# Reflection and refraction

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

For simplicity, consider a monochromatic compressional plane wave that impinges at normal incidence upon a flat layer boundary at z = 0. The incident energy is partitioned between a reflected and transmitted compressional plane wave. For this special case, there is only one stress component, Pzz, and one displacement component, w, which is only a function of z. The equation of wave motion (L-29c) for this special case takes the form

 ${\displaystyle \rho {\frac {\partial ^{2}w}{\partial t^{2}}}=(\lambda +2\mu ){\frac {\partial ^{2}w}{\partial z^{2}}},}$ (3a)

where ρ is density of the medium, and λ and μ are Lamé’s constants (equations L-19a,L-19b) associated with an isotropic solid. They are directly related to the compressional-wave velocity α by equation (L-35) which is rewritten below as

 ${\displaystyle \alpha ={\sqrt {\frac {\lambda +2\mu }{\rho }}}.}$ (3b)

A solution to equation (3a) can be written as

 ${\displaystyle w_{0}(z,t)=A_{0}\ \exp \left(i{\frac {\omega }{\alpha _{1}}}z-i\omega t\right),}$ (4a)

where w0 is the wave function for the incident compressional wave, A0 is its amplitude, ω is the angular frequency, α1 is the compressional-wave velocity in the upper layer (equation 3b), and z-axis is downward positive. Similar wave functions can be written for the reflected plane wave:

 ${\displaystyle w_{1}(z,t)=A_{1}\ \exp \left(-i{\frac {\omega }{\alpha _{1}}}z-i\omega t\right)}$ (4b)

and for the transmitted plane wave:

 ${\displaystyle w_{2}(z,t)=A_{2}\ \exp \left(i{\frac {\omega }{\alpha _{2}}}z-i\omega t\right),}$ (4c)

where α2 is the compressional-wave velocity in the lower layer.

Given the incident wave amplitude A0, we want to compute the reflected and refracted wave amplitudes, A1 and A2, respectively. Equations (4a,4b,4c) must be accompanied with boundary conditions at z = 0 that satisfy the continuity of displacement and stress. The continuity of displacement condition, w0 + w1 = w2 at the interface z = 0 gives the relation

 ${\displaystyle A_{0}+A_{1}=A_{2}.}$ (5)

The stress component Pzz can be specialized from Hooke’s law (equation L-18c) for the present case of a normal-incident compressional plane wave

 ${\displaystyle P_{zz}=(\lambda +2\mu ){\frac {\partial w}{\partial z}}.}$ (6)

The continuity of stress condition at the interface z = 0 gives the relation

 ${\displaystyle {\big (}P_{zz}{\big )}_{0}+{\big (}P_{zz}{\big )}_{1}={\big (}P_{zz}{\big )}_{2}.}$ (7)

Now, differentiate the wave functions of equations (4a,4b,4c) with respect to z, substitute into equation (7) and set z = 0 to obtain the following expression:

 ${\displaystyle \rho _{1}\alpha _{1}A_{0}-\rho _{1}\alpha _{1}A_{1}=\rho _{2}\alpha _{2}A_{2},}$ (8)

where ρ1 and ρ2 are the densities of the upper and lower layers, respectively.

We now have two equations, (5) and (8), and two unknowns, A1 and A2. By combining equations (5) and (8), we can derive the ratio of the reflected wave amplitude to the incident wave amplitude, which is called the reflection coefficient c, associated with the layer boundary as

 ${\displaystyle c={\frac {A_{1}}{A_{0}}}={\frac {\rho _{2}\alpha _{2}-\rho _{1}\alpha _{1}}{\rho _{2}\alpha _{2}+\rho _{1}\alpha _{1}}}.}$ (9)

Define the product of density and velocity as the seismic impedance, I = ρα. If there is a difference between the seismic impedances of the two layers, then a reflection occurs at the interface. If the upper layer has a higher impedance than the lower layer, the reflection coefficient becomes negative causing a phase reversal on the reflected waveform.

