Ghosting as a notch filter
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.
A notch filter discriminates against a very narrow band of frequencies.
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 .
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:
where is in meters. The graph of versus is shown in Figure 9.20a.
If air-gun sources are fired at 10-m depth, how will this affect the spectrum?
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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Also in this chapter
- Fourier series
- Space-domain convolution and vibroseis acquisition
- Fourier transforms of the unit impulse and boxcar
- Extension of the sampling theorem
- Alias filters
- The convolutional model
- Water reverberation filter
- Calculating crosscorrelation and autocorrelation
- Digital calculations
- Convolution and correlation calculations
- Properties of minimum-phase wavelets
- Phase of composite wavelets
- Tuning and waveshape
- Making a wavelet minimum-phase
- Zero-phase filtering of a minimum-phase wavelet
- Deconvolution methods
- Calculation of inverse filters
- Inverse filter to remove ghosting and recursive filtering
- Ghosting as a notch filter
- Wiener (least-squares) inverse filters
- Interpreting stacking velocity
- Effect of local high-velocity body
- Apparent-velocity filtering
- Complex-trace analysis
- Kirchhoff migration
- Using an upward-traveling coordinate system
- Finite-difference migration
- Effect of migration on fault interpretation
- Derivative and integral operators
- Effects of normal-moveout (NMO) removal
- Weighted least-squares