1. Environmental and Exploration Geophysics II tom.h.wilson wilson@geo.wvu.edu

2. Complex numbers Seismic Trace Attributes

3. Real and Imaginary

4. Given the in-phase and quadrature components, it is easy to calculate the amplitude and phase or vice versa. Interrelationships

5. The seismic trace is the “real” or in-phase component of the complex trace Seismic Data How do we find the quadrature component?

6. Recall Frequency Domain versus Time Domain Relationships

7. Time-domain wavelets Zero PhaseMinimum Phase Individual frequency components Amplitude spectrum Phase spectrum Amplitude and Phase Spectra Fourier Transform of a time series

8. The seismic response is a “real” time series This is its amplitude spectrum Seismic Trace and its Amplitude Spectrum

9. Symmetrical Asymmetrical The Fourier Transform of a real function, like a seismic trace, is complex, i.e., it has real and imaginary parts. The real part is even The imaginary part is odd Real and Imaginary Parts of the Fourier Transform

10. We assume that the complex time series s(t) = s r (t) + is i (t) exists and that s r (t) is the recorded seismic signal. For this to be so, the Fourier transform of s, defined as S(f), must have a real part (S r ) equal to the Fourier transform of s r and an imaginary part (S i ) equal to the Fourier transform of s i. Creating the quadrature Component

11. Amplitude and Phase Representation of the Real and Quadrature Traces Real Trace Complex Trace Tanner, Koehler, and Sheriff, 1979 The Spectrum of the Complex Trace

12. That relationship between S i and S r is written as S i (f) = H(f)S r (f) where H(f) is a step function having value -i for 0  and i for -  0. That function is referred to as a Hilbert transform, and the inverse Fourier transform of this imaginary step function in the frequency domain yields the real function h(t) in the time domain. Tanner, Koehler, and Sheriff, 1979 The time-domain Hilbert Transform

13. Multiplication in the frequency domain equals convolution in the time domain Fourier Transforms and Convolution The convolution integral Seismic Analog where S is the seismic signal or trace, w is the seismic wavelet, and r is the reflectivity sequence

14. Physical nature of the seismic response Seismic Response

15. The output is a superposition of reflections from all acoustic interfaces and the convolution integral is a statement of the superposition principle. Convolutional model Convolutional Model

16. Discrete form of the Convolution Integral 1) Folding 2) Shifting 3) Multiplication 4) Summation As defined by this equation, the process of convolution consists of 4 simple mathematical operations Discrete form of the convolution integral

17. Simple digital components

18. Folding and Shifting

19. Multiply and Sum Output sample 0

20. Output Sample 1

21. Output Sample 2

22. Computing the Quadrature Trace

23. Generating Attributes in Kingdom Suite

24. Instantaneous Frequency