Refractions and refraction 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 6.16a

Determine the traveltime curve for the refraction ${\mathit {SMNPQR}}$ and the refraction multiple ${\mathit {SMNTUWPQR}}$ in Figure 6.16a.

Solution

We assume that the velocities are known to three significant figures. Then, using equation (3.1a),

${\begin{aligned}\theta _{2}={\rm {sin}}^{-1}(2.00/4.20)=28.4^{0};\theta _{1}={\rm {sin}}^{-1}(2.80/4.20)=41.8^{\circ }\end{aligned}}$

The traveltimes can be obtained either graphically or by calculation. Calculating, we get for the refraction traveltimes $t_{R}$

${\begin{aligned}t_{R}=2SM/2.80+2MN/2.00+NP/4.20\\=2\times 0.75/2.80{\rm {\;cos\;}}41.8^{\circ }+2\times 3.25/(2.00{\rm {\;cos\;}}28.4^{\circ })\\+(x-2\times 0.75{\rm {\;tan\;}}41.8^{\circ }-2\times 3.25{\rm {\;tan\;}}28.3^{\circ })/4.20\\=x/4.20+2\times 0.75{\rm {\;cos\;}}41.8^{\circ }/2.80+2\times 3.25{\rm {\;cos\;}}28.4^{\circ }/2.00\\=x/4.20+3.26.\end{aligned}}$

Figure 6.16a.  Refraction multiple.

The critical distance (see equation (6.15a) is

${\begin{aligned}x^{'}=2(0.75{\rm {\;tan\;}}41.8^{\circ }+3.25{\rm {\;tan\;}}28.4^{\circ })=4.86\ {\rm {km}}.\end{aligned}}$

The traveltime curve for SMNTUWPQR is parallel to that for SMNPQR and displaced toward longer time by the amount $\Delta t$ where

${\begin{aligned}\Delta t=2TU/2.00-TW/4.20\\=2\times 3.25/(2.00{\rm {\;cos\;}}28.4^{\circ })-2\times 3.25({\rm {\;tan\;}}28.4^{\circ })/4.20=2.86\ {\rm {s}}.\end{aligned}}$

The critical distance for SMNTUWPQR is increased to

${\begin{aligned}x^{'}=2\times 0.75{\rm {\;tan\;}}41.8^{\circ }+4\times 3.25{\rm {\;tan\;}}28.4^{\circ }=8.37\ {\rm {km}}.\end{aligned}}$

The traveltime curves are plotted as curves (a) in Figure 6.16b.

Figure 6.16b.  Traveltime curves. Letters denote curves for respective parts (a), (b), (c).

Problem 6.16b

Determine the traveltime curves when both refractor and reflector dip $8^{\circ }$ down to the left, the depths shown in Figure 6.16a now being the slant distances from $S$ to the interfaces.

Solution

A combined graphical and calculated solution probably provides the easiest solution although Adachi’s method (see problem 11.5) could be used to give greater precision if the data accuracy warranted. A large-scale graph was used to achieve better accuracy; Figure 6.16c is a reduced-scale replica. The traveltime curves are shown in Figure 6.16b labeled (b).

The critical distance for the refraction is $SR_{1}=4.38$ km, and

${\begin{aligned}t_{R_{1}}=(1.00+0.18)/2.80+2\times 3.71/2.00=4.13\ {\rm {s}}.\end{aligned}}$

The $V_{2}$-layer outcrops at $R_{2}=5.39$ km,

${\begin{aligned}t_{R_{2}}=1.00/2.80+2\times 3.71/2.00+1.12/4.20=4.33\ {\rm {s}}.\end{aligned}}$

Figure 6.16c.  Geometry and raypaths for dip $8^{\circ }$ to the left.

The headwave has a different slope to the right of $R_{2}$. To plot the curve in this zone, we use point $R_{4}$ at the offset $x=8.79\ {\rm {km}}$. Then,

${\begin{aligned}t_{R_{4}}=1.00/2.80+(3.71+3.21)/2.00+4.71/4.20=4.94\ {\rm {s}},\end{aligned}}$

and the headwave curve is a straight line joining the traveltimes at the points $R_{2}$ and $R_{4}$.

