A spectral decomposition comparison

Brian H. Russell

We describe several methods that attempt to localize time versus frequency changes, which are referred to as spectral decomposition methods. We first review the limitations of the Fourier Transform, and then introduce the Short Time Fourier Transform, or STFT, in which events are localized by convolving a short window within the Fourier Transform. We will then introduce the Wavelet Transform, in which a wavelet dictionary is created using a dilated and translated “mother wavelet”. After this, we introduce the Basis Pursuit method, which again uses a wavelet dictionary. However, in the Basis Pursuit method, the solution is found using a linear programming algorithm. We finish by discussing the Amplitude versus Frequency, or AVF, approach, which analyzes the results of a collection of constant frequency traces created using each of our spectral decomposition methods. These methods are illustrated using both real and synthetically generated seismic traces.