Ghil, M., Cohn, S.E., Tavantzis, J., Bube, K., and Isaacson, E., 1981: Application of estimation theory to numerical weather prediction. Dynamical Meteorological: Data Assimilation Methods, L. Bengtsson, M. Gill and E. Kallen, Eds., Spring-Verlag, 139-224.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Zupanski and Mesinger 1995; to cite just a few) The most encouraging achievement occurred recently at ECMWF, where 4D Var has been implemented for operational use at the end of 1997. Parallel to variational methods, sequential methods have attracted extensive attention since the beginning of 80 s (Ghil et al. 1981). Most of meteorological applications of the Kalman lter use idealized one and two dimensional models (e.g. Cohn and Parrish 1991; Dee 1990; Todling and Cohn 1994; Daley 1992; Evensen 1994) Cohn et al. 1994) introduce a Kalman smoother to meteorology, called retrospective data assimilation, ....

Ghil, M., Cohn, S.E., Tavantzis, J., Bube, K., and Isaacson, E., 1981: Application of estimation theory to numerical weather prediction. Dynamical Meteorological: Data Assimilation Methods, L. Bengtsson, M. Gill and E. Kallen, Eds., Spring-Verlag, 139-224.

....rather only to the arrays k j . If we are only interested in the final result of the analysis, there is no need to obtain K explicitly; however, if we are particularly interested in investigating the influence of a certain observation on to distinct elements of the state vector (e.g. Ghil et al. [66]) it is necessary to calculate the complete gain matrix. The simplest way to recover the gain matrix, when using serial processing, is to do so after having obtained the analysis error covariance matrix by making use of the alternative expression for the gain matrix, K k = P a k H T k R ....

....to its error e a k , for all k. Using finite induction, start by showing that Efw a 1 (e a 1 ) T g = 0 and that Efw a 2 (e a 2 ) T g = 0 Then, assume that Efw a k (e a k ) T g = 0 is true, and show that Efw a k 1 (e a k 1 ) T g = 0 is satisfied. 5. Ghil et al. [66]) Consider the Kalman filter applied to the scalar, discrete time system: x k = ax k Gamma1 w k z k = x k v k where the noises w k and v k are white, normal with mean zero and variances q = const: and r = const: respectively. In this case, the Kalman filter reduces to the following ....

[Article contains additional citation context not shown here]

, S. Cohn, J. Tavantzis, K. Bube & E. Isaacson, 1981: Applications of estimation theory to numerical weather prediction. Bengtsson, L., M. Ghil, & E. Kall'en (Eds.): Dynamic Meteorology: Data Assimilation Methods. Springer--Verlag, 139--224. 182

....Q is a parameterization of the model error covariance matrix, and P a and P f are the analysis and forecast error covariance matrices, respectively. The application of the Kalman filter to atmospheric four dimensional data assimilation (4DDA) was first recognized by Cohn, Ghil, and coworkers [4]. This was recognized as an extension of the successful approaches that have been used for years in weather forecasting (e.g. optimal interpolation [5] 6] The application of the Kalman filter in climate analysis and weather forecasting is made considerably more difficult by the large number of ....

M. Ghil, S. E. Cohn, J. Tavantzis, K. Bube, and E. Isaacson. Applications of estimation theory to numerical weather prediction. In L. Bengstsson, M. Ghil, and E. Kallen, editors, Dynamic Meteorology: Data Assimilation Methods, page 330. Springer-Verlag, Berlin, 1981.

....of the shape or amplitude of Psi. When discretized with a onestep numerical scheme in time, this equation may be written on the form (1) and it has an error covariance propagation equation given by (3) A study of a simple linear system used for illustrational purposes has also been given by Ghil et al. 1981]. They discussed a simplified linear barotrophic model for onedimensional atmospheric wave propagation, where a modified Kalman filter, designed to remove fast wave modes, was compared to the standard Kalman filter. The time evolution and propagation of averaged errors in data rich areas (land) ....

Ghil, M., S. Cohn, J. Tavantzis, K. Bube, and E. Isaacson, Applications of estimation theory to numerical weather prediction, in Dynamic Meteorology: Data Assimilation Methods, edited by L. Bengtsson, M. Ghil, and E. Kall'en, pp. 139-- 224, Springer Verlag, New York, 1981.

....description we will move back and forth between Kalman filter and fourdimensional variational arguments. A number of authors have discussed the relationship between Kalman filtering and variational problems, see, for example Jazwinski(1970) Ghil and MalanotteRizzoli (1991) See also Daley(1991) Ghil et al. (1981), Gauthier, Courtier and Moll(1993) Rabier et al.(1993) We will review some of these results in the process of providing the ML, GML, GCV and UBR estimates. 5.1 ML, GML, GCV and UBR based on the Kalman Filter at a single time step Dee(1993) has described a ML estimate in this context. We first ....

Ghil, M., Cohn, S., Tavantzis, J., Bube, K. & Isaacson, E. (1981), Applications of estimation theory to numerical weather prediction, in L. Bengtsson, M. Ghil & E. Kallen, eds, `Dynamic Meteorology: data assimilation methods.', Springer-Verlag, pp. 139--284.

....the optimality of the filter for linear dynamics has motivated an extensive study of the method for lower dimensional and simpler models, e. g, Miller (1986) Miller (1989) Miller and Cane (1989) Miller (1990) Bennett and Budgell (1987) Budgell (1987) for some oceanographic applications and Ghil et al. 1981), Cohn and Parrish (1991) Cohn (1993) Daley (1992d, 1992a, 1992c, 1992b, 1993) Todling and Ghil (1994) and Dee (1991) for some meteorological applications. 3.2.2 Comparison of methods It is instructive to compare the Kalman filter with the representer method, and it is then convenient to ....

