chemical source inversion using assimilated constituent observations

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.