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
Speaking non-technically, a time domain graph shows how a signal changes over time, whereas a frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies.

2. Time domain & frequency 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
Speaking non-technically, a time domain graph shows how a signal changes over time, whereas a frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. 2

3. TD 30 minutes

4. TD 2 minutes

5. TD 10 seconds

6. FD 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)
A time-domain seismogram can be thought of as the sum of an infinite number of harmonic (single frequency) waves with varying amplitudes and phase. The phase is the time shift. In practice, we have to use a finite number of waves, of course.

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:

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
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.

29. Where to get seismic data