We have two types of reflected refractions: a typical path for the first type is $SMNP_{1}U_{2}WP_{4}R_{4}$, the reflection occurring at the shallow dipping interface, The second type, $SMNP_{5}R_{6}P_{7}R_{7}$, involves reflection at the surface. The first type exists between $R_{3}$ and $R_{4}$, and the curve is parallel to the head-wave curve to the right of $R_{3}$. The second type exists to the right of $R_{4}$ and the curve is parallel to the other reflected refraction. To plot the reflected-refraction curves, we need one point on each curve and then use the refraction-curve slope to the right of $R_{3}$. For the first type, we find the coordinates of $R_{5}$:

${\begin{aligned}x=6.80\ {\hbox{km}},\\t_{R_{5}}=1.00/2.80+(2\times 3.71+3.32+3.12)/2.00+1.12/4.20=7.55\ {\rm {s}}.\end{aligned}}$

For the second type we find coordinates of $R_{7}$:

${\begin{aligned}x=9.15\ {\rm {km}},\\t_{\rm {R_{7}}}=1.00/2.80+(3.71+3.30+2.97+2.78)/2.00+3.85/4.20=7.65\ {\rm {s}}.\end{aligned}}$

Problem 6.16c

What happens when the reflector dips $3^{\circ }$ to the left and the refractor $5^{\circ }$to the left?

Figure 6.16d.  Geometry and raypaths for $3^{\circ }$ and $5^{\circ }$ dips to the left.

Solution

A summary of the detailed graphical solution is as follows. The traveltime curves are shown in Figure 6.16b labeled (c). The various angles in Figure 6.16d are

${\begin{aligned}\theta _{c}=28.4^{\circ }\;,\;\theta _{2}=(28.4^{\circ }+5^{\circ }-3^{\circ })\;=30.4^{\circ },\\\theta _{1}={\rm {sin}}^{-1}[(2.80/2.00){\rm {\;sin\;}}30.4^{\circ }]=45.1^{\circ },\\\alpha _{1}={\rm {angle\ of\ approach}}\ =45.1^{\circ }+3^{\circ }=48.1^{\circ },\\\theta _{2}^{'}=(28.4^{\circ }-5^{\circ }+3^{\circ })\;=26.4^{\circ },\\\theta _{1}^{'}={\rm {sin}}^{-1}[(2.80/2.00){\rm {\;sin\;}}38.5^{\circ }]=38.5^{\circ },\\\alpha _{1^{'}}=38.5^{\circ }-3^{\circ }\;=35.5^{\circ }\end{aligned}}$

The refraction curve is a straight line though $R_{5}$ and $R_{7}$:

${\begin{aligned}R_{5}:x^{'}=4.64\ {\rm {km}}={\rm {critical\ distance}},\\t_{R_{5}}=(1.09+0.44)/2.80+(3.65+3.50)/2.00=4.20\ {\rm {km}};\\R_{7}:x=8.15\ {\rm {km}},\\t_{R_{7}}=(1.09+0.44)/2.80+(3.64+3.35)/2.00+3.80/4.20\\=4.95\ {\rm {s}}.\end{aligned}}$

The incident angle at $W$ is $\theta _{3}=24.4^{\circ }$, which is less than the critical angle, so that no refraction will be generated there, only a reflection. However, the refraction that starts at $N$ will give rise to upgoing rays which will be reflected, giving a reflected refraction, such as the ray that ends at $R_{8}$. The traveltime curve is parallel to the refraction curve and exists beyond $R_{6}$ whose coordinates are

${\begin{aligned}x=7.79\ {\rm {km}},\\t_{R_{6}}=(1.09+0.48)/2.80+(3.64+3.50+3.42+3.36)/2.00=7.52\ {\rm {s}}.\end{aligned}}$

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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

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