Fourier Theory in Seismic Processing (From Liner and Ikelle and Amundsen) Temporal aliasing Spatial aliasing

Also with notes from pt

Fourier series Periodic functions and signals may be expanded into a series of sine and cosine functions

The Nyquist Frequency The Nyquist frequency is equal to one-half of the sampling frequency. The Nyquist frequency is the highest frequency that can be measured in a signal.

The Fourier Transform A transform takes one function (or signal) and turns it into another function (or signal)

The Fourier Transform A transform takes one function (or signal) and turns it into another function (or signal) Continuous Fourier Transform:

The Fourier Transform The input signal gives the proper weight to all the cosines and sines Continuous Fourier Transform:

Famous Fourier Transforms Sinc function Square wave

The Fourier Transform

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Mathematica Plot

A transform takes one function (or signal) and turns it into another function (or signal) The Discrete Fourier Transform: Discrete Fourier Transform

Fast Fourier Transform The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform FFT algorithm published by Cooley & Tukey in 1965 In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!

Sampling Real Data

5*sin (2  4t) Amplitude = 5 Frequency = 4 Hz seconds A sine wave

5*sin(2  4t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second seconds Sampling duration = 1 second A sine wave signal

An undersampled signal

Sample Rates What is the fewest number of times I need to sample this waveform per second? ? ? ?

Sample Rates

What is the fewest number of times I need to sample this waveform per second? At least twice per wavelength or period! OTHERWISE ….

Undersampled waveforms True frequency (f -true) Amplitude Reconstructed frequency (f -aliased)

Oversampled waveforms = True frequency (f -true) Amplitude Reconstructed frequency frequency is unaliased Nyquist frequency Nyquist frequency = 1 / twice the sampling rate Minimum sampling rate must be at least twice the desired frequency E.g., 1000 samples per second for 500Hz, 2000 samples per second for 1000 Hz

Oversampled waveforms Amplitude Nyquist frequency In practice we are best oversampling by double the required minimum i.e samples per second for a maximum of 500 Hz i.e., 2000 samples per second for a maximum of 1000 Hz Oversampling is relatively cheap.

Spatial frequency, or wavenumber (k) is the number of cycles per unit distance. One spatial cycle or wavenumber = frequency/velocity. Each wavenumber must be sampled at least twice per wavelength (two CMP’s per wavelength) Spatial aliasing IN PRACTICE each wavenumber must be sampled at least four times per minimum wavelength (two CMP’s per wavelength)

Spatial aliasing However, dip (theta) as well as frequency and velocity event changes the number of cycles per distance, so Liner, 9.7,p.192

Spatial aliasing For aliasing NOT to occur, delta(t) must be less than T/2

Spatial aliasing

Geophone Spacing and Spatial Aliasing K=0

1/4 wavelength shift per trace total shift across array=3/4 wavelength K=+ or -ve?

1/4 wavelength shift per trace total shift across array=3/4 wavelength K=?

1/2 wavelength shift per trace total shift across array=3/2 wavelength K=0

3/4 wavelength shift per trace total shift across array=2 1/4 wavelength