Figure 11.2-2  A long, deep-water seismic section that shows internal reflections within the water layer caused by density contrast associated with temperature and salinity changes in the water layer. (Data courtesy IFP.)

An impedance contrast at a layer boundary often is largely caused by velocity contrast. Nevertheless, conditions exist for which density contrast can be significant in giving rise to reflections. Figure 11.2-2 illustrates one such case. The internal reflections within the water layer occur because of changes in water temperature and salinity that causes variations in water density.

A typical reflection coefficient for a strong reflector is about 0.2. The reflection coefficient associated with a hard water bottom is about 0.3. Note that it is the impedance contrast, and not the density or velocity contrasts, that gives rise to reflection energy.

In the foregoing discussion, a normal-incident compressional plane wave was considered. If the same compressional plane wave was incident at an oblique angle to the interface, then derivation of the reflected and transmitted wave functions gets complicated, and we find that the reflection coefficient changes with angle of incidence. Moreover, at non-normal incidence, the incident compressional-wave energy is partitioned at the interface into four components: reflected compressional, reflected shear, transmitted compressional, and transmitted shear waves.

For simplicity, consider a 2-D compressional plane wave that impinges on a layer boundary with an angle of incidence φ0 as depicted in Figure 11.2-3a. The incident wavefront is denoted by AC. Point A at the layer boundary acts as a Huygens’ secondary source and generates its own spherical wavefronts associated with compressional and shear waves propagating in both upper and lower media with the corresponding velocities. In Figure 11.2-3a, only the reflected compressional wavefront and raypath are shown. By the time the incident wavefront at C reaches the reflecting interface at B, the spherical wavefront associated with the reflected compressional wave reaches D, so that AD = CB and the tangential line DB becomes the wavefront for the reflected compressional wave. Since the incident and the reflected compressional waves travel with the same velocity, the angle of incidence φ0 is equal to the angle of reflection φ1.

Figure 11.2-3

Reflection and refraction of an incident P-wave at a layer boundary. Medium parameters: ρ is density, α is P-wave velocity, β is S-wave velocity. (a) Reflected P-wave; (b) reflected S-wave; (c) refracted P-wave; (d) refracted S-wave; (e) raypaths associated with the incident P wave, and reflected and refracted P- and S-waves. The radius of the circular wavefront associated with Huygens’ secondary source at A on the layer boundary is CB for the reflected wave, (β1/α1) for the reflected S-wave, (α2/α1)CB for the refracted P-wave, and (β2/α1) for the refracted S-wave. The relationship between the angles in (e) is given by Snell’s law (equation 10).

Now consider the reflected shear wave as shown in Figure 11.2-3b. By the time the incident wavefront at C reaches the reflecting interface at B, the spherical wavefront associated with the reflected shear wave reaches D, so that AD = (β1/α1)CB, and the tangential line DB becomes the wavefront for the reflected shear wave. The angle for the reflected shear wave ψ1 is no longer equal to the angle of incidence φ0.

Huygens’ principle can also be applied to describe the refracted wave at the interface. Refer to Figures 11.2-3c and 11.2-3d and note that the compressional incident plane wave at A acts as a Huygens’ secondary source and generates its own compressional wavefront that travels into the lower medium with velocity α2 and shear wavefront that travels into the lower medium with velocity β2. By the time the incident wavefront at C in Figure 11.2-3c reaches the layer boundary at B, the spherical wavefront associated with the refracted compressional wave reaches D, so that AD = (α2/α1)CB, and the tangential line DB becomes the wavefront for the refracted compressional wave. Similarly, by the time the incident wavefront at C in Figure 11.2-3d reaches the layer boundary at B, the spherical wavefront associated with the refracted shear wave reaches D, so that AD = (β2/α1)CB, and the tangential line DB becomes the wavefront for the refracted shear wave.

