State and Parameter Estimation in Stochastic Dynamical Models

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1 State and Parameter Estimation in Stochastic Dynamical Models Timothy DelSole George Mason University, Fairfax, Va and Center for Ocean-Land-Atmosphere Studies, Calverton, MD June 21, collaboration with Xiaosong Yang (GMU/COLA; now at GFDL)

2 Stochastic Parameterizations Involve Tunable Parameters Berner et al Parameters for spectral AR model for streamfunction. In principle, AR parameters should vary with wavenumber. Parameters for generating noise with power law behavior. Perturbations weighed by dissipation rate implied by numerical dissipation, wave drag, convection. Shutts 2005 Cellular automaton stochastic backscatter scheme. CA involves numerous parameters (life time, conditions for birth and death, survival rules, spatial smoothing). Perturbations weighed by dissipation rate implied by numerical dissipation, wave drag, convection. Buizza et al Multiplicative noise perturbs parameterized physics tendencies. Random numbers drawn from uniform distribution [ 0.5, 0.5]. Random numbers constant over boxes. Random numbers constant for 6 time steps.

3 How Can Parameters in Stochastic Parameterizations Be Estimated?

4 How Can Parameters in Stochastic Parameterizations Be Estimated? Adjoint parameter estimation Carrera and Neuman, 1986, Water Resour. Res. Bennett, 1992, Inverse Methods in Physical Oceanography Navon, 1997, Dyn. Atmos. Oceans The Augmentation Method Jazwinski, 1970, Stochastic Processes and Filtering Theory Gelb, 1974, Applied Optimal Estimation Anderson, 2001, Mon. Wea. Rev. Gershgorin, Harlim, Majda, 2010, J. Comput. Phys.

5 Parameter Estimation with the Kalman Filter x: State vector b: Parameter vector Usual method: augment state vector with unknown parameters: ( ) x z = b Assume parameter update model is b t = b t 1 Jazwinski, 1970; Anderson, 2001 b t = b t 1 + w t Friedland & Grabousky 1982 db/dt = ab t + k + dw t Gershgorin, Harlim, Majda, 2010 How Does This Work? Variations in b cause variations in x. The covariance between these variables can be used to infer one from the other.

6 The Update Equations for Augmented State Vectors Typically, only observations of the state are available: H z = ( H x 0 ) Interpolation Operator In this case, the Kalman Filter equations decouple: ) µ a x = µ f x + K x (o H x µ f x State ( ) µ a b = µf b + K b o H x µ f x Parameter State update is exactly the same as in state-only assimilation. State can be updated with existing data assimilation system. Parameter update has same structural form as state update.

7 Illustration with Modified Lorenz96 Model dx i dt = (x i+1 x i 2 ) x i 1 x i f i, 1 + d i true values of d i and f i are chosen randomly. Note that d i is a multiplicative parameter. Parameter update model b f t = βb f t 1 + (1 β)ba t 1. Localization and inflation applied to state and parameters i = 1, 2,..., observations (every other grid point is observed). Augmented state vector has 120 elements: x = ( x 1 x 2... x 40 f 1... f 40 d 1... d 40 ) T

8 Additive and Multiplicative Parameter Estimation Estimate f i and d i (additive and multiplicative). Compare with Imperfect f i = 0, d i = 0 Perfect f i and d i equal to their true values RMSE Imperfect Augmented Perfect Ensemble Size 2 Yang and DelSole

9 Estimation in Stochastic Models Using Augmented KF Consider the simplest possible stochastic model x t = φx t 1 + βw t, where φ < 1 and w is standardized Gaussian white noise. Determinitic Parameter, Augmented Determinitic Parameter, GML 0.8 a) 0.8 b) Estimated φ Estimated φ Estimated β Stochastic Parameter, Augmented c) Estimated β Stochastic Parameter, GMLE d) Assimilation Time Assimilation Time

