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Time domain & frequency domain

Time domain & frequency domain. Objectives: 1) to be able to analyze time series in both the time and frequency domains, while being aware of potential pitfalls 2) to get an idea of some of the interesting seismic data time series and what you can do with them.

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Time domain & frequency domain

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  1. Time domain & frequency domain Objectives: 1) to be able to analyze time series in both the time and frequency domains, while being aware of potential pitfalls 2) to get an idea of some of the interesting seismic data time series and what you can do with them

  2. Time domain & frequency domain Time domain : Every point on the time domain plot represents the amplitude at a particular time frequency domain: Every point on a frequency spectrum represents the power or amount of energy at that frequency over a finite time window

  3. TD 30 minutes

  4. TD 2 minutes

  5. TD 10 seconds

  6. FD 0 100 Hz

  7. FD 0 20 Hz

  8. FD 0 10 Hz

  9. Fourier Transform (FFT) • An efficient way to convert from time domain to frequency domain • Used to investigate the frequency content of a signal • The Fourier transform integral: • Needs to be converted to a discrete form for use with real digital data

  10. Fourier Transform (FFT) • Based on the discrete Fourier transform: For function f(t) with N samples at t intervals,

  11. Fast Fourier Transform (FFT) • Based on the discrete Fourier transform: For function f(t) with N samples at t intervals, So we have a sum of harmonic waves with varying amplitudes and phase delays

  12. Fast Fourier Transform (FFT)

  13. Fast Fourier Transform (FFT)

  14. Fast Fourier Transform (FFT) • Each component is described by the amplitude and phase for each frequency

  15. Fourier Transform (FFT) • Based on the discrete Fourier transform: For function f(t) with N samples at t intervals, Gives a function of frequencies from: 0 -> (N-1) The second half of the values are angular frequencies that are the “mirror” of the frequencies in the first half

  16. Fast Fourier Transform (FFT) • Based on the discrete Fourier transform: For function f(t) with N samples at t intervals, Gives a function of frequencies from: 0 -> (N-1) The second half of the values are angular frequencies that are the “mirror” of the frequencies in the first half greater than the Nyquist frequency: (N/2) These are "aliased"

  17. Fourier Transform (FFT) • Based on the discrete Fourier transform: For function f(t) with N samples at t intervals, Gives a function of frequencies from: 0 -> (N-1) The second half of the values are angular frequencies that are the “mirror” of the frequencies in the first half Frequencies greater than the Nyquist frequency: (N/2) are "aliased"

  18. Aliasing • Ground motion is continuous (analog) • To examine digital data, we sample the continuous data • Aliasing results from inadequate sample rate for the frequency of the signal

  19. Nyquist Frequency • Limit of resolvable frequencies for a given sample rate • fN=1/(2t) • Best case scenario - only see frequencies this high when samples are ideally placed

  20. Nyquist Frequency

  21. Nyquist Frequency

  22. Nyquist Frequency

  23. Aliasing • Ground motion is continuous (analog) • To examine digital data, we sample the continuous data • Aliasing results from inadequate sample rate for the frequency of the signal For a javascript animation, see: http://www.michaelbach.de/ot/mot_wagonWheel/index.html

  24. More on FFT • The highest frequency that can be resolved (Nyquist) depends on the sampling rate • The resolution (spacing between frequencies) depends on the number the number of samples in time, N • To increase the resolution, you can pad your time series with zeros • Form of the inverse FFT is similar to FFT • It is relatively easy to back and forth between time domain and frequency domain

  25. Seismic data at volcanoes A. Highly varied 1. signals at many frequencies 2. process at many time scales B. results in large files! 1. sampled at 100 sps 2. must be careful not to get bogged down with 10s of Gb of data C. example from Stromboli 1. LP and VLP - two sources nearly coincident in time, but not space

  26. (a) Hour-long record of the east component of velocity for a station on Stromboli, about 400 m southeast of the vents. (b) Band-pass filtered record of (a). Two repeating events were identified suggesting a repetitive, non-destructive source process. (after Chouet et al., JGR 2003)

  27. Figure from Garcés, M. A., M. T. Hagerty, and S. Y. Schwartz (1998), Magma acoustics and time-varying melt properties at Arenal Volcano,Costa Rica, Geophys. Res. Lett., 25(13), 2293–2296.

  28. Where to get seismic data • http://www.iris.washington.edu/data/ • http://www.iris.edu/data/tutorial.htm • http://www.iris.edu/forms/webrequest.htm • http://www.iris.edu/SeismiQuery/assembled.phtml

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