Ghost amplitude and energy
If the source depth is (where is the wavelength) and in equation (6.7a), discuss the conditions under which the amplitude of is zero.
The low-velocity layer (LVL) is discussed in problem 4.16, ghosts in problem 3.8. When a ghost is superimposed on a downgoing wave, it affects not only the waveshape but also the directivity. In Figure 6.7a, is a point source at a depth and is the image point (see problem 4.1) for energy reflected at the surface. For a ghost arriving at point , the virtual path is . If the source emits the wave and the reflection coefficient at the surface is , the combined primary wave plus ghost at point is
Using the identity , we get
When , , ; also , so we get
Transmissivities and are defined in problem 3.6 where equation (3.6c) shows that
Absorption is discussed in problem 2.18.
Equation (6.7a) gives for the amplitude of the primary wave plus ghost,
For , , that is, , .... For , , i. e., , and the waves are traveling horizontally. When and , so there are no appropriate values of .
For a source below the base of the LVL, compare the amplitude and energy of ghosts generated at the base of the LVL and at the surface of the ground, given that the velocities and densities just below and within the LVL are , , and , respectively.
We assume small incidence angles so that equations (3.6a,b) are valid. Then
At the base of the LVL,
Assuming equal (unit) amplitudes for waves leaving the source in different directions, the ghost produced at the base of the LVL has amplitude and energy 0.50. The ghost produced at the surface has amplitude and energy . The amplitude of the ghost from the base of the LVL is times that of the ghost from the surface while the ratio of the energies is
Assume that the LVL is in thickness and that for the LVL; now what are the ratios of the ghost amplitudes and energies?
The surface ghost has to travel a distance farther than the ghost from the base of the LVL during which its amplitude is reduced by the factor . The previous amplitude was , so with absorption this becomes , and energy becomes 0.22 . The ratios of the amplitudes and energies in part (b) now become and , the dB values being 8.9 and 18.0 dB, respectively.
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Also in this chapter
- Characteristics of different types of events and noise
- Horizontal resolution
- Reflection and refraction laws and Fermat’s principle
- Effect of reflector curvature on a plane wave
- Diffraction traveltime curves
- Amplitude variation with offset for seafloor multiples
- Ghost amplitude and energy
- Directivity of a source plus its ghost
- Directivity of a harmonic source plus ghost
- Differential moveout between primary and multiple
- Suppressing multiples by NMO differences
- Distinguishing horizontal/vertical discontinuities
- Identification of events
- Traveltime curves for various events
- Reflections/diffractions from refractor terminations
- Refractions and refraction multiples
- Destructive and constructive interference for a wedge
- Dependence of resolvable limit on frequency
- Vertical resolution
- Causes of high-frequency losses
- Ricker wavelet relations
- Improvement of signal/noise ratio by stacking