Refractions and refraction multiples

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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 6.16a

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

Solution

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

{\displaystyle {\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 ${\displaystyle t_{R}}$

{\displaystyle {\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

{\displaystyle {\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 ${\displaystyle \Delta t}$ where

{\displaystyle {\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

{\displaystyle {\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 ${\displaystyle 8^{\circ }}$ down to the left, the depths shown in Figure 6.16a now being the slant distances from ${\displaystyle S}$ 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 ${\displaystyle SR_{1}=4.38}$ km, and

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

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

{\displaystyle {\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 ${\displaystyle 8^{\circ }}$ to the left.

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

{\displaystyle {\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 ${\displaystyle R_{2}}$ and ${\displaystyle R_{4}}$.

We have two types of reflected refractions: a typical path for the first type is ${\displaystyle SMNP_{1}U_{2}WP_{4}R_{4}}$, the reflection occurring at the shallow dipping interface, The second type, ${\displaystyle SMNP_{5}R_{6}P_{7}R_{7}}$, involves reflection at the surface. The first type exists between ${\displaystyle R_{3}}$ and ${\displaystyle R_{4}}$, and the curve is parallel to the head-wave curve to the right of ${\displaystyle R_{3}}$. The second type exists to the right of ${\displaystyle R_{4}}$ 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 ${\displaystyle R_{3}}$. For the first type, we find the coordinates of ${\displaystyle R_{5}}$:

{\displaystyle {\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 ${\displaystyle R_{7}}$:

{\displaystyle {\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 ${\displaystyle 3^{\circ }}$ to the left and the refractor ${\displaystyle 5^{\circ }}$to the left?

Figure 6.16d.  Geometry and raypaths for ${\displaystyle 3^{\circ }}$ and ${\displaystyle 5^{\circ }}$ 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

{\displaystyle {\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 ${\displaystyle R_{5}}$ and ${\displaystyle R_{7}}$:

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

{\displaystyle {\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}}}

Figure 6.17a.  Reflections where the second and third reflectors converge; zero-phase wavelet.

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