Chemical Source Inversion Using Assimilated Constituent Observations

1. Chemical Source Inversion Using Assimilated Constituent Observations Andrew Tangborn Global Modeling and Assimilation Office

2. How are data assimilation and chemical source inversion related? 1. Underdetermined systems – fewer constraints than unknowns. 2. Use Bayesian methods – require error statistics. 3. If errors are normally distributed and unbiased – optimal scheme results in weighted least squares estimate.

3. How are they different? 1. Different goals: state estimate vs. source estimate. 2. Use different errors: source errors vs. model and State errors.

4. Example: Kalman Filtering (KF) and Green’s function (GF) Inversion Inputs chemical tracer observations winds chemical source/sinks Initial state Error estimates

5. Algorithms KF: Kalman gain observation operator ca = cf + K(co – Hcf) observations analysis forecast

6. KF: Where K = PfHT (R+HPfHT)-1 is a weighted by obs error (R) and forecast error (HPfHT) covariances. The forecast error covariance Pf = MPaMT + Q is evolved in time from analysis error using the discretized model (winds) M and added model error Q

7. GF: Green’s function Inverse of source error cov. xinv = (GTXG+W)-1 (GTXco+Wz) Inverse of R obs first guess source inverted source G: calculated by running model forward using unit sources at each grid point.

8. Differences between KF and GF KF: Initial condition (analysis) used in forecast contains earlier observation information. GF: Initial condition does not explicitly contain observation data. KF: Uses model (wind) and observation errors. State errors are propagated in time. GF: Uses source and observation errors. State errors are never calculated.

9. Combing KF and GF • Carry out KF assimilation of tracer observations. • Use both the analysis and analysis error covariance as the observations and observation error in the GF co = ca X = (Pa)-1 • Now X contains information on model errors (including wind errors) and co is spread to all grid points through assimilation.

10. Numerical Experimentsadvection diffusion in 2p x 2p domainwith constant and random source errors Model source True Source

11. Tracer Fields True field Model solution Analysis field

12. Source InversionError Standard Deviation No Assimilation With assimilation

13. Conclusions • Data Assimilation can add information to source inversion. • Improvements likely come through improved and more complete error covariance information and spreading observation information to more grid points.