42 Seismic Imaging 2.2 Amplitudes of refracted waves Amplitudes of refracted waves, also called head waves, are not commonly used in seismic refraction studies. Nikrouz (2016) has conducted an interesting synthesis of the relationship between head wave amplitudes and seismic refraction velocities, showing through field studies that variations in amplitudes are associated with velocity changes in the refractor: the higher the contrast in the refractor wave velocity, the lower the head wave amplitude and vice versa (Palmer, 2001). Heelan (1953) and Werth (1967) have shown that the amplitude of a head wave for a thick reflector with a plane horizontal interface is K.G.F(t) where: • K is the head coefficient which is a function of seismic velocities and densities in the upper layer and the refractor. Werth (1967) expressed K as: K m m m = + ( ) + − ( ) [ ] − ( ) + − ( ) + + 2 1 2 2 1 1 2 1 2 2 2 2 2 1 2 2 2 2 ρχγ λ γ λ ρ γ γ γ γ ρ ρχλ χ / λ γ 1 2 2 2 1 2+ ( ) m γ =V V P P 1 2 / ρ=ρ ρ 1 2 / m V V V V S P S P = ( ) −( ) ρ 1 2 1 2 2 2 1 2 / / χ= −γ ( ) 1 2 1/2 λ γ 1 1 2 1 2 2 1 2 = − ( ) V V P S / / λ γ 2 1 2 2 2 2 1 2 = − ( ) V V P S / / where VP1, VS1 and ρ1 are, respectively, the P-wave velocity, the S-wave velocity and the density in the upper medium 1; and VP1, VS2 and ρ2 are, respectively, the P-wave velocity, the S-wave velocity and the density in the lower medium 2. The variations of K for a set of elastic parameters can be decreased as the contrast in velocities VP2 / VP1 is increased (Figure 2.4-a). Figures 4-b to d show variations of K versus VS2 / VS1 for VP2 / VP1 ranging from 1.1 to 2, in 3 cases: VP1 = 1,700 m/s and VS1 = 500 m/s, VP1 = 2,000 m/s and VS1 = 1,000 m/s, VP1 = 2,400 m/s and VS1 = 1,600 m/s. • G is the geometric spreading component given by 1/((rL3)1/2) where r is the shot-receiver distance and L is the distance the wave has travelled within the refractor. The geometric spreading component is the major contributor of head wave amplitude. • F(t) is the displacement potential of the incident pulse.
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