Dynamic Meteorology: Data Assimilation Methods pp 139-224 | Cite as

# Applications of Estimation Theory to Numerical Weather Prediction

## Abstract

Numerical weather prediction (NWP) is an initial-value problem for a system of nonlinear partial differential equations (PDEs) in which the initial values are known only incompletely and inaccurately. Data at initial time can be supplemented, however, by observations of the system distributed over a time interval preceding it. Estimation theory has been successful in approaching such problems for models governed by systems of ordinary differential equations and of linear PDEs. We develop methods of sequential estimation for NWP.

A model exhibiting many features of large-scale atmospheric flow important in NWP is the one governed by the shallow-fluid equations. We study first the estimation problem for a linearized version of these equations. The vector of observations corresponds to the different atmospheric quantities measured and space-time patterns associated with conventional and satellite-borne meteorological observing systems. A discrete Kalman-Bucy (K-B) filter is applied to a finite-difference version of the equations, which simulates the numerical models used in NWP.

The specific character of the equations’ dynamics gives rise to the necessity of modifying the usual K-B filter. The modification consists in eliminating the high-frequency inertia-gravity waves which would otherwise be generated by the insertion of observational data. The modified filtering procedure developed here combines in an optimal way dynamic initialization (i.e., elimination of fast waves) and four-dimensional (space-time) assimilation of observational data, two procedures which traditionally have been carried out separately in NWP. Comparisons between the modified filter and the standard K-B filter have been made.

The matrix of weighting coefficients, or filter, applied to the observational corrections of state variables converges rapidly to an asymptotic, constant matrix. Using realistic values of observational noise and system noise, this convergence has been shown to occur in numerical experiments with the linear system studied; it has also been analyzed theoretically in a simplified, scalar case. The relatively rapid convergence of the filter in our simulations leads us to expect that the filter will be efficiently computable for operational NWP models and real observation patterns.

Our program calls for the study of the asymptotic filter’s dependence on observation patterns, noise levels, and the system’s dynamics. Furthermore, the covariance matrices of system noise and observational noise will be determined from the data themselves in the process of sequential estimation, rather than be assigned predetermined, heuristic values. Finally, the estimation procedure will be extended to the full, nonlinear shallow-fluid equations.