10 Why Augmentation Fails for Stochastic Parameters x t = φx t 1 + β t w t, Distribution of x t for fixed β t and fixed x t 1 is x t β t, x t 1 N(φx t 1, β 2 t σ 2 w ). Ensemble mean of x t is independent of β t. Variations in β t affect the ensemble spread, not the mean. It can be shown that cov[x t, β t ] = 0 if cov[x 0, β 0 ] = 0. Vanishing covariance implies x t and β t are independent (under normal distribution). Independence implies KF cannot estimate β t from x t

11 Bayes Theorem p(βx oθ) p(o xβθ) p(x βθ) p(β Θ) Posterior likelihood forecast prior where o: Observation at time t Θ: All observations up to time t 1. x: State variable β: Variance Parameter in stochastic-dynamical model.

12 Log of the Posterior 2 log p(βx oθ) = (o Hx) T R 1 (o Hx) + log R + M o log 2π+ Likelihood (x µ f T P f 1 (x µ f + log P f + M x log 2π+ Forecast ( ) T ( ) β µβ Σ 1 β β µβ + log Σβ + M β log 2π Prior

13 Adjoint Parameter Estimation Adjoint parameter estimation in data assimilation (Navon 1997) is based on minimizing the functional J = (o Hx) T R 1 (o Hx) + (x µ f ) T P f 1 (x µ f ) + ( β µ β ) T Σ 1 β ( β µβ ), This is the non-constant part of the posterior provided the forecast covariance P f is fixed. But in stochastic parameter estimation, P f is not constant!

14 Stochastic Parameter Estimation Requires Varying P f Setting the derivative of the posterior to zero and solving gives the (generalized) maximum likelihood estimate. posterior x = 0 Standard Kalman Filter update (for fixed β) posterior β = 0 New nonlinear equation to solve for β The key difference from past studies (e.g., adjoint methods) is that I do not assume that P f / β vanishes.

15 Estimating Derivatives of Covariance Matrices Generate ensemble with fixed β + β, another with fixed β β: P f β = Pf (β + β) P f (β β) 2 β

16 Connections GMLE does not distinguish stochastic and deterministic parameters deterministic parameters characterized by µ f / β 0. stochastic parameters characterized by P f / β 0 Augmentation is equivalent to GMLE of deterministic parameters, if Σ β is interpreted as the spread of the parameter ensemble.

17 Stochastic Parameter Estimation x t = φx t 1 + βw t, Determinitic Parameter, Augmented Determinitic Parameter, GMLE 0.8 a) 0.8 b) Estimated φ Estimated φ Estimated β Stochastic Parameter, Augmented c) Estimated β Stochastic Parameter, GMLE d) Assimilation Time Assimilation Time

18 Parameter Estimation in Stochastic Lorenz Model Slightly modified version of Hansen and Penland (2006) model: dx = a(x y)dt dy = (rx xz y)dt + r s x dw dz = (xy bz)dt Determinitic Parameter, Augmented 11 a) Determinitic Parameter, GMLE 11 b) Estimated a 10 9 Estimated a Stochastic Parameter, Augmented Stochastic Parameter, GMLE Estimated r s c) Estimated r s d) Assimilation Time Assimilation Time

19 Two-Stage Ensemble Generation State-parameter estimation can be achieved in two stages. 1. State-only data assimilation produces ensemble X a. 2. Ensemble is corrected to account for parameter estimation: X aa = X a ( I + δ + ww T ) where ( w = X at f 1 Pf P P f 1 (µ β a µ f ) P f 1 µ f k β k ) δ + = even more complicated! This algorithm takes advantage of an existing ensemble filter.

20 Summary Proposed deriving state and parameter estimates for stochastic dynamical models from generalized maximum likelihood theory. Stochastic parameter estimation requires accounting for dependence of forecast covariance on the parameter. Solution obtained in two-stages: first a standard Kalman Filter, followed by correction to take into account parameter update. Proposed solution outperforms augmentation methods for estimating stochastic parameters. We show that augmentation method is useless for stochastic parameter estimation (contrary to statements in the literature). Method requires generating new ensembles for each stochastic parameter being estimated. More innovative methods?