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. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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.

3. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Information feedback loops between CTMs and observations: data assimilation and targeted meas. Optimal analysis state Chemical kinetics Aerosols CTM Transport Meteorology Emissions Observations Data Assimilation Targeted Observ. Improved: forecasts science field experiment design models emission estimates Improved: forecasts science field experiment design models emission estimates

4. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Continuous forward model Discrete forward model Continuous adjoint model Computational adjoint model adj discr adj discr In the 4D-Var approach D.A. is formulated as an optimization problem (gradient based algorithms)

5. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Adjoints of stiff chemical kinetics: formulation, challenges, and automatic implementation Optimal analysis state Chemical kinetics Aerosols CTM Transport Meteorology Emissions Observations Data Assimilation Targeted Observ. Improved: forecasts science field experiment design models emission estimates Improved: forecasts science field experiment design models emission estimates

6. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting KPP automatically generates simulation and direct/adjoint sensitivity code for chemistry #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; 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 equation REAL*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 K P [Damian et.al., 1996; Sandu et.al., 2002] Chemical mechanism Simulation code

7. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Sparse Jacobians, Hessians, and sparse linear algebra routines are automatically generated by KPP SAPRC-99 79 spc./211 react. NZ=839, NZLU=920 SAPRC-99. NZ = 848x2 (0.2%) #HESSIAN [ ON | OFF ] HessVar(…), HessVar[TR]_Vec(…) #JACOBIAN [ ON | OFF | SPARSE ] JacVar(…), JacVar[TR]_SP_Vec(…) KppDecomp(…), KppSolve[TR](…)

8. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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]

9. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Singular perturbation analysis is relevant for studying the method behavior on stiff systems If RK with invertible coefficient matrix A and R(∞) = 0; and the cost function depends only on the non-stiff variable y Then λ z = 0 and λ y are solved with the same accuracy as the original method, within O(ε). A similar conclusion holds for continuous RK adjoints. [Sandu, Daescu, 2003] Singular perturbation test problem : Distinguish between derivatives w.r.t. stiff/non-stiff variables

10. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Rosenbrock methods and their adjoints are efficiently implemented by KPP Discrete Adjoint (T≈2.3T fwd ) Continuous Adjoint (T≈1.2T fwd ) Rosenbrock Method (T fwd ) [Sandu et.al., 2002]

11. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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]

12. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Adjoints for Integral-PDE aerosol dynamic equations: formulation and challenges Optimal analysis state Chemical kinetics Aerosols CTM Transport Meteorology Emissions Observations Data Assimilation Targeted Observ. Improved: forecasts science field experiment design models emission estimates Improved: forecasts science field experiment design models emission estimates

13. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Populations of aerosols (particles in the atmosphere) are described by their mass density m+m’ m m’ Coagulation m Growth Aerosol dynamic equation - IPDE

14. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Adjoint aerosol dynamic models are needed to solve inverse problems [Sandu et. al., 2005; Henze et. al., 2004] Continuous adjoint equation Observations of density in each bin allow the recovery of initial distribution and of parameters

15. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Discrete adjoint models for numerical advection: formulation and challenges Optimal analysis state Chemical kinetics Aerosols CTM Transport Meteorology Emissions Observations Data Assimilation Targeted Observ. Improved: forecasts science field experiment design models emission estimates Improved: forecasts science field experiment design models emission estimates

16. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Inconsistency, instability of discrete adjoint advection schemes when the forward scheme pattern changes CFL<0.87 CFL<0.81 unstable Von Neumann stability regions for different upwindings upwind direction change upwind direction change discretization change discretization change Forward flux-limiter Status change Forward flux-limiter Status change

17. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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 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]

18. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting The 4D-Var tools have been implemented in the parallel adjoint STEM and are being applied to real data Optimal analysis state Chemical kinetics Aerosols CTM Transport Meteorology Emissions Observations 4D-Var Data Assimilation Targeted Observ. Improved: forecasts science field experiment design models emission estimates Improved: forecasts science field experiment design models emission estimates

19. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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]

20. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Assimilation of P3-B observations taken during the Trace-P campaign on March 07, 2001 P3-B observations: O 3 (8%); NO, NO 2 (20%); HNO 3, PAN, RNO 3 (100%) Control: Init. conc. 50 species (sensitivity) Optimization problem with ~ 5 million variables Parallel adjoint STEM, 2-level checkpointing NO 2 [Chai et.al., 2004] O3O3

21. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Assimilation of AIRNOW O 3 surface observations for July 20, 2004 adjusts predictions considerably Observations: circles, color coded by O 3 mixing ratio Surface O 3 (forecast)Surface O 3 (analysis)

22. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Ensemble-based chemical data assimilation can complement variational techniques Optimal analysis state Chemical kinetics Aerosols CTM Transport Meteorology Emissions Observations Ensemble Data Assimilation Targeted Observ. Improved: forecasts science field experiment design models emission estimates Improved: forecasts science field experiment design models emission estimates

23. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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 AR model of background errors N ∆t ≈ lifetime of the species B: flow dependent inexpensive to compute has full rank McMillian Reservoir, DC

24. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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 [Liao and Sandu, 2005] Cause: stiff transient Solution: TLM + ADJ chem. projections

25. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting TESVs are shaped by both meteorology and chemistry, as seen for different sections NO 2 O3O3 #1 #2 σ 2 Trace-P, March 1-2, 2001 σ 2 Trace-P, March 1-2, 2001

26. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Ensemble-based chemical data assimilation considerably improves the solution O 3 original error O 3 error w/ analysis NO 2 original error NO 2 error w/ analysis 24h, March 1-2, 2001 50 members Perturbed I.C., B.C., and emissions AR + TESV O3 + NO2 obs (red) 24h, March 1-2, 2001 50 members Perturbed I.C., B.C., and emissions AR + TESV O3 + NO2 obs (red)

27. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Results for species not directly observed also show marked improvements after assimilation CO original error CO error w/ analysis PAN original error PAN error w/ analysis HCHO original error HCHO error w/ analysis

28. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting The model can be used to place the observations in the locations of maximum informational benefit Optimal analysis state Chemical kinetics Aerosols CTM Transport Meteorology Emissions Observations Data Assimilation Targeted Observations Improved: forecasts science field experiment design models emission estimates Improved: forecasts science field experiment design models emission estimates

29. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting Targeted observations are deployed to areas where they provide maximum of information (criterion based on SVs) [Liao and Sandu, 2005] Verification: Korea, ground O 3 0 GMT, Mar/4/2001 O3O3 NO 2 HCHO

30. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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

31. June 20, 2005 Chemical D.A. and Data Needs for Air Quality Forecasting 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)