A New MCMC Algorithm For Seismic Parameter

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A New MCMC Algorithm For Seismic Parameter
Estimation and Uncertainty Analysis Tiancong
Hong Mrinal K. Sen Institute for
Geophysics The Jackson School of Geosciences The
University of Texas at Austin Feb. 27, 2007
A New MCMC Algorithm For Seismic Parameter
Estimation and Uncertainty Analysis
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OUTLINE

• Objectives
• Inverse problem
• Bayesian inference and uncertainty analysis
• MCMC algorithms
• Applications of a new MCMC
• Conclusion

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OBJECTIVES

• A powerful MCMC algorithm for nonlinear
  high-dimensional
• seismic inverse problems

• Handling model parameterization issue

• Leading to an accurate quantitative uncertainty
  analysis

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• Bayesian inference framework

Forward problem
Inference problem
Posterior Probability Density (PPD)
(Tarantola 2005)

• All consistent models are fully characterized by
  p(m)

• p(m) provides not only the likely values of
  model parameters
• but also the likelihoods of those values to be
  taken by the
• parameters

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• Bayesian inference a means to handle
  situations where
• information is incomplete or inaccurate and a
  deterministic
• reasoning cannot be made and the plausibility
  of competing
• models and uncertainties are assessed by the
  constructed
• PPD (Gregory 2005). Back to seismic

• Bayes Theorem

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• Prior information

Prior summarizes available information
independent of data

• Likelihood Function

i). Depends on noise/error distribution
ii). It is safe to assume the most conservative
Gaussian error
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• PPD

i). The denominator is the integral over entire
model space, analytical calculation is
impractical for high-dimensional case
ii). Need methods to derive the PPD while
circumventing direct calculation of the
denominator integration (MCMC)
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• Mean, model covariance and marginal PPD

Mean
Covariance
Marginal PPD

• All these fall into the general form

• Efficient, accurate method is needed (MCMC)

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• Constructs a Markov chain that has the target
  PPD as its
• stationary distribution

• Markov chain given the present state, the
  future states are
• independent of the past states

• A Markov chain has a stationary distribution if
  and only if

i). Irreducibility every state is reachable
from other states ii). Aperiodicity a state can
recur at each next step, it helps stop the chain
from oscillating between different states
in a regular periodic movement
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• Most common MCMCs Metropolis-Hastings, Gibbs
  sampler

• Easy to implement
• Take very long time to converge (NOT efficient)

• Genetic Algorithm, and Simulated Annealing have
  been proved
• to converge to a stationary distribution, and
  therefore can
• be used as sampling tools to construct PPD

• Efficient
• Only approximate the PPD (NOT accurate)

Accuracy and efficiency still remain critical
issues!!!!
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Original Features

• Multi-scaling takes advantage of benefits from
  both the coarse
• scale and the fine scale

• Further improve the efficiency and accuracy by
  multiple chains
• and exchanging information between for
  intelligent realizations

• Use Multi-scaling to overcome the model
  parameterization
• issues

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• 9000 samples are drawn by the MCMC
• Distribution of sample points matches the
  contours of true PPD
• More samples over the peak regions

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i). Seismic waveform inversion in (x, t) domain
ii). The calculated data from 80-sample well logs
are used as observed gathers for inversion
of Vp, Vs, and density
iii). 1D earth model is parameterized into 4
scales 10, 20, 40 and 80 layers
iv). A reflectivity algorithm is used for
modeling of seismic wave propagation
v). Low frequency trends of well logs are used as
prior information to define model space
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• Goodness of fit between observed and synthetic
  data

• Likelihood function (Fitness function)

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• Inverted Vp structures

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• Histories of fitness values

• Fast convergence and very good estimation are
  obtained
• Multi-scaling improves accuracy and efficiency
  by maximizing the fitness
• function

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• Inverted seismograms

• Coarse scales result in bad datafit, but speed
  up convergence
• Multi-scaling facilitates the fine scale models
  to the actual earth model

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• Uncertainty Analysis

Marginal PPDs of Vp
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• Uncertainty Analysis

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i). The new multiscale GA based MCMC improves
estimation accuracy and efficiency
ii). Multiscaling is particularly attractive in
addressing layer definition in seismic
waveform inversion
iii). The new MCMC provides both the
best-fitting models and a convenient way
to quantify uncertainties
iv). The inverted Vp, Vs and density can be
further used for accurate estimation of
reservoir petrophysics and corresponding
uncertainty analysis