The prototype carbon inverse problem: estimation of regional CO 2 sources and sinks from global atmospheric [CO 2 ] measurements David Baker NCAR / Terrestrial Sciences Section 23 July 2006

Outline Description of the source/sink problem CO 2 measurement availability thru time Method determined by computational issues The hierarchy of inverse methods used: Batch least-squares The (full) Kalman filter/smoother The use of the adjoint: what it can (and can’t) do The ensemble Kalman filter/smoother Variational data assimilation

TransCom3 Regions (22) and Measurement Sites (78)

Application: CO 2 flux estimation Discretize problem: solve for monthly fluxes for 22 regions Measurement equations: Hx=z –x – regional CO 2 fluxes, monthly avg –z – CO 2 measurements, monthly avg –H – transport matrix; columns are Green’s functions of a 1-month pulse of unitized CO 2 flux from each region calculated using the transport model

0 HH Form of the Batch Measurement Equations fluxes concentrations Transport basis functions

Batch least-squares Optimal fluxes found by minimizing where giving

Monthly fluxes Deseasonalized fluxes IAV (subtract mean) IAV w/ errors Calculation of Interannual Variability (IAV) -- Europe (Region #11) 13 transport models [PgC/year]

Total flux IAV (land + ocean) [PgC/yr]

Ocean Flux IAV Land flux IAV

[PgC/yr]

Flux IAV [PgC/yr], 11 land regions

Flux IAV [PgC/yr], 11 ocean regions

“Representativeness” or “Discretization” Error Incorrect assumptions about space/time patterns across large space/time blocks can cause large biases in inversions To fix this, solve at as fine a resolution possible, and transform final estimate to coarser scales post-inversion, if necessary Sounds good, but poorly-known correlations must still be used in the fine-scale inversions to constrain the results Resolution of robust results ultimately set by measurement density and precision

Computational considerations Transport model runs to generate H: –22 regions x 16 years x 12 months x 36 months = 152 K tracer-months (if using real winds) –22 x 12 x 36 = 9.5 K tracer-months (climatological winds) Matrix inversion computations: O(N 3 ) –N= 22 regions x 16 years x 12 months = 4.4 K Matrix storage: O (N*M) MB –M = 78 sites x 16 years x 12 months = 15 K

Present  Future New measurements augmenting the weekly flasks: More continuous analyzers More aircraft flights More towers (both tall and eddy-flux) Ships/planes of opportunity Low-cost in situ analyzers CO 2 -sondes, tethered balloons, etc. Satellite-based column-integrated CO 2, CO 2 profiles Higher frequency, more sensitive to continental air Most importantly: MORE DATA, better coverage  Will permit more detail to be estimated

Eddy-flux tower sites

Growth of measurement density and likely solution resolution ~22 regions, monthly, ~60 sites, 1987-present: N ~50 regions, weekly, ~100 sites, c. 2000: 10·N ~200 regions, daily?, ~200 continuous sites, ~2005: 300·N ~60000 regions (1°x1°), hourly, ~250 meas/hour, from satellites launched ~2008: 2,000,000·N Upshot: batch least squares not feasible for much longer

Three approaches to handling these computational limitations: 1) Break span into smaller blocks (Bousquet, et al 1999) year spin-up time limits this, due to required overlaps 2) Solve in the (smaller) measurement space -- i.e., O(M 3 ) (Peylin, et al 2005) 3) Use the adjoint to the transport model to efficiently generate H basis functions in time-reverse mode (Kaminski, et al, 1999; Roedenbeck, et al 2005) 2) and 3) allow one to efficiently solve for many fluxes when the number of measurements is small(er), but both break down when both the number of measurements and fluxes are large

Computational considerations Transport model runs to generate H: –22 regions x 16 years x 12 months x 36 months = 152 K tracer-months (if using real winds) –22 x 12 x 36 = 9.5 K tracer-months (climatological winds) Matrix inversion computations: O(N 3 ) –N= 22 regions x 16 years x 12 months = 4.4 K Matrix storage: O (N*M) MB –M = 78 sites x 16 years x 12 months = 15 K Three key ideas to get around these limitations: Use sequential estimators instead of batch inversion Use indirect (iterative) solvers instead of direct ones Solve for only an approximate covariance, not the full-rank one

0 HH Kalman Filter/Smoother

0 HH

Kalman Filter Equations Measurement update step at time k: Dynamic propagation step from times k to k+1: Put multiple months of flux in state vector x k, method becomes effectively a fixed-lag Kalman smoother

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)

Traditional (full-rank) Kalman filter  ensemble Kalman filter Full KF only a partial fix due to still-large P matrices Ensemble KF (EnKF): full covariance matrix replaced by approximation from an ensemble EnKF can be thought of as a reduced-rank square root KF with each column of the reduced-rank P 1/2 being given by a state deviation (dx) from an ensemble of independently propagated and updated states x xMatrix operations involving P in the basic KF equations replaced with dot products of state deviations (dx) summed over the ensemble See Peters, et al (2005) for implementation

Ensemble KF vs. Variational DA Ensemble KF still a relatively new method -- details still being worked out: –Proper method for introducing measurement and dynamical errors into ensemble –Proper means for handling time/space correlations in measurements –Proper way to apply it to time-dependent CO 2 fluxes Variational data assimilation is more developed –Pros: Wide use in numerical weather prediction Based on a minimization -- can always be made to converge –Con*: -- Requires an adjoint model * Since ensemble smoothers need an adjoint, too, for true retrospective analyses (those for which a fixed-lag smoother won’t work well), this is not necessarily a con in those cases

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), h is a vector of measurement operators 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,  is the transition matrix (representing the transport model), and G distributes the current fluxes into the bottom layer of the model

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 as: The gradient vector is given by  F/  u i and  F/  x o o, so the optimal u and x 0 may then be found using one’s favorite descent method. I have been using the BFGS method, since it conveniently gives an estimate of the leading terms in the covariance matrix.