From the geometry of reflected and refracted ray-paths shown in Figure 11.2-3e, Snell’s law of refraction can be deducted as

 ${\displaystyle {\frac {\sin \varphi _{0}}{\alpha _{1}}}={\frac {\sin \varphi _{1}}{\alpha _{1}}}={\frac {\sin \varphi _{2}}{\alpha _{2}}}={\frac {\sin \psi _{1}}{\beta _{1}}}={\frac {\sin \psi _{2}}{\beta _{2}}}.}$ (10)

Note that for all four cases in Figure 11.2-3, the horizontal distance AB at the interface is the same for the incident and reflected or the refracted wave. If AC is set to the wavenumber along the propagation path of the incident wave, then AB is the horizontal wavenumber which is invariant as a result of reflection or refraction. Actually, Snell’s law given by equation (10) is a direct consequence of this physical observation.

If the compressional velocity α1 of the upper layer is less than the compressional velocity α2 of the lower layer, then there exists an angle of incidence such that no refracted compressional energy is transmitted into the lower layer. Instead, the refracted energy travels along the interface and is refracted back to the upper layer with an angle equal to the angle of incidence. This angle is called the critical angle of incidence for compressional waves and is given by

 ${\displaystyle \sin \varphi _{c}={\frac {\alpha _{1}}{\alpha _{2}}}.}$ (11a)

The critically refracted wave is often called the head wave and is the basis for refraction statics (refraction statics corrections).

If the compressional velocity α1 of the upper layer is less than the shear velocity β2 of the lower layer, then there exists an angle of incidence such that no refracted shear energy is transmitted into the lower layer. Again, the refracted energy travels along the interface and is refracted back to the upper layer with an angle which is called the critical angle of P-to-S conversion given by

 ${\displaystyle \sin \psi _{c}={\frac {\alpha _{1}}{\beta _{2}}}.}$ (11b)

For the general case of non-normal incidence, boundary conditions at the interface involve not only principal stress and strain components but also the shear stress and strain components. Again, by using the requirement that the stress and displacement must be continuous at the interface, a set of equations can be derived to compute the amplitudes of the reflected and refracted P- and S-wave components associated with an incident compressional source (the Zoeppritz equations):

 ${\displaystyle \cos \varphi _{1}A_{1}+{\frac {\alpha _{1}}{\beta _{1}}}\sin \psi _{1}B_{1}+{\frac {\alpha _{1}}{\alpha _{2}}}\cos \varphi _{2}A_{2}-{\frac {\alpha _{1}}{\beta _{2}}}\sin \psi _{2}B_{2}=\cos \varphi _{1},}$ (12a)

 ${\displaystyle -\sin \varphi _{1}A_{1}+{\frac {\alpha _{1}}{\beta _{1}}}\cos \psi _{1}B_{1}+{\frac {\alpha _{1}}{\alpha _{2}}}\sin \varphi _{2}A_{2}+{\frac {\alpha _{1}}{\beta _{2}}}\cos \psi _{2}B_{2}=\sin \varphi _{1},}$ (12b)

 ${\displaystyle -\cos 2\psi _{1}A_{1}-\sin 2\psi _{1}B_{1}+{\frac {\rho _{2}}{\rho _{1}}}\cos 2\psi _{2}A_{2}-{\frac {\rho _{2}}{\rho _{1}}}\sin 2\psi _{2}B_{2}=\cos 2\psi _{1},}$ (12c)

and

 ${\displaystyle \sin 2\varphi _{1}A_{1}-{\frac {\alpha _{1}^{2}}{\beta _{1}^{2}}}\cos 2\psi _{1}B_{1}+{\frac {\rho _{2}\beta _{2}^{2}\alpha _{1}^{2}}{\rho _{1}\beta _{1}^{2}\alpha _{2}^{2}}}\sin 2\varphi _{2}A_{2}+{\frac {\rho _{2}\alpha _{1}^{2}}{\rho _{1}\beta _{1}^{2}}}\cos 2\psi _{2}B_{2}=\sin 2\varphi _{1}.}$ (12d)
Figure 11.2-4
Figure 11.2-5  Partitioning of a unit-amplitude incident P-wave energy into four components — reflected and refracted P- and S-waves [1].).

