1. Monte Carlo Sampling to Inverse Problems Wojciech Dębski Inst. Geophys. Polish Acad. Sci. debski@igf.edu.pl 1 st Orfeus workshop: Waveform inversion

2. Introduction The most often encountered seismological (geophysical) inverse problems can be stated as a parameter estimation problem: having a given set of data and knowing the relation between the data and model parameters, what are the values of the sought parameters? Today, the question ``what are the values'' should be understood not only in terms of obtaining the numerical values but also as the task of estimating their uncertainties In this presentation some aspects of the modern, probabilistic approach to inverse problem which can deal with this task is presented. Theoretical aspects are illustrated by some applications.

3. Direct and Indirect measurements

4. Error analysis

5. Source of uncertainties Direct measurements Indirect measurements

6. A posteriori PDF Tarantola and Vallet (1982) have shown how to manage different source of uncertainties and join them into the final error estimates – the a posteriori PDF

7. Construction a posteriori PDF Bayesian Inverse theory solves the inverse problem by building the a posteriori probability distribution σ(m) over the model space M which describes the probability of a given model being the true one σ(m) = const. f(m) L(m, d)‏ L(m, d) = exp( -|| d - d (m) ||)‏ obs

8. Solving invers problems

9. Sampling a posteriori PDF Grid search Near neighborhood algorithm Blind random sampling Guided Monte Carlo (SA, GA,...)‏ Markov Chain Monte Carlo

10. Ilustration – back projection D m d

11. Data errors

12. Theoretical errors

13. A priori uncertainties

14. Null space No information about m in data

15. Tomography imaging Maximum Likelihood Average model

16. Tomography imaging - errors

17. Tomography – PDF

18. Inspecting PDF

19. Source time function inversion

21. Conclusions Probabilistic (Bayesian) approach allows an exhaustive error analysis. MCMC is an efficient sampling technique which can be used within the probabilistic inversion. Solution of any inverse task must include an estimation of inversion errors