Page 37 - Acquisition and Processing of Marine Seismic Data

28 1. INTRODUCTION

amplitudes of reflected and transmitted waves The relationship between the incidence and
can be calculated by Zoeppritz equations given reflection angles is governed by Snell’s Law,
by Eq. (12.8). In the case of vertical incidence which states that the incidence angle is equal
as shown in Fig. 1.18B, Zoeppritz equations to the reflection angle. The transmitted ray also
can be simplified in such a way that the reflected changes direction with respect to the normal
and transmitted wave amplitudes relative to the vector of the interface for nonnormal incidence
incident wave amplitude are associated with the angles. The direction of the transmitted ray is
reflection and transmission coefficients, respec- determined by the ratio of the velocities: If the
tively. The reflection coefficient of a given inter- velocity of the underlying medium is faster than
face is that of the upperlying medium, the transmitted
ray is bent toward the horizontal, which is the
Z 2 Z 1
k R ¼ (1.5) most common case. The opposite occurs in the
Z 2 + Z 1
case of a higher velocity upperlying medium.
and the transmission coefficient is The relationship between the incidence and
transmission angles based on the geometry
2Z 1
k T ¼ (1.6) and parameters in Fig. 1.18A is given by
Z 2 + Z 1
sinθ 1 sinθ 2
where Z 1 and Z 2 are the acoustic impedances of ¼ (1.9)
the upper- and underlying medium, respec- V 1 V 2
tively. The amplitude of the reflected portion Specifically, the transmitted wave travels along
of the incident wave is the interface with the propagation velocity of
underlying medium when θ 2 ¼ 90 degees. In
A R ¼ k R A 0 (1.7)
this case, sin(θ 2 ) ¼ 1 and Eq. (1.9) becomes
and the transmitted wave amplitude is
V 1 (1.10)
A T ¼ k T A 0 (1.8) sinθ C ¼ V 2
where A 0 is the amplitude of the incident where the incidence angle θ 1 is termed the criti-
wave. The sum of the reflected and transmitted cal angle and usually denoted by θ C . The inci-
amplitudes equals to the incident wave dent P wave with the incidence angle of θ C
amplitude. traveling along the boundary with a velocity
According to Eq. (1.7), amplitude and polar- of V 2 is known as a refracted wave, or a
ity of the reflected wave depend only on acoustic head wave.
impedances of both media, Z 1 and Z 2 . If the dif- The ray paths and appearance of the reflected
ference between acoustic impedances is high, signal, direct wave and head wave on the syn-
then it will produce a high reflection coefficient thetic seismograms are schematically illustrated
value, and hence a large reflected wave ampli- in Fig. 1.19. By a certain distance, indicated by
tude. In addition, if Z 2 < Z 1 , the reflection coef- X K in Fig. 1.19, direct waves arrive first at the
ficient is negative, resulting in a reverse polarity receivers because the ray path they travel is
reflection with respect to the incident wave the shortest. After crossing distance X K , the head
polarity. The changes in acoustic impedance waves catch the direct arrivals and become the
are predominantly caused by velocity variations waves that arrive first at the receivers because
within the upper- and underlying medium, they travel at the velocity of the underlying
since the changes in medium densities are medium V 2 , and hence they are faster, since
negligible as compared to the variations in V 2 > V 1 . By the critical distance X C in Fig. 1.19,
velocity. we cannot observe head waves on the