The 8th EnKF Data Assimilation Workshop




Analysis of Iterative Ensemble Smoothers for solving inverse problems

Geir Evensen
International Research Institute of Stavanger


Talk: Evensen_WS_Montreal.pdf

Please see this pdf for a nicely rendered version of this abstract.


Sakov et al. (2012) introduced the iterative EnKF (iEnKF) to improve the solution of sequential assimilation problems when the dynamical model and the measurement operator are highly nonlinear. The iEnKF solves the inverse problem for $ {\mathbf{x}}_i$ given a forward model $ {\mathbf{g}}({\mathbf{x}})$ and observations $ {\mathbf{d}}$,

$\displaystyle {\mathbf{x}}_{i+1}$ $\displaystyle = {\mathbf{g}}({\mathbf{x}}_i),$   $\displaystyle {\mathbf{d}}= {\mathbf{h}}({\mathbf{x}}_{i+1}) + {\mathbf{e}}.$ (1)

Since we assume the model to be perfect, we can compute the solution at time $ {\mathbf{x}}_{i+1}$ as soon as we have updated $ {\mathbf{x}}_i$ using the data at time $ t_{i+1}$. The iEnKF reduces errors in the case with strong nonlinearities in the model or the measurement operator, compared to a non-iterative EnKF update.

The formulation above is identical to the inverse problem solved for parameter estimation in the reservoir engineering community except the model is written as a prediction $ {\mathbf{y}}$ given a parametrization $ {\mathbf{x}}$ and measurements of $ {\mathbf{y}}$ (Evensen, 2018),

$\displaystyle {\mathbf{y}}$ $\displaystyle = {\mathbf{g}}({\mathbf{x}}),$   $\displaystyle {\mathbf{d}}= {\mathbf{h}}({\mathbf{y}}) + {\mathbf{e}}.$ (2)

The iterative ensemble smoother (IES) by Chen and Oliver (2012), Chen and Oliver (2013) and the ensemble smoother with multiple data assimilation (ESMDA) by Emerick and Reynolds (2012) are now becoming standard for parameter estimation in reservoir models. However, IES and ESMDA assume the model is perfect, and they update only the uncertain parameters before they rerun the ensemble simulation to obtain the final result. If we do not take significant model errors into account, there is a risk for an unphysical update of some uncertain parameters, that compensates for the neglected errors.

Sakov et al. (2018) extended the iEnKF to work for an imperfect model containing additive errors,

$\displaystyle {\mathbf{x}}_{i+1}$ $\displaystyle = {\mathbf{g}}({\mathbf{x}}_i) + {\mathbf{q}}_{i+1},$   $\displaystyle {\mathbf{d}}= {\mathbf{h}}({\mathbf{x}}_{i+1}) + {\mathbf{e}}.$ (3)

We will discuss basic assumptions and properties of iterative ensemble methods for solving the inverse problem, and we will illustrate how they can reduce the impact of nonlinearity in ensemble methods. Further, we will explain how also IES and ESMDA can be reformulated to account for model errors and illustrate how mainly ESMDA is a viable alternative to the standard EnKF (and iEnKF) in the presence of strong nonlinearity.



Bibliography

Chen, Y., and D. S. Oliver, Ensemble randomized maximum likelihood method as an iterative ensemble smoother, Math. Geosci., 44, 1-26, 2012.

Chen, Y., and D. S. Oliver, Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification, Computational Geosciences, 17, 689-703, 2013.

Emerick, A. A., and A. C. Reynolds, History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations, Computational Geosciences, 16, 639-659, 2012.

Evensen G (2018) Analysis of iterative ensemble smoothers for solving inverse problems. Computat. Geosci., https://doi.org/10.1007/s10596-018-9731-y

Sakov, P., D. S. Oliver, and L. Bertino, An iterative EnKF for strongly nonlinear systems, Mon. Weather Rev., 140, 1988-2004, 2012.

Sakov, P., J.-M. Haussaire, and M. Bocquet, An iterative ensemble kalman filter in the presence of additive model error, Q. J. R. Meteorol. Soc., 2018, doi: 10.1002/qj.3213.