3-D Tomographic Reconstruction

Transcription

1 Mitglied der Helmholtz-Gemeinschaft 3-D Tomographic Reconstruction of Atmospheric Trace Gas Concentrations for Infrared Limb-Imagers February 21, 2011, Nanocenter, USC Jörn Ungermann

2 Tomographic trace gas retrievals Outline Forward problem Trace gasses in the atmosphere Measurement principle 2-D/3-D Tomography Inverse problem Minimisation and iterative solvers Regularisation Diagnostics Numerical example Air-borne 3-D tomography including advection February 21, 2011, Nanocenter, USC Slide 2

3 FORWARD PROBLEM February 21, 2011, Nanocenter, USC Slide 3

4 Atmospheric trace gas concentrations Small scale structures (C. Varotsosa et al., Comparison of vertical ozone profiles as deduced from remote and in situ sensing techniques, doi: / ) February 21, 2011, Nanocenter, USC Slide 4

5 Atmospheric trace gas concentrations Small scale structures February 21, 2011, Nanocenter, USC Slide 5

6 Radiative transport for infrared atmospheric limb-emission sounding February 21, 2011, Nanocenter, USC Slide 6

7 Radiative transport for infrared atmospheric limb-emission sounding Temperature Pressure Trace gas concentration February 21, 2011, Nanocenter, USC Slide 7

8 Radiative transport for infrared atmospheric limb-emission sounding February 21, 2011, Nanocenter, USC Slide 8

9 Radiative transport for infrared atmospheric limb-emission sounding A simplified model: Trace gas concentration February 21, 2011, Nanocenter, USC Slide 9

10 Atmospheric trace gas concentrations Exemplary conventional retrieval February 21, 2011, Nanocenter, USC Slide 10

11 Tomographic retrievals 2-D tomography from satellite-borne instruments (Riese et al., GLObal limb Radiance Imager for the February 21, 2011, Nanocenter, USC Atmosphere (GLORIA): Scientific objectives, 2005) Slide 11

12 Tomographic retrievals 2-D tomography from satellite-borne instruments February 21, 2011, Nanocenter, USC Slide 12

13 Tomographic retrievals 3-D tomography from air-borne instruments February 21, 2011, Nanocenter, USC Slide 13

14 Tomographic retrievals 3-D tomography from air-borne instruments February 21, 2011, Nanocenter, USC Slide 14

15 Tomographic retrievals 3-D tomography from air-borne instruments February 21, 2011, Nanocenter, USC Slide 15

16 Tomographic retrievals 3-D tomography from air-borne instruments February 21, 2011, Nanocenter, USC Slide 16

17 INVERSION February 21, 2011, Nanocenter, USC Slide 17

18 Remote sensing of the atmosphere An inverse problem temperature, trace gasses, etc. spectral radiances (NASA) February 21, 2011, Nanocenter, USC Slide 18

19 Retrieval An inverse problem F is a forward model that maps a given atmospheric state onto synthetic measurements x is the vector of unknowns (state space) b are modeled parameters, which are assumed to be known y is a vector of measurements (measurement space) Invert Often ill-posed! y is the vector of true measurements ε is a vector of errors February 21, 2011, Nanocenter, USC Slide 19

20 Retrieval for large-scale inverse problems By means of minimizing a cost function S ε x a S a covariance matrix of the measurement errors a priori expectation about the state of the atmosphere a priori covariance of the state of the atmosphere The cost function J shall be minimized: Open issue: Use of different regularization norms (l p, TV, )? February 21, 2011, Nanocenter, USC Slide 20

21 Retrieval for large-scale inverse problems Truncated Quasi-Newton / Conjugate Gradients Newton iteration: February 21, 2011, Nanocenter, USC Slide 21

22 Retrieval for large-scale inverse problems Truncated Quasi-Newton / Conjugate Gradients Levenberg-Marquardt: Newton iteration: Fʹ(x i ) S a -1 S ε -1 Only 1 to 2 percent of entries occupied Sparse for autoregressive or Tikhonov approach Diagonal, block-diagonal or block-toeplitz matrix February 21, 2011, Nanocenter, USC Slide 22

23 Retrieval for large-scale inverse problems Computation of Jacobian matrix Split forward model F into m independent functions f i Use adjoint model based on operator overloading to calculate gradient of each f i (STCE of RWTH Aachen, Germany) Assemble F Total cost: O(F)! February 21, 2011, Nanocenter, USC Slide 23

