Extended ensemble Kalman filters for high-dimensional hierarchical - PowerPoint PPT Presentation

Extended ensemble Kalman filters for high-dimensional hierarchical state-space models Matthias Katzfuss Department of Statistics Texas A&M University Joint work with Jon Stroud (Georgetown) and Chris Wikle (Missouri) Matthias Katzfuss (Texas A&M) Extended EnKFs 1 / 37

Outline Outline 1 Data assimilation 2 Review of the EnKF 3 Extended EnKFs 4 Numerical examples 5 Conclusions Matthias Katzfuss (Texas A&M) Extended EnKFs 2 / 37

Data assimilation Outline 1 Data assimilation 2 Review of the EnKF 3 Extended EnKFs 4 Numerical examples 5 Conclusions Matthias Katzfuss (Texas A&M) Extended EnKFs 3 / 37

Data assimilation Data assimilation Data assimilation (DA): sequentially infer the true state of a system, by combining (noisy) observations with an evolution operator M state time 0 1 2 3 Matthias Katzfuss (Texas A&M) Extended EnKFs 4 / 37

Data assimilation Data assimilation Data assimilation (DA): sequentially infer the true state of a system, by combining (noisy) observations with an evolution operator M state data M time 0 1 2 3 Matthias Katzfuss (Texas A&M) Extended EnKFs 4 / 37

Data assimilation Data assimilation Data assimilation (DA): sequentially infer the true state of a system, by combining (noisy) observations with an evolution operator M state data update M time 0 1 2 3 Matthias Katzfuss (Texas A&M) Extended EnKFs 4 / 37

Data assimilation Data assimilation Data assimilation (DA): sequentially infer the true state of a system, by combining (noisy) observations with an evolution operator M state data update M M time 0 1 2 3 Matthias Katzfuss (Texas A&M) Extended EnKFs 4 / 37

Data assimilation Data assimilation Data assimilation (DA): sequentially infer the true state of a system, by combining (noisy) observations with an evolution operator M state data update M M update time 0 1 2 3 Matthias Katzfuss (Texas A&M) Extended EnKFs 4 / 37

Data assimilation Data assimilation Data assimilation (DA): sequentially infer the true state of a system, by combining (noisy) observations with an evolution operator M state data update M M update update M time 0 1 2 3 Matthias Katzfuss (Texas A&M) Extended EnKFs 4 / 37

Data assimilation Data assimilation: Goals and examples Goals: • State inference, initialization for forecasting, model calibration, . . . Examples of environmental applications: • weather forecasting • climate studies • pollution monitoring In geophysical applications, the state typically consists of one or more (discretized) spatial fields, the state dimension is often massive, and the evolution model is highly complex and nonlinear Reviews: e.g., Wikle and Berliner, 2007; Evensen, 2009; Nychka and Anderson, 2010; Houtekamer and Zhang, 2016 Matthias Katzfuss (Texas A&M) Extended EnKFs 5 / 37

Data assimilation Overview of data assimilation methods Best methods (in terms of accuracy and computation time) for data assimilation as a function of state dimension and the degree of nonlinearity: Degree of Nonlinearity ??? Strongly nonlinear EnKF Moderately PF nonlinear UKF EnKF Mildly EKF nonlinear KF Linear 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 Dimension KF = Kalman filter; PF = particle filter; EKF = extended KF; UKF = unscented KF; EnKF = ensemble KF Here we only consider methods for sequential, probabilistic DA, not variational methods (e.g., 4DVAR; Talagrand and Courtier 1987) Matthias Katzfuss (Texas A&M) Extended EnKFs 6 / 37

Data assimilation Notation: The state-space model (SSM) From a statistical perspective: DA is filtering in a SSM SSM with additive Gaussian error in discrete time t = 1 , 2 , . . . : v t ∼ N m t ( 0 , R t ) , y t = H t x t + v t , x t = M t ( x t − 1 ) + w t , w t ∼ N n ( 0 , Q t ) , where • y t is the m t -dimensional vector of observations at time t • x t is the n -dimensional state of primary interest • H t is the observation matrix • M t ( · ) is the (possibly nonlinear) evolution operator • errors v t and w t are mutually and serially independent • initial state: x 0 ∼ N n ( µ 0 | 0 , Σ 0 | 0 ) • no unknown parameters (for now) Note that this SSM is very general Matthias Katzfuss (Texas A&M) Extended EnKFs 7 / 37

Data assimilation The Kalman filter (KF; Kalman, 1960) Filtering: At each time t , find cond. distrib. of x t given y 1: t = { y 1 , . . . , y t } If M t is linear (i.e., M t ( x t − 1 ) = M t x t − 1 ), can use KF: For t = 1 , 2 , . . . : • x t − 1 | y 1: t − 1 ∼ N n ( µ t − 1 | t − 1 , Σ t − 1 | t − 1 ) computed at t − 1 • Forecast step: x t | y 1: t − 1 ∼ N n ( µ t | t − 1 , Σ t | t − 1 ), where Σ t | t − 1 := M t Σ t − 1 | t − 1 M ′ µ t | t − 1 := M t µ t − 1 | t − 1 , t + Q t • Update step: x t | y 1: t ∼ N n ( µ t | t , Σ t | t ), where µ t | t := µ t | t − 1 + K t ( y t − H t µ t | t − 1 ) , Σ t | t := ( I n − K t H t ) Σ t | t − 1 t + R t ) − 1 is the n × m t Kalman gain and K t := Σ t | t − 1 H ′ t ( H t Σ t | t − 1 H ′ KF is exact, but infeasible if M t nonlinear and/or n and m t large Matthias Katzfuss (Texas A&M) Extended EnKFs 8 / 37

