Inverse modeling in a linear algebra framework

State vector x (dimension n) Observation vector y (dimension m)

1

2

n

x

x

x

1

2

m

y

y

y

correlationsbetween vectorelements

Use m observations to constrain a n-dimensional state vector

Individual observations influenced By multiple state vector elements

General Bayesian solution to the inverse problem

State vector 1( ,... ) Tnx xx

Observation vector 1( ,... ) Tmy yy

Forward model y F(x)

Prior xA with prior error covariance matrix SA

Observation ywith observational error covariance matrix SO

prior pdf P(x) observational pdf P(y | x)

ln ( ) ln ( ) ln ( ) P P Px | y x y | x

ln ( | ) Px x y 0solve

pdfs for vectors and error covariance matrices

1/2/2

1 1( ) exp[ ( [ ]) ( [ ])]

2(2 )T

nP E E

-1x x - x S x - x

S

Gaussian pdf for vector x with expected value E[x] and error

Error covariance S:

1 1

1

var( ) cov( , )

[ ]

cov( , ) var( )

nT

n n

E

S εε

1 2( , ,... )Tn ε

Optimal estimate solution

ln ( ) ( ) TP -1A A Ax x x ) S (x - x

1ln ( ) ( ) ( ) TP Oy | x y F(x) S y F(x)

prior pdf

observational pdf

Minimize cost function:

-1 1( ) ( ) ) ( ) ( ) A A A Ox x x S (x - x y F(x) S y F(x)T TJ

solve -1 1( ) 2 ( ) 2 ( ) x A A x Ox S x x F S F(x) - y 0T

J

Linear forward model allows analytical solution

y = F(x) = Kx where

1 1 1

1

/ /

/ /

n

m m n

y x y x

y x y x

K Is the Jacobian matrix

ˆ A Ax x + G(y - Kx )Solution: 1( )T T A A OG S K KS K S

-1 -1 1ˆ ( T O AS K S K + S )

with

gain matrix

posterior error covariance matrix

ˆ ( ( ) n A Ox x I A) x x + Gε

Relate solution to true value:

with A = GKaveraging kernel matrixsmoothing

errorobservational

errortruth

A little more on the averaging kernel matrix

A describes the sensitivity of the retrieval to the true state

1 1 1

1

ˆ ˆ/ /ˆ

ˆ ˆ/ /

n

n n n

x x x x

x x x x

xA

x

and hence the smoothing of the solution:

Analytical inversion gives A as part of the solution:

1 ˆ( ) T T -1A A O n AA = GK = S K KS K S K I - SS

The trace of A gives the # of independent pieces of information in the inversion – also called the degrees of freedom for signal (DOFS)

ˆ ( ( ) n A Ox x I A) x x + Gεsmoothing

errorobservational

errortruth

Before making any observations, A can diagnose the utility of an observing systemfor constraining the state vector

Example 1: constraining Asian CO emissions using aircraft

Prior bottom-up emission inventory

Observed CO concentrations (229 flight hours, March-April 2001)

Forward model: GEOS-Chem CTM

Emission state vector (n = 5) selected from candidates using averaging kernel matrix

1. China (fuel) 2, China (fires), 3. Korea+Japan4. Southeast Asia 5. Rest of world

Palmer et al. [2003]

Source: combustion Sink: atmospheric oxidation (lifetime 2 months)

Error characterization for the inversion

• Prior error on bottom-up inventory (SA): activity rates emission factors

1. China (anthropogenic) 109 ± 42 Tg CO yr-1 2. China (biomass burning 19 ± 103. Korea + Japan 19 ± 4 4. Southeast Asia 136 ± 545. Rest of World 1888 ± 355

• Observational error covariance matrix:

RO I MS = + S + SS

• Observational error = 20-30% (next slide), with spatial correlation scale 200 km

– Instrument error (precision) = 1%– Representation error = 5% (from observed fine-scale variability of CO)

instrumentrepresentation

forward model

Forward model error 20-30%

Residual error method to construct observational error covariance matrix

Compare observations to forward model simulation with prior sources

AO Ay - F(x ) y - Fε - (x= )difference mean bias to be corrected

In inversion

Residual defines observational error:

Model simulation of CO with prior sources – aircraft observations

Residual errorof 20-30%

Averaging kernel matrix

1

1 1 1

ˆ ˆ/

ˆ ˆ/

/

/

n n

n

n

x x

x x x

x

x

xChina fuel (CHBFFF)

Korea +Japan (KRJP)Southeast Asia (SEA)

China biomass burning (CHBB)

Rest of World (RW)

CHBFFF KRJP SEA CHBB RW

• Very strong constraint on China fuel, Southeast Asia, Rest of World

• Strong constraint on China biomass burning

• Weak constraint on Korea +Japan

Example 2: satellite remote sensing of carbon monoxide

MOPITT thermal infrared instrument On NASA Terra satellite

CO columns from MOPITT (March-April 2001)

IR absoption spectrumof Earth’s atmosphere

Atmospheric sounding in the thermal IR

absorbing gasdz

z

atmospherictransmittance L

Lo 1

B(l,To)

