Data assimilation (DA) is a method used (perhaps most importantly in the weather forecasting process) in which observations of the current (and possibly, past) state of a system are combined with the results from a mathematical model to produce an analysis, which is considered as 'the best' estimate of the current state of the system. In addition to weather forecasting, other uses include trajectory estimation for the Apollo program.

Weather forecasting applications

Data assimilation can be considered as a method of including observations of variables like temperature, and atmospheric pressure in the model used to predict weather.

In weather forecasting there are 2 main types of data assimilation: 3 dimensional (3DDA) and 4 dimensional (4DDA). In 3DDA only those observations are used available at the time of analyses. In 4DDA the past observations are included (thus, time dimension added).

History of Data Assimilation in Weather forecasting

The first data assimilation methods were called the "objective analyses" (e.g., Cressman algorithm). This was in contrast to the "subjective analyses", when (in the past practice) numerical weather predictions (NWP) forecasts were arbitrarily corrected by meteorologists. The objective methods used simple interpolation approaches, and thus were the kind of 3DDA methods.

The similar 4DDA methods, called "nudging" also exist (e.g. in MM5 NWP model). They are based on the simple idea of Newtonian relaxation. The idea is to add in the right part of dynamical equations of the model the term, proportional to the difference of the calculated meteorological variable and the observation value. This term, that has a negative sign "keeps" the calculated state vector closer to the observations.

The breakdown in the field of data assimilation was achieved by L. Gandin (1963) who introduced the "statistical interpolation" (or "optimal interpolation" ) method. His work developed the previous ideas of Kolmogorov. That method is the 3DDA method and is a kind of regression analyses, which utilizes the information about the spatial distributions of covariance functions of the errors of the "first guess" field (previous forecast) and "true field". These functions are never known. However, the different approximations were assumed.

In fact the optimal interpolation algorithm is the reduced version of the Kalman filtering (KF) algorithm, when the covariance matrices are not calculated from the dynamical equations, but are pre-determined in advance. When this was recognised the attempts to introduce the KF algorithms as a 4DDA tool for NWP models were done. However, this was (and remains) a very difficult task, since the full version of KF algorithm requires solution of the enormous large number of additional equations. In connection with that the special kind of KF algorithms (suboptimal KF's) for NWP models were developed.

Another significant advance in the development of the 4DDA methods was utilizing the optimal control theory (variational approach) in the works of Le Dimet and Talagrand, 1986, based on the previous works of G. Marchuk. The significant advantage of the variational approaches is that the meteorological fields satisfy the dynamical equations of the NWP model and at the same time they minimize the functional, characterizing their difference from observations. Thus, the problem of constrained minimization is solved. The 3DDA variational methods also exist (e.g., Sasaki, 1958).

As was shown by Lorenc, 1986, the all abovementioned kinds of 4DDA methods are in some limit equivalent, i.e. under some assumptions they minimize the same cost function. However, in practical applications these assumptions are never fulfilled, the different methods perform differently and generally it is not clear, what approach (Kalman filtering or variational) is better.

Future Development in NWP

The rapid development of the various data assimilation methods for NWP models is connected with the two main points in the field of numerical weather prediction:

1. Utilizing the observations currently seems to be the most promising chance to improve the quality of the forecasts at the different spatial scales (from the planetary scale to the local city, or even street scale) and time scales.
2. The number of different kinds of available observations (sodars, radars, satellite) is rapidly growing.

The main question is: can the principal limit of the predictability of the weather forecast models be overcome (at least, to some extend) with the help of data assimilation?

Other applications of Data Assimilation

Data assimilation methods are currently also used in other environmental forecasting problems, e.g. in hydrological forecasting. Basically, the same types of DA methods as those described above, are in use there.

Data assimilation is a part of the challenge for every forecasting problem.

References

• R. Daley, Atmospheric data analysis, Cambridge University Press, 1991.
• MM5 community model homepage

Weather prediction | Numerical climate and weather models | Estimation theory | Control theory