Geophysical%20Inverse%20Problems

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Geophysical Inverse Problems

• Guy Masters
• Cider 2008

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Anatomy of a GIP

• Existence -- mainly of mathematical interest
• Uniqueness -- nope -- at least for linear inverse
  problems
• Construction -- what we are all obsessed with
• Evaluation -- what we should be obsessed with

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Types of GIPs

• Linear -- (rare -- usually uninteresting)
• Mildly non-linear (linearizable)
• Strongly non-linear (not discussed here -- model
  space search algorithms, e.g. GA, EP, etc)

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Are all data really independent?
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Rank and winnow

• Take combinations of data which are orthogonal
• Rank in order of decreasing precision
• Reject data combinations which have excessively
  large errors (from 200 original data get about 60
  useful combinations)

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What about model evaluation?
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Resolving kernel for density at 3000km
0.1 error
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0.5 error
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In practice -- too much data -- therefore
parameterize
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(With generalized inverse, can compute errors and
resolution)
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Matrices are usually too big to do direct
inversionUse iterative inversion
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Example -S-wave mantle tomography

• Inversion of large travel-time, surface wave, and
  free oscillation data sets
• Voxel parameterization -- 4 degree lateral
  dimension, 100km thick in upper mantle and 200km
  thick in lower mantle
• 50,000 model parameters, 400,000 data

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Shear velocity -- -1 isovelocity surfaces
Includes S and SS cluster analysis data
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How well resolved?

• Biggest issue is data coverage and mixture of
  data types
• Less important is theory (e.g. banana-donut
  kernels) -- important for some kinds of
  structures
• What happens without surface waves?

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Checkerboard tests
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Typical errors from a monte-carlo type
calculation (order of magnitude smaller
than the models)
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Adding diffracted phases
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Diffracted Waves

• Coverage of CMB poor in existing models
• Diffracted waves travel along base of mantle
• Lots of data

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Coverage of S and P
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Coverage with Diffracted
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Models
S Model
P Model
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S Resolution
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P Resolution
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Final comments

• Choice of final model is still somewhat
  subjective (would be less so if we really knew
  the errors on our data)
• Your ability to recover structure depends mainly
  upon data coverage. Smoothing is still necessary
  -- leads to tradeoff between resolution and error
• You get to investigate this yourself on Monday!

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(Beware the tomography police!)