Inverse Modeling - Atmospheric Chemistry - Lecture Slides, Slides of Chemistry

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INVERSE MODELING OF ATMOSPHERIC COMPOSITION DATA

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THE INVERSE MODELING PROBLEM

Optimize values of an ensemble of variables (

state vector

x

) using observations:

THREE MAIN APPLICATIONS FOR ATMOSPHERIC COMPOSITION:1.^

Retrieve atmospheric concentrations (

x) from observed atmospheric

radiances (

y) using a radiative transfer model as forward model

2.^

Invert sources (

x) from observed atmospheric concentrations (

y) using a

CTM as forward model

3.^

Construct a continuous field of concentrations (

x) by assimilation of sparse

observations (

y) using a forecast model (initial-value CTM) as forward model

a priori estimate

xa

a

observation vector

y

forward model^ y = F(x)

+

“MAP solution”“optimal estimate”

“retrieval”

ˆ^

x +

Bayes’theorem

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SIMPLE LINEAR INVERSE PROBLEM FOR A SCALAR

use single measurement used to optimize a single source

a priori

bottom-up estimate

xa

 a

Monitoring sitemeasuresconcentration

y

Forward model gives

y = kx

^

“Observational error”



^ ^

-^ instrument •^ fwd model

y = kx

^ 

2

2 2

2 (^

)

(^

)

ln

(^

|^

)^

ln

(^

|^

)^

ln

( )

a a x^

x

y^

kx

P x

y

P y

x

P x

^ 



^

^



^

Max of

P

( x|y

)^ is given by minimum of cost function

2

2 2

2 (^

)

(^

)

( )

a a x^

x

y^

kx

J x

 

 ^



Solution:

ˆ^

(^

a^

a

x^

x^

g^

y^

kx

^

^

^

where

g

is a

gain factor

2 2

2

2 a k a

g^

k

  ^



^



^ 2

2

2

1

(^

(^

/^

)^

)

a^

k 

^

^

 ^

^



^



Alternate expression of solution:

(

)^

a

y^

kx

x^

ax

a x

g





^

^



^

^

^





where

a = gk

is an

averaging kernel



solve for

/^

dJ

dx

Assume random Gaussian errors, let

x

be the true value. Bayes’ theorem:

Variance of solution:

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GENERALIZATION:

CONSTRAINING

n

SOURCES WITH

m

OBSERVATIONS

n^1 j^

ij^

i

i

y^

k x  ^

Linear forward model: A cost function defined as

,^

1

1

2

2

1

1

,^

,

(^

(^

(^

n j^

ij^

i

n^

m

i^

a i

i

n

i^

j

a i

j

y^

k x

x^

x

J x

x









^



^

is generally not adequate because it does not account for correlation betweensources or between observations. Need vector-matrix formalism:

1

1

(^

)^

(^

T^

T

n^

m

x^

x^

y^

y

^

x^

y^

y^

Kx

Jacobian matrix

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GAUSSIAN PDFs FOR VECTORS

A priori

pdf for

x

: Scalar

x

Vector

2 2 (^

)

1

( )

exp[

]

2

2

a a

a

x^

x

P x



^





^



1

,^

1

,^

,

1

,^

,^

,

var(

)^

cov(

,^

cov(

,^

)^

var(

a^

a^

n^

a n

a^

n^

a n

n^

a n

x^

x^

x^

x^

x^

x

x^

x^

x^

x^

x^

x

^

^

^

^

^

^

^

^

^

S^ a

^

1

2 ln

( )

(^

)^

(^

)

T

P

c



^

^





a^

a^

a

x^

x^

x^

S^

x^

x 1/ 2 (^1) / 2

1

( )

exp[

2

(

)

T

n

a

P

^

^

- a^

a^

a

x^

(x - x ) S

(x - x )]

S

1

(^

T ) n

x^

x

x

where

S

is thea

a priori

error covariance matrix describing error statistics on (

x-x

)a^

In log space:

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OBSERVATIONAL ERROR COVARIANCE MATRIX

^

^

i^

m

y^

F x

ε^

+^

observation

true value

fwd model errorinstrument error

observational error

i^

m

ε^

= ε

+

ε

Observational error covariance matrix

1

1

1 var(

)^

cov(

,^

)

cov(

,^

)^

var(

n )

n^

n





^



^





^



^



 ^



^



^



S



^







is the sum of the instrument and fwd model error covariance matrices:

i^

m

ε^

ε^

ε

S

= S

+ S

How well can the observing system constrain the

true value

of

x

?

1

2

2 ln

(^

)^

(^

)^

(^

)

T

P

c

 

^

^

^

^



y | x

y^

Kx

S

y^

Kx

Corresponding pdf, in log space:

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PARALLEL BETWEEN VECTOR-MATRIX AND SCALAR SOLUTIONS:

Scalar problem

Vector-matrix problem

^

^



2

2

2

2

2

2

2

1

ˆ^

(^

(^

/^

ˆ^

a^

a

a a a

a

x^

x^

g y

kx

k

g^

k

k

x^

ax

a x

g

a^

gk

   ^



^

^

^

 

^



^

^

^

^

^

^

^

MAP solution:

