Applied Seismology: Introduction and Principles
4.3 Raypath geometry in layered ground
4.3 Raypath geometry in layered
Box 4.6 Reflection and transmission coefficients (see Figure 4.5)
For normal and low angles (<20◦) of incidence:
Reflection coefficient
R=A1/A0=(Z2−Z1)/(Z2+Z1) R≤ ±1.
Transmission coefficient
T=A2/A0=2Z1/(Z2+Z1).
Z1and Z2are the acoustic impedances of the first and second layers, respectively. Z = Vρ, where V and ρare the seismic velocity and density of a given layer; A0, A1 and A2 are the relative amplitudes of the incident, reflected and transmitted rays, respectively.
Of the incident energy, the proportions of energy reflected (ER) and transmitted (ET) are given by:
Reflected energy
ER=(Z2−Z1)2/(Z2+Z1)2 Transmitted energy
ET=4Z1Z2/(Z2+Z1)2 Note that ER+ET=1.
For the derivation of these formulae, see Telford et al. (1990, p. 156).
transmitted (T=1 and ET=1), suggesting that there is no contrast in acoustic impedance across the interface (i.e. Z1=Z2) In such a situation there still may be differences in both velocity and density between the two materials. For example, if the seismic velocities and densities for layers 1 and 2 are 1800 m/s, 1.6 Mg/m3and 1600 m/s, 1.8 Mg/m3, respectively, there would be no contrast in acoustic impedance as Z1=Z2=2880 Mg/m2s.
In the above discussion, it has been assumed that the reflection from an interface arises from a point. In reality, it is generated from a finite area of the reflector surface as defined by the first Fresnel zone (Figure 4.6). The second and subsequent Fresnel zones can be ignored in the case of normal incidence. Effectively, the incident wavefront has a discrete footprint on the reflector surface.
The reflection coefficient for a given interface is thus the average response over the first Fresnel zone.
Furthermore, surface roughness of the interface also becomes important if the amplitude of the roughness is of the same order or greater than the quarter-wavelength of the incident wave. The rougher the surface, the more it behaves as a specular reflector, returning rays at a very wide variety of angles. The amount of energy reaching the surface is therefore much reduced and the observed
Source
h
r
First Fresnel zone h + λ/4
λ/4
Figure 4.6 The first Fresnel zone on a reflector at a depth h below the source of the incident spherical wave.
reflection coefficient is much less than that predicted for a given contrast in acoustic impedances. The radius (r) of the first Fresnel zone is related to the depth of the reflector below the source (h) and the wavelength of the incident wave (λ), such that r2≈λh/2 (Box 4.7). As two-way travel times are considered, the quarter- wavelength is used as opposed to classical optics where only a half- wavelength is used. For a more detailed discussion, see McQuillin et al. (1984), Yilmaz (2001), Knapp (1991) and Eaton et al. (1991).
It is clear from this that the first Fresnel zone becomes larger as a result of increasing depth and decreasing frequency (i.e. greater wavelengths). The effect that this has on horizontal resolution is considered in Chapter 6 for seismic reflection surveying, and in Chapter 12 for ground penetrating radar. The determination of Fresnel zones for travel-time measurements has been discussed in detail by Hubral et al. (1993).
4.3.2 Reflection and refraction of obliquely incident rays
In the case of an incident wave impinging obliquely on an interface across which a contrast in acoustic impedance exists, reflected and
Box 4.7 First Fresnel zone (see Figure 4.6)
The radius (r) of the first Fresnel zone is given by:
r2=λh/2+λ2/16≈λh/2 r≈(λh/2)1/2=(V/2)(t/f )1/2
where h is the distance between the source and the reflector, and λis the wavelength of the incident wave of frequency f , with propagation speed V in the material and a reflector at a two-way travel time t on a seismic section.
P
SV
SV V2 > V1
P
P
VP1, VS1, ρ1
VP2, VS2, ρ2
r φ3
φ1
φ2
φ1
Figure 4.7 Geometry of rays associated with a P-wave (shown as a ray incident obliquely on a plane interface), and converted vertically polarised S-waves (SV; shown as a ray). VPand VSare the respective P- and S-wave velocities andρis the density.
Suffixes 1 and 2 depict the layer number.
transmitted waves are generated as described in the case of normal incidence. At intermediate angles of incidence, reflected S-waves generated by conversion from the incident P-waves (Figure 4.7) may have larger amplitudes than reflected P-waves. This effect is particularly useful in the study of deep reflection events in crustal studies where very large offsets (source-receiver distances) are used.
