Ensemble Kalman Filter¶
An implementation of the Ensemble Kalman filter is provided for sequential data assimilation. Use is through the Ensemble_Kalman class.
The EnKF aproximates the probability distribution of the solution through a finite ensemble of members, (see also Markov Chain Monte Carlo). This estimate is updated at each prediction step in a Bayesian fashion, with the predicted ensemble forming the prior. Upon incorporating the measurement data, we have a posterior distribution.
Classes¶
- Ensemble_Kalman : Implements the filter.
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class
EnKF.Ensemble_Kalman(x_meas, Nx, ensemble_size)[source]¶ This class implements the update step of the ensemble Kalman filter. Once initialised, the point of access for the user is through the apply method.
Attributes
- Nx : The spatial domain size
- N : the number of ensemble members
- H : the measurement operator such that
\[d = H\psi\]where \(d\) is the vector of observations, \(H\) is the measurement operator, and \(\psi\) is the full solution vector.
Methods
- _ensemble_perturbation : Computes the ensemble perturbation
- _perturb_observations : Computes the perturbed observations and
- ensemble of perturbations
- apply: Applies the EnKF to a predicted solution for some measurement
Parameters
- x_meas : The indices where measurements are known on the solution grid
- Nx : The number of gridpoints
- ensemble_size : The number of ensemble members
See Also
- Evensen, The Ensemble Kalman Filter: theoretical formulation and practical implementation. 2003. Ocean Dynamics. DOI:10.1007/s10236-003-0036-9.
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apply(A, d)[source]¶ Applies the EnKF, updating the predicted ensemble matrix with the measured data. The result is the posterior distribution, i.e. the best estimate for the state of the system.
Let n be the size of the model state vector, N be the number of ensemble members, and m be the number of measurements.
Parameters
- A : the prior distribution (real, size n x N)
- d : the measurement vector (real, size m)
Returns
- The posterior distribution after data has been assimilated. This matrix is suitable for continued use in MCMC solvers.
