Comparison of adjoint and analytical approaches for solving atmospheric chemistry inverse problems Monika Kopacz 1, Daniel J. Jacob 1, Daven Henze 2, Colette Heald 3, David G. Streets 4, Qiang Zhang 5 October 11, Harvard, 2. CalTech, 3. UC Berkeley, 4. Argonne NL, 5. Tsinghua University, China

Problems in atmospheric chemistry CO, CO 2, NO, NO 2, CH 4, Hg etc. Emissions (pollution and natural) Chemistry (CO  CO 2, CH 4  CO, NO x + CO  O 3 …) Transport: using assimilated meteorology (from GEOS) (CO 2 ) uptake ocean CO 2, Hg Atmosphere as simulated by a Chemical Transport Model (CTM) Solves continuity equation

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

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

Satellite observations revolutionize tropospheric chemistry Number of constraints (x) limited by the inverse methodology 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??? Measurement of Pollution In the Troposphere (MOPITT)

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

Analytic vs. adjoint solution How the analytical approach becomes infeasible… Not computing Jacobian matrix explicitly  fortran code used to represent it Using reverse mode  efficient 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… Assumptions we can’t avoid… Gaussian errors Linearization of nonlinear processes  using gradient descent ( )

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 concentrationaverage satellite (MOPITT) concentration a priori emission inventory

Inversion comparison setup Heald et al, 2004this work Inversion approach Optimized vector size MOPITT observations forward CTM a priori cost function a posteriori cost function % decrease analytical 11 Feb 21-April 10, 2001 ~21,000 obs GEOS-Chem v4.3 ~35,000 ~28,000 ~23% adjoint 3006 (144x91) Feb 21-April 10, 2001 ~21,000 obs GEOS-Chem v6.5 ~29,000 ~22,000 ~23% 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.

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) All error covariance: set to zero A priori source error variance: from emissions inventory for each country and source type Includes model error, representation error and instrument error

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

Comparison: coarse (adjoint) vs. averaged detail (adjoint) source estimates analytical adjoint 1. C. China (ChCE) SE Asia (SEAs) Philippines (Ph) Indonesia (Id) India (In) rest of world (RoW) oxidation source 1.11 analytical adjoint 1. W. China (ChW) S. China (ChSE) N. China (ChNE) Japan (Jp) Korea (Ko) Europe (EU) regions (state vector elements) from Heald et al, 2004 Potentially affected by aggregation error

Analytical biasAdjoint bias a priori a posteriori

Limitations and challenges of using the adjoint for inverse modeling Do we have enough observations??? 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? fwd + adj run for simple CO chemistry (69 days): 4h L-BFGS (Liu and Nocedal, 1989)

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Adjoint model development Based on GEOS-Chem forward model (GEOS-3, v ) 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