Recent Developments in Computational Tools for Chemical Data Assimilation

1. Recent Developments in Computational Tools for Chemical Data Assimilation A. Sandu: Virginia Tech G.R. Carmichael: University of Iowa J.H. Seinfeld: Caltech D. Daescu: Portland State University NASA, NOAA, EPA for Trace-P and ICARTT data NSF CAREER ACI-0093139 NSF ITR AP&IM-0205198

2. Prediction is difficult … “There is no reason anyone would want a computer in their home.” Ken Olson, President, Chairman, and Founder of Digital Equipment Corp. (DEC), 1977. Chemical D.A. and Data Needs for Air Quality Forecasting

3. Information feedback loops between CTMs and observations: data assimilation and targeted meas. Transport Meteorology Optimal analysis state Chemical kinetics Observations CTM Data Assimilation Aerosols Targeted Observ. • Improved: • forecasts • science • field experiment design • models • emission estimates Emissions Chemical D.A. and Data Needs for Air Quality Forecasting

4. Continuous forward model Continuous adjoint model adj discr discr Computational adjoint model Discrete forward model adj In the 4D-Var approach D.A. is formulated as an optimization problem (gradient based algorithms) Chemical D.A. and Data Needs for Air Quality Forecasting

5. Adjoints of stiff chemical kinetics: formulation, challenges, and automatic implementation Transport Meteorology Optimal analysis state Chemical kinetics Observations CTM Data Assimilation Aerosols Targeted Observ. • Improved: • forecasts • science • field experiment design • models • emission estimates Emissions Chemical D.A. and Data Needs for Air Quality Forecasting

6. K P P KPP automatically generates simulation and direct/adjoint sensitivity code for chemistry Simulation code Chemical mechanism SUBROUTINE FunVar ( V, F, RCT, DV )INCLUDE 'small.h'REAL*8 V(NVAR), F(NFIX) REAL*8 RCT(NREACT), DV(NVAR)C A - rate for each equationREAL*8 A(NREACT)C Computation of equation rates A(1) = RCT(1)*F(2) A(2) = RCT(2)*V(2)*F(2) A(3) = RCT(3)*V(3) A(4) = RCT(4)*V(2)*V(3) A(5) = RCT(5)*V(3) A(6) = RCT(6)*V(1)*F(1) A(7) = RCT(7)*V(1)*V(3) A(8) = RCT(8)*V(3)*V(4) A(9) = RCT(9)*V(2)*V(5) A(10) = RCT(10)*V(5)C Aggregate function DV(1) = A(5)-A(6)-A(7) DV(2) = 2*A(1)-A(2)+A(3)-A(4)+A(6)-&A(9)+A(10) DV(3) = A(2)-A(3)-A(4)-A(5)-A(7)-A(8) DV(4) = -A(8)+A(9)+A(10) DV(5) = A(8)-A(9)-A(10)END #INCLUDE atoms #DEFVAR O = O; O1D = O; O3 = O + O + O; NO = N + O; NO2 = N + O + O; #DEFFIX O2 = O + O; M = ignore; #EQUATIONS{ Small Stratospheric }O2 + hv = 2O : 2.6E-10*S; O + O2 = O3 : 8.0E-17; O3 + hv = O + O2 : 6.1E-04*S; O + O3 = 2O2 : 1.5E-15; O3 + hv = O1D + O2 : 1.0E-03*S; O1D + M = O + M : 7.1E-11; O1D + O3 = 2O2 : 1.2E-10; NO + O3 = NO2 + O2 : 6.0E-15; NO2 + O = NO + O2 : 1.0E-11; NO2 + hv = NO + O : 1.2E-02*S; [Damian et.al., 1996; Sandu et.al., 2002] Chemical D.A. and Data Needs for Air Quality Forecasting

7. Sparse Jacobians , Hessians, and sparse linear algebra routines are automatically generated by KPP #JACOBIAN [ ON | OFF | SPARSE ] JacVar(…), JacVar[TR]_SP_Vec(…) KppDecomp(…), KppSolve[TR](…) #HESSIAN [ ON | OFF ] HessVar(…), HessVar[TR]_Vec(…) SAPRC-99. NZ = 848x2 (0.2%) SAPRC-99 79 spc./211 react. NZ=839, NZLU=920 Chemical D.A. and Data Needs for Air Quality Forecasting

