Transcript Document

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
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
TD 30 minutes
TD 2 minutes
TD 10 seconds
FD 0 100 Hz
FD 0 20 Hz
FD 0 10 Hz
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:
F( ) 

 f t e
it
dt

• Needs to be converted to a discrete form for
use with real digital data
Fourier Transform (FFT)
• Based on the discrete Fourier transform:
For function f(t) with N samples at t intervals,
N 1
F(k )  t  f nt e
n 0

ikt
Fast Fourier Transform (FFT)
• Based on the discrete Fourier transform:
For function f(t) with N samples at t intervals,
N 1
F(k )  t  f nt e
ikt
n 0
So we have a sum of harmonic waves with varying amplitudes
and phase delays

F( )  Re F( )  Im F( )
2
ImF( )
  tan1

ReF( )
2
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT)
• Each component is described by the amplitude and
phase for each frequency
Fourier Transform (FFT)
• Based on the discrete Fourier transform:
For function f(t) with N samples at t intervals,
N 1
F(k )  t  f nt e
ikt
n 0

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
Fast Fourier Transform (FFT)
• Based on the discrete Fourier transform:
For function f(t) with N samples at t intervals,
N 1
F(k )  t  f nt e
ikt
n 0

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"
Fourier Transform (FFT)
• Based on the discrete Fourier transform:
For function f(t) with N samples at t intervals,
N 1
F(k )  t  f nt e
ikt
n 0

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"
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
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
Nyquist Frequency
Nyquist Frequency
Nyquist Frequency
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
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
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
(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)
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.
Seismic data availabilty
Volcano
Data source
Fuego
INSIVUMEH
Fuego
Fuego
Fuego
MTU campaign 5 stations of bb5 days in Jan 2008
MTU campaign 1 bb
7 days in Jun-July 2008
MTU campaign 10 bb stations 20 days in Jan 2009
Pacaya
Pacaya
INSIVUMEH
Type
short period
network
short period
network
Dates
~2007 to ~ present with some
gaps!
~2007 to ~ present with some
gaps!
MTU campaign 5 stations of bb5 days in Jan 2008
short period
~2007 to ~ present with some
Santiaguito INSIVUMEH
network
gaps!
MTU-NMT-UNC
Santiaguito campaign
5 bb stations 5 days in Dec 2008
Where to get seismic data
A.
B.
C.
D.
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