Environmental and Exploration Geophysics II tom.h.wilson

Complex numbers Seismic Trace Attributes

Real and Imaginary

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

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

Recall Frequency Domain versus Time Domain Relationships

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

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

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

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

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

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

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

Physical nature of the seismic response Seismic Response

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

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

Simple digital components

Folding and Shifting

Multiply and Sum Output sample 0

Output Sample 1

Output Sample 2

Computing the Quadrature Trace

Generating Attributes in Kingdom Suite

Instantaneous Frequency