8. Runge-Kutta methods and their adjoints are well suited for inverse chemical kinetic problems RK Method Continuous Adjoint Discrete Adjoint [Hager, 2000] Consistency: The discrete adjoint of RK method of order p is an order p discretization of the adjoint equation. (Proof using elementary differentials of transfer functions). Note: BDF adjoints with variable step are not consistent with continuous adjoint equation. [Sandu, Daescu, 2003] Chemical D.A. and Data Needs for Air Quality Forecasting

9. Singular perturbation analysis is relevant for studying the method behavior on stiff systems Singular perturbation test problem: If RK with invertible coefficient matrix A and R(∞) = 0; and the cost function depends only on the non-stiff variable y Thenλz= 0 and λy are solved with the same accuracy as the original method, within O(ε). A similar conclusion holds for continuous RK adjoints. [Sandu, Daescu, 2003] Distinguish between derivatives w.r.t. stiff/non-stiff variables Chemical D.A. and Data Needs for Air Quality Forecasting

10. Rosenbrock methods and their adjoints are efficiently implemented by KPP Rosenbrock Method (Tfwd) Continuous Adjoint (T≈1.2Tfwd) Discrete Adjoint (T≈2.3Tfwd) [Sandu et.al., 2002] Chemical D.A. and Data Needs for Air Quality Forecasting

11. Second order adjoints provide Hessian-vector products useful in optimization and analysis RK & TLM Methods (KPP) First and Second Order RK Discrete Adjoints (KPP) [Sandu et. al., 2005] Chemical D.A. and Data Needs for Air Quality Forecasting

12. Adjoints for Integral-PDE aerosol dynamic equations: formulation and challenges Transport Meteorology Optimal analysis state Chemical kinetics Observations CTM Data Assimilation Aerosols Targeted Observ. • Improved: • forecasts • science • field experiment design • models • emission estimates Emissions Chemical D.A. and Data Needs for Air Quality Forecasting

13. Growth Coagulation m+m’ m m’ m Populations of aerosols (particles in the atmosphere) are described by their mass density Aerosol dynamic equation - IPDE Chemical D.A. and Data Needs for Air Quality Forecasting

14. Adjoint aerosol dynamic models are needed to solve inverse problems Continuous adjoint equation Observations of density in each bin allow the recovery of initial distribution and of parameters [Sandu et. al., 2005; Henze et. al., 2004] Chemical D.A. and Data Needs for Air Quality Forecasting

15. Discrete adjoint models for numerical advection: formulation and challenges Optimal analysis state Transport Meteorology Chemical kinetics Observations CTM Data Assimilation Aerosols Targeted Observ. • Improved: • forecasts • science • field experiment design • models • emission estimates Emissions Chemical D.A. and Data Needs for Air Quality Forecasting

16. Inconsistency, instability of discrete adjoint advection schemes when the forward scheme pattern changes discretization change upwind direction change Forward flux-limiter Status change Von Neumann stability regions for different upwindings CFL<0.87 unstable CFL<0.81 Chemical D.A. and Data Needs for Air Quality Forecasting

17. Loss-of-information hinders the optimization process Instability and the inconsistency do not seem to affect the optimization for small Courant numbers Loss of information hinders the optimization process: • solution collapses into a sink • exits the domain through an outflow boundary • propagates too slow to reach one of the observation sites [Sandu et. al., 2005] Chemical D.A. and Data Needs for Air Quality Forecasting

18. The 4D-Var tools have been implemented in the parallel adjoint STEM and are being applied to real data Transport Meteorology Optimal analysis state Chemical kinetics Observations CTM 4D-Var Data Assimilation Aerosols Targeted Observ. • Improved: • forecasts • science • field experiment design • models • emission estimates Emissions Chemical D.A. and Data Needs for Air Quality Forecasting

19. High performance computing: parallel adjoint STEM-3 Distributed, 2-level checkpointing scheme • Chemistry: KPP • Forward: sparse Rosenbrock, RK, LMM • DDM sensitivity (Rosenbrock, LMM) • Discrete adjoints: Rosenbrock • Continuous adjoints: Rosenbrock, RK, LMM • Aerosols: 0-D, not yet 3-D • Transport: • Forward: upwind FV, FD, FE • Adjoints for linear upwind FD • Parallelization: with PAQMSG [Sandu et.al., 2003, 2004;Carmichael et. al., 2003, 2004] Chemical D.A. and Data Needs for Air Quality Forecasting

