Lagrangian Data Assimilation and Its Application to Geophysical Fluid Flows
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1 Lagrangian Data Assimilation and Its Application to Geophysical Fluid Flows Laura Slivinski June, 3 Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 / 3
2 Data Assimilation Setup: Given dynamical system ẋ = f (x) (deterministic or stochastic) Uncertainty in initial conditions x() Would like to estimate state at specific time: x(t k ) Observations y = h(x) + noise Application of Bayes Rule: p(x y) p(x)p(y x).5 prior likelihood posterior observation Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 / 3
3 Lagrangian Data Assimilation Suppose we want to estimate the Eulerian flow field x F, but the observations are of Lagrangian positions of passive drifters x D. z = h(x, y, t) ẋ D,i = x F (x D,i, t).5 z.5 y 6 y 4 x 4 6 x Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 3 / 3
4 Lagrangian Data Assimilation One approach to Lagrangian data assimilation: Append drifter position x D to flow state vector x F : x = Observation operator has simple, linear form: H = [ I] ( xf Sequential filters: Forecast (evolve previous estimate forward under dynamical system) Analysis (update current estimate with observation) x D ) Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 4 / 3
5 Traditional filters: Ensemble Kalman Filter (EnKF) Represent probability distribution with an ensemble of state vectors Evolve each ensemble member forward under model until next observation time When an observation is available, update each ensemble member according to the traditional Kalman analysis step Drawback: tends to impose Gaussianity at each assimilation step Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 5 / 3
6 Traditional filters: Particle Filter (PF) Represent probability distribution with weighted ensemble of state vectors, called particles When observation is available, update each particle s weight according to Bayes Rule Need to resample to avoid weight converging on one particle Drawback: necessary number of particles increases exponentially with state dimension (curse of dimensionality) Figure: weight histograms: dim=, N=5 Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 6 / 3
7 EnKF vs PF: Non-Gaussian Prior Lagrangian data assimilation leads to non-gaussian priors Flow may solve linear system, but drifters solve nonlinear system: ẋ F = f (x F ) [linear or nonlinear] ẋ D = g(x F, x D ) [always nonlinear] r =.6667 f r = f obs likelihood prior EnKF posterior PF posterior Bayes posterior N = Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 7 / 3
8 Hybrid PF-EnKF EnKF on high-dimensional Eulerian state x F PF on low-dimensional, highly nonlinear Lagrangian part x D Ensemble: {x F i, x D i,j, w i,j } i=...ne, j=...m Update weights via standard particle filter update, and at resampling times, update x F according to EnKF analysis. Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 8 / 3
9 Linear Shallow Water Equations System: u = v h x v = u h y ḣ = u x v y Solution with two modes: u(x, y, t) = sin(x) cos(y)u + cos(y)u (t) v(x, y, t) = cos(x) sin(y)u + cos(y)v (t) h(x, y, t) = sin(x) sin(y)u + sin(y)h (t) Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 9 / 3
10 Results u(x, y, t) = sin(x) cos(y)u + cos(y)u (t) v(x, y, t) = cos(x) sin(y)u + cos(y)v (t) h(x, y, t) = sin(x) sin(y)u + sin(y)h (t) PF: N = 4 EnKF: N = 4 Hybrid: N e = 4, M = PF EnKF hybrid u.8.6 u.8.6 v h x.7 y Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 / 3
11 Results Figure: Ensemble Kalman filter Laura Slivinski (Brown University) Lagrangian Data Assimilation Figure: Hybrid filter June, 3 / 3
12 Summary and Future Work Hybrid filter combines advantages of PF and EnKF while avoiding disadvantages of each in Lagrangian DA case More computationally intensive than EnKF, but more accurate when drifters encounter saddle point Future: high dimensional nonlinear shallow water equations, drifter deployment experiments (Salman, Ide, Jones) Figure: Salman et al., 8 Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 / 3
13 References A. Apte, C.K.R.T. Jones, A.M. Stuart (8) A Bayesian approach to Lagrangian data assimilation Tellus A 6, G. Evenson (3) The Ensemble Kalman Filter: theoretical formulation and practical implementation Ocean Dynamics 53, H. Salman (8) A hybrid grid/particle filter for Lagrangian data assimilation (I & II) Q. J. R. Meteorol. Soc. 34, H. Salman, K. Ide, C.K.R.T. Jones (8) Using flow geometry for drifter deployment in Lagrangian data assimilation Tellus A 6(), P.J. van Leeuwen (9) Particle Filtering in Geophysical Systems Monthly Weather Review 37, Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 3 / 3
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