Ibrahim Hoteit

1. Should we be using Data Assimilation to Combine Seismic Imaging and Reservoir Modeling? Earth Sciences and Engineering Applied Math and Computational Sciences Ibrahim Hoteit KAUST, CSIM, May 2010

4. Publisher: VDM Verlag Date: September 11, 2009

5. Outline • What is Data Assimilation? • Methods of Data Assimilation Least-Squares & Bayesian methods • Ensemble Kalman Filtering An Example (ocean - but related - application …) • Discussion

6. What is Data Assimilation? • Estimate the 4D state of a dynamical system: Atmosphere, Ocean, Hydrology, Reservoir, … • Sources of Information • Observations … but too sparse and noisy • Numerical models … but imperfect • Data Assimilation • Combines model and observations to make the best possibleestimate of the state of a dynamical system • It is an inverse problem … with model dynamics as (weak or strong) constraint

7. Context in Reservoir-Seismic 6 4D Seismic Time Reservoir model Poroelasticity model

8. Formulation of the Assimilation Problem • A dynamical model • and an observation model • state, transition operator • data, observational operator • & represent model and obs. errors • TwoSchools: • Least-Squares (Deterministic) & Bayesian Theory (Stochastic)

9. Least-Squares Formulation • control vector (any model parameter) • Optimization constrained by model dynamics • Run adjoint backward to compute gradients • Non-convex optimization! • Adjust a set of “control variables” to the fit model trajectory to available data over a given period of time:

10. Recursive Bayesian Formulation 9 • Determine the pdf of the state given all observations up to the estimation time • Forecast step: Integrate analysis pdfwith the model • Correction step: Update forecast pdfwith the new obs

11. Least-Squares vs. Bayesian • Same solution when the system is linear and perfect • Geosciences applications: • Nonlinear and imperfect system • Huge dimension (~ 108) and costly models •  Different solutions! • Which one is more appropriate? • Least-Squares • Requires adjoint models • Non-convopti when nonlinear • Not suitable for forecasting • Bayesian • Only forward models • Better with nonlinear systems • Estimates of uncertainties

12. A Demonstration: Bayesian Assimilation with a Real Problem • Predicting the evolution of loop current in the Gulf of Mexico (GOM) • Funded by BP, in collaboration with Anderson (NCAR) & Heimbach (MIT) • 1/10o ocean model of GOM • state dimension ~ 2.107 • Datasets: Satellite SSH and SST • Weekly forecasts of the GOM circulation

13. Ocean GCM Models

14. Bayesian Solution - Kalman Filtering • State pdf is Gaussian for linear models with Gaussian noise • Need to determine mean and covariance of the pdf only • In this case, the Kalman filter recursively provides the BLUE estimate of the state given previous observations • Two steps: • Forecast step to propagate estimate and uncertainties • Analysis step to correct forecast with the new observation

15. Kalman Filter Algorithm Analysis Model Observation Forecast • Nonlinear model ? • Dimension of the state ~ 2.107?

16. Ensemble Kalman Filtering (EnKF) • Monte-Carlo Approach: Represents uncertainties by an • ensemble of vectors • Update the ensemble instead of : • Solves storage problem • Reduces computational burden to reasonable level • Suitable for nonlinear systems • Combines nonlinear forecast step of the Particle filter and linear analysis step of the Kalman filter (Hoteit et al., MWR - 2010) …

17. EnKF Algorithms Analysis Forecast data + KF Model New Analysis Ensemble Analysis Ensemble Forecast Ensemble Time

18. Weekly SSH Estimates Satellite SSH Forecast Analysis 50 ensemble members

19. EnKFs – Pros • Reasonable cost and flexible • Only forward models are required • Ability to integrate multivariate/multisources data models • Propagate information to non-observed variables • Propagate uncertainties in time • …. Still have room for improvements

20. Working on (Frontiers of DA) • Generalize EnKFs to nonlinear observations • Hoteit et al. (MWR, 2008 - 2010) & Luo et al. (Physica-D, 2010) • State-parameter estimation - Dual theory • Hoteit et al. (MWR, submitted - 2010) • Nonlinear smoothers to assimilate future data • Account for model errors (problem dependent!)

21. Discussion • Bayesian assimilation methods were proven efficient to constrain large dimensional nonlinear models with measurements • A framework to integrate different/multi observation models • A framework to deal with different uncertainties • Should we be using data assimilation in seismic imaging and reservoir modeling? • If we want to use 4D seismic data to constrain the reservoir state, the answer is definitely yes!

22. THANK YOU Other group members – Bayesian Assimilation: X. Luo (Postdoc, PhD University of Oxford, UK) U. Altaf (Postdoc, PhD Delft University of Technology, Netherlands) W. Wang (PhD student jointly with Prof. Sun, KAUST) Int. collaborators: D.-T. Pham (CNRS, France) B. Cornuelle (Scripps, USA) G. Triantafyllou & G. Korres (HCMR-Greece) J. Anderson (NCAR, USA) and P. Heimbach (MIT, USA) C. Dawson & C. Jackson (UT-Austin, USA) A. Kohl (University of Hamburg, Germany)

23. Low-Rank Deficiency of EnKFs • Difficulties: • Underestimated error covariance matrices • Not enough degrees of freedom to fit the data • Amplification by an ‘inflation’ factor: • Localization: • of the observations impact (distance-dependent analysis) • of the covariance matrix (using a Schur product) • of the analysis subspace

24. What’s Next … • Define a theoretical framework for ensemble Kalman filters and generalize their analysis step to nonlinear systems • State-Parameter estimation • Dual Kalman filtering for assimilation into coupled models • Dynamical consistency of EnKF analysis • Nonlinear smoothers