Ghosting as a notch filter

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	9
Pages	295 - 366
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	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 $G(z)(1-z^{n})$.

Figure 9.20a.  Ghosting versus source depth.

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.

Continue reading

Previous section	Next section
Inverse filter to remove ghosting and recursive filtering	Autocorrelation
Previous chapter	Next chapter
Reflection field methods	Geologic interpretation of reflection data
Table of Contents (book)

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
• Semblance
• 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
• Autocorrelation
• 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

External links

find literature about Ghosting as a notch filter