24 Retrieval for large-scale inverse problems An exemplary 3-D Jacobian matrix 99% empty February 21, 2011, Nanocenter, USC Slide 24

25 Retrieval for large-scale inverse problems Truncated Quasi-Newton / Conjugate Gradients Evaluation of Levenberg-Marquardt step: Use sparse matrix representations and Conjugate Gradients (etc.) to solve LES by cheap evaluations of February 21, 2011, Nanocenter, USC Slide 25

26 Retrieval for large-scale inverse problems Performance of iterative linear equation system solvers Performance for exemplary 3-D problem Method iterations A T r 2 κ CG CGLS LSQR PCG , PCGLS , PLSQR , Open issues: Good matrix-free preconditioners? (currently Jacobi) LSQR or other solvers? February 21, 2011, Nanocenter, USC Slide 26

27 Retrieval for large-scale inverse problems Regularization L 0 L x 1 α Identity matrix, scaled with standard deviations scaled 1st order finite difference matrix Tuning parameters (α 0 = 0.1, α 1x = α 1y = 0.8 km/ppb, α 1z = km/ppb) Open issues: Regularisation across boundary layers (Mumford-Shah?) Selection of tuning parameters February 21, 2011, Nanocenter, USC Slide 27

28 Retrieval for large-scale inverse problems Linearised diagnostics Diagnosis is based on first order Taylor approximation: with x t being the true state, x f the result and A quantifies measurement content and resolution G quantifies influence of uncertainties G and A can be evaluated row-by row using CG Open issue: how to efficiently provide full diagnosis February 21, 2011, Nanocenter, USC Slide 28

29 Retrieval for large-scale inverse problems Diagnostics - resolution February 21, 2011, Nanocenter, USC Slide 29

30 NUMERICAL EXAMPLE February 21, 2011, Nanocenter, USC Slide 30

31 Tomographic retrievals 3-D tomography from air-borne instruments February 21, 2011, Nanocenter, USC Slide 31

32 Tomographic 3-D ozone retrieval Characteristics Measurement grid: images Combine detector pixels to one super-pixel observations ( pencil beams) Pan step: 4 degrees, flight path: circle, radius: 200 km Atmospheric grid: ~ 10 km 10 km 0.25 km unknowns Initial guess: polar profile, a priori: mid-latitude profile Regularization: L-curve-optimal 3-D Tikhonov SP-Noise: offset: 1.875e-6 W/m 2 sr cm -1, gain: 0.1% February 21, 2011, Nanocenter, USC Slide 32

33 Tomographic 3-D ozone retrieval Atmospheric situation February 21, 2011, Nanocenter, USC Slide 33

34 Tomographic 3-D ozone retrieval Perfect wind speed knowledge February 21, 2011, Nanocenter, USC Slide 34

35 Tomographic 3-D ozone retrieval AVK (0.00E, 46.00N, 12.00km) February 21, 2011, Nanocenter, USC Slide 35

36 Tomographic 3-D ozone retrieval AVK (0.00E, 44.00N, 12.00km) February 21, 2011, Nanocenter, USC Slide 36

37 Tomographic 3-D ozone retrieval Perfect wind speed knowledge - resolution February 21, 2011, Nanocenter, USC Slide 37

38 Tomographic trace gas retrievals Summary Tomography is an new but expanding topic in atmospheric sciences (for limb sounding but also for nadir sounding) Large scale 3-D tomographic retrievals are feasible by employing sparse matrices, iterative solvers, and adjoint methods, 3-D structures of atmospheric tracers can be reproduced using a simple model and synthetic measurements Using synthetic GLORIA measurements, a vertical resolution of ~ m and a horizontal resolution of ~30 km seems achievable February 21, 2011, Nanocenter, USC Slide 38

39 Tomographic trace gas retrievals Open issues Regularisation across boundary layers (Mumford-Shah?) Selection of regularization tuning parameters Use of different regularization norms (l p, TV, )? Solving linear equation systems Good matrix-free preconditioners? (currently Jacobi) LSQR or similar algorithms? Different optimizers Registration of images (on-line correction) Optimal selection of frequencies Optimal scanning pattern Convergence in presence of local minima February 21, 2011, Nanocenter, USC Slide 39