Advanced Concepts in Data Assimilation and Kalman Filtering

Data Assimilation Spring 2014

Outlines Concept 2/29

Concept 1960s: How best we can use information from the model and various source of data to produce an estimate that is better than the model or data alone. Data Fusion Integration of multiple data and knowledge into a consistent, accurate, and useful representation. Reduction technique with improved confidence. Different source of permeability. Data Reconciliation and Validation Uses process information and mathematical methods to correct measurements in industrial processes. Extracts accurate and reliable information about the state of process from raw measurement data. Data Assimilation 3/29

Concept Model Observation Criterion Data Assimilation Methods Assimilated/Fitted Model Uncertainty Assessment Prediction Sensitivity Data Assimilation 3/29

DA methods Autonomous/ Non-autonomous Linear/ Nonlinear Structure 1D Discrete Time Model Space 2D Continuous 3D Deterministic Stochastic Static Dynamic Dynamic Static Data Assimilation 3/29

Kalman Filter continuously provide the best estimate of the system state at the time of the last measurement Estimation Filtering Smoothing Prediction Data Assimilation 3/29

Kalman Filter Assumptions: Linear dynamic system Initial condition is Gaussian with known mean and covariance Model error follows a white Gaussian distribution Measurement noise follows a white Gaussian distribution Initial condition, Model error, and Measurement noise are uncorrelated ( ) = = = ( ) = + x M x w ; w ~ N 0, Q + + + k 1 k k k 1 k N 1 k ( k ) = + z H x v v ; ~ 0, R + k k k k Q 1 if j otherwise if j otherwise k k = R = k T j E w w k 0 = k , = k T j E v v k 0 x ~ N m P 0 0 0 T E x w 0 0 T E x v 0 0 T j E v w 0 Data Assimilation 3/29

Kalman Filter Derivation: e ( ) = = f x z M H x x k w = + + f f x x K H x x v v k k 1 k 1 k k k k k k k ( ) ( ) ( )( M ) k T T + ( ) = x E x x x x tr E x x = x x k k k = + + + M x K H M x x H w v k k k k k k k k + x k 1 k 1 k k k 1 k 1 k 1 k k k k k 1 k 1 = = f x M x = x x k f k k 1 k f k 1 k k k I ( ) e x x = ) + M ( I K H e K H M e K H w K v k k 1 k 1 k k k 1 k 1 k k k k k ( ) = + M x x w = + M e w K v k k k 1 k 1 k k k k 1 k 1 k 1 k ( ) T = + M e w = P E e e k 1 k 1 k k k k ( ) e T ( )( ) ( = f f f P E e T ( ) ) T = + + + T k T k I K H E M e w M e w I K H K E v v K k k k k k k 1 k 1 k k 1 k 1 k k k k k ( )( ) T k ( ) ) ( ) T = + + P E M e w M e w = k + + T k T k I K H M P M Q I K H K R K k 1 k 1 k k 1 1 k k k k 1 k 1 1 k k k k k ( ( ) T = + f T k ; I K H P I K H K R K = + T k z M P M Q k k k k k k k + k 1 k 1 1 k = + = + f f f T k T k T k f T k P K H P P H K K D K D H P H R = f f x x K H x k k k k k k k k k k k k k k k k k T k k k = + f f T k 1 f f T k 1 f T k 1 P P P H D H P K P H D D K P H D k k k k k k k k k k innovation,a new information, residual nomaly MVE k = f T k 1 K P H D k k ( ) = f P I K H P 1 k k k k = + f T k f T k P H H P H R Data Assimilation 3/29 k k k k

Kalman Filter Interpretation of Kalman gain: = = = 11 22 k nn ( = + = = + + + ii ii ii ii P R P R n m 1 = + f f K P P R H k k k k = I f f f P P P k = Diag , , , 11 22 nn ( ( ) f f f f P Diag P , P , , P + + + f f f P R P R P R k 11 22 nn R 11 11 22 22 nn nn ) R Diag R , R , , ) + f f f x x K z H x I K x K z k k k k k k k k k k f R P f x x z ii ii i k , i k , k f f Estimated covariance is independent of observations Data Assimilation 3/29

Kalman Filter Example: Data Assimilation 3/29

Kalman Filter Advantages: Optimum for linear, non-autonomous, dynamical systems Sequential Estimation of states and parameters Data from different sources Asynchronous observations Considering model error and observation noise Data Assimilation 3/29

Nonlinear Dynamics Extended Kalman Filter (EKF) Nonlinear Filters (PF, UnKF) Ensemble Kalman Filter (EnKF) 1 N ( ) T T ( ) N ( ) i ( ) i = = f + f f f f P E e e e e + + + + k 1 k 1 k 1 k 1 k 1 N 1 = i 1 ( ) ( ) i ( ) i ( ) i = + f M w + + k k 1 k 1 Data Assimilation 3/29

Emerick, A. A., "History Matching and Uncertainty Characterization Using Ensemble-based Methods," University of Tulsa, 2012. Gibbs, B. P., Advanced Kalman filtering, least-squares and modeling: a practical handbook: John Wiley & Sons, 2011. Phale, H. A., "Data Assimilation Using the Ensemble Kalman Filter with Emphasis on the Inequality Constraints," University of Oklahoma, 2010. Jafarpour, B., "Oil reservoir characterization using ensemble data assimilation," Massachusetts Institute of Technology, 2008. Park, K., Modeling uncertainty in metric space: Stanford University, 2011. Dumedah, G. and Coulibaly, P., "Evaluating forecasting performance for data assimilation methods: The ensemble Kalman filter, the particle filter, and the evolutionary-based assimilation," Advances in Water Resources, vol. 60, pp. 47-63, 2013. Lewis, J. M., Lakshmivarahan, S., and Dhall, S., Dynamic data assimilation: a least squares approach vol. 13: Cambridge University Press, 2006 Pasetto, D., Camporese, M., and Putti, M., "Ensemble Kalman filter versus particle filter for a physically-based coupled surface subsurface model," Advances in Water Resources, vol. 47, pp. 1-13, 10// 2012. Myrseth ,I., S trom, J., and Omre, H., "Resampling the ensemble Kalman filter," Computers & Geosciences, vol. 55, pp. 44-53, 2013. . 28/29

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