Ghosting as a notch filter

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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 . The reflection coefficient at the surface is –1, so, from problem 9.19, the ghost is , where the two-way traveltime through the water layer is . Therefore, the recorded signal is .

Figure 9.20a.  Ghosting versus source depth.

The recorded signal will be zero whenever . Using Euler’s formula (Sheriff and Geldart, 1995, problem 15.12a), we get

so when , that is, when The two-way travel traveltime for a source depth is . Taking the water velocity as m/s, we arrive at the result:


(9.20a)

where is in meters. The graph of versus 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 . 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.

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