These are the Zoeppritz equations which can be solved for the four unknowns, the reflected compressional-wave amplitude A1, the reflected shear-wave amplitude B1, the refracted compressional-wave amplitude A2, and the refracted shear-wave amplitude B2. Equations (12a,12b,12c,12d) have been normalized by the incident-wave amplitude A0 = 1 (the Zoeppritz equations).

Framework for derivation of the Zoeppritz equations. For details see the Zoeppritz equations.

From Snell’s law (equation 10), given the incident angle for the compressional wave and specifying the compressional- and shear-wave velocities, the angles of reflection and refraction can be computed. Substitution into equation (12) yields the required wave amplitudes. These wave amplitudes, of course, depend on the angle of incidence (Figure 11.2-3e).

Figure 11.2-4 outlines the framework for deriving the Zoeppritz equations based on an earth model that comprises two layers separated by a horizontal interface. Details are left to the Zoeppritz equations. Starting with the equations of motion and Hooke’s law, derive the wave equation for elastic waves in isotropic media in which elastic properties are invariant in any spatial direction at any given location. Then, use the equations of continuity which state that the vertical and tangential stress and stress components coincide at layer boundary, plane-wave solutions to the wave equation and Snell’s law that relates propagation angles to wave velocities to derive the equations for computing the amplitudes of the reflected and transmitted P- and S-waves.

Refer to Figure 11.2-5 for a specific case of the partition of energy of an incident compressional-wave amplitude into four components. Note the significant changes at critical angles of refraction for the compressional and shear wave. It is important to keep in mind that the shape of these curves varies greatly with different situations of medium parameters. Also, note that at normal incidence no P-to-S conversion takes place, and equations (12a,12b,12c,12d) reduce to the special case described by equations (5) and (8).

Figure 11.2-6  An example of modeling of a CMP gather based on Zoeppritz equations. (Courtesy Hampson-Russell.)

From a practical standpoint, the angle-dependency of reflection coefficients implies that the reflection amplitude associated with a reflecting boundary varies with source-receiver separation as well as the depth of the reflector. For sufficiently deep reflectors and the typical source-receiver separations used in practice, the amplitude for the reflected compressional wave is nearly constant or slowly varying with offset (left of the critical angle on the curve corresponding to the reflected compressional-wave energy in Figure 11.2-5). It is this slowly varying portion of the P-to-P reflection curve that we have to detect from CMP data with limited offset range and in the presence of noise. Note also from Figure 11.2-5 that the largest P-to-S conversion occurs beyond the critical angle, corresponding to large source-receiver separations.

Reflection amplitude variations with angle of incidence, and therefore with offset, can be modeled using the Zoeppritz equations. For modeling the reflection amplitudes, you need well-log curves for the P-wave velocity, S-wave velocity, and density. Then, using equation (13a) compute the P-to-P reflection amplitudes as a function offset. Figure 11.2-6 shows a modeled CMP gather using well data. A sonic log was first converted to a blocky form to simplify the modeled amplitudes on the gather. Then, using the Gardner relation (equation 34a), a density profile was derived. Also, using a ratio of P-wave to S-wave velocity, a shear-wave velocity profile was generated. The objective in this modeling exercise was to see the effect of a change in Poisson’s ratio at some depth on the amplitude variation with offset. The Poisson’s ratio profile shows a change at approximately 650 ms at which time we observe a marked variation of amplitudes with offset on the CMP gather.

 ${\displaystyle \rho =k\alpha ^{1/4},}$ (34a)

## References

1. Richards, 1961, Richards, T. C., 1961, Motion of the ground on arrival of reflected longitudinal and transverse waves at wide-angle reflection distances: Geophysics, 27, 277–297.