31 1. Wave propagation The wave can be characterized equally well by its: • frequency f, expressed in Hz (or in cycles/second), • Wavelength λ, expressed in m, and • Wavenumber k, expressed in cycles/meter (c/m). Frequency is the reciprocal of the period T ( f = 1/T ). Wavelength is the distance traveled by a wavefront during a time equal to the period T (λ = VT =V / f, V being the propagation velocity). Wavenumber k is related to the wavelength λ in the same way that frequency f is related to the period T (k=1 / λ). Velocity V can be expressed in the time-distance domain (t, x) or in the frequency – wavenumber domain ( f, k): V = x / t = f /k. The translation from the (t, x) domain to the ( f, k) domain and vice versa is accomplished by the forward and inverse Fourier transforms. The amplitude representation of a (t, x) record in the ( f, k) domain is called a two–dimensional amplitude spectrum. Waves can be separated by different filters (Mari, 2015) such as: • Frequency filter ( f ), • Wavenumber filter (k), • Apparent velocity filter in the ( f, k) domain (Va = f /k). Other filters can also be used, such as: • SVD (Singular Value Decomposition) filter, • polarization filter, • … Figure 1.8 shows an example of an in-line shot point obtained with an impulse source. The receiver spread is composed of 48 single geophones. The distance between 2 adjacent geophones is 5 m. The time sampling interval is 0.25 ms and the recording length is 0.5 s. The data are presented both in the time distance domain and in the frequency - wavenumber domain. The wave separation is done in the ( f, k) domain by apparent velocity filters. The 2D amplitude spectrum of the raw data (Figure 1.8-a) shows that some of the energy is located in the negative wavenumbers. The waves associated with this energy propagate with negative apparent velocities. The main energetic wave fields are the direct waves, and the low and high apparent velocity pseudo-Rayleigh waves (Figures 1.8-b and c). Figure 1.8-d shows the wavefield associated with negative wavenumbers. These waves are converted refracted waves. The refracted wave is shown in Figure 1.8-e. The reflected waves, shown in Figure 1.8-f, are characterized by small wavenumbers and, consequently, very high apparent velocities.
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