# Reflection coefficients and transmission coefficients

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 8 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

We will deal with the 1D case, so we treat only variations along the depth axis, which points straight down into the earth. As we saw in Chapter 3, a plane wave is a wave whose wavefronts are infinite parallel planes normal to the direction of travel. Suppose a plane P-wave travels vertically downward in a homogeneous isotropic medium. This downgoing incident wave encounters a horizontal interface separating two media. The upper medium has density ${\displaystyle \rho _{1}}$ and seismic wave velocity ${\displaystyle v_{1}}$, and the lower medium has density ${\displaystyle {\rho }_{2}}$ and seismic wave velocity ${\displaystyle v_{2}}$. The result is that a portion of the wave energy will be reflected at the interface, and the remainder will be transmitted.

This splitting of the incident wave into a reflected wave and a transmitted wave at the interface is caused by the abrupt change in rock density and/or velocity. For the normal-incidence case that we treat here, the reflected and transmitted waves have the same shape and breadth as did the incident wave, but they differ from it in amplitude. The ratio of the amplitude of the reflected wave to that of the incident wave is termed the reflection coefficient. Similarly, the ratio of the amplitude of the transmitted wave to that of the incident wave is called the transmission coefficient. However, the reflection coefficient defined for a wave in which the amplitude is measured in terms of particle velocity is different from the reflection coefficient for a wave in which amplitude is measured in terms of pressure. The same statement holds for the transmission coefficient. Let us now establish this difference.

A reflector is characterized by a contrast in acoustic impedance, which gives rise to a seismic reflection (O’Doherty and Anstey, 1971[1]). Reflectivity refers to the reflection coefficient. The amplitudes that are required in the definitions of reflection and transmission coefficients can be obtained by solving equations that express the continuity of displacement and stress at the boundary. We consider the case of normal incidence on an interface separating two media of densities ${\displaystyle \rho _{1}}$ and ${\displaystyle \rho _{2}}$ and velocities ${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$. The impedances are, respectively, ${\displaystyle Z_{1}={\rho }_{1}v_{1}}$ and ${\displaystyle Z_{2}={\rho }_{2}v_{2}}$. Let an incident downgoing wave in a formation with impedance ${\displaystyle Z_{1}}$ strike a formation with impedance ${\displaystyle Z_{2}}$ (Figure 1).

Figure 1.  Reflection and transmission for a downgoing wave at an interface. All the wave directions are vertical, but they are depicted as slanting lines for visual clarity.

Both the particle-velocity attribute and the pressure attribute of the wave motion must be continuous across the interface, as expressed by the two equations ${\displaystyle D_{1}+U_{1}=D_{2}}$ and ${\displaystyle d_{1}+u_{1}=d_{2}}$. As we have seen, the Berkhout convention gives ${\displaystyle d_{1}=Z_{1}D_{1},u_{1}=-Z_{1}U_{1}}$, and ${\displaystyle d_{2}=Z_{2}D_{2}}$. Thus, the two equations can be written as

 {\displaystyle {\begin{aligned}D_{1}+U_{1}=D_{2}\mathrm {\ \ and\ \ } Z_{\rm {1}}D_{1}-Z_{1}U_{1}=Z_{2}D_{2}.\end{aligned}}} (4)

If the amplitudes of the waves are measured in terms of particle velocity (denoted by the subscript pv on the coefficient), then the reflection coefficient is defined as ${\displaystyle {\varepsilon }_{pv}=U_{1}/D_{1}}$ and the transmission coefficient as ${\displaystyle {\tau }_{pv}=D_{2}/D_{1}}$. Using equations 4, we obtain the expression for the particle-velocity reflection coefficient given by

 {\displaystyle {\begin{aligned}{\varepsilon }_{pv}={\frac {U_{1}}{D_{\rm {1}}}}={\frac {Z_{1}-Z_{2}}{Z_{1}+Z_{2}}}.\end{aligned}}} (5)

If the amplitudes of the downgoing incident wave are measured in terms of pressure (denoted by the subscript pres on the coefficient), then the corresponding pressure reflection coefficient is

