1. Seismic Reflection Data Processing and Interpretation A Workshop in Cairo 28 Oct. – 9 Nov. 2006 Cairo University, Egypt Dr. Sherif Mohamed Hanafy Lecturer Title: 1D Fourier Transform

2. 1D Fourier Transform Theory and Practice Real Data Examples Using SU and MatLab

3. Simple Harmonic Motion A simple harmonic motion is fully described in time domain by its amplitude, frequency, and phase difference. A simple harmonic motion is fully described in time domain by its amplitude, frequency, and phase difference. A simple harmonic motion is fully described in space domain by its amplitude, wavelength, and phase shift. A simple harmonic motion is fully described in space domain by its amplitude, wavelength, and phase shift.

4. A wave in time domain

5. A wave in space domain

6. The connection between time and space domains is Velocity

7. Simple Harmonic Motion Where A is the amplitude w is the angular frequency t is the time

8. Consider the following simple harmonic motions

14. Adding some of these simple harmonic motions together will give us a more complex harmonic motions

20. If we have the sinusoidal wave in time domain, could we know the frequencies making it? Yes, using Fourier transform

21. What dose “transform” means? Transform is taking a group of data as input to give another group of data as output. The output results can not be calculated unless all the input is available and used at once

22. Jean Baptiste Joseph Fourier (1768- 1830), a French mathematician and physicist said that; “Any signal in the time domain is the summation of a specific number of simple sinusoidal waves”

23. Fourier Transform is given by :- Inverse Fourier Transform is given by :-

24. Discrete Fourier Transform Discrete Inverse Fourier Transform

25. Note

26. DFT

27. Fast Fourier Transform If number of data is = 2 n, where n is a positive integer number. Then we can use fast Fourier transform FFT is incredibly more efficient, often reducing the computation time by hundreds

29. Practical solution of FFT 1. Transform the 1 signal of N points into N signals of 1 point

30. With the help of binary numbers, things are much easier

31. 2. Find the frequency spectra of the 1 point time domain signals Nothing could be easier; the frequency spectrum of a 1 point signal is equal to itself

32. 3. Combine the N frequency spectra in the exact reverse order that the time domain decomposition took place Unfortunately, the bit reversal shortcut is not applicable, and we must go back one stage at a time. In the first stage, 16 frequency spectra (1 point each) are synthesized into 8 frequency spectra (2 points each). In the second stage, the 8 frequency spectra (2 points each) are synthesized into 4 frequency spectra (4 points each), and so on. The last stage results in the output of the FFT, a 16 point frequency spectrum

33. Synthetic Examples

36. Real Examples

39. End of this lecture Thank You for you attention All examples on this lecture is based on my work