Conditions for successful data assimilation

Transcription

1 Conditions for successful data assimilation Matthias Morzfeld *,**, Alexandre J. Chorin *,**, Peter Bickel # * Department of Mathematics University of California, Berkeley ** Lawrence Berkeley National Laboratory # Department of Statistics University of California, Berkeley Probabilistic Approaches to Data Assimilation for Earth Systems, Banff, Canada Tuesday, 8th of February 3

2 Introduction Goal Use a mathematical model to make predictions about a physical process Problem Small errors grow quickly and become large errors 5 Solution Use noisy data to update the model

3 Solution of the data assimilation problem Goal Compute the random variable x z p(x z) Conditional mean Z E(x z) = xp(x z)dx is the minimum mean square error estimate Methods Kalman filter and its variants for linear Gaussian models Variational data assimilation computes the most likely state given the data Particle filters (Monte Carlo methods) construct empirical estimate of the conditional pdf Question: What are the conditions under which DA can be successful? 3

4 Assumptions Question: What are the conditions under which DA can be successful? To answer the question we analyze the conditional pdf (which depends on errors in model and data) Assumptions: Linear Gaussian synchronous model x n+ = Ax n + w n, w n N (,Q), iid z n+ = Hx n+ + v n+, v n N (,R), iid, independent of w n Initial state Gaussian x N (µ, ) Kalman formalism gives us the conditional pdf 4

5 Kalman filter Model and data: x n+ = Ax n + w n, w n N (,Q), iid z n+ = Hx n+ + v n+, v n N (,R), iid, independent of w n Kalman update: X n =AP n A T + Q K n =X n H T (HX n H T + R) P n+ =(I K n H)X n In steady state : P n+ = P n = P =(I KH)X X = AXA T AXH T (HXH T + R) HXA T + Q (Discrete Algebraic Ricatti Equation) 5

6 Conditional pdf in steady state In steady state : x n N (µ n,p) j are eigenvalues of P µ n x n µ n Distance from mean (most likely state): x n r = q (x n µ n ) T (x n µ n ) E(r) mx j= j A mx j= j mx j= j.5 var(r) A mx j= mx j= j j 6

7 Conditional pdf in steady state = O()! E(r) =O(m / ), var(r) =O() Samples of posterior pdf collect on a thin shell O() This is problematic There is not enough information in model and data to make reliable conclusions about the state This is unphysical Similar experiments are expected to have similar outcomes O(m / ) 7

8 Conditional pdf in steady state Large posterior covariance: Sample collect on a thin shell There is not enough information in model and data to make reliable conclusions about the state Small posterior covariance: Samples collect on a low dimensional ball There is enough information in model and data to make reliable conclusions about the state O() O(m / ) 8

9 The effective dimension O() O(m / ) vs Effective dimension: Definition: Frobenius norm of steady state covariance matrix P Interpretation: effective dimension must be well bounded or else data assimilation is hopeless (regardless of the algorithm) P F = q X 9

10 r Boundedness of the effective dimension: example Put: A = H = I, Q = qi, R = ri Steady state covariance:.8.4 P = p q +4qr q I. Effective dimension:.6 P F = p m p q +4qr q. m = m = m = q.8 In general: small Frobenius norm of Q and R lead to small effective dimension

11 Why is this realistic? Small effective dimension Khinshin s theorem: bounded energy implies small Frobenius norms of Q and R Smoothness of errors implies small Frobenius norm of Q and R Errors with spherical symmetries (error in one component leads to errors in all other components) lead to small Frobenius norms of Q and R Energy Modes / p m

12 Why is this realistic? Energy / p m Modes In general: correlations lead to small Frobenius norm of Q and R and therefore to a small effective dimension Probability mass is concentrated on a lower dimensional manifold due to correlations in the errors Error models in literature typically show strong correlations

13 Effective dimension: summary r O() O(m / ) vs Energy Modes Effective dimension must be bounded or else data assimilation is hopeless (independent of the DA algorithm).8.4 Bounded effective dimension induces balance condition between errors in model and data..6 In practice, the effective dimension is often small (correlations in errors). m = m = m = q.8 3

14 Agenda. Introduction. What can be expected in general? 3. How good are particle filters? 4. How good is 4D-Var or particle smoothing?

