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Monte Carlo Sampling to Inverse Problems

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A posteriori PDF ... the a posteriori probability distribution ... Sampling a posteriori PDF. Grid search. Near neighborhood algorithm. Blind random sampling ... – PowerPoint PPT presentation

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Title: 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
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