Data assimilation The Kalman filter (KF; Kalman, 1960) Filtering: At each time t , find cond. distrib. of x t given y 1: t = { y 1 , . . . , y t } If M t is linear (i.e., M t ( x t − 1 ) = M t x t − 1 ), can use KF: For t = 1 , 2 , . . . : • x t − 1 | y 1: t − 1 ∼ N n ( µ t − 1 | t − 1 , Σ t − 1 | t − 1 ) computed at t − 1 • Forecast step: x t | y 1: t − 1 ∼ N n ( µ t | t − 1 , Σ t | t − 1 ), where Σ t | t − 1 := M t Σ t − 1 | t − 1 M ′ µ t | t − 1 := M t µ t − 1 | t − 1 , t + Q t • Update step: x t | y 1: t ∼ N n ( µ t | t , Σ t | t ), where µ t | t := µ t | t − 1 + K t ( y t − H t µ t | t − 1 ) , Σ t | t := ( I n − K t H t ) Σ t | t − 1 t + R t ) − 1 is the n × m t Kalman gain and K t := Σ t | t − 1 H ′ t ( H t Σ t | t − 1 H ′ KF is exact, but infeasible if M t nonlinear and/or n and m t large Matthias Katzfuss (Texas A&M) Extended EnKFs 8 / 37

Data assimilation The Kalman filter (KF; Kalman, 1960) Filtering: At each time t , find cond. distrib. of x t given y 1: t = { y 1 , . . . , y t } If M t is linear (i.e., M t ( x t − 1 ) = M t x t − 1 ), can use KF: For t = 1 , 2 , . . . : • x t − 1 | y 1: t − 1 ∼ N n ( µ t − 1 | t − 1 , Σ t − 1 | t − 1 ) computed at t − 1 • Forecast step: x t | y 1: t − 1 ∼ N n ( µ t | t − 1 , Σ t | t − 1 ), where Σ t | t − 1 := M t Σ t − 1 | t − 1 M ′ µ t | t − 1 := M t µ t − 1 | t − 1 , t + Q t • Update step: x t | y 1: t ∼ N n ( µ t | t , Σ t | t ), where µ t | t := µ t | t − 1 + K t ( y t − H t µ t | t − 1 ) , Σ t | t := ( I n − K t H t ) Σ t | t − 1 t + R t ) − 1 is the n × m t Kalman gain and K t := Σ t | t − 1 H ′ t ( H t Σ t | t − 1 H ′ KF is exact, but infeasible if M t nonlinear and/or n and m t large Matthias Katzfuss (Texas A&M) Extended EnKFs 8 / 37

Data assimilation The Kalman filter (KF; Kalman, 1960) Filtering: At each time t , find cond. distrib. of x t given y 1: t = { y 1 , . . . , y t } If M t is linear (i.e., M t ( x t − 1 ) = M t x t − 1 ), can use KF: For t = 1 , 2 , . . . : • x t − 1 | y 1: t − 1 ∼ N n ( µ t − 1 | t − 1 , Σ t − 1 | t − 1 ) computed at t − 1 • Forecast step: x t | y 1: t − 1 ∼ N n ( µ t | t − 1 , Σ t | t − 1 ), where Σ t | t − 1 := M t Σ t − 1 | t − 1 M ′ µ t | t − 1 := M t µ t − 1 | t − 1 , t + Q t • Update step: x t | y 1: t ∼ N n ( µ t | t , Σ t | t ), where µ t | t := µ t | t − 1 + K t ( y t − H t µ t | t − 1 ) , Σ t | t := ( I n − K t H t ) Σ t | t − 1 t + R t ) − 1 is the n × m t Kalman gain and K t := Σ t | t − 1 H ′ t ( H t Σ t | t − 1 H ′ KF is exact, but infeasible if M t nonlinear and/or n and m t large Matthias Katzfuss (Texas A&M) Extended EnKFs 8 / 37

Review of the EnKF Outline 1 Data assimilation 2 Review of the EnKF 3 Extended EnKFs 4 Numerical examples 5 Conclusions Matthias Katzfuss (Texas A&M) Extended EnKFs 9 / 37

Review of the EnKF The ensemble Kalman filter (EnKF; Evensen 1994) Approximate version of the KF for large or nonlinear SSMs, in which the state distribution is represented by a sample or “ensemble” Assume ensemble x (1) t − 1 | t − 1 , . . . , x ( N ) t − 1 | t − 1 is sample from filtering distribution at time t − 1. For i = 1 , . . . , N : 1. Forecast Step: Apply the evolution model: x ( i ) t | t − 1 = M t ( x ( i ) t − 1 | t − 1 ) + w ( i ) w ( i ) t , ∼ N n ( 0 , Q t ) t 2. Update Step: Update by applying a linear “shift”: x ( i ) t | t = x ( i ) y ( i ) − H t x ( i ) t | t − 1 + � K t ( � t | t − 1 ) t where � K t is an approximation of the Kalman gain, and y ( i ) = y t + v ( i ) � is a perturbed observation t t Matthias Katzfuss (Texas A&M) Extended EnKFs 10 / 37