Inverse Methods - Optimal Estimation 2

We now use the criterion of optimality to determine Κ k . Since we will assume we know p(yo ) and p(xt ) , we will choose a value for Κ k which minimizes the cost function J (equation 12) for the minimum variance Bayes estimate. Specifically Jk = Ε[(νka )T νka ] = trace [Ρka ] (19) Evaluating ∂Jk / ∂Κ k = 0 and solving for the so-called “Kalman Gain” matrix Κ k we have Κ k = Ρkf ΗkT [Ηk Ρkf ΗkT + Rk ]−1 (20) Substituting (20) into (18) then yields Ρka = [Ι − Κ k Ηk ]Ρkf (21) Finally, using the state space equation (7) x(t) = M(t, t o )x(t o ) + η(t, t o ) we then obtain the estimates of xkf needed in (15) and Ρkf needed in (21) xkf = Mk −1xka−1 (22) Ρkf = Μk −1Ρka−1ΜkT−1 + Qk −1 (23) where Qk −1 = Ε[ηk −1ηkT−1 ] , and xka −1 and Ρka −1 are the optimal outputs from the previous iteration of the filter. From our earlier discussion (Section 3), Q could represent random forcing in the system model due to transport model errors. To use the filter we must provide initial (a priori) estimates for x and P. Then from any prior output estimates (xka −1 , Ρka −1 ) , we use measurement k information ( yko ,Rk ) and model information ( Hk , Qk ) together with equations (22), (23), (20), (15), and (21) to provide outputs xka and Ρka for inputs to the next step. The filter equations are summarized in Table 1. Some intuitive concepts regarding the DKF are useful in understanding its operation. First, from equation (20), the gain matrix Κ k → Ηk−1 (its “maximum” value) as the measurement error covariance (noise) matrix R k → 0, and Κ k → Ρkf ΗkTRk−1 (its “minimum” value) as R k → ∞ . Since the update in the state vector xka − xkf varies linearly with Κ k , it is clear that measurements noisy enough so that R k much exceeds Ηk Ρkf ΗkT , will contribute much less to improvement of the state vector estimation. In this respect we can usefully consider Ηk Ρkf ΗkT as the error covariance matrix for the measurement estimates yk . This emphasizes the importance of the weighting of the data inherent in R k and the distortions created if erroneous R k are used. Note that R k can include model error, mismatch error, and instrumental error as noted earlier. Second, using (21), and recognizing that the maximum value of Κ kHk = Ι , we a k see Ρ ≤ Ρkf with equality occurring for infinitely noisy measurements. Hence, the error covariance matrix Ρk (whose diagonal elements are the variances of the state vector element estimates) decreases by amounts sensitively dependent on the measurement errors. Third, we note from (23), that random forcings η in the system (state-space) model [equation (7)], which are represented here by Q, will increase the extrapolated error covariance matrix Ρkf by amounts depending on the relative values of Qk −1 and the extrapolation matrix Μk −1Ρka−1ΜkT−1 in the absence of system (statespace) model noise. The inclusion of Q lessens the influence (or memory) of previous iterations in the filter operation. In the extreme, sufficiently large values of Q will prevent the capability of even non-noisy measurements to decrease Ρk and hence increase the confidence in the state vector estimate. In other words excellent (nonnoisy) measurements are of little use if the system (state-space) model is very noisy (e.g., through random variations η introduced by random transport errors). Table 1: Kalman Filter Equations* Definition Equation Measurement equation (model) yko = Hk xkt + εk ; System (state) equation (model) xk = Mk −1xk −1 + ηk −1 State update xka − xkf = Κ k (yko − yk ) Error Update Ρka = (1 − Κ k Ηk )Ρkf Kalman gain update Κ k = Ρkf ΗkT (Ηk Ρkf ΗkT + Rk )−1 State time extrapolation xkf = Μk −1xka −1 Error time extrapolation Ρkf = Μk −1Ρka −1ΜkT−1 + Qk −1 System random forcing covariance Qk = Ε(ηk ηkT ) Measurement error covariance Rk = Ε(εk εk T ) Estimation error covariance Ρk = Ε(νk νk T ) Input measurement matrix = Ηk = ∂yk / ∂xk Input system random forcing covariance = Qk Input state extrapolation = Μk Input measurement yko Input measurement error covariance = Rk Filter iteration − − − → (k − 1)f , → estimate yk = Hk xkf → (k − 1)a , → extrapolate → (k)f , → − − − ___________________________________________________________________ *A superscript a or superscript f denotes respectively the value before (f) or after (a) an update of an estimate using measurements, and k denotes the measurement number. In general, errors are assumed random with zero mean and measurement and estimation errors are uncorrelated. Kalman Filter Notes on slide notation: (+) = a, (-) = f, z = y, (^) = model Image removed due to copyright considerations. See: Plate 1. Prinn, R.G., Measurement equation for trace chemicals in fluids and solution of its inverse, in Inverse Methods in Global Biogeochemical Cycles, ed. P. Kasibhatla et al., Geophysical Monographs, 114, American Geophysical Union, pgs. 3-18, 2000. 11 0 200 P(mb) 500 1000 Advection Chemistry Eddy 90S 30S Diffusion 0 τj = chemical lifetime = (kj[OH])-1 30N Emissions, Ej 90N Ocean sink (effective lifetime of 85 years) CH3CCl3 (g) CH3CCl3 (aq) hydrolysis products Xj = state vector Ej [CH3CCl3]j - ocean sink = τj [M]j Image removed due to copyright considerations. See: Figure 3. Prinn, R.G., Measurement equation for trace chemicals in fluids and solution of its inverse, in Inverse Methods in Global Biogeochemical Cycles, ed. P. Kasibhatla et al., Geophysical Monographs, 114, American Geophysical Union, pgs. 3-18, 2000. Image removed due to copyright considerations. See: Figure 4. Prinn, R.G., Measurement equation for trace chemicals in fluids and solution of its inverse, in Inverse Methods in Global Biogeochemical Cycles, ed. P. Kasibhatla et al., Geophysical Monographs, 114, American Geophysical Union, pgs. 3-18, 2000. Smoother Combines The Forward And Backwards Filters Image removed due to copyright considerations. See: Figure 5. Prinn, R.G., Measurement equation for trace chemicals in fluids and solution of its inverse, in Inverse Methods in Global Biogeochemical Cycles, ed. P. Kasibhatla et al., Geophysical Monographs, 114, American Geophysical Union, pgs. 3-18, 2000.