# Ghost reflection

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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

An explosion in a lower medium (medium 2) initiates an upgoing pulse and a downgoing pulse. The upgoing pulse is reflected at the interface of medium 2 with an upper medium 1, and the downgoing reflected pulse (called the ghost reflection) forms a composite wavelet with the primary downgoing pulse (Figure 3). What will this composite wavelet look like? Suppose that the upper layer is very soft — for example, air, with $Z_{1}=0$ . The upgoing reflection coefficients are Figure 3.  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.

 {\begin{aligned}{\varepsilon }_{U{,pv}}={\frac {Z_{2}-0}{{0}+Z_{2}}}=1\mathrm {\ and\ } {\varepsilon }_{U},pres={\frac {0-Z_{2}}{{0}+Z_{2}}}=-1.\end{aligned}} (10)

On both sides of the shot point (i.e., above and below it), the pressure starts out as a high positive value, and the particle velocity starts out at a large magnitude. There is a polarity difference, however, between the particle velocity above the shot and the particle velocity below the shot because the explosion forces material away from the shot point. Therefore, we must consider particle velocity as a directional quantity. According to one convention, the particle velocity of the downgoing explosive wave is taken as negative, and the particle velocity of the upgoing explosive wave is taken as positive. Thus, the energy flux of the upgoing wave from the explosion is the negative of the energy flux of the downgoing explosive wave because the energies of the two waves are flowing in opposite directions. In summary, the explosion produces both (1) an upgoing wave (positive particle velocity, positive pressure), and (2) a downgoing wave (negative particle velocity, positive pressure).

The upgoing direct wave is reflected at the interface (the surface). The downgoing reflected wave is the ghost reflection, or “ghost.” The particle-velocity reflection coefficient is +1, and the pressure reflection coefficient is –1. Thus, the ghost (a downgoing wave) has positive particle velocity (because the upgoing particle reflection coefficient is +1) and negative pressure (because the upgoing pressure reflection coefficient is –1). The composite downgoing waveform in terms of particle velocity has the form of a negative direct pulse followed by a positive ghost pulse. The composite downgoing waveform in terms of pressure has the form of a positive direct pulse followed by a negative ghost pulse.

Thus, whether we measure pressure or particle velocity, the ghost reflection is of opposite polarity to that of the direct pulse. The delay time between the direct pulse and the ghost is equal to the two-way traveltime from the shot to the upper interface. If this delay is represented by T discrete time units and if the ideal explosive pulse is a spike of unit amplitude, then the composite wavelet for particle velocity has the form (–1,0, 0, …,0, 1,0, 0, …), and the composite wavelet for pressure has the form (1, 0, 0, …, 0, –1, 0, 0, …), where there are T – 1 zeros between the primary spike and the ghost spike. Figure 4a shows the downgoing direct-wave/ghost-wave complex with amplitude in terms of pressure. Figure 4b shows the corresponding complex in terms of particle velocity. In these figures, the spike series have been convolved with an arbitrary waveshape. Figure 4.  (a) The ghost complex for pressure. (b) The ghost complex for particle velocity.

Suppose that a receiver that is sensitive to particle velocity is placed on the water bottom. Would such a receiver be effective? For a rigid sea bottom (i.e., $Z_{2}=\infty$ for the layer below sea bottom), the reflection coefficients are

 {\begin{aligned}{\varepsilon }_{pv}={\frac {Z_{1}-Z_{2}}{Z_{1}+Z_{2}}}=-1\mathrm {and} \\{\varepsilon }_{\rm {pres}}={\frac {Z_{2}-Z_{1}}{Z_{1}+Z_{2}}}={1},\end{aligned}} (11)

and

 {\begin{aligned}\tau _{pv}=1+\varepsilon _{pv}=1-1=0\;\;{\text{and}}\\\tau _{\text{pres}}=1+\varepsilon _{\text{pres}}=1+1=2.\end{aligned}} (12)

The coefficient ${\tau }_{pv}={0}$ shows that when a downgoing wave strikes a hard bottom, the particle velocity cannot be transmitted. Because the particle velocity is continuous across the interface, it follows that the particle velocity at the bottom must be zero. Thus, a receiver that is sensitive to particle velocity and is dragging on the bottom from a cable would lose its effectiveness for a rigid sea bottom.

Let us discuss a hydrophone at the water surface. Why must the hydrophone (which is sensitive to pressure) be kept below the surface of the water? Consider the water surface as a free surface ($Z_{1}={0}$ for air). The upgoing reflection coefficients are

 {\begin{aligned}{\varepsilon }_{U,p\nu }={\frac {Z_{2}-Z_{1}}{Z_{1}+Z_{2}}}={1}\mathrm {\ and\ } {\varepsilon }_{U},pres={\frac {Z_{1}-Z_{2}}{Z_{1}+Z_{2}}}=-1.\end{aligned}} (13)

The upgoing transmission coefficients for particle velocity and pressure are

 {\begin{aligned}{\tau }_{U,pv}={1}+{\varepsilon }_{U,pv}={1+l=2}\mathrm {\ and\ } {\tau }_{U{\rm {pres}}}={l}+{\varepsilon }_{U{\rm {,pres}}}={1}-{1=0}.\end{aligned}} (14)

We see from the above two relations that the particle velocity at the surface is twice the particle velocity of the upgoing incident wave, whereas the pressure at the surface is zero. Thus, the pressure-sensitive hydrophone must be kept below the surface of the water because its effectiveness diminishes as the streamer floats to the surface.

What are the Knott-Zoeppritz equations? Thus far, we have limited ourselves to normal incidence. In the more general case of a plane wave that is incident at an arbitrary angle, reflected P- and S-waves and transmitted P- and S-waves will be generated. The amplitudes of all these waves can be found from the Knott-Zoeppritz equations (Aki and Richards, 2002; Sheriff, 2002). These equations are basic to amplitude-variation-with-offset (AVO) studies.