Monte Carlo Sampling to Inverse Problems

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Monte Carlo Sampling to Inverse Problems

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

1st Orfeus workshop Waveform inversion
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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.

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Direct and Indirect measurements

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Error analysis
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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
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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) )?
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Solving invers problems
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Sampling a posteriori PDF

• Grid search
• Near neighborhood algorithm
• Blind random sampling
• Guided Monte Carlo (SA, GA,...)?
• Markov Chain Monte Carlo

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Ilustration back projection
d
D
m
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Data errors
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Theoretical errors
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A priori uncertainties
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Null space

• No information about m in data

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Tomography imaging
Average model
Maximum Likelihood
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Tomography imaging - errors
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Tomography PDF
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Inspecting PDF
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Source time function inversion
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Source time function inversion
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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