Chapter 6: Characterization of Seismic Signals by Statistical Random Variables: II. Complex-valued Signals
Society of Exploration Geophysicists eBooks · 2018
In this chapter, as in Chapter 5, the focus is the characterization of seismic signals by statistical random variables. This chapter, however, considers complex-valued signals rather than real-valued signals. In petroleum seismology the complex-valued signals generally occur because we often work in the Fourier domain. In other words, it is sometimes desirable to Fourier-transform seismic wavefields with respect to time and/or space in order to take advantage of the computational efficiency of fast Fourier transforms (FFT) and because differentiations in time and space (t and x) can be converted into simple functions of frequency and wavenumber. If the Fourier transform is limited to time only, the transform domain is characterized as the frequency-space (f-x). If the Fourier transform is performed with respect to both time and space, the transformed domain is characterized as the frequency-wavenumber (f-k). If the windowed-Fourier transform is used instead of the Fourier transform, the f-x domain, for example, becomes the t-f-x domain. The signals in these domains contain complex values. In these domains, using statistical analysis and modern statistical techniques, such as the ICA-based data separation methods decribed in Chapter 5, requires the definitions of statistical averages of random variables and vectors and the computation of the gradients of objective functions associated with independent component analysis (ICA) solutions. One option is to split complex-valued signals into real and imaginary parts and separately determine the statistical averages of the real and imaginary parts using the results discussed in Chapter 5. This approach often leads to complicated formulae that are difficult to interpret, especially for techniques such as ICA, which require computations of the complex-valued gradient and the Hessian matrix. The other option is to conduct derivations with respect to the original complex-valued variables. The latter option is chosen in this chapter.