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Fourier Theory in Seismic Processing

This text delves into the principles of Fourier theory as applied to seismic processing. It discusses the critical concepts of temporal and spatial aliasing, emphasizing the importance of sampling rates. The Nyquist criterion indicates that the minimum sampling rate must be at least twice the desired frequency. For instance, 1000 samples per second are necessary for a 500 Hz signal. Furthermore, to avoid spatial aliasing, each wavenumber should be sampled at least twice per wavelength, ideally four times in practice. The text underscores the significance of careful sampling strategies in obtaining accurate seismic data.

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Fourier Theory in Seismic Processing

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  1. Fourier Theory in Seismic Processing (From Liner and Ikelle and Amundsen) • Temporal aliasing • Spatial aliasing

  2. Sample Rates What is the fewest number of times I need to sample this waveform per second? ? ? ?

  3. Sample Rates

  4. Sample Rates

  5. Sample Rates

  6. Sample Rates What is the fewest number of times I need to sample this waveform per second? At least twice per wavelength or period! OTHERWISE ….

  7. Undersampled waveforms Amplitude True frequency (f -true) Reconstructed frequency (f -aliased)

  8. Oversampled waveforms Nyquist frequency Amplitude Reconstructed frequency frequency is unaliased = True frequency (f -true) Nyquist frequency = 1 / twice the sampling rate Minimum sampling rate must be at least twice the desired frequency E.g., 1000 samples per second for 500Hz, 2000 samples per second for 1000 Hz

  9. Oversampled waveforms Nyquist frequency Amplitude In practice we are best oversampling by double the required minimum i.e. 1000 samples per second for a maximum of 500 Hz i.e., 2000 samples per second for a maximum of 1000 Hz Oversampling is relatively cheap.

  10. Spatial aliasing Spatial frequency, or wavenumber (k) is the number of cycles per unit distance. One spatial cycle or wavenumber = frequency/velocity. Each wavenumber must be sampled at least twice per wavelength (two CMP’s per wavelength) IN PRACTICE each wavenumber must be sampled at least four times per minimum wavelength (two CMP’s per wavelength)

  11. Spatial aliasing However, dip (theta) as well as frequency and velocity event changes the number of cycles per distance, so Liner, 9.7,p.192

  12. Spatial aliasing For aliasing NOT to occur, delta(t) must be less than T/2

  13. Spatial aliasing

  14. Geophone Spacing and Spatial Aliasing K=0

  15. 1/4 wavelength shift per trace total shift across array=3/4 wavelength K=+ or -ve?

  16. 1/4 wavelength shift per trace total shift across array=3/4 wavelength K=?

  17. 1/2 wavelength shift per trace total shift across array=3/2 wavelength K=0

  18. 3/4 wavelength shift per trace total shift across array=2 1/4 wavelength

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