibility of applying the proposed algorithm in an unstable dynamical system. Since the system is unstable, the open-loop estimate is skipped and the closed-loop iteration should be initialized with a reasonable gain matrix. Using a diagonal matrix as the starting point, the closed-loop iteration has been shown to converge. Moreover, the data assimilation system with the constant gain matrix, estimated using the proposed algorithm seems to do a reasonably good job in tracking the true solution. This result also demonstrates the potential use of the proposed algorithm for unstable systems.

In the experiment using the DCSM, the two forecasts were generated by running the model twice for the same period and by perturbing both runs using statistically independent perturbations drawn from a known distribution. In practice, however, this distribution is unknown. Therefore, in the next study, we are going to explore the possibility of using two wind fields produced by two different atmospheric models to generate proxies for the forecast error of the DCSM. The two wind fields are considered as two realizations of a random atmospheric process. Using the algorithm proposed in this paper, the two forecasts will be generated by running the model twice, each run will use one of the two wind fields as input without any artificially generated perturbations nor assuming any distribution. With this approach explicit modeling of model error covariance is avoided.