1. Carbon Cycle Data Assimilation with a Variational Approach (“4-D Var”) David Baker CGD/TSS with Scott Doney, Dave Schimel, Britt Stephens, and Roger Dargaville 24 Sept 2004

2. Outline The problem: estimate CO 2 sources and sinks at fine space/time scales (2° x 2.5°, hourly/daily) Method: –Why use 4-D Var? (Kalman) filtering, smoothing, and variational methods – pros and cons –Mathematical background of 4-D Var applied to atmospheric trace gases Some 4-D Var results using simulated truth Additional topics to ponder: –100 descent iterations  100 ensemble members? –Error estimates: 4-D Var vs. ensemble filters

4. 0 HH Transport: surface fluxes  concentrations fluxes concentrations Transport basis functions

5. Present  Future Shift towards newer instruments/platforms: More continuous analyzers, new cheap in situ analyzers Aircraft, towers (flux & tall), ships/planes of opportunity CO 2 -sondes, tethered balloons, etc. Satellite-based column-integrated CO 2, maybe CO 2 profiles Higher frequency with better spatial coverage -- will permit more detail to be estimated More sensitive to continental air, detailed flow features -- synoptic meteorology, diurnal cycle must be resolved Solve for the fluxes at the resolution of the transport model  2° x 2.5°, 25 levels, daily/hourly time step  With current inversion techniques, computations grow as O(N 3 )… more efficient techniques required (iterative vs. direct inversions, adjoint allows efficient gradient computation, minimal storage)

6. For retrospective analyses, a 2-sided smoother gives more accurate estimates than a 1-sided filter. The 4-D Var method is 2-sided, like a smoother. (Gelb, 1974)

7. Variational Data Assimilation vs. Ensemble (Kalman) filter Pros: Greater accuracy achieved with 2-sided smoother than 1-sided filter Initial transients reduced Cons: Adjoint model required [Correlations are pre-specified, rather than calculated, as with a Kalman filter]

8. 4-D Var Data Assimilation Method Find optimal fluxes u and initial CO 2 field x o to minimize subject to the dynamical constraint where x are state variables (CO 2 concentrations), v are independent variables used in model but not optimized, z are the observations, R is the covariance matrix for z, u o is an a priori estimate of the fluxes, P uo is the covariance matrix for u o, x o is an a priori estimate of the initial concentrations, P xo is the covariance matrix for x o

9. 4-D Var Data Assimilation Method Adjoin the dynamical constraints to the cost function using Lagrange multipliers Setting  F/  x i = 0 gives an equation for i, the adjoint of x i: The adjoints to the control variables are given by  F/  u i and  F/  x o o as: The optimal u and x o may then be found iteratively by

10. ° ° ° ° 00 22 11 33 x2x2 x1x1 x3x3 x0x0 Adjoint Transport Forward Transport Forward Transport Measurement Sampling Measurement Sampling “True” Fluxes Estimated Fluxes Modeled Concentrations “True” Concentrations Modeled Measurements “True” Measurements Assumed Measurement Errors Weighted Measurement Residuals  /(Error) 2 Adjoint Fluxes =  Flux Update 4-D Var Iterative Optimization Procedure Minimum of cost function J

11. Truth Prior Estimate (30 descent steps) OSSE fluxes, snapshot for Jan 1st

12. Prior - TruthEstimate - Truth

15. Future Plans for CO 2 Assimilate remotely-sensed data Finer resolution (1º x 1º, or regional) Improve predictive capability of carbon cycle models (in two steps) by –Tying fluxes to remotely-sensed patterns –Estimating parameters in ocean and land biosphere models using remotely-sensed fields directly as data

16. Atmospheric transport model NASA/GSFC DAO ‘PCTM’ model: –Lin-Rood advection –Vertical diffusion –Simple cloud convection Driven by saved wind & mixing fields from DAO analyses 6-hourly winds interpolated to 15 minute time step 2º x 2.5º resolution, 25 vertical levels Adjoint : Coded manually; straight-forward because of –Linearity of CO 2 transport –Simplicity of vertical mixing routines Runs as fast as forward code