# Difference between revisions of "Refractions and refraction 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 6.16a

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{align} \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{align}

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

\displaystyle \begin{align} 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{align}

The critical distance (see equation (6.15a) is

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

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{align} \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{align}

The critical distance for SMNTUWPQR is increased to

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

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

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

\displaystyle \begin{align} t_{R_{2} } =1.00/2.80+2\times 3.71/2.00+1.12/4.20=4.33\ {\rm s}. \end{align} 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{align} t_{R_{4}} = 1.00/2.80+(3.71+3.21)/2.00+4.71/4.20=4.94\ {\rm s}, \end{align}

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{align} 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{align}

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

\displaystyle \begin{align} 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{align}

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

\displaystyle \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 $\displaystyle W$ 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{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} Figure 6.17a.  Reflections where the second and third reflectors converge; zero-phase wavelet.