Monika Kopacz 1 , Daniel J. Jacob 1 , Daven Henze 2 ,

1. Comparison of adjoint and analytical approaches for solving atmospheric chemistry inverse problems Monika Kopacz1, Daniel J. Jacob1, Daven Henze2, Colette Heald3, David G. Streets4, Qiang Zhang5 October 11, 2006 1. Harvard, 2. CalTech, 3. UC Berkeley, 4. Argonne NL, 5. Tsinghua University, China

2. Problems in atmospheric chemistry Atmosphere as simulated by a Chemical Transport Model (CTM) Solves continuity equation Chemistry (CO CO2, CH4 CO, NOx+ CO O3 …) CO, CO2, NO, NO2, CH4, Hg etc. Emissions (pollution and natural) Transport: using assimilated meteorology (from GEOS) (CO2) uptake CO2, Hg ocean

3. Problems in atmospheric chemistry (estimating emissions) 2 approaches to emissions estimation Bottom up emissions estimates Top-down emissions estimates Creating detailed emissions inventories at a model resolution Inverse modeling atmospheric observations need for accurate emission estimates for regulatory purposes

4. Current inverse modeling standard in atmospheric chemistry Emissions = P(x) Observations = P(y) Forward Model (GEOS-Chem) 2°x2.5° resolution P(y|x) Bayes statistics  inverse model: “bottom up” emission inventories Approach: least squares min . • Assume Gaussian distributions • sparse observations Size of x ~ O(10) emission regions analytical solution = … Number of constraints (x) limited by the number of observations

5. Satellite observations revolutionize tropospheric chemistry Measurement of Pollution In the Troposphere (MOPITT) • Advantages of satellite instruments (recent tropospheric measurements): • offer dense, daily global coverage (from 1 to 16 days orbit repeat) • observations over/close to the sources not background like surface station in remote atmosphere and oceans • can constrain more emissions regions??? Number of constraints (x) limited by the inverse methodology

6. set explicitly compute Jacobian matrix: Solve explicitly for Solving an inverse problem Objective: minimize compute Analytical method Adjoint method compute using adjoint model use optimization algorithm to iteratively find MAP solution

7. Analytic vs. adjoint solution How the analytical approach becomes infeasible… Increasing the size of the optimized vector O(10)  O(10^5) Constructing full Jacobian matrix K Inverting large matrices ( ) How an adjoint addresses problems of the analytical approach… Not computing Jacobian matrix explicitly  fortran code used to represent it Using reverse mode  efficient ( ) Assumptions we can’t avoid… Gaussian errors Linearization of nonlinear processes  using gradient descent

8. Method comparison project Comparison objective: Perform a (adjoint)inversion similar to a previous (analytical) inversion using the same observations, emissions inventory, time frame, error characterization and forward model (but not resolution!) Inversion objective: Constrain Asian CO emissions during the Spring 2001 average model CO concentration average satellite (MOPITT) concentration a priori emission inventory ≠

9. Inversion comparison setup Heald et al, 2004: Comparative inverse analysis of satellite (MOPITT) and aircraft (TRACEP) observations to estimate Asian sources of carbon monoxide, J. of Geophys. Res.

10. Constructing error covariance Observational error Obs. error variance: Relative Residual Error (RRE) method, ie. Computing deviation from an ensemble mean error (model bias due to error in sources) A priori source error variance: from emissions inventory for each country and source type % All error covariance: set to zero Includes model error, representation error and instrument error

11. CO constraints using adjoint inversion A posteriori emissions scaling factors Blue: a priori overestimate East India, Southeast Asia, Philippines Red: a priori underestimate China, Northern India

12. Comparison: coarse (adjoint) vs. averaged detail (adjoint) source estimates 11 regions (state vector elements) from Heald et al, 2004     analytical adjoint 1. C. China (ChCE) 1.83 1.34 2. SE Asia (SEAs) 0.63 0.67 3. Philippines (Ph) 0.89 0.73 4. Indonesia (Id) 0.96 0.90 5. India (In) 0.50 0.68 6. rest of world (RoW) 1.16 0.91 oxidation source 1.11 analytical adjoint 1. W. China (ChW) 2.38 1.16 2. S. China (ChSE) 0.31 1.18 3. N. China (ChNE) 0.76 1.02 4. Japan (Jp) 1.88 0.99 Korea (Ko) 1.02 5. Europe (EU) 0.75 1.00 Potentially affected by aggregation error

13. Analytical bias Adjoint bias a priori a posteriori

14. Limitations and challenges of using the adjoint for inverse modeling Do we have enough observations??? fwd + adj run for simple CO chemistry (69 days): 4h How computationally efficient is it? If we use adjoint approach for inversion, what is the best optimization algorithm, considering the requirements of: - quick convergence - accuracy - non-negative solution, ie. will not yield negative emissions! - no bias? L-BFGS (Liu and Nocedal, 1989)

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17. Adjoint model development Based on GEOS-Chem forward model (GEOS-3, v6-05-07) • advection • deep convection • turbulent mixing • CO chemistry • CO sources • integration with MOPITT observations Daven Henze at Caltech Harvard Note: The adjoint model also contains aerosol thermodynamics (full chemistry adjoint), wet and dry deposition and aerosol emissions components all developed by Daven Henze