Intro to Spectral Analysis and Matlab Q: How Could you quantify how much lower the tone of a race car is after it passes you compared to as it is coming towards you? How would you set the experiment up?

Running the Experiment. Data is often recorded in the time domain. The stored dataset is called a timeseries. It is a set of time and amplitude pairs.

Frequency Domain (Do a Fourier Transform on Timeseries) We have converted to the Frequency Domain. This dataset is called a Spectra. It is a set of frequency and Amplitude pairs.

Time Domain What’s the Frequency? What’s the Period? What will this look like in the Frequency Domain?

What’s the new (red) period? How Does its amplitude Compare to the 1 s signal?

Power Spectral Densities Secondary Microseism (~8 s) Primary Microseism (~ 16 s)

QSPA PSD PDF

The Mysterious Case of HOWD

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?

Aliasing FFT will give wrong frequency

Nyquist frequency 1/2 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 and iFFT Must be careful about aliasing –Always sample at least 2X highest frequency of interest

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

Exercise plots

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 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 Sediment site 110 m/s /2.5 Hz = 44 m wavelength Basin thickness = 11 m Peat Bog 80 m/s /1 Hz = 80 m Basin thickness = 20 m

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

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

Zoomed in once

Zoomed in again

Triggered earthquakes