Statistical Journey to the Center of the Earth

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Statistical Journey to the Center of the Earth

• Ping Ma
• Department of Statistics
• Center for the Advanced Study
• University of Illinois Urbana-Champaign

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Development of Earth Science

• From theory-based science to data-driven science
• Hundreds of earthquakes occur throughout the
  world every day

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Objective

• To learn about Earth structure and dynamics, and
  the physical processes controlling earthquakes
  and volcanoes.

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Current status of research

• Methods that are geophysical sound, mathematical
  elegant, statistical rigorous are still of
  rarity.

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Earthquake Waves
Van der Hilst et al (2007)
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Isochrons and image point gathers
Van der Hilst et al (2007)
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Generalized Radon Transform in Nutshell

• Waveform measurement at time
  and receiver coordinate
• Travel time function
  with source coordinate
• Objective get image at a depth with coordinate
• GRT consists of integrating over
  a set of curves

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Generalized Radon Transform in action

• Factors need to be taken into account
• (1) source-receiver geometry
• slowness vectors and their projections
• (2) properties of Earths mantle and crust
• Elasticity and anisotropy
• stiffness and mass density

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Generalized Radon Transform in action

• Start with inhomogeneous wave equation of body
  force
• The solution is the Greens function
• Approximations are applied to get computable
  formulae.

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Interpretation of Image

• Geophysical view the stack of isochrons.
• Mathematical view weight average of wave
  functions.
• Statistical view the strength of the reflectors.

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(Wang et al., JGR, 2006)
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Statistical image processing

• Let data speak

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Statistical considerations

• The images are smooth since we start with wave
  equation
• The images are correlated
• Automatic

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Mixed-Effect Model

• The image point gathers at angle and depth
  is

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Penalized Hendersons likelihood
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Reproducing Kernel Hilbert Space

• The general Hilbert space
• Reproducing Kernel Hilbert space
• The evaluation functional is continuous

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Solution in Reproducing Kernel Hilbert Space
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Model Selection

• We may choose them through minimization of
  Generalized Cross-Validation (GCV)
• The asymptotic optimality of GCV was shown by Gu
  and Ma (2005 Annals of Statistics) in a
  decision-theoretic framework

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Bayesian confidence intervals

• Uncertainty estimates for estimated image over
  the whole depth range rather at single depth
• Bayesian confidence intervals have a certain
  across-the function coverage properties.

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(Wang et al., JGR, 2006)
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(Ma et al., JGR, 2007)
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Two-D image
Juxtaposition
(Ma et al., JGR, 2007)
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Reference

• Wang, P., de Hoop, M. V., van der Hilst, R. D.,
  Ma, P., and Tenorio, L. (2006) Imaging of
  structure at and near the core mantle boundary
  using a generalized Radon transform 1.
  Construction of image gathers, Journal of
  Geophysical Research, 111, B12304.
• Ma, P., Wang, P., Tenorio, L., de Hoop, M. V.,
  and van der Hilst, R. D. (2007) Imaging of
  structure at and near the core mantle boundary
  using a generalized Radon transform 2.
  Statistical inference of singularities, Journal
  of Geophysical Research , 112, B08303.
• van der Hilst, R.D., de Hoop. M.V., Wang, P.,
  Shim,S.-H., Ma, P., Tenorio, L. (2007)
  Seismo-stratigraphy and thermal structure of
  Earth's core-mantle boundary region, Science,
  315(5820), 1813-1817.

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Acknowledgement

• Beckman Fellowship, Center for the Advanced Study
  at University of Illinois at Urbana-Champaign
• NSF DMS 0723759 CMG