At an interface between two rock layers there is gen- erally a change of propagation velocity resulting from the difference in physical properties of the two layers.
At such an interface, the energy within an incident seis- mic pulse is partitioned into transmitted and reflected pulses. The relative amplitudes of the transmitted and reflected pulses depend on the velocities and densities
of the two layers, and the angle of incidence on the interface.
3.6.1 Reflection and transmission of normally incident seismic rays
Consider a compressional ray of amplitude A0normally incident on an interface between two media of differ- ing velocity and density (Fig. 3.8). A transmitted ray of amplitude A2 travels on through the interface in the same direction as the incident ray and a reflected ray of amplitude A1 returns back along the path of the incident ray.
The total energy of the transmitted and reflected rays must equal the energy of the incident ray. The relative proportions of energy transmitted and reflected are de- termined by the contrast in acoustic impedance Zacross the interface.The acoustic impedance of a rock is the prod- uct of its density (r) and its wave velocity(v); that is,
It is difficult to relate acoustic impedance to a tangible rock property but, in general, the harder a rock, the higher is its acoustic impedance. Intuitively, the smaller the contrast in acoustic impedance across a rock inter- face the greater is the proportion of energy transmitted through the interface. Obviously all the energy is trans- mitted if the rock material is the same on both sides of the
Z=rv
Input spike
20 ms
After 1 s
After 2 s After 3 s
After 4 s After 5 s
Fig. 3.7 The progressive change of shape of an original spike pulse during its propagation through the ground due to the effects of absorption. (After Anstey 1977.)
interface, and more energy is reflected the greater the contrast. From common experience with sound, the best echoes come from rock or brick walls. In terms of physical theory, acoustic impedance is closely analogous to elec- trical impedance and, just as the maximum transmission of electrical energy requires a matching of electrical im- pedances, so the maximum transmission of seismic ener- gy requires a matching of acoustic impedances.
The reflection coefficient Ris a numerical measure of the effect of an interface on wave propagation, and is calcu- lated as the ratio of the amplitude A1of the reflected ray to the amplitude A0 of the incident ray
To relate this simple measure to the physical properties of the materials at the interface is a complex problem. As we have already seen, the propagation of a P-wave de- pends on the bulk and shear elastic moduli, as well as the density of the material. At the boundary the stress and strain in the two materials must be considered. Since the materials are different, the relations between stress and strain in each will be different. The orientation of stress and strain to the interface also becomes important. The formal solution of this physical problem was derived early in the 20thcentury, and the resulting equations are named the Zoeppritz equations (Zoeppritz 1919; and for explanation of derivations see Sheriff & Geldart 1982). Here, the solutions of these equations will be accepted. For a normally incident ray the relationships are fairly simple, giving:
R v v
v v
Z Z
Z Z
= -
+ = -
+
r r
r r
2 2 1 1 2 2 1 1
2 1
2 1
R=A A1 0
where r1,v1,Z1and r2,v2,Z2are the density, P-wave velocity and acoustic impedance values in the first and second layers, respectively. From this equation it follows that -1£R£ +1. A negative value of Rsignifies a phase change of p(180°) in the reflected ray.
The transmission coefficient Tis the ratio of the ampli- tude A2 of the transmitted ray to the amplitude A0 of the incident ray
For a normally incident ray this is given, from solution of Zoeppritz’s equations, by
Reflection and transmission coefficients are some- times expressed in terms of energy rather than wave am- plitude. If energy intensity Iis defined as the amount of energy flowing through a unit area normal to the direc- tion of wave propagation in unit time, so that I0,I1 and I2 are the intensities of the incident, reflected and trans- mitted rays respectively, then
and
¢ = =
( + )
T I
I
Z Z
Z Z
2 1
1 2
2 1
2
4
¢ = = -
+ È ÎÍ
˘ R I ˚˙
I
Z Z
Z Z
1 0
2 1
2 1
2
T Z
Z Z
= +
2 1
2 1
T =A2 A0
Incident ray, amplitude A0
Reflected ray, amplitude A1
Transmitted ray, amplitude A2
v1, ρ1
v2, ρ2 ρ2v2 = ρ/ 1v1 Fig. 3.8 Reflected and transmitted rays associated with a
ray normally incident on an interface of acoustic impedance contrast.
where R¢and T¢are the reflection and transmission coef- ficients expressed in terms of energy.
If Ror R¢ =0, all the incident energy is transmitted.
This is the case when there is no contrast of acoustic im- pedance across an interface, even if the density and ve- locity values are different in the two layers (i.e.Z1=Z2).
If Ror R¢ = +1 or -1, all the incident energy is reflected.
A good approximation to this situation occurs at the free surface of a water layer: rays travelling upwards from an explosion in a water layer are almost totally reflected back from the water surface with a phase change of p(R
= -0.9995).