15 Review of importance sampling Direct Monte Carlo sampling Suppose we are interested in x ~ p(x) and want to compute the expected value of x. The Monte Carlo approximation is: E(x) = Z xp(x)dx N Importance sampling NX i= Suppose we can evaluate p(x) and want to compute the expected value of x. E(x) = Z xp(x)dx = x i w i / p(x i) (x i ), X wi = x i (x) Z, x i p(x) x p(x) N (x) (x)dx X x i w i i= Problem: it is not easy to obtain samples directly Replace direct samples with weighted samples 5

16 Review of particle filters Particle filters Apply importance sampling recursively to the conditional pdf p(x :n+ z :n+ )=p(x :n z :n ) p(xn+ x n )p(z n+ x n+ ) p(z n+ z :n ) This requires importance function that factorizes (x :n+ z :n+ )= (x ) Recursion for weights W n+ j / Ŵ n j n+ Y k= k (x k x :k,z :k ) p(x n+ j Xj n)p(zn+ X n+ j ) n+ (X n+ j Xj :n,z :k ) If variance of unnormalized weights is large, then the particle filter collapses (see papers by Snyder, Bickel, Anderson,...) 6

17 Review of the collapse of particle filters The SIR particle filter Importance function: n+ = p(x n+ x n ) Weights: Condition for collapse in high dimensions is large norm of * : W n+ j / p(z n+ X n+ j ). = H(Q + AP A T )H T R Collapse can also happen in low dimensions: * as shown by Snyder, Bickel, Anderson et al. 7

18 Review of the collapse of particle filters The optimal particle filter Importance function: n+ = p(x n+ x n,z n+ ) Weights: W n+ j / p(z n+ X n j ) Condition for collapse in high dimensions is large norm of * : = HAP A T H T HQH T + R Collapse is avoided in low dimensions: * as shown by Snyder 8

19 How good can we get with particle filters? Example revisited: Optimal filter: A = H = I, Q = qi, R = ri F = p p q +4qr m (q + r) q 3 Data assimilation possible Data assimilation not possible 3 Optimal particle filter works.5.5. r m = m = q q 9

20 How good can we get with particle filters? Example revisited: Optimal filter vs. SIR filter 3 Data assimilation possible Data assimilation not possible 3 Optimal particle filter works 3 SIR filter works r m = m = m = q q q.

21 How good can we get with particle filters? The general linear case: Using matrix bounds we find the balance conditions: A F H F P F apple H F Q F + R F Optimal filter H F Q F + A F P apple R F SIR filter Balance condition is easy in simple cases, but delicate in general The nonlinear/non-gaussian case: Correlations can be expected for realistic noise models Balance conditions must be worked out in each particular case Optimal filter hard/impossible to implement, while SIR remains easy to use

22 Agenda. Introduction. What can be expected in general? 3. How good are particle filters? 4. How good is 4D-Var or particle smoothing?

23 Strong constraint 4D-Var Perfect model assumption: Conditional pdf: p(x z :n ) / exp x T µ x µ nx z j HA j x T R z j HA j x Covariance: x n+ = Ax n z n+ = Hx n+ + v n+, = + j= nx (A j ) T H T R HA j j= Strong constraint 4D-Var can only be successful if the Frobenius norm of the covariance is small v n N (,R), iid, independent of w A 3

24 r How good can we get with 4D-Var? Example revisited: Norm of covariance matrix: F = p m + r r m = m = m =.5..5 σ 4

25 Weak constraint 4D-VAr Model and data: x n+ = Ax n + w n, w n N (,Q), iid z n+ = Hx n+ + v n+, v n N (,R), iid, independent of w n Conditional pdf: p(x :n z :n ) / exp x T µ x µ nx z j Hx j T R z j Hx j Covariance: = j= + A T A AT Q Q A Q + A T Q A + H T R H A T Q A T Q Q A Q + H T R H A C A. 5

26 r How good can we get with 4D-Var? Strong constraint 4D-Var Boundedness of covariance matrix induces balance condition between errors in prior and in the data.6. Weak constraint 4D-Var Boundedness of covariance matrix induces balance condition between errors in prior, in the model and in the data Particle smoothing Same balance condition as in 4D- Var, but choice of importance function is critical.8.4 m = m = m =.5..5 σ 6

27 Conclusions Numerical data assimilation hopeless unless effective dimension is small (probability mass is concentrated on a low dimensional manifold) Boundedness of effective dimension induces balance condition between errors in model and data In practice, effective dimension often small because correlations in errors Particle filters can work in high dimensions, provided their implementation is sound Variational data assimilation requires well boundedness of covariance matrix, i.e. balance condition between errors in prior, model and data Analysis for linear Gaussian case only. Nonlinear/non-Gaussian problems must be analyzed in each particular case. 7

28 Fin Thank you! 8

29 m How good can we get with particle filters? Example revisited: Optimal filter vs. SIR filter Optimal filter works SIR and optimal filter work 4 None of the two filters work q / r 4 9