1 1

ˆ^

(^

(^

ˆ^

(^

ˆ^

(^

T^

T

T







^

^



^

^

^

^

^

^

^

a^

a

a^

a 1

(^1) a

n^

a

x^

x^

G y

Kx

G

S K

KS K

S

S^

K S

K

S

x^

Ax

I^

A x

G

ε

A

GK

Gain factor: A posteriori error:Averaging kernel:Jacobian matrix

G

 x/ y

K =

y/ x

sensitivity of observations to true state

Gain matrix

sensitivity of retrieval to observations

Averaging kernel matrix

^

^

A

x/ x

sensitivity of retrieval to true state

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A LITTLE MORE ON THE AVERAGING KERNEL MATRIX

A

describes the sensitivity of the retrieval to the true state

1

1

1

ˆ^1

ˆ

/^

/

ˆ

ˆ^

ˆ

/^

n /

n^

n^

n

x^

x^

x^

x

x^

x^

x^

x

^

^

^



^



^

^



^



^



^

^



^

^

^



^



x

A

x



^







ˆ^

(^

)









n^

a

x^

Ax

I^

A x

G

ε

and hence the smoothing of the solution:

smoothing error

retrieval error

MAP retrieval gives

A

as part of the retrieval:

1

(^

)

T^

T







a^

a

A = GK = S K

KS K

S

K

Other retrieval methods (e.g., neural network, adjoint method) do

not

provide

A

# pieces of info in a retrieval = degrees of freedom for signal (DOFS) =

trace(A)

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INVERSE ANALYSIS OF MOPITT AND TRACE-P (AIRCRAFT) DATA

TO CONSTRAIN ASIAN SOURCES OF CO

TRACE-P CO DATA

(G.W. Sachse)

Bottom-upemissions(customizedfor TRACE-P)

Fossil and biofuel

Daily biomass burning(satellite fire counts)

GEOS-Chem CTM

MOPITT CO

Inverseanalysis

validation

chemicalforecaststop-downconstraints

OPTIMIZATION OF

SOURCES

Streets et al. [2003]

Heald et al. [2003a]

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COMPARE TRACE-P OBSERVATIONS

WITH CTM RESULTS USING

A PRIORI

SOURCES

Model is low north of 30

oN: suggests Chinese source is low

Model is high in free trop. south of 30

oN: suggests biomass burning source is high

-^ Assume that Relative Residual Error (RRE) after bias is removeddescribes the observational error variance (20-30%) •^ Assume that the difference between successive GEOS-Chem COforecasts during TRACE-P (

t^ +48h and o

t^ o

+ 24 h) describes the covariant

error structure (“NMC method”)

Palmer et al. [2003], Jones et al. [2003]

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COMPARATIVE INVERSE ANALYSIS OF ASIAN CO SOURCES

USING DAILY MOPITT AND TRACE-P DATA

-^

MOPITT has higher information content than TRACE-P because it observes sourceregions and Indian outflow

-^

Don’t trust a posteriori error covariance matrix; ensemble modeling indicates 10-40%error on

a posteriori

sources

Heald et al. [2004]

CO observations from Spring 2001, GEOS-Chem CTM as forward model

TRACE-P Aircraft CO

MOPITT CO Columns

4 degreesof freedom

10 degreesof freedom

(from validation)

“Ensemble modeling”: repeat inversion with different values of forward modeland covariance parameters to span uncertainty range

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Analytical solution to inverse problem

1

ˆ^

(^

)^

(^

)

T^

T













a^

a^

a^

a

x^

x^

S K

KS K

S

y^

Kx

requires (iterative) numerical construction of the Jacobian matrix

K

and matrix operations of dimension

(mxn)

; this limits the size of

n , i.e.,

the number of variables that you can optimizeAddress this limitation with Kalman filter (for time-dependent

x

)

or with adjoint method

-^

1

(^

)^

(^

T

J

 

^

^

^

x^

a^

a

x^

S^

x^

x^

K S

Kx - y



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ADJOINT INVERSION (4-D VAR)

a

^2

^1

^3

x^2 x^1

x^3 x a Minimum of cost function

J

-^

1

( )

2

(^

)^

2

(^

( )

)

T

J

 

^

^

^

^



x^

a^

a

x^

S^

x^

x^

K S

F x - y

0

Solve

numerically rather than analytically1. Starting from

a priori

x

calculate , (^) a

(^

) J 

x^

xa

2. Using an optimization algorithm (BFGS),

get next guess

x

1

3. Calculate

(^

) J  x^

x^1

, get next guess

x

2

4. Iterate until convergence

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NUMERICAL CALCULATION OF COST FUNCTION GRADIENT

-^

1

(^

)^

(^

T

J

 

^

^

x^

a^

a

x^

S^

x^

x^

K S

F x - y

Adjoint model is applied to error-weighted difference between model and obs…but we want to avoid explicit construction of

K

( )^

( )^

( -1)

(1)^

(0)

(0)

( -1)

( -2)

(0)

(0)

...

i^

i^

i

i^

i

^

^

^

^



^

 ^

^

^

^



y^

y^

y^

y^

y

K

x^

y^

y^

y^

x

( )^

( -1)

(1)^

(0)

(0)

(1)^

(^ -1)

(^ )

( -1)

( -2)

(0)

(0)

(0)

(0)

(^ -2)

(^ -1)

...

...

T^

T^

T^

T^

T

i^

i^

n^

n

T

i^

i^

n^

n

^

^

^

^

^

^

^

^

^



^

^

^

^

^

^

^



^



^

^

^

^

^

^

^

^

^



^

^

^

^

^

^

^

^

^



^

^

^

^

^

^

^



^

^

^

^

^

^

^

^

^



y^

y^

y^

y^

y^

y^

y^

y

K

y^

y^

y^

x^

x^

y^

y^

y

and since (

AB)

T^ = B

T A

T ,

Apply transpose of tangent linear model to the adjoint forcings; for time interval[ t^0

, t ], start from observations at n

t^ n

and work backward in time until

t^0

,^ picking up

Construct new observations (adjoint forcings) along the way.

tangent linear model

( )^

(^

/ i^

i 

^

y^

y

describing evolution of concentration field over time interval [

ti-

, t

] i

Sensitivity of

y

)^ (i to x

) ( 0 at time

t^0

can then be written

of forward model

adjoint

“adjoint forcing”

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