# Water reverberations

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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

The water-reverberation problem in marine seismic operations can be described as follows: Consider just two interfaces — the water surface and the water bottom — with a separation given by the two-way traveltime T. The reflectivity is ${\displaystyle \{\varepsilon _{0},\;0,\;0,\;...,\;0,\;\varepsilon _{T}\}}$. For definiteness, let us assume that T = 3. Then the reflectivity is ${\displaystyle \{\varepsilon _{0},\;0,\;0,\;\varepsilon _{3}\}}$. The two-sided autocorrelation gt of the reflectivity is

 {\displaystyle {\begin{aligned}\left(g_{3}{,0,0,\ }g_{0}{,0,0,\ }g_{3}\right)=\left({\varepsilon }_{0}{\varepsilon }_{2}{\ ,\ 0,0,\ }{\varepsilon }_{0}^{2}+{\varepsilon }_{2}^{2}{,0,0,\ }{\varepsilon }_{0}{\varepsilon }_{3}\right).\end{aligned}}} (16)

First, take the zero-lag term and the right-hand side of this autocorrelation, and then replace the zero-lag term by unity. The result is

 {\displaystyle {\begin{aligned}\left({1,0,0,\ }g_{3}\right)=\left({1,\ 0,0,\ }{\varepsilon }_{0}{\varepsilon }_{3}\right).\end{aligned}}} (17)

As a Z-transform, this result is

 {\displaystyle {\begin{aligned}{1+}g_{3}Z^{3}={1+}{\varepsilon }_{0}{\varepsilon }_{3}Z^{3}.\end{aligned}}} (18)

We will show now that this Z-transform represents the feedback loop that produces a reverberating wavetrain.

The surface (the water-air interface) has the reflection coefficient ${\displaystyle {\varepsilon }_{0}}$. Because the surface is a strong reflector, this reflection coefficient is close to unity in magnitude. The reflection coefficient for a wave striking the water surface from below is ${\displaystyle -{\varepsilon }_{0}}$.

The water-bottom interface also is a strong reflector, with the reflection coefficient ${\displaystyle \varepsilon _{3}}$. Thus, the water layer acts as an imperfect energy trap in which a seismic pulse is reflected successively between its two interfaces. Energy that bounces back and forth between two interfaces is called a reverberation.

Let us examine the dynamics of this water reverberation. Our approach is applicable to any reverberation occurring between two interfaces, not just to the water reverberation occurring between the water-surface layer and the water-bottom layer encountered in marine seismic work.

In summary, the source and receiver locations are just below the water surface. Source energy travels down to the water bottom, from which it is reflected. The reflected upgoing energy proceeds toward the water surface, where it is recorded as a primary reflection. However, this upgoing energy also is reflected from the surface and then continues to bounce back and forth in the water layer. These multiple reflections within the water layer make up the water-layer reverberation. Such reverberations are generally undesirable because they obscure reflections from deeper horizons. Figure 8 shows the raypaths for the water reverberation. (For clarity, the raypaths have been drawn as slanting lines, although in our normal-incidence model, they are perpendicular to the two interfaces.)

Figure 8.  The downward pass through the water layer.

In our example, the integer T = 3 represents the two-way traveltime parameter in the water layer. The source is a unit impulse at A, which is a point just below the surface. The pulse travels downward in the water and arrives at the water bottom. There, it is reflected. The resulting upgoing pulse at B has value ${\displaystyle {\varepsilon }_{3}}$ and arrives at the water surface C at time 3. The receiver records this upgoing pulse as the primary reflection. However, this upgoing pulse then is reflected downward. In the case of an upcoming incident wave, the water surface has a reflection coefficient of –${\displaystyle {\varepsilon }_{3}}$, so the downgoing pulse at D has the value ${\displaystyle -{\varepsilon }_{0}{\varepsilon }_{3}}$. This downgoing pulse travels downward and is reflected from the water bottom at E. It returns to the surface at F. The receiver records this upgoing pulse as the first multiple reflection, with the value ${\displaystyle {\varepsilon }_{3}\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right)}$ occurring at time 2T = 6. This repeats to produce the second multiple reflection with the value ${\displaystyle {\varepsilon }_{3}{\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right)}^{2}}$ occurring at time 3T = 9. The process keeps repeating itself, each time producing another multiple reflection. Because we omit the shot, we write the trace as

 {\displaystyle {\begin{aligned}h={\ }\left({0,0,0,\ }{\varepsilon }_{3}{,0,0,\ }{\varepsilon }_{3}\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right){,\ 0,0,\ }{\varepsilon }_{1}{\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right)}^{2}{\ ,\ 0,0,,\ }\dots \right),\end{aligned}}} (19)

where the first 0 is at time 0, the second 0 is at time 1, the third 0 is at time 2, the ${\displaystyle \varepsilon _{3}}$ is at time 3, and so on. In other words, the shot occurs at time 0, the first nonzero value ${\displaystyle {\varepsilon }_{3}}$ (the primary) occurs at time T = 3, the second nonzero value (the first multiple) occurs at time 6, the third nonzero value (the second multiple) occurs at time 9, and so on. The successive nonzero values (after the primary) are separated by the time parameter T = 3. For that reason, T is called the cycle time of the reverberation. Thus, the Z-transform of the impulsive trace is

${\displaystyle H\left(Z\right)={\varepsilon }_{3}Z^{3}+{\varepsilon }_{3}\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right)Z^{6}+{\varepsilon }_{3}{\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right)}^{2}Z^{9}+\dots }$

 {\displaystyle {\begin{aligned}={\varepsilon }_{3}Z^{3}\left[{1}+\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right)Z^{3}+{\left(-{\varepsilon }_{0}{\varepsilon }_{3}\right)}^{2}Z^{6}+\dots \right]={\frac {{\varepsilon }_{3}Z^{3}}{{1}+{\varepsilon }_{0}{\varepsilon }_{3}Z^{3}}}.\end{aligned}}} (20)

The geometric series within square brackets is convergent and summable because the autocorrelation coefficient ${\displaystyle g_{3}=\varepsilon _{0}\;\varepsilon _{3}}$ is less than one in magnitude. Now comes an important point. We recognize that the multiple-generating factor ${\displaystyle {\varepsilon }_{0}{\varepsilon }_{3}}$ that occurs in the above expression for the trace X is the autocorrelation coefficient ${\displaystyle g_{3}}$. Thus, the Z-transform becomes

 {\displaystyle {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{3}Z^{3}}{{1}+g_{3}Z^{3}}}.\end{aligned}}} (21)

We therefore have shown that the primary reflection is represented by the feedforward term ${\displaystyle {\varepsilon }_{3}Z^{3}}$, whereas the reverberation is represented by the feedback term ${\displaystyle 1+g_{3}Z^{3}}$ (Figure 9).

Figure 9.  Block diagram of the feedback system that generates reverberation.