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Intro to Spectral Analysis and Matlab

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Time domain Seismogram - particle position over time Time Amplitude

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

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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

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Sine wave in time

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Spectra of infinite sine wave

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Two sine waves in time

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Spectra of 2 infinite sine waves

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Spectra of discrete, finite sine waves

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To create arbitrary seismogram Becomes integral in the limit Fourier Transform –Computer: Fast Fourier Transform - FFT

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Time domain, single spike in time

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Spectra of a single spike in time

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Sampling Frequency Digital signals aren’t continuous –Sampled at discrete times How often to sample? –Big effect on data volume

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How many samples/second are needed?

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Are red points enough?

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Aliasing FFT will give wrong frequency

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Nyquist frequency 1/2 sampling frequency

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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

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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

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Exercise plots

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Sine_wave column 2

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Sine_wave column 2 and 3

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Sine_wave column 2 and 3 sum

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Spectra, column 2

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Spectra, columns 2, 3

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Spectra, column 2, 3, 2 and 3 sum

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Multi_sine, individual columns

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Multi_sine spectra

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Spike in time

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Spike in time, frequency

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Rock, sed, bog time series

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Rock spectra

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

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Spectral ratio sed/rock

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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

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Station LKWY, Utah raw Filtered 2-19 Hz Filtered twice

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Station LKWY, Utah raw Filtered 2-19 Hz Filtered twice

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Zoomed in once

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Zoomed in again

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Triggered earthquakes

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Vibrationdata 1 Unit 5 The Fourier Transform. Vibrationdata 2 Courtesy of Professor Alan M. Nathan, University of Illinois at Urbana-Champaign.

Vibrationdata 1 Unit 5 The Fourier Transform. Vibrationdata 2 Courtesy of Professor Alan M. Nathan, University of Illinois at Urbana-Champaign.

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