Geophysical tutorial; the significance of negative frequencies in Fourier transforms

Negative frequencies play a fundamental role in Fourier transforms, although they are often neglected in practice when displaying spectra of seismic traces. We can study their role quite easily when looking at the relationship of a seismic trace and its analytic counterpart that has no negative frequency components. This analytic trace, comprising the original trace and its Hilbert transform, is well known and used frequently to calculate attributes for seismic interpretation. Using Python code snippets and a simple Ricker wavelet as exemplary seismic signal, this tutorial explores and demonstrates the significance of negative frequencies in both a mathematical and a graphical way.