Analytical Hilbert-Transform Attributes of Ricker and Gabor Wavelets

The Ricker and Gabor wavelets and their Hilbert transforms (HTs) are widely used in many physical/engineering fields. In seismology, they are commonly utilized to describe source functions involved in the computation of synthetic seismograms in acoustic or elastic media or, inversely, the extraction of source wavelets from real seismic traces. We provide analytical formulas of the Ricker and Gabor wavelets’ HTs, yielding analytic expressions of the envelope, instantaneous frequency, and phase attributes of these wavelets. We analyze the plausibility of these wavelets by revealing the regions of positive instantaneous frequency and nonzero envelope.