G. Evensen, `Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model', J. Geophys. Res., 97, 17905 -- 17924 (1992).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....n and m Theta m variance covariance matrices of the first guess and the observational errors. We assume moreover that the first guess and the observational noises are uncorrelated. Equations 1 to 2 constitute the estimation problem as introduced in the analysis step of data assimilation (see Evensen, 1992 for a more detailed introduction) Note that in many applications, the unbiasedness assumption is only a coarse approximation. In fact, the major problem lies in the modeling of C f k , which has to be estimated at each time step. The solution comes from the utilization of the physical model. ....

Evensen, G. (1992). Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model. Journal of Geophysical Research, 97(17) : 905-924.

....is the optimal solution of the fixed lag smoothing problem. The perfect model assumption of 4D VAR is not made. The FLKS may be generalized to nonlinear dynamics and observations, either in the way that the so called extended Kalman filter generalizes the KF (e.g. Jazwinski 1970, Chapter 9; Evensen 1992, 1993; Daley 1994; Miller et al. 1994) or by the use of self consistent moment closure (e.g. Cohn 1993, and references therein) The FLKS can also be used for identifying model parameters (Heemink 1990) There are a variety of ways of deriving the FLKS, and a variety of algebraically ....

Evensen, G., 1992: Using the extended Kalman filter with a multilayer quasi--geostrophic ocean model. J. Geophys. Res. Oceans, 97, 17905--17924.

....in oceanography at the end of the 1980s [20] The KF has been mostly applied to quasi linear tropical ocean situations [6, 17, 39] but it was extended to the nonlinear case [22] with model linearization. This extended Kalman filter (EKF) has been examined with different dynamical models by Evensen [14], Gauthier et al. [19] and Miller et al. [28] A common conclusion from these studies is that the EKF has an apparent closure problem since the error evolution is computed using the model tangent linear operator, and thus, if the model is in an unstable regime there is no nonlinear saturation of ....

....equation is nonlinear: Z n f n (Z n , q n ) 20) and the covariance matrix is propagated by successive linearizations of the model f n , when the Taylor expansion of the model is generally stopped at the first order. Developments to higher orders can be necessary in some cases, e.g. see [14] for a limitation of the EKF due to neglecting higher order moments, but this is rarely done due to the increasing complexity of the equations and the numerical load associated with storage and evolution of the higher order statistics. Here two alternative approaches are used, one based on a ....

Evensen G 1992 Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model J. Geophys. Res. 97 17905--24

....been mostly applied to quasi linear tropical ocean situations (Fukumori [17] Cane et al. 6] Verron et al. 39] but it was extended to the non linear case (Jazwinski [22] with model linearization. This Extended Kalman Filter (EKF) has been examined with different dynamical models by Evensen [14], Gauthier et al. 19] 2 and Miller et al. 28] A common conclusion from these studies is that the EKF has an apparent closure problem since the error evolution is computed using the model tangent linear operator, and thus, if the model is in an unstable regime there is no non linear ....

....equation is non linear Z n = f n (Z n ; q n ) 9 and the covariance matrix is propagated by successive linearizations of the model f n , when the Taylor expansion of the model is generally stopped at the first order. Developments to higher orders can be necessary in some cases, e.g. see Evensen [14] for a limitation of the EKF due to neglecting higher order moments, but this is rarely done due to the increasing complexity of the equations and the numerical load associated with storage and evolution of the higher order statistics. Here two alternative approaches are used, one based on an ....

G. Evensen. Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model. Journal of Geophysical Research, 97(C11):17905--17924, 1992.

....method similar to the original Kalman filter where the error covariance statistics evolve forward in time due to a linearized version of the dynamical model. It has been shown in a number of studies that this linearization may be too severe when the model dynamics become too nonlinear, e.g. Evensen (1992) and Miller et al. 1994) This has initiated the search for new filtering techniques which are designed to handle strongly nonlinear dynamical models in a consistent manner. One such method, the ensemble Kalman filter (EnKF) was proposed by Evensen (1994) It was derived as a Markov Chain Monte ....

....Evensen (1997b) 3.1 Ensemble Kalman filter The EnKF was designed to resolve two major problems related to the use of the EKF with nonlinear dynamics in large state spaces. The linearization used in the error covariance equation has been shown to be invalid in a number of applications, e.g. by Evensen (1992) and Miller et al. 1994) In fact, the equation (9) is no longer the fundamental equation for the error evolution when the dynamical model is nonlinear. In this case, by using (9) one neglects contributions from higher order statistical moments, as a statistical closure approximation. For a ....