° ° ° ° 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

BFGS Minimization Approach Variable metric method -- accumulates 2nd order information (approximate covariance matrix) as minimization proceeds Search direction: s k = H k  k H k approximates [  2 J/  u 2 ] -1 = P u Precondition with H 0 =P u° H never stored, but reconstructed from p and q as needed

“Soft” Dynamical Constraint Allows for dynamical errors Add dynamical mismatches  i to J as Note that Then solve for  i from i and add these to the propagated state

Atmospheric Transport Model Parameterized Chemical Transport Model (PCTM; Kawa, et al, 2005) –Driven by reanalyzed met fields from NASA/Goddard’s GEOS-DAS3 scheme –Lin-Rood finite volume advection scheme –Vertical mixing: diffusion plus a simple cloud convection scheme –Exact adjoint for linear advection case Basic resolution 2  x 2.5 , 25  layers,  t  30 min, with ability to reduce resolution to –4  x 5 ,  t  60 min –6  x 10 ,  t  120 min <  we’ll use this in the example –12  x 15 ,  t  180 min Measurements binned at  t resolution

Data Assimilation Experiments Monday: Test performance of 4DVar method in a simulation framework, with dense data (6°x10°, lowest level of model, every  t, 1 ppm (1  ) error) –Case 1 -- no data noise added, no prior –Case 2 -- w/ data noise added, no prior –Case 3 -- w/ data noise added, w/ prior –Case 4 -- no data noise added, w/ prior Tuesday: with Case 3 above, do OSSEs for –Case 5 -- dense data, but OCO column average –Case 6 -- OCO ground track and column average –Case 7 -- extended version of current network –Case 8 -- current in situ network Possibilities for projects: examine the importance of –Data coverage and accuracy vs. targeted flux resolution –Prior error pattern and correlation structures –Measurement correlations in time/space –Errors in the setup assumptions (“mistuning”) –Effect of biases in the measurements

How to run the 4D-Var code Home directory: /project/projectdirs/m598/dfb/4DVar_Example/scripts/case1/BFGS Work directories: /scratch/scratchdirs/dfb/case1/work_fwd & /scratch/scratchdirs/dfb/case1/work_adj Submit batch job by typing ‘llsubmit runBFGS2_LL’ while in /project/projectdirs/m598/dfb/4DVar_Example/scripts/case1/BFGS/ This executes the main driver script, found in BFGSdriver4d.F, in same directory, which controls setting up all the files and running FWD and ADJ inside the minimization loop The scripts that execute the FWD and ADJ runs of the model are found in …/4DVar_Example/scripts/case1/, named run.co2.fvdas_bf_fwd(adj)_trupri997_hourly Check progress of job by typing ‘llqs’ Jobs currently set up to do a 1 year-long run (360 days), solving for the fluxes in 5-day long chunks, at 6.4  x10  resolution, with  t =2 hours

How to monitor job while running In /scratch/scratchdirs/dfb/case1/work_fwd/costfuncval_history/temp –Column 1 -- measurement part of cost function –Column 2 -- flux prior part of cost function –Column 4 -- total cost function value –Column weighted mismatch from true flux –Column unweighted mismatch from true flux Columns 4, 10, and 11 ought to be decreasing as the run proceeds Columns X and Y give the iteration count and 1-D search count

How to view detailed results A results file in netCDF format written to: /scratch/scratchdirs/dfb/case1/work_fwd/estim_truth.nc sftp this to davinci.nersc.gov (rename it, so that you don’t overwrite another group’s file) Pull up an X-window to davinci and ssh -X davinci.nersc.gov On davinci, ‘module load ncview’ Then ‘ncview estim_truth.nc’ Click on a field to look at it Hint: set ‘Range’ to +/- 2e-8 for most fields

Other code details The code for the FWD and ADJ model is in../4DVar_Example/src_fwd_varres and src_adj_varres Measurement files are located in /scratch/scratchdirs/dfb/case1/meas Two files controlling the tightness of the prior and whether or not noise is added to the measurements are /scratch/scratchdirs/dfb/case1/work_fwd/ferror and /scratch/scratchdirs/dfb/case1/work_fwd/measnoise_on.dat

Monday’s Experiment 2-hourly measurements in the lowest model level at 6.4  x 10 , 1 ppm error (1  ) Iterate 30 descent steps, 1-year-long run, starts 1/1 4 cases –Case 1 -- No measurement noise added, no prior –Case 2 -- Add measurement noise added, no prior –Case 3 -- Add noise, and apply a prior –Case 4 -- No noise, but apply a prior Designed to test the method and understand the impact of data errors and the usefulness of the prior Case 3 is the most realistic and will be used to do OSSEs for several possible future networks for Tuesday’s problem set

Tuesday’s Experiment Use Case 3 from above to test more-realistic measurement networks: –Current in situ network –Extended version of current network –OCO satellite –Hourly 6.4  x 10  column measurements Essentially an “OSSE” (observing system simulation experiment) -- tells you how well your instrument should do in constraining the fluxes. Only gives the random part of the error, not biases

OCO Groundtrack, Jan 1st (Boxes at 6  x 10  ) Across 1 day 5 days 2 days