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David Baker NCAR / Terrestrial Sciences Section 23 July 2006

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The prototype carbon inverse problem: estimation of regional CO2 sources and sinks from global atmospheric [CO2] measurements

David Baker

NCAR / Terrestrial Sciences Section

23 July 2006

Description of the source/sink problem

CO2 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)

Discretize problem: solve for monthly fluxes for 22 regions

Measurement equations: Hx=z

x – regional CO2 fluxes, monthly avg

z – CO2 measurements, monthly avg

H – transport matrix; columns are Green’s functions of a 1-month pulse of unitized CO2 flux from each region calculated using the transport model

Form of the Batch Measurement Equations

0

H

fluxes

Transport basis functions

concentrations

Optimal fluxes found by minimizing

where

giving

Calculation of Interannual Variability (IAV) -- Europe (Region #11)

[PgC/year]

Monthly fluxes

13 transport models

Deseasonalized fluxes

IAV (subtract mean)

IAV w/ errors

Total flux IAV (land + ocean)

[PgC/yr]

Land flux IAV

[PgC/yr]

Ocean

Flux IAV

[PgC/yr]

Flux IAV [PgC/yr],

11 land regions

Flux IAV [PgC/yr],

11 ocean regions

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

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(N3)

N= 22 regions x 16 years x 12 months = 4.4 K

Matrix storage: O (N*M) --- 66 MB

M = 78 sites x 16 years x 12 months = 15 K

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

CO2-sondes, tethered balloons, etc.

Satellite-based column-integrated CO2, CO2 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

~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

1) Break span into smaller blocks (Bousquet, et al 1999) -- 1-2 year spin-up time limits this, due to required overlaps

2) Solve in the (smaller) measurement space -- i.e., O(M3) (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

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(N3)

N= 22 regions x 16 years x 12 months = 4.4 K

Matrix storage: O (N*M) --- 66 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

Kalman Filter/Smoother

0

H

Kalman Filter/Smoother

0

H

Measurement update step at time k:

Dynamic propagation step from times k to k+1:

Put multiple months of flux in state vector xk, 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)

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 P1/2 being given by a state deviation (dx) from an ensemble of independently propagated and updated states x

Matrix 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 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 CO2 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 CO2 field xo to minimize
- subject to the dynamical constraint
- where
- x are state variables (CO2 concentrations),
- h is a vector of measurement operators
- z are the observations,
- R is the covariance matrix for z,
- uo is an a priori estimate of the fluxes,
- Puo is the covariance matrix for uo,
- xo is an a priori estimate of the initial concentrations,
- Pxo is the covariance matrix for xo,
- 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

SettingF/xi = 0 gives an equation for i, the adjoint of xi:

The adjoints to the control variables are given byF/uiandF/xoas:

The gradient vector is given by F/uiandF/xoo, so the optimal u and x0 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.

4-D Var Iterative Optimization Procedure

0

x0

°

Forward

Transport

Forward

Transport

Estimated

Fluxes

“True”

Fluxes

1

2

Modeled

Concentrations

“True”

Concentrations

°

x1

°

Measurement

Sampling

Measurement

Sampling

x2

Modeled

Measurements

“True”

Measurements

Assumed

Measurement

Errors

D/(Error)2

3

x3

°

Weighted

Measurement

Residuals

Flux

Update

Adjoint

Transport

Adjoint

Fluxes

=

Minimum of cost function J

Variable metric method -- accumulates 2nd order information (approximate covariance matrix) as minimization proceeds

Search direction: sk= Hkk

Hk approximates [2J/u2]-1= Pu

Precondition with H0=Pu°

H never stored, but reconstructed from p and q as needed

Allows for dynamical errors

Add dynamical mismatches ito J as

Note that

Then solve for i from iand add these to the propagated state

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 2x 2.5, 25 layers, t30 min, with ability to reduce resolution to

4x 5, t60 min

6x 10, t120 min < we’ll use this in the example

12x 15, t180 min

Measurements binned at t resolution

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

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 6.4x10 resolution, with t =2 hours

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 10 -- weighted mismatch from true flux

Column 11 -- 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

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

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

2-hourly measurements in the lowest model level at 6.4x 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

Use Case 3 from above to test more-realistic measurement networks:

Current in situ network

Extended version of current network

OCO satellite

Hourly 6.4x 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

Across 1 day

OCO Groundtrack,

Jan 1st

(Boxes at 6x 10)

2 days

5 days