Evensen, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97(C11), 17,905--17,924, 1992.

....provide reliable results when used with OGCMs. More advanced data assimilation techniques apply time dependent and dynamically consistent error statistics. This requires the forward integration of an error covariance equation for the error statistics, e.g. by using an Extended Kalman Filter (EKF) [6,7], or, as a better alternative, one can integrate an ensemble of ocean states as is done in the recently proposed Ensemble Kalman Filter (EnKF) 8,9,12] The recent developments related to so called advanced methods like the EnKF, and the significant improvement of available computer resources, now ....

G. Evensen, Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97 (C11), 17,905--17,924, 1992.

....Kalman filter integrates an approximate equation for the error covariance matrix where all higher order statistical moments are neglected. This linearization has proven to be inconsistent for strongly nonlinear dynamics and it will in many cases be a too simplified closure approximation, see Evensen, 1992, Miller et al. 1994) The ensemble Kalman filter integrates an ensemble of model states from which the error covariances can be calculated. This is equivalent to a Monte Carlo method for integrating Kolmogorov s equation which is the fundamental equation for evolution of error statistics. No ....

....the probability density will approach zero at a rate proportional to N Gamma 1 2 . For practical ensemble sizes, say O(100) the errors will be dominated by statistical noise, not by closure problems or unbounded error variance growth as have been observed in the extended Kalman filter (see Evensen 1992, 1994b) When the Monte Carlo method is applied one first calculates a best guess initial condition based on available information from data and statistics. The model solution calculated from this initial state is denoted the central forecast. The uncertainty in the best guess initial condition ....

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Evensen, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97(C11), 17,905--17,924, 1992.

.... Ensemble Kalman Filter (EnKF) was introduced by Evensen (1994b) as an alternative to the traditional extended Kalman filter (EKF) which has been shown to be based on a too severe statistical linearization or closure approximation to be useful for some cases with strongly nonlinear dynamics (see Evensen (1992), Miller et al. 1994) Gauthier et al. 1993) and Bouttier (1994) If the dynamical model is written as a stochastic differential equation, one can derive the Fokker Planck or Kolmogorov s equation for the time evolution of the probability density function which contains all the information ....

....covariances while the ensemble mean is used as the best guess trajectory. For nonlinear dynamics the so called extended Kalman filter may be used and is given by the evolution equations (20) and (21) with the n.l. terms neglected. This makes the extended Kalman filter unstable is some situations (Evensen, 1992), while the EnKF is stable. In addition, there is no need in the EnKF for a tangent linear operator or its adjoint, and this makes the EnKF very easy to implement for practical applications. An inherent assumption in all Kalman filters is that the errors in the analysis step are Gaussian to a good ....

Evensen, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97(C11), 17,905--17,924, 1992.

....to a number of initial value problems, although the method is rather expensive because an equation for the spatial error covariances must be integrated. Further, for nonlinear dynamics the solutions will no longer be optimal because linearized equations are used to propagate the error statistics (Evensen, 1992). For more advanced nonlinear primitive equation models, additional problems occur. First of all, it is not clear that the adjoint operator exists for all models. If it really exists, and can be found, it is still necessary to have a well posed formulation for both the forward and backward model. ....

....and backward model. Oceanographic models are often regional with open boundaries, and are often ill posed in the sense that the boundary conditions are unknown and may be over or under specified (Bennett and Chua, 1994) Note that the adjoint of the tangent linear operator is approximate, and in Evensen (1992) it was found that because of the linearization dynamical instabilities failed to saturate and it was not clear how this could be handled. The problems arising from strongly nonlinear models suggest the use of direct substitution algorithms where one avoids the actual integration of the forward ....

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Evensen, G. (1992), Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys.

....is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble. Introduction The implementation of the extended Kalman filter for data assimilation in a multilayer quasi geostrophic (QG) model has previously been discussed by Evensen [1992, 1993] The main result from Evensen s work [1992] is the finding of an apparent closure problem in the error covariance evolution equation. The extended Kalman filter applies a closure scheme where third and higher order moments in the error covariance evolution equation are discarded. This ....

....and clusters of workstations where each processor integrates a few members of the ensemble. Introduction The implementation of the extended Kalman filter for data assimilation in a multilayer quasi geostrophic (QG) model has previously been discussed by Evensen [1992, 1993] The main result from Evensen s work [1992] is the finding of an apparent closure problem in the error covariance evolution equation. The extended Kalman filter applies a closure scheme where third and higher order moments in the error covariance evolution equation are discarded. This simple closure technique results in a unbounded error ....

