Monte Carlo Sampling to Inverse Problems

1
Monte Carlo Sampling to Inverse Problems

• Wojciech Debski
• Inst. Geophys. Polish Acad. Sci.
• debski_at_igf.edu.pl

1st 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 s(m) over the model space M which
describes the probability of a given model being
the true one
s(m) const. f(m) L(m, d)?
obs
L(m, d) exp( - d - d (m) )?
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
D
m
11
Data errors
12
Theoretical errors
13
A priori uncertainties
14
Null space

• No information about m in data

15
Tomography imaging
Average model
Maximum Likelihood
16
Tomography imaging - errors
17
Tomography PDF
18
Inspecting PDF
19
Source time function inversion
20
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