Comparison of adjoint and analytical approaches for solving atmospheric chemistry

inverse problems

Monika Kopacz1, Daniel J. Jacob1, Daven Henze2, Colette Heald3, David G. Streets4, Qiang Zhang5

October 11, 2006

1. Harvard, 2. CalTech, 3. UC Berkeley, 4. Argonne NL, 5. Tsinghua University, China

Problems in atmospheric chemistry

CO, CO2, NO, NO2, CH4, Hg etc.

Emissions (pollution and natural)

Chemistry (CO CO2, CH4 CO, NOx+ CO O3 …)

Transport: using assimilated meteorology (from GEOS)

(CO2) uptake

ocean

CO2, Hg

Atmosphere as simulated by a Chemical Transport Model (CTM)

( )n

n P Lt

U

Solves continuity equation

Problems in atmospheric chemistry (estimating emissions)

need for accurate emission estimates for regulatory purposes

Inverse modeling atmospheric observations

Bottom up emissions estimates

Top-down emissions estimates

Creating detailed emissions inventories at a model resolution

2 approaches to emissions estimation

Current inverse modeling standard in atmospheric chemistry

1 1( ) ( ( ) ) ( ( ) ) ( ) ( )T Ta a aJ

x F x y S F x y x x S x x

Forward Model (GEOS-Chem)2°x2.5° resolution

“bottom up” emission inventories

P(y|x)

Bayes statistics inverse model:

Emissions = P(x) Observations = P(y)

P( | ) P( )P( | ) =

P( )

y x xx y

y

Approach:• Assume Gaussian distributions

least squares min .

• sparse observations Size of x ~ O(10) emission regions

Number of constraints (x) limited by the number of observations

analytical solution = …x̂

Satellite observations revolutionize tropospheric chemistry

Number of constraints (x) limited by the inverse methodology

Advantages of satellite instruments (recent tropospheric measurements):

• offer dense, daily global coverage (from 1 to 16 days orbit repeat)

• observations over/close to the sources not background like surface station in remote atmosphere and oceans

• can constrain more emissions regions???

Measurement of Pollution In the Troposphere (MOPITT)

ˆ ) T -1 -1 -1 T -1a x x a x ax = x + ( F S F + S ) F S (F( x y)

Solving an inverse problem

1 1( ) ( ( ) ) ( ( ) ) ( ) ( )T Ta a aJ

x F x y S F x y x x S x xminimize

T 1 1x( ) = 2( 2 ( )aJ

x ax F(x)) S (F(x) y) + S x xcompute

Objective:

Analytical method Adjoint method

( ) xJ xcompute

use optimization algorithm to iteratively find

ˆ xSolve explicitly for

ˆ x

explicitly compute Jacobian matrix:

F( ) = x x K

( ) = 0xJ xset

MAP solution

T 1 1 1 T 1a aˆ + ( + ) ( )a

x x K S K S K S y Kx

using adjoint model

Analytic vs. adjoint solution

T 1 1 1 T 1a aˆ + ( + ) ( )a

x x K S K S K S y Kx

How the analytical approach becomes infeasible…

Not computing Jacobian matrix explicitly fortran code used to represent itUsing reverse mode efficient

Increasing the size of the optimized vector O(10) O(10^5)Constructing full Jacobian matrix KInverting large matrices

How an adjoint addresses problems of the analytical approach…

Assumptions we can’t avoid…Gaussian errors Linearization of nonlinear processes using gradient descent

( )

T 1 1x( ) = ( ( )x aJ

ax F(x)) S (F(x) y) + S x x( )

Method comparison project

Comparison objective: Perform a (adjoint) inversion similar to a previous (analytical) inversion using the same observations, emissions inventory, time frame, error characterization and forward model (but not resolution!)

Inversion objective: Constrain Asian CO emissions during the Spring 2001

≠

average model CO concentration average satellite (MOPITT) concentration

a priori emission inventory

Inversion comparison setup

Heald et al, 2004 this work

Inversion approachOptimized vector sizeMOPITT observations

forward CTMa priori cost functiona posteriori cost function% decrease

analytical

11Feb 21-April 10, 2001~21,000 obsGEOS-Chem v4.3~35,000~28,000

~23%

adjoint

3006 (144x91)Feb 21-April 10, 2001~21,000 obsGEOS-Chem v6.5~29,000 ~22,000

~23%

Heald et al, 2004: Comparative inverse analysis of satellite (MOPITT) and aircraft (TRACEP) observations to estimate Asian sources of carbon monoxide, J. of Geophys. Res.

Constructing error covariance

%

Observational errorObs. error variance: Relative Residual Error (RRE) method, ie. Computing deviation from an ensemble mean error (model bias due to error in sources)

All error covariance: set to zero

aS

S

A priori source error variance: from emissions inventory for each country and source type

S Includes model error, representation error and instrument error

CO constraints using adjoint inversion

Red: a priori underestimate

China, Northern India

Blue: a priori overestimate

East India, Southeast Asia, Philippines

A posteriori emissions scaling factors

Comparison: coarse (adjoint) vs. averaged detail (adjoint) source estimates

analytical adjoint1. C. China (ChCE) 1.83 1.342. SE Asia (SEAs) 0.63 0.673. Philippines (Ph) 0.89 0.734. Indonesia (Id) 0.96 0.905. India (In) 0.50 0.686. rest of world (RoW) 1.16 0.91 oxidation source 1.11

analytical adjoint1. W. China (ChW) 2.38 1.162. S. China (ChSE) 0.31 1.18 3. N. China (ChNE) 0.76 1.024. Japan (Jp) 1.88 0.99 Korea (Ko) 1.025. Europe (EU) 0.75 1.00

11 regions (state vector elements) from Heald et al, 2004

Potentially affected by aggregation error

Analytical bias Adjoint bias

a priori

a posteriori

Limitations and challenges of using the adjoint for inverse modeling

Do we have enough observations???

How computationally efficient is it?

If we use adjoint approach for inversion, what is the best optimization algorithm, considering the requirements of: - quick convergence - accuracy - non-negative solution, ie. will not yield negative emissions! - no bias?

fwd + adj run for simple CO chemistry (69 days): 4h

L-BFGS (Liu and Nocedal, 1989)

END

Adjoint model development

Based on GEOS-Chem forward model (GEOS-3, v6-05-07)

• advection

• deep convection

• turbulent mixing

• CO chemistry

• CO sources

• integration with MOPITT observations

Daven Henze at Caltech

Harvard

Note: The adjoint model also contains aerosol thermodynamics (full chemistry adjoint), wet and dry deposition and aerosol emissions components all developed by Daven Henze