20. Assimilation of P3-B observations taken during the Trace-P campaign on March 07, 2001 P3-B observations: O3 (8%); NO, NO2 (20%); HNO3, PAN, RNO3 (100%) Control: Init. conc. 50 species (sensitivity) Optimization problem with ~ 5 million variables Parallel adjoint STEM, 2-level checkpointing O3 NO2 [Chai et.al., 2004] Chemical D.A. and Data Needs for Air Quality Forecasting

21. Assimilation of AIRNOW O3 surface observations for July 20, 2004 adjusts predictions considerably Observations: circles, color coded by O3 mixing ratio Surface O3 (forecast) Surface O3 (analysis) Chemical D.A. and Data Needs for Air Quality Forecasting

22. Ensemble-based chemical data assimilation can complement variational techniques Transport Meteorology Optimal analysis state Chemical kinetics Observations CTM Ensemble Data Assimilation Aerosols Targeted Observ. • Improved: • forecasts • science • field experiment design • models • emission estimates Emissions Chemical D.A. and Data Needs for Air Quality Forecasting

23. AR model of background errors accounts for flow-dependent correlations and is inexpensive • Background error representation considerably impacts assimilation results • Typically estimated empirically from multiple model runs (NMC) • “Correct” mathematical models of background errors are of great interest McMillian Reservoir, DC • AR model of background errors • N ∆t ≈ lifetime of the species • B: • flow dependent • inexpensive to compute • has full rank Chemical D.A. and Data Needs for Air Quality Forecasting

24. Singular vectors characterize the directions of maximum error growth (under different measures) • Measures: E=C=[I,0]  TESV E=I, C=Pa-1  HSV • Understand areas of maximum sensitivity to uncertainty • Construct the subspace of initial perturbations for EnKF Cause: stiff transient Solution: TLM + ADJ chem. projections [Liao and Sandu, 2005] Chemical D.A. and Data Needs for Air Quality Forecasting

25. TESVs are shaped by both meteorology and chemistry, as seen for different sections O3 NO2 σ2 Trace-P, March 1-2, 2001 #1 #2 Chemical D.A. and Data Needs for Air Quality Forecasting

26. Ensemble-based chemical data assimilation considerably improves the solution NO2 original error O3 original error O3 error w/ analysis NO2 error w/ analysis 24h, March 1-2, 2001 50 members Perturbed I.C., B.C., and emissions AR + TESV O3 + NO2 obs (red) Chemical D.A. and Data Needs for Air Quality Forecasting

27. Results for species not directly observed also show marked improvements after assimilation PAN original error HCHO original error CO original error PAN error w/ analysis HCHO error w/ analysis CO error w/ analysis Chemical D.A. and Data Needs for Air Quality Forecasting

28. The model can be used to place the observations in the locations of maximum informational benefit Transport Meteorology Optimal analysis state Chemical kinetics Observations CTM Data Assimilation Targeted Observations Aerosols • Improved: • forecasts • science • field experiment design • models • emission estimates Emissions Chemical D.A. and Data Needs for Air Quality Forecasting

29. Targeted observations are deployed to areas where they provide maximum of information Verification: Korea, ground O3 0 GMT, Mar/4/2001 (criterion based on SVs) O3 HCHO NO2 [Liao and Sandu, 2005] Chemical D.A. and Data Needs for Air Quality Forecasting

30. Dynamic integration of chemical data and atmospheric models is an important, growing field • Assimilation of chemical observations into CTMs: • improves reanalysis of fields • model forecast skills • provides top-down estimate of emission inventories • Current state of the art: • the tools needed for 4d-Var chemical data assimilation are in place: • adjoints for stiff systems, aerosols, transport; singular vectors • parallelization and multi-level checkpointing schemes • models of background errors • their strengths demonstrated using real (field campaign) data • ambitious science projects are ongoing • Emerging needs and research directions: • hybrid methods (combining ensemble and variational approaches) • second order adjoints and optimization • reduced order models (POD, singular vectors) • algorithms for targeted observations Chemical D.A. and Data Needs for Air Quality Forecasting

31. Quote of the day “Persons pretending to forecast the future shall be considered disorderly under section 901(3) of the criminal code and liable to a fine of $250 and/or 6 months in prison.” Section 889, New York State Code of Criminal Procedure(after M.D. Webster) Chemical D.A. and Data Needs for Air Quality Forecasting