Optimized design of frequencydomain acoustic waveform tomography experiments

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Optimized design of frequency-domain acoustic
waveform tomography experiments

• Hansruedi Maurer and Stewart GreenhalghETH
  Zurich, Switzerland

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Outline

• Waveform sensitivities
• Experimental design of waveform tomography
  experiments
• Suitable data representations
• Choice of temporal frequencies
• Temporal frequencies vs. spatial sampling
• Tests with synthetic data
• Conclusions

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Time-domain sensitivities
Acoustic waveform inversion

• Solution of the forward problem
• finite differences
• finite elements
• spectral elements
• ...

Solution of the inverse problem
Although not necessarily computedexplicitly, the
sensitivities containedin J are essential for
the solution ofthe inverse problem.
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Frequency-domain inversions
Frequency-domain sensitivities

• Seismic data generally band-limited
• Computationally less demanding compared with
  time-domain inversions
• Stationary problem to be solved

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Optimized design of frequency-domain acoustic
waveform inversions

• How can we set up an experiment, with which
  maximum subsurface information can be extracted
  at minimal (field efforts AND computational
  expenses) costs?

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Eigenvalue spectra and model resolution
Threshold
Eigenvalue
Eigenvalue index
Resolved model space
Null space
Characterizes informationthat can be extracted
frominversions!
Characterizes data information content!
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Test model
P-wave velocity 2000 m/sFinite element
mesh 0.15 mInversion mesh 0.30
mFrequencies 100, 200, 500, 750, 1000, 1250,
1500 HzSource-receiver spacings 0.25, 0.5, 1,
2, 5 m
30 m
20 m
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Data representation

• Complex-valued spectra
• Hartley spectra
• Amplitude
• Phase

Which representation is most appropriate?
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Data representation
If possible, full complex-valued spectra should
be considered!
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Choice of temporal frequencies

• Single frequency is likely inappropriate
• Many frequencies offer better resolution, but at
  computationally higher costs
• Large bandwidth may be difficult to achieve

Which combination of frequencies offer best
benefit-to-cost-ratio?
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Choice of temporal frequencies
If the bandwidth is sufficiently wide,
highfrequencies are becoming particularly useful!
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Model resolution of single and cummulative
frequencies

• Model resolution of single frequencies is
  substantially lower than those of cummulative
  frequencies
• Acoustic waveform inversion may be capable to
  resolve features outside of the tomographic plane!

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Temporal frequencies vs. spatial sampling

• Frequency content is more important than spatial
  sampling
• Trade-off between selection of temporal
  frequencies and spatial sampling exist

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Inversion tests

• Cumulative frequencies superior to single
  frequency inversion
• Trade-off between temporal frequencies and
  spatial sampling confirmed
• Features outside of tomographic plane resolved

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Conclusions

• Complex-valued spectra offer substantially more
  information than other data representations.
• Combination of some bandwidth and high
  frequencies is most useful.
• Trade-off between choice of temporal frequencies
  and spatial sampling exists.