1. INVERSE MODELING OF ATMOSPHERIC COMPOSITION DATA Daniel J. Jacob See my web site under “educational materials” for lectures on inverse modeling atmospheric transport and chemistry models Notation and terminology in this lecture follows that of Rodgers (2000)

2. THE INVERSE MODELING PROBLEM Optimize values of an ensemble of variables (state vector x ) using observations: THREE MAIN APPLICATIONS FOR ATMOSPHERIC COMPOSITION: 1.Retrieve atmospheric concentrations ( x ) from observed atmospheric radiances ( y ) using a radiative transfer model as forward model 2.Invert sources ( x ) from observed atmospheric concentrations ( y ) using a CTM as forward model 3.Construct a continuous field of concentrations ( x ) by assimilation of sparse observations ( y ) using a forecast model (initial-value CTM) as forward model a priori estimate x a +  a observation vector y forward model y = F(x) +  “MAP solution” “optimal estimate” “retrieval” Bayes’ theorem

3. BAYES’ THEOREM: FOUNDATION FOR INVERSE MODELS P(x) = probability distribution function (pdf) of x P(x,y) = pdf of (x,y) P(y|x) = pdf of y given x a priori pdf observation pdf normalizing factor (unimportant) a posteriori pdf Maximum a posteriori (MAP) solution for x given y is defined by solve for  P(x,y)dxdy  Bayes’ theorem

4. SIMPLE LINEAR INVERSE PROBLEM FOR A SCALAR use single measurement used to optimize a single source a priori bottom-up estimate x a  a Monitoring site measures concentration y Forward model gives y = kx “Observational error”   instrument fwd model y = kx   Max of P(x|y) is given by minimum of cost function Solution: where g is a gain factor Alternate expression of solution: where a = gk is an averaging kernel  solve for Assume random Gaussian errors, let x be the true value. Bayes’ theorem: Variance of solution:

5. GENERALIZATION: CONSTRAINING n SOURCES WITH m OBSERVATIONS Linear forward model: A cost function defined as is generally not adequate because it does not account for correlation between sources or between observations. Need vector-matrix formalism: Jacobian matrix

6. JACOBIAN MATRIX FOR FORWARD MODEL Consider a non-linear forward model y = F(x) Use of vector-matrix formalism requires linearization of forward model and linearize it about x a : is the Jacobian of F evaluated at x a with elements If forward model is non-linear, K must be recalculated iteratively for successive solutions K T is the adjoint of the forward model (to be discussed later) Construct Jacobian numerically column by column: perturb x a by  x i, run forward model to get corresponding  y

7. GAUSSIAN PDFs FOR VECTORS A priori pdf for x : Scalar x Vector where S a is the a priori error covariance matrix describing error statistics on ( x-x a ) In log space:

8. OBSERVATIONAL ERROR COVARIANCE MATRIX observation true value instrument error fwd model errorobservational error Observational error covariance matrix is the sum of the instrument and fwd model error covariance matrices: How well can the observing system constrain the true value of x ? Corresponding pdf, in log space:

9. MAXIMUM A POSTERIORI (MAP) SOLUTION Bayes’ theorem: MAP solution: miminize cost function J : Solve for Analytical solution: with gain matrix bottom-up constraint top-down constraint

10. PARALLEL BETWEEN VECTOR-MATRIX AND SCALAR SOLUTIONS: Scalar problemVector-matrix problem MAP solution: Gain factor: A posteriori error: Averaging kernel: Jacobian matrixsensitivity of observations to true state Gain matrix sensitivity of retrieval to observations Averaging kernel matrix sensitivity of retrieval to true state

11. A LITTLE MORE ON THE AVERAGING KERNEL MATRIX A describes the sensitivity of the retrieval to the true state and hence the smoothing of the solution: smoothing error retrieval error MAP retrieval gives A as part of the retrieval: Other retrieval methods (e.g., neural network, adjoint method) do not provide A # pieces of info in a retrieval = degrees of freedom for signal (DOFS) = trace(A)

12. APPLICATION TO SATELLITE RETRIEVALS Here y is the vector of wavelength-dependent radiances (radiance spectrum); x is the state vector of concentrations; forward model y = F(x) is the radiative transfer model Illustrative MOPITT averaging kernel matrix for CO retrieval MOPITT retrieves concentrations at 7 pressure levels; lines are the corresponding columns of the averaging kernel matrix trace(A) = 1.5 in this case; 1.5 pieces of information

