Inverse modeling in a linear algebra framework State vector x (dimension n) Observation vector y (dimension m) correlations between vector elements 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 Observation vector Forward model Prior x A with prior error covariance matrix S A Observation y with observational error covariance matrix S O prior pdf P(x)observational pdf P(y | x) solve

pdfs for vectors and error covariance matrices Gaussian pdf for vector x with expected value E[x] and error Error covariance S :

Optimal estimate solution prior pdf observational pdf Minimize cost function: solve

Linear forward model allows analytical solution y = F(x) = Kx where Is the Jacobian matrix Solution:with gain matrix posterior error covariance matrix Relate solution to true value: with A = GK averaging kernel matrix smoothing error observational error truth

A little more on the averaging kernel matrix A describes the sensitivity of the retrieval to the true state and hence the smoothing of the solution: Analytical inversion gives A as part of the solution: The trace of A gives the # of independent pieces of information in the inversion – also called the degrees of freedom for signal (DOFS) smoothing error observational error truth Before making any observations, A can diagnose the utility of an observing system for constraining the state vector

Example 1: constraining Asian CO emissions using aircraft Prior bottom-up emission inventory MOPITT CO March-April 2001 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+Japan 4. 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 (S A ): activity rates  emission factors 1.China (anthropogenic)109 ± 42 Tg CO yr -1 2.China (biomass burning19 ± 10 3.Korea + Japan19 ± 4 4.Southeast Asia136 ± 54 5.Rest of World1888 ± 355 Observational error covariance matrix: 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) instrument representation 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 differencemean bias to be corrected In inversion Residual defines observational error: Model simulation of CO with prior sources – aircraft observations Residual error of 20-30%

Averaging kernel matrix China 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 spectrum of Earth’s atmosphere

Atmospheric sounding in the thermal IR absorbing gas dz z atmospheric transmittance L LoLo 1 B(,T o ) B(,T(z))d  Satellite measures Observed spectra contain information on vertical profile n(z) but problem is generally underconstrained Blackbody function

MOPITT retrieval Observation vector: radiances in the 4.6 µm channel CO transmittances CO 2, O 3, N 2 O transmittances Typical top-of-atmosphere observed 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 Typical Ideal Lines of different colors represent different rows of the matrix 0 1 Averaging kernel CO vertical profile true smoothed by avker MOPITT (symbols)

Example 3: would a satellite CO 2 sensor to constrain CO 2 fluxes gain from an added capability to measure CO at the same time? CO 2 :CO error correlations in GEOS-Chem state vector of carbon fluxes CO 2 -only inversion Joint CO 2 :CO inversion CO 2 :CO error correlation H. Wang et al. [2009]

The CO 2 :CO combination is useful in non-growing season and for fires, less so in growing season Ratio of posterior error variances better than 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 Difficult for state vector: xAxA packet 1 S O,1 packet 2 S O,2 Assume that observations have limited zones of influence Full Jacobian must still be constructed Chemical data assimilation with simple mapping forward model (Kalman filter)

Analytical data assimilation using Kalman filter prior x A,0 ± S a,0 toto time observations state vector y0y0 Apply evolution model M for [t 0, t 1 ]: t1t1 y1y1 Apply evolution model for [t 1, t 2 ]… etc. simple mapping Observation at t 0; initialization of forecast for [t 0, t 1 ] Forecast for t 1 Forecast for t 2 Observation at t 1 ; initialization of forecast for [t 1, t 2 ]

Example: observation system simulation experiment (OSSE) for geostationary observation of ozone air quality TEMPO Sentinel-4 GEMS Geostationary constellation to be launched in TEMPO will include first ozone observation in the weak visible Chappuis bands: surface ~3 km air scattering thermal contrast 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 observed model Downwelling of high background ozone over the Intermountain West 4 th highest annual 8-h average ozone, current standard: 75 ppb proposed: ppb Spring ozone trend, 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 Pressure, hPa 0 1 Sensitivity DOFS Use a CTM to produce A virtual atmosphere Sample this virtual atmosphere on TEMPO observing schedule Use TEMPO averaging kernel matrix to simulate what TEMPO would see TEMPO synthetic ozone data

2 nd step: assimilate synthetic TEMPO data with separate CTM Forecast = nested GEOS-Chem “Truth” = AM3-Chem model ppbv 0.5 o x0.5 o 0.5 o x0.67 o 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