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 $t_{R}$

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

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.

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,

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

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?

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \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{align}

The refraction curve is a straight line though Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_{5} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_{7} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

The incident angle at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): W is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_{6} whose coordinates are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

Continue reading

Previous section	Next section
Reflections/diffractions from refractor terminations	Destructive and constructive interference for a wedge
Previous chapter	Next chapter
Geometry of seismic waves	Characteristics of seismic events
Table of Contents (book)

Also in this chapter

External links

find literature about
Refractions and refraction multiples

Refractions and refraction multiples

Problem 6.16a

Solution

Problem 6.16b

Solution

Problem 6.16c

Solution

Continue reading

Also in this chapter

External links

Navigation menu

Search