 {\displaystyle {\begin{aligned}{\varepsilon }_{\rm {pres}}={\frac {u_{\rm {1}}}{d_{1}}}={\frac {Z_{2}-Z_{1}}{Z_{1}+Z_{2}}}.\end{aligned}}} (6)

We see that one is the negative of the other; that is, ${\displaystyle {\varepsilon }_{\rm {pres}}=-{\varepsilon }_{pv}}$. The transmission coefficient ${\displaystyle \tau }$ is always equal to ${\displaystyle 1+\varepsilon }$. Thus, the particle-velocity transmission coefficient and the pressure transmission coefficient are, respectively,

 {\displaystyle {\begin{aligned}{\tau }_{pv}={1}+{\varepsilon }_{pv}={\frac {2Z_{1}}{Z_{1}+Z_{2}}}\mathrm {and} {\tau }_{\rm {pres}}={1}+{\varepsilon }_{\rm {pres}}={\frac {2Z_{2}}{Z_{1}+Z_{2}}}.\end{aligned}}} (7)

We note that the particle-velocity reflection coefficient and the pressure reflection coefficient have different signs. As a physical example, suppose a downgoing compressional wave of magnitude 1.0 hits a layer with higher impedance. For simplicity, let us assume that the wave strikes a layer with infinitely great impedance. This gives rise to an upgoing compressional wave of magnitude 1.0. Both reflection coefficient formulas predict this. The pressure-reflection-coefficient formula is equal to +1. The reflected upgoing wave, as recorded by a hydrophone, would retain the same amplitude as does the incident downgoing wave. We note that pressure measurements are scalars and are independent of the wave’s direction of travel. The particle-velocity reflection coefficient is equal to –1.0. The reflected upgoing wave, as recorded by a geophone, would have the negative amplitude of the incident downgoing wave. We note that particle-velocity measurements are vectors and are dependent on the wave’s direction of travel. Thus, the particle velocity of the reflected upgoing wave has the opposite sign to that of the particle velocity of the incident downgoing wave. The two formulas for the reflection coefficient describe wave action as it is measured by the different recording devices.

The reflection and transmission coefficients usually refer to a downgoing incident wave. What are the respective coefficients for an upgoing incident wave? Let the coefficient subscript U denote an upgoing incident wave. We want to determine the reflection coefficient ${\displaystyle \varepsilon _{U}}$ and transmission coefficient ${\displaystyle \tau _{U}}$ for an upgoing incident wave in the lower medium with impedance ${\displaystyle Z_{2}}$, when the wave strikes an interface with the upper medium with impedance ${\displaystyle Z_{1}}$ (Figure 2).

Figure 2.  Reflection and transmission for an upgoing wave at an interface. All the wave directions are vertical, but they are depicted as slanting lines for visual clarity.

Now the incident wave is upgoing, and the change of impedance is from ${\displaystyle Z_{2}}$ to ${\displaystyle Z_{1}}$.

Thus, the roles of ${\displaystyle Z_{1}}$ and ${\displaystyle Z_{2}}$ are interchanged, so the upgoing coefficients become

 {\displaystyle {\begin{aligned}\varepsilon _{U,pv}=-\varepsilon _{pv}={\frac {Z_{2}-Z_{1}}{Z_{1}+Z_{2}}}\;\;{\text{and}}\;\varepsilon _{U,{\text{pres}}}=-\varepsilon _{\text{pres}}={\frac {Z_{1}-Z_{2}}{Z_{1}+Z_{2}}},\end{aligned}}} (8)

and

 {\displaystyle {\begin{aligned}\tau _{U,pv}=1+\varepsilon _{U,pv}={\frac {2Z_{2}}{Z_{1}+Z_{2}}}\;\;{\text{and}}\;\;\tau _{\text{U,pres}}=1+\varepsilon _{U,{\text{pres}}}={\frac {2Z_{1}}{Z_{1}+Z_{2}}}.\end{aligned}}} (9)