# Water reverberations - book

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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 11 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The water-reverberation problem in marine seismic operations can be described by the following normal-incidence model: The water-air interface is a strong reflector, with a reflection coefficient of nearly –1. The water-bottom interface is also a strong reflector, with reflection coefficient ${\displaystyle \rho }$, which is less than unity in magnitude. The water layer acts as an imperfect energy trap in which a seismic pulse is reflected successively between its two interfaces. Seismic energy from the source first encounters the water layer on its way downward. The transmitted energy proceeds toward the deep horizons, which reflect it. Upon reflection, the energy returns in an upward direction, and it again encounters the water layer. Multiple reflections within the water layer appear on the seismic trace as reverberations, which obscure the reflections from the deep horizons (Robinson, 1967[1]; Sengbush, 1983[2], p. 103-123).

Figure 7.  Raypaths for the downward pass through the water layer. (For clarity, the raypaths have been drawn as slanting lines, although in our normal-incidence model, they are perpendicular to the two interfaces.)

Let the integer N represent the two-way traveltime parameter in the water layer (Figure 7). A unit source impulse is generated at the surface at time zero. The impulse travels downward in the water and is reflected at the water bottom. Thus, an upgoing pulse of size ${\displaystyle \rho }$ is generated, and it arrives at the water surface at discrete time parameter N. There it is reflected downward and becomes a secondary source. The water surface has a reflection coefficient of –1 as seen from below, so the secondary impulse has the value ${\displaystyle -\rho }$ and occurs at time parameter N. This secondary impulse travels the same path as did the initial impulse, thereby producing another secondary source with its impulse ${\displaystyle \rho ^{2}}$ occurring at time parameter 2N. This repeats again and again. The total result is the one-pass minimum-delay reverberation train

 {\displaystyle {\begin{aligned}q=(1,0,0,\ldots 0,-\rho ,0,0,\ldots ,\ 0,{\rho }^{2},0,0,\ldots \ 0,-{\rho }^{3},0,0,\dots ),\end{aligned}}} (1)

where the successive nonzero values are separated by the discrete time parameter N. That is, ${\displaystyle N-{1}}$ zeros separate any two consecutive nonzero values. For this reason, N is called the cycle time of the reverberation. The one-pass dereverberation filter is the inverse of the one-pass reverberation train; that is, the one-pass dereverberation filter is the minimum-delay operator

 {\displaystyle {\begin{aligned}q^{-1}=({\rm {l}},0,0,\ldots 0,\rho ).\end{aligned}}} (2)

The analysis just given treats the water-reverberation effect for one-way travel (down only). The water layer acts as a filter, and the seismic energy passes through this filter twice — once as it goes down to the deep reflecting horizon and again as it returns to the surface. Hence, we have, in effect, a situation in which the source wavelet passes through two cascaded sections of the water layer. Thus, the two-pass reverberation train is the convolution of the one-pass reverberation train with itself. That is, the two-pass reverberation train is the minimum-delay signal

 {\displaystyle {\begin{aligned}b=q*q=\ \left({1,0,0,\ldots ,0,}-2\rho {,\ 0,0,\ldots ,0,3}{\rho }^{2}{,0,0,\ldots ,0,}-{4}{\rho }^{3}{,0,0,\ }\dots \right).\end{aligned}}} (3)

Its inverse gives the two-pass minimum-delay dereverberation filter

 {\displaystyle {\begin{aligned}a=b^{-1}=({\rm {l}},0,0,\ldots ,{0,2}\rho ,0,0,\ldots ,0,{\rho }^{2}).\end{aligned}}} (4)

The convolution of a and b gives a unit spike. When the inverse filter a is convolved with our idealized trace, it eliminates the water-layer reverberations on the trace.

Figure 8.  Amplitude spectra of reverberations and of a dereverberation operator.

Suppose the water depth is 200 ft (60 m). Let the velocity of the water be 4800 ft/s (1460 m/s). Thus, the two-way time in the water layer is ${\displaystyle N={2}\left({200}\right){/}4800=0.083s}$ (Figure 8). The amplitude spectrum of the reverberation train is periodic, with resonant peaks occurring at 6, 18, 30, 42, 54, 66, 78, … Hz. The separation between resonant peaks is ${\displaystyle 1/N={12Hz}}$. The amplitude spectrum of the dereverberation operator is the reciprocal of the amplitude spectrum of the reverberation train. The notches occur at 6, 18, 30, 42, 54, 66, 78, … Hz.

In practice, reverberations often are generated by a more complicated physical situation than the one we have described, so the simple dereverberation operator a given above is not adequate. The received seismic trace is made up of many reflections, each represented by its reverberation train. All of these reverberation signals overlap to various degrees. Thus, usually it is not feasible to obtain a direct measurement of the shape of any individual reverberation train. In such a situation, the method of predictive deconvolution can be used to attenuate the reverberations and thus to better delineate the deep reflections, which we want to see.

## References

1. Robinson, E. A., 1967, Multichannel time series analysis with digital computer programs: Holden Day Press.
2. Sengbush, R. L., 1983, Seismic exploration methods: International Human Resources Development Company (IHRDC).