# Amplitude variation with offset for seafloor multiples

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 6 181 - 220 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 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) ${\displaystyle R}$ in problem 3.6, the critical angle in problem 4.18. A refraction does not exist to the left of ${\displaystyle Q}$ (in Figure 4.18a); ${\displaystyle OQ=2h{\rm {\;tan\;}}\theta _{c}=x^{'},x^{'}}$ being the critical distance. The reflectivity ${\displaystyle R}$ (see equation 3.6a) is generally maximum when ${\displaystyle \theta =\theta _{c}}$ (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, ${\displaystyle \theta _{c}={\rm {sin}}^{-1}(1.50/2.59)=35.4^{\circ }}$. If we assume that the source and receiver are both at depths of 15 m (the conventional streamer depth), the critical distance ${\displaystyle x^{'}}$ is

{\displaystyle {\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 ${\displaystyle -R}$. Taking the amplitude of the incident wave as unity, the amplitudes of the multiples become ${\displaystyle -R}$, ${\displaystyle R^{2}}$, ${\displaystyle -R^{3}}$, ${\displaystyle R^{4}}$. 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,

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

where ${\displaystyle Z_{t}=\rho _{i}V_{i}}$ (see problem 3.6). Taking ${\displaystyle \rho _{1}=1.03\ {\hbox{g}}/{\hbox{cm}}^{3}}$, ${\displaystyle Z_{1}=1.03\times 1.50=1.54}$, we get ${\displaystyle Z_{2}=3.9}$. Since ${\displaystyle V_{2}=2.59\ {\rm {km/s}}}$, we have only one unknown, ${\displaystyle \rho _{2}}$, and we get ${\displaystyle \rho _{2}\approx 1.5}$ ${\displaystyle {\rm {g/cm}}^{3}}$, 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.