Ghil, M., Cohn, S., Tavantzis, J., Bube, K., and Isaacson, E. (1981), Applications of estimation theory to numerical weather prediction, in Dynamic Meteorology: Data Assimilation Methods, edited by Bengtsson, L., Ghil, M., and Kall'en, E., pp. 139--224, Springer Verlag, New York.

....can be used. Again, in order to save computations one ought to approximate solutions using projection ideas for which the simulation does not diverge too much from the true solution. Note that these equations are now being used to solve very large scale problems in very diverse areas [16] [13] besides control. 7. Basic numerical tools The topics discussed in previous sections all point to the significant role linear algebra problems play in the control, optimization and model reduction of multivariable linear systems. Over the years, numerous algorithms have been developed in that ....

M. Ghil, S. Cohn, J. Tavantzis, K. Bube, E. Isaacson, Applications of estimation theory to numerical weather prediction, in Dynamic Meteorology Data Assimilation Methods, L. Bengtsson, M. Ghil, E. Kallen (Eds.), Appl. Math. Sc. Vol. 36 (Springer Verlag, Berlin), 1981, pp.139-224.

....prediction model itself. If these Dee, D.P. A simple scheme for tuning. uncertainties can be quantified, then it is possible to produce an estimate of the atmospheric state which is as accurate as possible in a statistical sense. This can then be used to initialize a prediction (Cohn, 1982; Ghil et al. 1981). The statistical approach is conceptually no more complex than the variational approach. However, when attempting to apply it to the meteorological data assimilation problem one is faced with an obstacle of a more fundamental nature. Very little is known about statistical properties of the ....

Ghil, M., S. E. Cohn, J. Tavantzis, K. Bube, and E. Isaacson, 1981: Applications of estimation theory to numerical weather prediction. In: Dynamic meteorology: Data assimilation methods, eds.

.... and Lonnberg 1986; Lonnberg and Hollingsworth 1986; Daley 1991; Bartello and Mitchell 1992) Advanced statistical data assimilation techniques aim to improve the accuracy of forecast error statistics by taking into account the effect of model dynamics on the evolution of forecast errors (Ghil et al. 1981; Dee 1991; Cohn and Todling 1996) The point of departure in covariance modeling is complete knowledge of the means. Most often it is simply assumed that the forecast model as well as the observing instruments are unbiased; that is, the mean errors are zero or they have been removed. See, ....

....of this article is to present a rigorous, yet practical, method for estimating forecast bias in an atmospheric data assimilation system. The method is fully consistent with the state space approach of estimation theory, originally presented in the context of atmospheric data assimilation by Ghil et al. 1981). This theory requires explicit assumptions on statistics of observation errors and on forecast errors, possibly including unknown systematic (i.e. nonzero mean) components. From these assumptions it is then possible to derive a consistent set of algorithms for estimating forecast bias and for ....

Ghil, M., S. Cohn, J. Tavantzis, K. Bube, and E. Isaacson, 1981: Applications of estimation theory to numerical weather prediction. Pp. 139--225 in: Bengtsson, L., M. Ghil, and E. Kall'en, Dynamic Meteorology: Data Assimilation Methods, Springer-Verlag, New York, 330pp.

....the impact of assimilating data past the analysis time and to understand the dependence of the FLKS analysis error variance upon the lag number and upon the system parameters. The recursive FLKS equations can be solved in closed form in this case. Our analysis extends the scalar KF analysis of Ghil et al. 1981) and complements the spectral analysis of Daley and M enard (1993) and M enard and Daley (1994) a. General relations All vectors and matrices now become scalars, and they are denoted by the respective non boldface letters. Assume the propagator A and the variances Q and R are time ....

.... 1 ; 3.56) p a 1 = 8 : 0 if jAj 1 1 Gamma A Gamma2 if jAj 1 : 3. 57) Thus, both p f kjk Gamma1 and p a kjk asymptote to zero for dissipative (jAj 1) and conservative (jAj = 1) perfect models, while for unstable (jAj 1) perfect models the asymptotic values are nonzero (cf. Ghil et al. 1981, Equations 4.7c,d) Substituting (3.52) 3.55) into (3.40) and (3.41) and simplifying, and also recalling (3.51) we have for k 0, p a kj1 = 8 : 1 GammaA 2 )p a 0 1 GammaA 2 (1 Gammap a 0 ) A 2k if jAj 1 0 if jAj 1 : 3.58) That is, both conservative and unstable ....

[Article contains additional citation context not shown here]

Ghil, M., S. Cohn, J. Tavantzis, K. Bube and E. Isaacson, 1981: Applications of estimation theory to numerical weather prediction. In Bengtsson, L., M. Ghil, and E. Kall'en (eds.): Dynamic Meteorology: Data Assimilation Methods. Springer--Verlag, pp. 139--224.

..... 17 6 Summary and Conclusions 18 7 Appendix 20 iii 1 Introduction This article introduces one of the current research efforts at the Data Assimilation Office (DAO) of the NASA Goddard Space Flight Center to use the Kalman filter (e.g. Ghil et al. 1981) for atmospheric data assimilation. At present, a full implementation of the Kalman filter in a four dimensional data assimilation (4DDA) context is impossible. Considerable research needs to be undertaken before any implementation could be used operationally. Many open questions need to be ....

Ghil, M., S. E. Cohn, J. Tavantzis, K. Bube, and E. Isaacson, 1981: Applications of estimation theory to numerical weather prediction. Dynamic Meteorology: Data Assimilation Methods, L. Bengtsson, M. Ghil, and E. Kallen, Eds.

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