Values of reflection coefficient Rfor interfaces be- tween different rock types rarely exceed±0.5 and are typically much less than±0.2.Thus, normally the bulk of seismic energy incident on a rock interface is transmitted and only a small proportion is reflected. By use of an em- pirical relationship between velocity and density (see also Section 6.9), it is possible to estimate the reflection coefficient from velocity information alone (Gardner et al. 1974, Meckel & Nath 1977):
Such relationships can be useful, but must be applied with caution since rock lithologies are highly variable and laterally heterogeneous as pointed out in Section 3.4.
3.6.2 Reflection and refraction of obliquely incident rays
When a P-wave rayis obliquely incident on an inter- face of acoustic impedance contrast, reflected and trans- mitted P-wave rays are generated as in the case of normal incidence. Additionally, some of the incident com- pressional energy is converted into reflected and trans- mitted S-wave rays (Fig. 3.9) that are polarized in a vertical plane. Zoeppritz’s equations show that the am- plitudes of the four phases are a function of the angle of incidence q. The converted rays may attain a significant magnitude at large angles of incidence. Detection and identification of converted waves can be difficult in seis- mic surveys, but they do have potential to provide more constraints on the physical properties of the media at the interface. Here consideration will be confined to the P-waves.
In the case of oblique incidence, the transmitted P- wave ray travels through the lower layer with a changed direction of propagation (Fig. 3.10) and is referred to as a refracted ray.The situation is directly analogous to the be- R =0.625ln(v1 v2
)
haviour of a light ray obliquely incident on the boundary between, say, air and water and Snell’s Law of Refraction applies equally to the optical and seismic cases. Snell de- fined the ray parameter p=sini/v, where iis the angle of inclination of the ray in a layer in which it is travelling with a velocity v.The generalized form of Snell’s Law states that, along any one ray, the ray parameter remains a constant.
For the refracted P-wave ray shown in Fig. 3.10, therefore
or
sinq1 sinq
1
2
v = v2
Reflected S
Reflected P
Refracted S
Refracted P Incident P
θ
v1 v2 > v1
Fig. 3.9 Reflected and refracted P- and S-wave rays generated by a P-wave ray obliquely incident on an interface of acoustic impedance contrast.
Reflected P
Refracted P Incident P θ1 θ1
v1 v2 > v1
θ2
Fig. 3.10 Reflected and refracted P-wave rays associated with a P-wave rays obliquely incident on an interface of acoustic impedance contrast.
Note that if v2>v1 the ray is refracted away from the normal to the interface; that is,q2>q1. Snell’s Law also applies to the reflected ray, from which it follows that the angle of reflection equals the angle of incidence (Fig.
3.10).
3.6.3 Critical refraction
When the velocity is higher in the underlying layer there is a particular angle of incidence, known as the critical angleqc, for which the angle of refraction is 90°. This gives rise to a critically refracted ray that travels along the interface at the higher velocity v2.At any greater angle of incidence there is total internal reflection of the incident energy (apart from converted S-wave rays over a further range of angles).The critical angle is given by
so that
The passage of the critically refracted ray along the top of the lower layer causes a perturbation in the upper layer that travels forward at the velocity v2, which is greater than the seismic velocity v1 of that upper layer. The situation is analogous to that of a projectile travelling through air at a velocity greater than the velocity of sound in air and the result is the same, the generation of a shock wave.This wave is known as a head wavein the seis- qc=sin-1(v1 v2
)
sinqc sin
v1 v2 v2
90 1
= ∞
= sin
sin q q
1 2
1 2
=v v
θc v1
v2 > v1
Ray paths Head wave
generated in overlying layer
Wavefront expanding in lower layer
A B
Fig. 3.11 Generation of a head wave in the upper layer by a wave propagating through the lower layer.
Reflection wavefront
Faulted layer
Diffraction wavefront Fig. 3.12 Diffraction caused by the
truncated end of a faulted layer.
mic case, and it passes up obliquely through the upper layer towards the surface (Fig. 3.11). Any ray associated with the head wave is inclined at the critical angle ic. By means of the head wave, seismic energy is returned to the surface after critical refraction in an underlying layer of higher velocity.
3.6.4 Diffraction
In the above discussion of the reflection and transmission of seismic energy at interfaces of acoustic impedance contrast it was implicitly assumed that the interfaces were continuous and approximately planar. At abrupt discontinuities in interfaces, or structures whose radius of curvature is shorter than the wavelength of incident waves, the laws of reflection and refraction no longer apply. Such phenomena give rise to a radial scattering of incident seismic energy known as diffraction. Common sources of diffraction in the ground include the edges of faulted layers (Fig. 3.12) and small isolated objects, such as boulders, in an otherwise homogeneous layer.
Diffracted phases are commonly observed in seismic recordings and are sometimes difficult to discriminate from reflected and refracted phases, as discussed in Chapter 4.