REF: https://en.wikipedia.org/wiki/Data_assimilation
Data assimilation is the process by which observations of a system are incorporated into the model state of a numerical model of that system. A numerical model determines how a model state at a particular time changes into the model state at a later time. Even if the numerical model were a perfect representation of an actual system (which of course can rarely if ever be the case) in order to make a perfect forecast of the future state of the actual system， the initial state of the numerical model would also have to be a perfect representation of the actual state of the system. Applications of data assimilation arise in many fields of geosciences, perhaps most importantly in weather forecasting and hydrology.
A frequently encountered problem is that the number of observations of the actual system available for analysis is orders of magnitude smaller than the number of values required to specify the model state. The initial state of the numerical model cannot therefore be determined from the available observations alone. Instead, the numerical model is used to propagate information from past observations to the current time. This is then combined with current observations of the actual system using a data assimilation method.
Most commonly this leads to the numerical modelling system alternately performing a numerical forecast and a data analysis. This is known as analysis/forecast cycling. The forecast from the previous analysis to the current one is frequently called the background.
The analysis combines the information in the background with that of the current observations, essentially by taking a weighted mean of the two; using estimates of the uncertainty of each to determine their weighting factors. The data assimilation procedure is invariably multivariate and includes approximate relationships between the variables. The observations are of the actual system, rather than of the model's incomplete representation of that system, and so may have different relationships between the variables from those in the model. To reduce the impact of these problems incremental analyses are often performed. That is the analysis procedure determines increments which when added to the background yield the analysis. As the increments are generally small compared to the background values this leaves the analysis less affected by 'balance' errors in the analysed increments. Even so, some filtering, known as initialisation, may be required to avoid problems, such as the excitement of unphysical wave like activity or even numerical instability, when running the numerical model from the analysed initial state.
As an alternative to analysis/forecast cycles, data assimilation can proceed by some sort of continuous process such as nudging, where the model equations themselves are modified to add terms that continuously push the model towards the observations.
REF: Data-Driven Numerical Modelling in Geodynamics: Methods and Applications
Dynamic processes in the Earth’s interior and on its surface can be described by geodynamic models. These models can be presented by a mathematical problem comprising a set of partial differential equations with relevant conditions at the model boundary and at the initial time. The mathematical problem can be then solved numerically to obtain future states of the model. Meanwhile the initial conditions in the geological past or some boundary conditions at the present are unknown, and the question of how to “find” the conditions with a sufficient accuracy attracts attention in the field of geodynamics. One of the mathematical approaches is data assimilation or the use of available data to reconstruct the initial state in the past or boundary conditions and then to model numerically the dynamics of the Earth starting from the reconstructed conditions.
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