Amplitude variation with offset for seafloor multiples

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	6
Pages	181 - 220
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 6.6a

Assume that Figure 6.6a shows relative amplitudes correctly (divergence having been allowed for). The water depth is 0.42 km and the velocity below the seafloor is 2.59 km/s. If the reflection coefficient is maximum at the critical angle, on which traces would you expect the maximum amplitude for the first, second, third, and fourth multiples?

Background

Divergence or geometrical spreading and multiples were discussed in problem 3.8, the reflection coefficient (reflectivity) $R$ in problem 3.6, the critical angle in problem 4.18. A refraction does not exist to the left of $Q$ (in Figure 4.18a); $OQ=2h{\rm {\;tan\;}}\theta _{c}=x^{'},x^{'}$ being the critical distance. The reflectivity $R$ (see equation 3.6a) is generally maximum when $\theta =\theta _{c}$ (see Sheriff and Geldart, 1995, section 3.3).

Figure 6.6a.  Record showing seafloor multiples.

Solution

We interpret the five more-or-less equally spaced events in Figure 6.6a as a primary reflection followed by four multiples. We assume that the amplitude maxima occur when the angle of incidence equals the critical angle. Hence the offsets for the amplitude maxima correspond to multiples of the critical distance.

Taking the water velocity as 1.50 km/s, $\theta _{c}={\rm {sin}}^{-1}(1.50/2.59)=35.4^{\circ }$. If we assume that the source and receiver are both at depths of 15 m (the conventional streamer depth), the critical distance $x^{'}$ is

${\begin{aligned}x^{'}=2z{\rm {\;tan\;}}\theta _{c}=2(420-15){\rm {\;tan\;}}35.4^{0}=580\ {\rm {m}}.\end{aligned}}$

The maximum amplitudes occur at offsets of approximately 600, 1200, 1800, 2400, and 3000 m, that is, at 600 m intervals, close to the calculated critical distance.

Problem 6.6b

What should be the ratio of the amplitudes of successive multiples on the short-offset trace? How do these calculations compare with observations? What unaccounted for factors affect this comparison?

Solution

The ratio of successive amplitudes should be $-R$. Taking the amplitude of the incident wave as unity, the amplitudes of the multiples become $-R$, $R^{2}$, $-R^{3}$, $R^{4}$. The measured amplitudes on the shortest traces are 32 (at offset 425 m), 12, 4.5, 2.0, 1.2 mm, the first two measurements not being very accurate. The ratios of the successive measurements are 2.7, 2.7, 2.2, 1.7, the average value being about 2.3. Assuming the value of 2.3,

${\begin{aligned}R=1/2.3=0.43=(Z_{2}-Z_{1})/(Z_{2}+Z_{1}),\end{aligned}}$

where $Z_{t}=\rho _{i}V_{i}$ (see problem 3.6). Taking $\rho _{1}=1.03\ {\hbox{g}}/{\hbox{cm}}^{3}$, $Z_{1}=1.03\times 1.50=1.54$, we get $Z_{2}=3.9$. Since $V_{2}=2.59\ {\rm {km/s}}$, we have only one unknown, $\rho _{2}$, and we get $\rho _{2}\approx 1.5$ ${\rm {g/cm}}^{3}$, a reasonable value (but nevertheless of questionable accuracy in view of the many uncertainties involved).

The above comparison would be affected principally by time-dependent factors that have not been taken into account, such as absorption, transmission losses, peg-leg multiples (problem 3.8), etc. Losses of these kinds are very small for wave travel in water. Amplitude changes may also have occurred during data processing.

Continue reading

Previous section	Next section
Diffraction traveltime curves	Ghost amplitude and energy
Previous chapter	Next chapter
Geometry of seismic waves	Characteristics of seismic events
Table of Contents (book)

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

External links

find literature about Amplitude variation with offset for seafloor multiples