covmats.CovViaEnsemble

class covmats.CovViaEnsemble(*args, **kwargs)

Represents a covariance matrix as an ensemble of realizations.

For a given ensemble with shape (\(N_{s}\), \(N_{e}\)), the number of points and the number of members in the ensemble respectively, the covariance matrix \(\mathbf{\Sigma_{ss}}\) is approximated from the ensemble in the standard way of EnKF [Evensen, 2007, Aanonsen et al., 2009]:

\[\mathbf{\Sigma_{ss}} = \frac{1}{N_{e} - 1} \sum_{j=1}^{N_{e}}\left(s_{j} - \overline{s}\right)\left(s_{j} - \overline{s^{l}} \right)^{T}\]

Or by defining a matrix of anomalies \(\mathbf{A} = \mathbf{S} - \overline{\mathbf{S}}\) with shape (\(N_{s}\), \(N_{e}\)):

\[\mathbf{\Sigma_{ss}} = \frac{1}{N_{e} - 1} \mathbf{A}^{T}\mathbf{A}\]

Note

Practically, the dense covariance matrix is never built, only the anomalies matrix \(\mathbf{A}\) is used. The product between the inverse of the covariance matrix and a vector \(\mathbf{x} = \mathbf{\Sigma_{ss}}^{-1}\mathbf{b}\) is obtained solving the system \(\mathbf{A}^{T}\mathbf{Ax} = \mathbf{b}\), using gmres, where only anomalies matrix vector products are required.

References

1. Geir Evensen. Data Assimilation - The Ensemble Kalman Filter. Springer Berlin Heidelberg, 2007. ISBN 978-3-642-03710-8. doi:10.1007/978-3-642-03711-5.

2. Sigurd Aanonsen, Geir Nævdal, Dean Oliver, Albert Reynolds, and Brice Vallès. The Ensemble Kalman Filter in Reservoir Engineering–a Review. SPE Journal - SPE J, 14:393–412, September 2009. doi:10.2118/117274-PA.

__init__(ensemble: ndarray[tuple[Any, ...], dtype[float64]]) → None

Initiate the instance.

Parameters:

ensemble (NDArrayFloat) – Ensemble of realization with shape (\(N_{e}\), \(N_{s}\)).

Properties

H	Hermitian adjoint.
T	Transpose this linear operator.
anomalies	Return the matrix of anomalies.
covariance	Explicit dense representation of the covariance matrix.
ensemble
log_pdet	Log of the pseudo-determinant of the covariance matrix.
n_ens	Return the number of members in the ensemble.
n_pts	Number of points in the domain (n).
ndim
precision	Explicit dense representation of the precision matrix with shape (n, n).
rank	Rank of the covariance matrix.
shape	Shape of the covariance matrix (n, n).
subspace_size	Subspace size of the covariance matrix.

Methods

adjoint	Hermitian adjoint.
colorize	Perform a colorizing transformation on data.
dot	Matrix-matrix or matrix-vector multiplication.
from_cholesky	Representation of a covariance provided via choleksy factorization.
from_diagonal	Representation of a covariance provided via diagonal.
from_eigendecomposition	Representation of a covariance provided via eigendecomposition.
from_precision	Return a representation of a covariance from its precision matrix.
get_diagonal	Return the diagonal entries of the matrix (variances).
get_trace	Return the trace of the covariance matrix (sum of diagonal elements).
matmat	Matrix-matrix multiplication.
matvec	Matrix-vector multiplication.
rmatmat	Adjoint matrix-matrix multiplication.
rmatvec	Adjoint matrix-vector multiplication.
sample_mvnormal	Draw samples from the multivariate normal N(0, A).
solve	Solve A^{T}Ax = b, with A, the anomalies matrix instance.
todense	Explicit dense representation of the covariance matrix with shape (n, n).
transpose	Transpose this linear operator.
whiten	Perform a whitening transformation on data.