13. Dynamic Retrievals: Techniques  12.3 Recently Developed Analysis  12.3.2 Bayesian approach

12.3.3 Nudging

The last four-dimensional data assimilation technique to be discussed is nudging, a method of dynamical relaxation. Robert Kistler, now an employee of NMC, was the first to apply nudging in a shallow fluid model in his M.S. thesis in 1974. Davies and Turner (1977) were the first Europeans to use nudging as a data assimilation tool. Seaman (1990) has recently reviewed the subject area. The British Meteorological Office uses a version of nudging for operational data assimilation (Lorenc et al. 1991).

In nudging, the model is pushed gently toward observed values or a gridded analysis in such a way that gravity wave noise is minimized. For example, let a be some prognostic variable of the model. The prognostic equation may be written as

$${da \over dt} = F(a,t) + G(t) \sum_i^N w_i(a_i - a)$$ (12.48)

 

where the term on the left hand side of the equation is the model tendency, F(a,t) is the model forcing, and the final term in the nudging term. G(t) is the nudging coefficient, w_i is an analysis weight, a_i is an observed value, and a is the interpolated model value. The nudging term should be large enough to be noticed by the model, but not so large that it dominates over other terms in the equation. For example, in the horizontal equation of motion, the nudging term may be as large in magnitude as the horizontal advection term, smaller than the Coriolis and horizontal pressure gradient terms, and larger than the vertical advection term. The nudging term is time dependent; it forces the model every time step--more when the observations are current and less at earlier and later times. The advantages of nudging are:

• Approximate balance is maintained in the model.
• Physical processes are easily accommodated in the model.
• Asynoptic data are incorporated at the appropriate times.

The main disadvantage of nudging is that it lacks a solid theoretical foundation, though some engineers would dispute this.

We summarize here the most important properties of recently developed four-dimensional data assimilation techniques:

• Adaptive (Kalman) filtering is a generalization of optimal interpolation which incorporates a prediction model explicitly and updates the covariance statistics after each assimilation cycle.
• The Bayesian approach to data assimilation has a good theoretical foundation. It helps to explain how the more common methods of objective analysis are related and points the way to more sophisticated methods.
• The adjoint method is one possible way to fit a model to data at several different times. Application of this method to high resolution models with full physics is very difficult and must await, at the very least, greater computing power.
• Nudging is a more economical method of four-dimensional data assimilation which shows promise but seems to lack a good theoretical basis.

Acknowledgment. Much of this material was derived from an earlier work by the author (Schlatter, 1988). That publication, a conference preprint, contains a comprehensive annotated bibliography on the subject of meteorological data assimilation, organized by sub-topics. The bibliography, current through 1987, may be of value to readers wishing to pursue the subject further.

  REFERENCES

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Bergthorsson, P., and B. Doos, 1955: Numerical weather map analysis. Tellus, 7, 329-340.

Cressman, G.P., 1959: An operational objective analysis system. Mon. Wea. Rev., 87, 367-374.

Daley, R., 1991: Atmospheric Data Analysis, Cambridge University Press, New York, 457 pp.

Davies, H.C., and R.E. Turner, 1977: Updating prediction models by dynamical relaxation: An examination of the technique. Quart. J. Roy. Meteor. Soc., 103, 225-245.

Derber, J.C., D. Parrish, and S.J. Lord, 1991: The new global operational analysis system at the National Meteorological Center. Wea. Forecasting, 6, 538-547.

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Gandin, L.S., 1963: Objective Analysis of Meteorological Fields. Gidrometeorologicheskoe Izdatel'stvo, Leningrad, translated from Russian in 1965 by Israel Program for Scientific Translations, Jerusalem, 242 pp.

Gelb, A., (ed.) 1974: Applied Optimal Estimation. The Analytic Sciences Corporation, MIT Press, Cambridge, Massachusetts, 374 pp.

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Koch, S.E., M. DesJardins, and P.C. Kocin, 1983: An interactive Barnes objective map analysis scheme for use with satellite and conventional data. J. Climate Appl. Meteor., 22, 1487-1503.

Lewis, J.M., 1990: Introduction to the adjoint method of data assimilation: Parts I and II. Mesoscale Data Assimilation, 1990 Summer Colloquium, Advanced Study Program, National Center for Atmospheric Research, Boulder, Colorado, 297-319.

Lorenc, A.C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112, 1177-1194.

Lorenc, A.C., R.S. Bell, and B. MacPherson, 1991: The Meteorological Office analysis correction data assimilation scheme. Quart. J. Roy. Meteor. Soc., 117, 59-89.

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Parrish, D.F., and J.C. Derber, 1992: The National Meteorological Center's spectral statistical interpolation analysis system. Mon. Wea. Rev., 120, 1747-1763.

Petersen, D.P., 1968: On the concept and implementation of sequential analysis for linear random fields. Tellus, 20, 673-686.

Purser, R.J., 1984: A new approach to the optimal assimilation of meteorological data by iterative Bayesian analysis. Proceedings, 10th Conference on Weather Forecasting and Analysis, Clearwater Beach, Florida, American Meteorological Society, 102-105.

Schlatter, T.W., 1988: Past and present trends in the objective analysis of meteorological data for nowcasting and numerical forecasting. Preprints, Eighth Conference on Numerical Weather Prediction, Baltimore, Maryland, American Meteorological Society, J9-J25.

Seaman, N.L. 1990: Newtonian nudging: A four-dimensional approach to data assimilation. Mesoscale Data Assimilation, 1990 Summer Colloquium, Advanced Study Program, National Center for Atmospheric Research, Boulder, Colorado, 263-277.

Thiebaux, H.J., and M.A. Pedder: Spatial Objective Analysis with Applications in Atmospheric Science, Academic Press, New York, 299 pp.

Wahba, G. and J. Wendelberger, 1980: Some new mathematical methods for variational objective analysis using splines and cross validation. Mon. Wea. Rev., 108, 1122-1143.

   
 13. Dynamic Retrievals  12.3 Recently Developed Analysis  12.3.2 Bayesian Approach