### Keywords

anisotropY Covariance Assimilation Vorticity Expense## Preview

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### References

- Bengtsson, L., 1975: 4-Dimensional Assimilation of Meteorological Observations, GARP Publications Series, No. 15, World Meteorological Organization — International Council of Scientific Unions, CH-1211 Geneva 20, Switzerland, 76 pp.Google Scholar
- Bergman, K. H., 1979: Multivariate analysis of temperatures and winds using optimum interpolation. Mon. Wea. Rev. 107, 1423–1444.CrossRefGoogle Scholar
- Browning, G., A. Kasahara and H.-O. Kreiss, 1979: Initialization of the primitive equations by the bounded derivative method, NCAR Ms. 0501-79-4, National Center for Atmospheric Research, Boulder, Colo. 80307, 44 pp.Google Scholar
- Bube, K. P., and M. Ghil, 1980: Assimilation of asynoptic data and the initialization problem, this volume.Google Scholar
- Bucy, R. S., and P. D. Joseph, 1968: Filtering for Stochastic Processes with Applications to Guidance, Wiley-Interscience, New York, 195 pp.Google Scholar
- Charney, J., M. Halem and R. Jastrow, 1969: Use of incomplete historical data to infer the present state of the atmosphere, J. Atmos. Sci. 26, 1160–1163.CrossRefGoogle Scholar
- Chin, L., 1979: Advances in adaptive filtering, in Control and Dynamic Systems, Vol. 15, C. T. Leondes (ed.). Academic Press, 278–356.Google Scholar
- Curtain, R. F., and A. J. Pritchard, 1978: Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, Vol. 8, Springer-Verlag, New York, 297 pp.CrossRefGoogle Scholar
- Davis, M.H.A., 1977: Linear Estimation and Stochastic Control, Halsted Press, John Wiley and Sons, New York, 224 pp.Google Scholar
- Faller, A. J., and C. E. Schemm, 1977: Statistical corrections to numerical prediction equations II, Mon. Wea. Rev. 105, 37–56.CrossRefGoogle Scholar
- Fleming, R. J., T. M. Kaneshige and W. E. McGovern, 1979a: The global weather experiment I. The observational phase through the first special observing period. Bull. Amer. Met. Soc. 60, 649–659.CrossRefGoogle Scholar
- —, —, —, and T. E. Bryan, 1979b: The global weather experiment II. The second special observing period. Bull. Amer. Met. Soc. 60, 1316–1322.CrossRefGoogle Scholar
- Gelb, A. (ed.), 1974: Applied Optimal Estimation, The M.I.T. Press, Cambridge, Mass., 374 pp.Google Scholar
- Ghil, M., 1980: The compatible balancing approach to initialization, and four-dimensional data assimilation, Tellus 32, 198–206.CrossRefGoogle Scholar
- —, and R. Mosebach, 1978: Asynoptic variational method for satellite data assimilation, in Halem et al. (1978), pp. 3.32–3.49.Google Scholar
- —, R. C. Balgovind and E. Kalnay-Rivas, 1980: A stochastic-dynamic model for global atmospheric mass field statistics, NASA Tech. Memo. 82009, NASA Goddard Space Flight Center, Greenbelt, Md. 20771, 50 pp.Google Scholar
- —, M. Halem and R. Atlas, 1979: Time-continuous assimilation of remote-sounding data and its effect on weather forecasting. Mon. Wea. Rev. 107, 140–171.CrossRefGoogle Scholar
- Halem, M., M. Ghil, R. Atlas, J. Susskind and W. J. Quirk, 1978: The GISS sounding temperature impact test. NASA Tech. Memo. 78063, NASA Goddard Space Flight Center, Greenbelt, Md. 20771, 421 pp. [NTIS N7831667].Google Scholar
- Isaacson, E., and H. B. Keller, 1966: Analysis of Numerical Methods, Wiley, New York, 541 pp.Google Scholar
- Jazwinski, A. H., 1970: Stochastic Processes and Filtering Theory, Academic Press, New York, 376 pp.Google Scholar
- Jones, R. H., 1965a: Optimal estimation of initial conditions for numerical prediction, J. Atmos. Sci. 22, 658–663.CrossRefGoogle Scholar
- Kalman, R. E., 1960: A new approach to linear filtering and prediction problems. Trans. ASME, Ser. D, J. Basic Eng., 82, 35–45.Google Scholar
- —, and R. S. Bucy, 1961: New results in linear filtering and prediction theory. Trans. ASME, Ser. D.: J. Basic Eng., 83, 95–108.Google Scholar
- Leith, C. E., 1978: Objective methods for weather prediction, Ann. Rev. Fluid Mech., 10, 107–128.CrossRefGoogle Scholar
- —, 1980: Nonlinear normal mode initialization and guasi-geostrophic theory, J. Atmos. Sci., 37, 958–968.CrossRefGoogle Scholar
- Lorenz, E. N., 1969: The predictability of a flow which possesses many scales of motion, Tellus, 21, 289–307.CrossRefGoogle Scholar
- McPherson, R. D., K. H. Bergman, R. E. Kistler, G. E. Rasch and D. S. Gordon, 1979: The NMC operational global data assimilation system. Mon. Wea. Rev., 107, 1445–1461.CrossRefGoogle Scholar
- Miyakoda, K., and O. Talagrand, 1971: The assimilation of past data in dynamical analysis. I, Tellus, 23, 310–317.CrossRefGoogle Scholar
- Ohap, R. F., and A. R. Stubberud, 1976: Adaptive minimum variance estimation in discrete-time linear systems, in Control and Dynamic Systems, Vol. 12, C. T. Leondes (ed.). Academic Press, 583–624.Google Scholar
- Palmén, E., and C. W. Newton, 1969: Atmospheric Circulation Systems, Academic Press, New York, 603 pp.Google Scholar
- Parzen, E., 1960: Modern Probability Theory and Its Applications, Wiley, New York, 464 pp.Google Scholar
- Pedlosky, J., 1979: Geophysical Fluid Dynamics, Springer-Verlag, New York, 624 pp.Google Scholar
- Petersen, D. P., 1968: On the concept and implementation of sequential analysis for linear random fields, Tellus, 20, 673–686.CrossRefGoogle Scholar
- —, 1970: Algorithms for sequential and random observations, Meteorol. Mono., 11, 100–109.Google Scholar
- —, 1973a: Transient suppression in optimal sequential analysis, J. Appl. Meteor., 12, 437–440.CrossRefGoogle Scholar
- —, 1973b: A comparison of the performance of quasi-optimal and conventional objective analysis schemes, J. Appl. Meteor, 12, 1093–1101.CrossRefGoogle Scholar
- —, 1973c: Static and dynamic constraints on the estimation of space-time covariance and wavenumber-frequency spectral fields, J. Atmos. Sci., 30, 1252–1266.CrossRefGoogle Scholar
- —, 1976: Linear sequential coding of random space-time fields. Info. Sci., 10, 217–241.Google Scholar
- Phillips, N. A., 1971: Ability of the Tadjbakhsh method to assimilate temperature data in a meteorological system, J. Atmos. Sci., 28, 1325–1328.CrossRefGoogle Scholar
- —, 1976: The impact of synoptic observing and analysis systems on flow pattern forecasts. Bull. Amer. Met. Soc., 57, 1225–1240.Google Scholar
- Richtmyer, R. D., and K. W. Morton, 1967: Difference Methods for Initial-Value Problems, 2nd. ed., Wiley-Interscience, New York, 405 pp.Google Scholar
- Rutherford, I. D., 1972: Data assimilation by statistical interpolation of forecast error fields, J. Atmos. Sci., 29, 809–815.CrossRefGoogle Scholar
- Schlatter, T. W., 1975: Some experiments with a multivariate stiatistical objective analysis scheme. Mon. Wea. Rev., 103, 246–257.CrossRefGoogle Scholar
- —, G. W. Branstator and L. G. Thiel, 1976: Testing a global multivariate statistical objective analysis scheme with observed data. Mon. Wea. Rev., 104, 765–783.CrossRefGoogle Scholar
- —, —, —, 1977: Reply (to Comments by H. J. Thiebaux on “Testing a global…”), Mon. Wea. Rev., 105, 1465–1468.CrossRefGoogle Scholar
- Tadjbakhsh, I. G., 1969: Utilization of time-dependent data in running solution of initial value problems, J. Appl. Meteor., 8, 389–391.CrossRefGoogle Scholar
- Wiener, N., 1949: E’xtrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, Wiley, New York, 163 pp.Google Scholar