In general, the P-wave amplitude decreases slightly as the angle of incidence increases. This is equivalent to a decrease in P-wave amplitude with increasing offset. However, there are cases where this does not occur, such as when Poisson’s ratio changes markedly, perhaps as a result of gas infilling the pore space within a rock. This phenomenon has been reported by Ostrander (1984) for seismic field records obtained over gas reservoirs and can be used as an indicator of the presence of hydrocarbon gas.
When a P-wave is incident at an oblique angle on a plane surface, four types of waves are generated: reflected and transmitted P-waves and reflected and transmitted S-waves. The relative amplitudes of these various waves are described by Zoeppritz’s equations (Telford et al., 1990). The direction of travel of the transmitted waves is changed on entry into the new medium, and this change is referred to as refraction. The geometry of the various reflected and refracted waves relative to the incident waves is directly analogous to light and can be described using Snell’s Laws of refraction (Box 4.8).
These state that the incident and refracted rays, and the normal at the point of incidence, all lie in the same plane; for any given pair of
Box 4.8 Laws of reflection and Snell’s Laws of refraction (Figure 4.7)
Snell’s Laws:
sin i VP1 = sin r
VP2 = sinβ1
VS1 = sinβ2
VS2 =p
where i and r are the angles of incidence and refraction respec- tively, and V1and V2are the speeds of propagation in layers 1 and 2 respectively for P- and S-waves as indicated by the suffix, and where p is the raypath parameter. Conversely:
sin i sin r = V1
V2
In the case of critical refraction:
sin ic
sin 90◦ = V1
V2
Since sin 90◦=1, sin ic=V1/V2, where icis the critical angle.
Laws of reflection:
r
The angle of incidence equals the angle of reflection.r
The incident, reflected and refracted rays and the normal at the point of incidence all lie in the same plane.media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant. In its generalised form, Snell’s Law also states that for any ray at the point of incidence upon an interface, the ratio of the sine of the angle of incidence to the velocity of propagation within that medium remains a constant, which is known as the raypath parameter.
4.3.3 Critical refraction
When the angle of incidence reaches a particular value, known as the critical angle, the angle of refraction becomes 90◦. The refracted wave travels along the upper boundary of the lower medium, whose speed of propagation is greater than that of the overlying medium (i.e. V2>V1). The material at the interface is subject to an oscil- lating stress from the passage of the refracted wave, which in turn generates upward-moving waves, known as head waves, which may eventually reach the surface (Figure 4.8). The orientation of the ray associated with the head wave is also inclined at the critical angle (Figure 4.8). Critical refraction is discussed further in Chapter 5.
4.3.4 Diffractions
If a wave impinges upon a surface that has an edge to it, such as a faulted layer, then the wavefront bends around the end of the feature and gives rise to a diffracted wave (Figure 4.9). Similarly, boulders whose dimensions are of the same order as the wavelength of the
Lower velocity material P
Head wave
Wavefront in lower layer Higher-velocity material
ic ic
V1 V1
V2 V2
Figure 4.8 Critical refraction at a plane boundary and the generation of a head wave.
incident signal can also give rise to diffractions. The curvature of the diffraction tails are a function of the velocity of the host medium (Figure 4.10). While diffractions are usually considered as noise and attempts are made at resolving them through data processing, they can be used as an interpretational aid (see Chapter 6).
The reason that diffraction occurs is best explained by Huygens’
Principle of secondary wavefronts. The object causing the diffrac- tion acts as a secondary source of waves which spread out spherically from that point and can travel into areas where, according to ray theory, there should be no signals observed, such as the shadow zone shown in Figure 4.9. In the case of an isolated cause, such as a boulder, where the shot is located over the source of the diffrac- tion, a hyperbolic travel-time response is obtained (Figure 4.10;
see also Box 4.9). For comparison, the two-way travel time for a shot–receiver pair with increasing offset (i.e. starting with both at station 6, then shot–receiver at 5 and 7, 4 and 8, 3 and 9, etc.) is given in Figure 4.10.
Whereas a diffraction from a point source in a uniform-velocity field is symmetrical, a diffraction caused by the termination of a re- flector undergoes a 180◦phase change on either side of a diffracting edge (Trorey, 1970; see Figure 4.11).