Hi Rajesh,

Data assimilation is a process where by you gather observations in order to start a numerical model, so data assimilation defines the initial state of the model. This process is particularly important in weather forecasting and climate modelling. These are good examples of mathematical models. A mathematical model is a mathematical representations of a physical system, in this case the earth's climatology. The laws of nature which determine the system behaviour are expressed through mathematical expressions relating the different variables of the system. With the advent of fast processing, mathematical models have been developed to study a large number of physical systems in engineering, biology, economics, geophysics, astronomy, etc.

Hope it helps,

Augusto

I agree with Sanabria. For an example of representation of a physical model (aero-elastic energy harvester in this case) read the following link and note how justified assumptions are used to develop a reduced order model. Another option can be to represent the system in high fidelity equations and solve them numerically which may be much more time consuming and resource expensive.

Article Piezomagnetoelastic energy harvesting from vortex-induced vi...

Mathematical models can be a powerful tool for prediction: for example, they play an important part in weather forecasting and in disease/treatment planning. However, to be really effective, mathematical models need to be validated with data. Data assimilation methods are ways of combining mathematical models with data in order to provided predictions, taking account of the fact that both the data and the mathematical models may be imperfect. Many problems are inherently multi-scale in either time or space or both, for example, the dynamics may be modelled on a fine scale, but data is only available for average properties of the system.

Just to illustrate Samra's answer, remember that a Kalman filter -- which has an internal model -- is composed of two stages. In the first stage, the model is used to predict the next state, this is called propagation. The second stage happens when a new measurement is received. Then, the predicted result is compared to the recently arrived date and the prediction is adjusted based on the error. This last stage is called data assimilation. So, in short, data assimilation is an adjustment made based on data in order to better approximate the predictions to such data. In a Bayesian framework the data assimilation is the step where the posterior (after the new data is received) estimate/pdf is produced based on the prior (before the last measurement arrives) estimate/pdf.