Intro to Spectral Analysis and Matlab

Time domain

• Seismogram - particle position over time

Time

Amplitude

Frequency domain

• Why might frequency be as or more important than amplitude?– Filtering signal from noise– Understanding earthquake source, propagation effects– Ground shaking

Time domain <-> Frequency domain

• Possible to mathematically transform from time to frequency domain

• Relative importance of the frequencies contained in the time series

• Can completely describe the system either way.

• Goal of today’s lab– Begin to become familiar with describing seismograms in either

time or frequency domains– Will leave out most of the mathematics

Sine wave in time

Spectra of infinite sine wave

Spectra of infinite sine wave

Two sine waves in time

Spectra of 2 infinite sine waves

Spectra of discrete, finite sine waves

To create arbitrary seismogram

• Becomes integral in the limit • Fourier Transform

– Computer: Fast Fourier Transform - FFT

Time domain, single spike in time

Spectra of a single spike in time

Sampling Frequency

• Digital signals aren’t continuous– Sampled at discrete times

• How often to sample?– Big effect on data volume

How many samples/second are needed?

Are red points enough?

AliasingFFT will give wrong frequency

Nyquist frequency1/2 sampling frequency

Nyquist frequency

• Can only accurately measure frequencies <1/2 of the sampling frequency– For example, if sampling frequency is 200

Hz, the highest theoretically measurable frequency is 100 Hz

• How to deal with higher frequencies?– Filter before taking spectra

Summary• Infinite sine wave is spike in frequency

domain• Can create arbitrary seismogram by adding

up enough sine waves of differing amplitude, frequency and phase

• Both time and frequency domains are complete representations– Can transform back and forth - FFT

• Must be careful about aliasing– Always sample at least 2X highest frequency

of interest

Exercise plots

Sine_wave column 2

Sine_wave column 2

Sine_wave column 2 and 3

Sine_wave column 2 and 3 sum

Spectra, column 2

Spectra, columns 2, 3

Spectra, column 2, 3, 2 and 3 sum

Multi_sine, individual columns

Multi_sine, individual columns

Multi_sine spectra

Spike in time

Spike in time, frequency

Rock, sed, bog time series

Rock spectra

Rock (black), Sed (red), bog (blue)

Spectral ratio sed/rock

Basin Thickness

• 110 m/s /2.5 Hz = 44 m wavelength• Basin thickness = 11 m

• 80 m/s /1 Hz = 80 m• Basin thickness = 20 m

Station LKWY, Utah

raw

Filtered2-19 Hz

Filtered twice

Station LKWY, Utah

raw

Filtered2-19 Hz

Filtered twice

Zoomed in once

Zoomed in once

Zoomed in again

Triggered earthquakes