B(l,T(z))dt

Satellite measures

0

( ) ( , )

( , ) ( , ( ))

o oI B T L

dL zB T z dz

dz

Observed spectra contain information on vertical profile n(z) but problem is generally underconstrained

( , ) ( )

( ) exp[ ( )] exp[ ( )]

d z n z

L z z dL z d

optical depth

Blackbody function

MOPITT retrieval

• Observation vector: radiances in the 4.6 µm channel

CO transmittances

CO2, O3, N2Otransmittances

Typical top-of-atmosphereobserved radiance spectrum

• State vector: CO mixing ratios at 7 levels (surface, 850, 700, 500, 350, 250, 150 hPa)• Prior: climatological vertical profile

Typical MOPITT averaging kernel matrix

TypicalIdeal

Lines of different colors represent different rows of the matrix

0 1

Averaging kernel

CO vertical profile

true

smoothedby avker

MOPITT(symbols)

Example 3: would a satellite CO2 sensor to constrain CO2 fluxesgain from an added capability to measure CO at the same time?CO2:CO error correlationsin GEOS-Chem

state vector of carbon fluxes

CO2-only inversionJoint CO2:CO inversion

[ ] [ ][ ]

[ ] [ ]

T TT

T T

E EE

E E

CO2 CO2 CO2 CO

O,CO2 CO2 CO2 O,CO2:CO

CO2 CO CO2 CO2

ε ε ε εS ε ε S

ε ε ε ε

CO2:CO error correlation

H. Wang et al. [2009]

The CO2:CO combination is useful in non-growing season and for fires, less so in growing season

ˆ ˆ) ( )CO2:CO CO2S / Sdiag( diagRatio of posterior error variances

betterthan 30%improvement

H. Wang et al. [2009]

Analytical solution of inverse problem requires small matrices

Observation vector is no problem as uncorrelated packets can be ingested sequentially

0

0

O,1

O,2O

O,p

S

SS

S

Difficult for state vector:

xAˆ

1x ˆ2x x̂packet 1

SO,1

packet 2

SO,2

• Assume that observations have limited zones of influence• Full Jacobian must still be constructed

Chemical data assimilation withsimple mapping forward model(Kalman filter)

Analytical data assimilation using Kalman filter

prior xA,0 ± Sa,0

to

time observations state vector

y0ˆˆ 0 0x S

Apply evolution model M for [t0, t1]:

ˆ ˆ

ˆ

A,1 0 M

A,1 0 M

x = Mx ± S

S = MS M ST

t1y1 1 1

ˆˆ x S

Apply evolution model for [t1, t2]…etc.

simple mapping Observation at t0;

initialization of forecastfor [t0, t1]

Forecast for t1

Forecast for t2

Observation at t1;initialization of forecast

for [t1, t2]

Example: observation system simulation experiment (OSSE)for geostationary observation of ozone air quality

TEMPO Sentinel-4 GEMS

Geostationary constellation to be launched in 2018-19

Wavelength (nm)

O3

Ab

sorp

tio

n c

ross

sec

tio

n (

cm2 )

TEMPO will include first ozone observation in the weak visible Chappuis bands:

surface

~3 kmair

scatteringthermalcontrast

UV IR Vis

Ozone spectrum

Will TEMPO improve our ability to monitor/forecast ozone air quality in US?

Intermountain West: new frontier for US ozone air quality

EPA [2014]

Ozone over NE Pacific(INTEX-B, Apr-May 2006)

ppb

observedmodel Downwelling of high background ozone

over the Intermountain West

4th highest annual 8-h average ozone, 2010-2012

current standard: 75 ppbproposed: 60-70 ppb

Spring ozone trend, 1990-2010

Can we use TEMPO to monitor/forecast high ozone events in Intermountain West, and separate domestic from background influences?

First step: Build virtual model of TEMPO instrument, produce synthetic observations

Pre

ssur

e, h

Pa

0 1Sensitivity

DOFS

Use a CTM to produce A virtual atmosphere

Sample this virtual atmosphereon TEMPO observing schedule

Use TEMPO averaging kernel matrix to simulate what TEMPO would see

TEMPO synthetic ozone data

2nd step: assimilate synthetic TEMPO data with separate CTM

“True” model

AM3-Chem

InstrumentSpecification

Synthetic Observations

Forecastmodel

GEOS-Chem

Assimilation(Kalman filter)

OptimalEstimate

Compare to “True” model

Forecast = nested GEOS-Chem

“Truth” = AM3-Chem model

ppbv

0.5ox0.5o

0.5ox0.67o

Zoogman et al. (2014)

The “truth” and assimilation CTMs must be independent

Ability of GEOS-Chem to reproduce “true” surface ozone in Intermountain West

Free-running model

With assimilation of surface data

With assimilation of surface+TEMPO data

Ability of assimilation system to reproduce frequency of high-ozone days

3-month assimilation for April-June 2010

Ability of TEMPO assimilated data to observe stratospheric intrusions in the Intermountain West

Stratospheric ozone intrusion over New Mexico, 13 June 2010