13. INVERSE ANALYSIS OF MOPITT AND TRACE-P (AIRCRAFT) DATA TO CONSTRAIN ASIAN SOURCES OF CO TRACE-P CO DATA (G.W. Sachse) Bottom-up emissions (customized for TRACE-P) Fossil and biofuel Daily biomass burning (satellite fire counts) GEOS-Chem CTM MOPITT CO Inverse analysis validation chemical forecasts top-down constraints OPTIMIZATION OF SOURCES Streets et al. [2003] Heald et al. [2003a]

14. COMPARE TRACE-P OBSERVATIONS WITH CTM RESULTS USING A PRIORI SOURCES Model is low north of 30 o N: suggests Chinese source is low Model is high in free trop. south of 30 o N: suggests biomass burning source is high Assume that Relative Residual Error (RRE) after bias is removed describes the observational error variance (20-30%) Assume that the difference between successive GEOS-Chem CO forecasts during TRACE-P (t o +48h and t o + 24 h) describes the covariant error structure (“NMC method”) Palmer et al. [2003], Jones et al. [2003]

15. CHARACTERIZING THE OBSERVATIONAL ERROR COVARIANCE MATRIX FOR MOPITT CO COLUMNS Diagonal elements (error variances) obtained by residual relative error method Add covariant structure from NMC method Heald et al. [2004]

16. COMPARATIVE INVERSE ANALYSIS OF ASIAN CO SOURCES USING DAILY MOPITT AND TRACE-P DATA MOPITT has higher information content than TRACE-P because it observes source regions and Indian outflow Don’t trust a posteriori error covariance matrix; ensemble modeling indicates 10-40% error on a posteriori sources Heald et al. [2004] CO observations from Spring 2001, GEOS-Chem CTM as forward model TRACE-P Aircraft COMOPITT CO Columns 4 degrees of freedom 10 degrees of freedom (from validation) “Ensemble modeling”: repeat inversion with different values of forward model and covariance parameters to span uncertainty range

17. Analytical solution to inverse problem requires (iterative) numerical construction of the Jacobian matrix K and matrix operations of dimension (mxn); this limits the size of n, i.e., the number of variables that you can optimize Address this limitation with Kalman filter (for time-dependent x ) or with adjoint method 

18. BASIC KALMAN FILTER to optimize time-dependent state vector a priori x a,0 ± S a,0 toto time observations state vector y0y0 Evolution model M for [t 0, t 1 ]: t1t1 y1y1 Apply evolution model for [t 1, t 2 ]… etc.

19. ADJOINT INVERSION (4-D VAR) ° ° ° ° aa 22 11 33 x2x2 x1x1 x3x3 xaxa Minimum of cost function J Solve numerically rather than analytically 1. Starting from a priori x a, calculate 2. Using an optimization algorithm (BFGS), get next guess x 1 3. Calculate, get next guess x 2 4. Iterate until convergence

20. NUMERICAL CALCULATION OF COST FUNCTION GRADIENT Adjoint model is applied to error-weighted difference between model and obs …but we want to avoid explicit construction of K and since ( AB) T = B T A T, Apply transpose of tangent linear model to the adjoint forcings; for time interval [t 0, t n ], start from observations at t n and work backward in time until t 0, picking up new observations (adjoint forcings) along the way. Construct tangent linear model describing evolution of concentration field over time interval [ t i-1, t i ] Sensitivity of y (i ) to x (0 ) at time t 0 can then be written of forward model adjoint “adjoint forcing”

21. APPLICATION OF GEOS-Chem ADJOINT TO CONSTRAIN ASIAN SOURCES WITH HIGH RESOLUTION USING MOPITT DATA MOPITT daily CO columns (Mar-Apr 2001) Correction factors to a priori emissions with 2 o x2.5 o resolution A priori knowledge of emissions Kopacz et al. [2007] GEOS-Chem adjoint with 2 o x2.5 o resolution Fossil fuel Biomass burning number of observations Iterative approach to minimum of cost function

22. COMPARE ADJOINT AND ANALYTICAL INVERSIONS OF SAME MOPITT CO DATA Adjoint solution reveals aggregation errors, checkerboard pattern in analytical solution [Heald et al., 2004] Kopacz et al. [2007]