Data Assimilation

The goal of data assimilation is to optimally combine a numerical simulation with observations. In adamantine, experimental data from infra-red cameras and thermo-couples can be used to improve the simulation. We perform the data assimilation using the stochastic Ensemble Kalman filter method (EnKF). For an in-depth discussion about this algorithm, we recommend Data Assimilation by Mark Asch, Marc Bocquet, and Maëlle Nodet.

EnKF combines simulations and observations as follows:

\[x_i^a = x_i^f + K [y-H z_i^f],\]

with:

\[K = P^f H^T (H P^f H^T +R)^{-1},\]

where \(x_i^a\) is the \(i^{th}\) updated simulation of the ensemble, \(x_i^f\) is the \(i^{th}\) simulation of the ensemble, \(K\) is the Kalman gain, \(y\) is the observations (the experimental data), \(H\) is the observation matrix that maps the simulation to the observation, \(P\) is the simulation error covariance, and \(R\) is the obversation error covariance. \(R\) depends on the instruments. \(P^f\) is given by:

\[P^f = \frac{1}{m-1} \sum_{i=1}^M (x_i^f - \bar{x}^f) (x_i^f - \bar{x}^f)^T,\]

with:

\[\bar{x}^f = \frac{1}{m} \sum_{i=1}^m x_i^f.\]

\(m\) is the number of ensemble simulations.

The Bare plate L example demonstrates adamantine data assimilation capabilities.

As of version 1.0 of adamantine data assimilation is restricted to thermal simulations.