Fourier Theory in Seismic Processing From Liner and

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

Also with notes from http: //lcni. uoregon. edu/fft. p pt

Fourier series • Periodic functions and signals may be expanded into a series of sine and cosine functions http: //www. falstad. com/fourier/j 2/

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

, and

• Mathematica Plot

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

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

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

An undersampled signal

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

Sample Rates

Sample Rates

Sample Rates

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 ….

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

Amplitude Oversampled waveforms Nyquist frequency Reconstructed frequency = True frequency (f -true) frequency is unaliased 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 500 Hz, 2000 samples per second for 1000 Hz

Amplitude Oversampled waveforms Nyquist frequency In practice we are best oversampling by double the required minimum i. e. 1000 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 aliasing 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) 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