Inverse Methods - Examples

KALMAN FILTER: A REVIEW 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. REVIEW OF THE CH4 INVERSE PROBLEM Emissions from seven seasonally varying (3 wetland, 3 burning, rice) & two steady sources (animals & waste, coal & gas) to be optimally estimated as amendments to the (a priori) reference Rest of the images removed due to copyright considerations. See Figure 2. Chen, Y.-H. and R.G. Prinn, Estimation of atmospheric methane emissions between 1996-2001 using a 3D global chemical transport model, Journal of Geophysical Research, 111, D10307, doi:10.1029/2005JD006058, 2006. 5-YEAR AVERAGE SEASONAL CYCLES AND ERRORS (SELECTED AND ALL DATA SETS) Image removed due to copyright considerations. See Figure 7. Chen, Y.-H. and R.G. Prinn, Estimation of atmospheric methane emissions between 1996-2001 using a 3D global chemical transport model, Journal of Geophysical Research, 111, D10307, doi:10.1029/2005JD006058, 2006. CAPTURES EXPECTED SEASONAL CYCLES (RICE PEAKS EARLIER) SIGNIFICANT YEAR-TO-YEAR EMISSION VARIATIONS MONTHLY ANOMALIES (from 5-year mean annual cycles) using data from: (a) high frequency in situ sites and flask sites (blue lines), and (b) high frequency sites only (red lines) Images removed due to copyright considerations. See the 8 sub-figures on the left in Figure 9. Chen, Y.-H. and R.G. Prinn, Estimation of atmospheric methane emissions between 19962001 using a 3D global chemical ____________________ transport model, Journal of *Note different Geophysical Research, 111, D10307, vertical scale for each doi:10.1029/2005JD006058, 2006. process ANIMALS & WASTE Estimation of the steady fluxes and errors ENERGY (COAL,GAS,etc.) Samoa observations versus model before inversion Samoa observations versus model after inversion Model with optimized emissions simulates observations at almost all sites but Samoa is odd in 1997-1999 WAS THERE A (tropics-weighted) OH DECREASE IN 1997-1999?? Image removed due to copyright considerations. See Figure 12. Chen, Y.-H. and R.G. Prinn, Estimation of atmospheric methane emissions between 1996-2001 using a 3D global chemical transport model, Journal of Geophysical Research, 111, D10307, doi:10.1029/2005JD006058, 2006. Global weighted average OH inferred from AGAGE CH3CCl3 Image courtesy of AGAGE. The inferred circa-1998 OH minimum coincides with massive global wildfires and a strong El Nino. Summary: Interannual variability (Monthly Anomalies) Images removed due to copyright considerations. See the 8 sub-figures on the right in Figure 9. Chen, Y.-H. and R.G. Prinn, Estimation of atmospheric methane emissions between 1996-2001 using a 3D global chemical transport model, Journal of Geophysical Research, 111, D10307, doi:10.1029/2005JD006058, 2006. 32-33 Tg yr-1 Total Emission increase in 1998 with 8-17 Tg yr-1 due to Rice regions ! Northern/Tropical Wetland and Rice Region Emissions dominate the total variability! 1998 wetland Flux Anomalies Fluxes in Tg yr-1 Inversion Bottom-up* Northern Wetlands Tropical Wetlands 5-10 8.3-9.9 12 13 *wetland model driven by 1998 record temperature and large precipitation anomalies (Dlugokencky et al. (2001)) BUT Boreal Fires in Siberia may have also contributed to our deduced strong Northern wetlands increase! Summary: 5-year averages Image removed due to copyright considerations. See Figure 8. Chen, Y.-H. and R.G. Prinn, Estimation of atmospheric methane emissions between 1996-2001 using a 3D global chemical transport model, Journal of Geophysical Research, 111, D10307, doi:10.1029/2005JD006058, 2006. COMPARED TO PREVIOUS ESTIMATES: (1) ENERGY RELATED EMISSIONS SMALLER (RUSSIAN GAS LEAKS?) (2) RICE RELATED EMISSIONS LARGER (PROXIMAL WETLANDS OR TROPICAL ECOSYSTEMS?)