[Article contains additional citation context not shown here]

Evensen, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97(C11), 17,905--17,924, 1992.

....dynamics the extended Kalman filter may be applied, in which an approximate linearized equation is used for the prediction of error statistics. The implementation of the extended Kalman filter for data assimilation in a multilayer quasi geostrophic (QG) model has previously been discussed by Evensen (1992). The main result from this work is the finding of an apparent closure problem in the error covariance evolution equation. The extended Kalman filter applies a closure scheme where third and higher order moments in the error covariance evolution equation are discarded. This simple closure ....

....in a unbounded error variance growth caused by the linearization performed when higher order moments are neglected. Thus it has been shown that the error covariance equation is based on a too simplified closure approximation and this may lead to a nonphysical error variance evolution (see e.g. Evensen (1992), Miller et al. 1994) Gauthier et al. 1993) and Bouttier (1994) The Ensemble Kalman Filter (EnKF) was introduced by Evensen (1994b) as an alternative to the traditional extended Kalman filter (EKF) It was shown that if the dynamical model is written as a stochastic differential equation, ....

[Article contains additional citation context not shown here]

Evensen, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97(C11), 17,905--17,924, 1992.

....and can be used even with data located at inflow boundaries. However, the approximations used in the scheme may lead to loss of positive definiteness for the error covariance matrix and an algorithm must be used to ensure positive definiteness for long time integrations. Introduction As given by Evensen [1992] (hereafter called Part I) the general formulation of the extended Kalman filter with a multilayer quasi geostrophic (QG) model was discussed. Further, data assimilation experiments were performed on a square domain, using closed boundary conditions. One of the main topics considered was the ....

G. Evensen, Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97 (C11), 17,905--17,924, 1992.

....in which an approximate linearized equation is used for the prediction of error statistics. It has been shown that this equation is based on a too simplified closure assumption, where higher order statistical moments have been neglected, and this may lead to a nonphysical error variance evolution (Evensen, 1992, Miller et al. 1994) So far, only a few methods have proven successful for weak constraint inverse calculations when the model dynamics are nonlinear, see e.g. the review by Evensen, 1994a) The most promising methods used today are the representer method and variants of a substitution or ....

....of the error covariance matrix) in Figure 6. The variance of the prior estimate grows exponentially during the initial integration and thereafter starts to level off. The filter variance has the characteristic behavior as has been observed in previous applications with the extended Kalman filter [Evensen, 1992], and the ensemble Kalman filter [Evensen, 1994b] The ensemble smoother estimate has some of the characteristic behaviors of the experiments with the Kalman smoother with a linear quasi geostrophic model by Bennett and Budgell [1989] i.e. there are local minima of the variance at measurement ....

Evensen, G., 1992: Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97(C11), 17,905--17,924.

....method for linear and weakly nonlinear inverse problems, both in terms of the problem minimized (full weak constraint problem) and in terms of numerical cost. 3.6. DISCUSSION Note that the equation for b, 62) is similar to the one solved in the analysis scheme in the standard Kalman filter [15]. Actually the only difference is that here we have a covariance matrix with corresponding representers or influence functions in space and time, while in the Kalman filter the time dimension is excluded. Thus, the solution is sought for in a similar form. In the Kalman filter the representer or ....

G. Evensen. Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model. J. Geophys. Res., 97(C11):17,905--17,924, 1992.

....the assimilation interval by some amount but there is still an upper limit where the strong constraint formulation fails to track every transition. Other problems related to error variance prediction have also been observed with more realistic applications of the extended Kalman filter, e.g. Evensen (1992) and Gauthier et al. 1993) who used the extended Kalman filter with quasi geostrophic models. Evensen (1992) pointed out that by evaluating the model operator at an unstable, say sheared, background flow this resulted in unbounded error variance growth. So in general a more consistent closure is ....

....fails to track every transition. Other problems related to error variance prediction have also been observed with more realistic applications of the extended Kalman filter, e.g. Evensen (1992) and Gauthier et al. 1993) who used the extended Kalman filter with quasi geostrophic models. Evensen (1992) pointed out that by evaluating the model operator at an unstable, say sheared, background flow this resulted in unbounded error variance growth. So in general a more consistent closure is needed in the error covariance equation. An alternative may be to use the ensemble Kalman filter that was ....

Evensen, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97(C11), 17,905--17,924, 1992.

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G. Evensen, `Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model', J. Geophys. Res., 97, 17905 -- 17924 (1992).

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