# Ghosting as a notch filter

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 9 295 - 366 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 9.20a

The ghost reflection from the sea surface acts as a notch filter for receivers planted on the sea floor. Plot the notch frequency versus water depth.

### Background

A notch filter discriminates against a very narrow band of frequencies.

### Solution

When a reflected wave is recorded by receivers on the sea floor, a ghost produced by reflection at the surface will be superimposed on the primary reflection $G(z)$ . The reflection coefficient at the surface is –1, so, from problem 9.19, the ghost is $G(z)(-z^{n})$ , where the two-way traveltime through the water layer is $n\Delta$ . Therefore, the recorded signal is $G(z)(1-z^{n})$ .

The recorded signal will be zero whenever $z^{n}=+1$ . Using Euler’s formula (Sheriff and Geldart, 1995, problem 15.12a), we get

{\begin{aligned}z^{n}=e^{-2\pi fn\Delta }\\=\cos(2\pi fn\Delta )\\-j{\sin }(2\pi fn\Delta ),\end{aligned}} so $z^{n}=+1$ when $2\pi fn\Delta =2\pi$ , that is, when $f=1/n\Delta$ The two-way travel traveltime for a source depth $d$ is $n\Delta =2d/V_{W}$ . Taking the water velocity as $V_{W}=1500$ m/s, we arrive at the result:

 {\begin{aligned}f=V_{W}/2d=750/d,\end{aligned}} (9.20a)

where $d$ is in meters. The graph of $f$ versus $d$ is shown in Figure 9.20a.

## Problem 9.20b

If air-gun sources are fired at 10-m depth, how will this affect the spectrum?

### Solution

Energy leaving the source and reflected at the surface will produce a ghost delayed by $\Delta t=2\times 10/1500\approx 13\ {\hbox{ms}}$ . The source ghost will have opposite polarity to the primary wave, with a delay of 13 ms corresponding to a frequency of 77 Hz; thus, it will interfere destructively with the frequency of 77 Hz in the original signal. As a result, frequencies in a narrow band centered on 77 Hz will be attenuated.

Additional ghosting will occur at receivers located below the surface due to reflection at the surface of the upcoming wavelet; this effect can be calculated in the same way as above.