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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