pyesmda¶
- class pyesmda.ESMDA(obs: array, m_init: array, stdev_d: array, stdev_m: array, forward_model: Callable, forward_model_args: tuple = (), forward_model_kwargs: dict = {}, n_assimilation: int = 4, alpha: Optional[List[int]] = None, m_bounds: Optional[array] = None)[source]¶
Ensemble Smoother with Multiple Data Assimilations.
Implement the ES-MDA as proposed by Emerick, A. A. and A. C. Reynolds 1, 2.
- Parameters
dobs (np.array) – Obsevrations vector.
m_init (np.array) – Initial ensemble of N_{e} parameters vector.
stdev_d (np.array) – Standard deviation of the observations.
stdev_m (np.array) – Standard deviation of the parameters.
forward_model (callable) – Function calling the non-linear observation model (forward model) for all ensemble members and returning the predicted data for each ensemble member.
forward_model_args (tuple) – Additional args for the callable forward_model.
forward_model_kwargs (dict) – Additional kwargs for the callable forward_model.
n_assimilation (int, optional) – Number of data assimilations. The default is 4.
alpha (Union[List[int], None], optional) – Multiplication factor used to inflate the covariance matrix of the measurement errors. The default is None.
m_bounds (Union[List[int], None], optional) – Top and bottom bounds on the initial ensemble of N_{e} parameters vector. The default is None.
References
- 1(1,2)
Emerick, A. A. and A. C. Reynolds, Ensemble smoother with multiple data assimilation, Computers & Geosciences, 2012.
- 2
Emerick, A. A. and A. C. Reynolds. (2013). History-Matching Production and Seismic Data in a Real Field Case Using the Ensemble Smoother With Multiple Data Assimilation. Society of Petroleum Engineers - SPE Reservoir Simulation Symposium 2013. 2. 10.2118/163675-MS.
- property alpha¶
Get the alpha coefficients used by ES-MDA.
Single and multiple data assimilation are equivalent for the linear-Gaussian case as long as the factor \(\alpha_{l}\) used to inflate the covariance matrix of the measurement errors satisfy the following condition:
\[\sum_{l=1}^{N_{a}} \frac{1}{\alpha_{l}} = 1\]In practise, \(\alpha_{l} = N_{a}\) is a good choice 1.
- analyse(assimilation_iteration)[source]¶
Analysis step of the ES-MDA.
Update the vector of model parameters using
\[m^{l+1}_{j} = m^{l}_{j} + C^{l}_{MD}\left(C^{l}_{DD}+\alpha_{l+1} C_{D}\right)^{-1} \left(d^{l}_{uc,j} - d^{l}_{j} \right), \textrm{for } j=1,2,...,N_{e}.\]- Return type
None.
- approximate_covariance_matrices()[source]¶
Calculate Average and Covariance MD and Covariance DD.
The covariance matrices \(C^{l}_{MD}\) and \(C^{l}_{DD}\) are approximated from the ensemble in the standard way of EnKF 3, 4.
References
- property d_dim¶
Return the number of forecast data.
- forecast()[source]¶
Forecast step of ES-MDA.
Run the forward model from time zero until the end of the historical period from time zero until the end of the historical period to compute the vector of predicted data
\[d^{l}_{j}=g\left(m^{l}_{j}\right),\textrm{for }j=1,2,...,N_{e},\]where \(g(·)\) denotes the nonlinear observation model, i.e., \(d^{l}_{j}\) is the \(N_{d}\)-dimensional vector of predicted data obtained by running the forward model reservoir simulation with the model parameters given by the vector \(m^{l}_{j}\) from time zero. Note that we use \(N_{d}\) to denote the total number of measurements in the entire history.
- Return type
None.
- property m_bounds¶
Get the parameter errors covariance matrix.
- property m_dim¶
Return the length of the parameters vector.
- property n_assimilation¶
Return the number of assimilation to perfom.
- property n_ensemble¶
Return the number of ensemble members.
- pertrub(assimilation_iteration)[source]¶
Perturbation of the observation vector step of ES-MDA.
Perturb the vector of observations
\[d^{l}_{uc,j} = d_{obs} + \sqrt{\alpha_{l+1}}C_{D}^{1/2}Z_{d}, \textrm{for } j=1,2,...,N_{e},\]where \(Z_{d} \sim \mathcal{N}(O, I_{N_{d}})\).
- Return type
None.
- property stdev_d¶
Get the observation errors covariance matrix.
- property stdev_m¶
Get the parameter errors covariance matrix.