Chemical Source Inversion Using Assimilated

Constituent Observations

Andrew TangbornGlobal Modeling and Assimilation Office

How are data assimilation and chemical source inversion related?

1. Underdetermined systems – fewer constraints than unknowns.

2. Use Bayesian methods – require error statistics.

3. If errors are normally distributed and unbiased – optimal scheme results in weighted least squares estimate.

How are they different?

1. Different goals: state estimate vs. source estimate.

2. Use different errors: source errors vs. model and State errors.

Example: Kalman Filtering (KF) and Green’s function (GF) Inversion

Inputs chemical tracer observations winds chemical source/sinks Initial state Error estimates

Algorithms

KF: Kalman gain observation operator

ca = cf + K(co – Hcf) observations

analysis

forecast

KF:

Where K = PfHT (R+HPfHT)-1 is a weighted by obs error (R) and forecast error (HPfHT) covariances.

The forecast error covariancePf = MPaMT + Qis evolved in time from analysis error using the

discretized model (winds) M and added model error Q

GF:

Green’s function Inverse of source error cov.

xinv = (GTXG+W)-1 (GTXco+Wz)

Inverse of R obs first guess source

inverted source

G: calculated by running model forward using unit sources at

each grid point.

Differences between KF and GF

KF: Initial condition (analysis) used in

forecast contains earlier observation

information.

GF: Initial condition does not explicitly

contain observation data.

KF: Uses model (wind) and observation errors.

State errors are propagated in time.

GF: Uses source and observation errors.

State errors are never calculated.

Combing KF and GF

• Carry out KF assimilation of tracer observations.• Use both the analysis and analysis error

covariance as the observations and observation error in the GF

co = ca

X = (Pa)-1

• Now X contains information on model errors (including wind errors) and co is spread to all grid

points through assimilation.

Numerical Experimentsadvection diffusion in 2 x 2domain

with constant and random source errors

True Source Model source

Tracer Fields

True field Model solution Analysis field

Source InversionError Standard Deviation

No Assimilation With assimilation

Conclusions

• Data Assimilation can add information to source inversion.

• Improvements likely come through improved and more complete error covariance information and spreading observation information to more grid points.