1. Team 2: Final Report Uncertainty Quantification in Geophysical Inverse Problems

2. Statement of the Tomography Problem Observe the arrival time, given the first order physics Invert for the slowness, or density  (x,z)

3. Creating Synthetic Data Our ‘unknown’ earth model is a layered model Vertical/Horizontal observations are important! The observations are calculated as line integrals through the synthetic model.

4. Choose A Model There are many choices for model! Each choice leads to a different solution Each solution can be evaluated for goodness of fit. Haar wavelets provide an easy way to describe a region.

5. Our Model Has Errors! The travel times do not allow us to reconstruct all the details of the layers. We use covariance matrices are used to measure the uncertainty of the model. –Prior covariance matrix is used to account for the model uncertainty without considering the observed traveltimes. –Posterior covariance is accounts for the observed traveltimes.

6. Solving the Inverse Problem for a single choice of model

7. The Prior Distribution “The natural choice for a prior pdf is the distribution that allows for the greatest uncertainty while obeying the constraints imposed by the prior information, and it can be shown that this least informative pdf is the pdf that has maximum entropy” (Jaynes 1968, 1995, Papoulis 1984) From Malinverno, 2000

8. Prior PDF: Mean Surface

9. Prior PDF: Uncertainty Window

10. Colormap Uncertainty Mean

11. Prior PDF: Composite

12. The Data Prediction Matrix For each observation, we calculate the same line integral through our wavelet model. These are the columns. The better the model is, the closer these integrals match up with our observations

13. Data Prediction Matrix

14. Posterior PDF: 3D Histogram Surfaces The posterior mean is the best estimate for the unknown function. The posterior uncertainty allows us to put error bars on this estimate.

15. Posterior PDF: Mean Surface

16. Posterior PDF: Uncertainty Window

17. Posterior PDF: Composite

18. Other Results

19. Solving the Inverse Problem for many choices of model

20. Finding a Best Model Strategy: Check many models and find which ones fit the data the best! Each model has a neighborhood of hereditary models Optimization algorithm: check neighborhood for better model (a single neighbor is selected at random) Run for a fixed amount of time

21. Synthetic Data

22. Prior Mean

23. Prior Uncertainty

24. Prior Mean/Uncertainty Comp.

25. Prior Uncertainty Surfaces

26. Log Marginal Likelihood

27. Optimal Decimation

28. Observed vs. Predicted

29. Posterior Mean

30. Posterior Uncertainty

31. Posterior Mean/Uncertainty