Refractions and refraction multiples

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Problem 6.16a

Determine the traveltime curve for the refraction 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/":): \mathit{SMNPQR} and the refraction multiple 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/":): \mathit{SMNTUWPQR} in Figure 6.16a.

Solution

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

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 _{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 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/":): t_{R}

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} 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}

Figure 6.16a.  Refraction multiple.

The critical distance (see equation (6.15a) 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/":): \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 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/":): \Delta t where

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} \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

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^{'} =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 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/":): 8^{\circ} down to the left, the depths shown in Figure 6.16a now being the slant distances from 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/":): 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 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/":): SR_{1} =4.38 km, 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/":): \begin{align} t_{R_{1} } =(1.00+0.18)/2.80+2\times 3.71/2.00=4.13\ {\rm s}. \end{align}

The $ V_{2} $-layer outcrops 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/":): R_{2} =5.39 km,

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} 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 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/":): 8^{\circ} to the left.

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

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} 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 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_{2} 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_{4} .

We have two types of reflected refractions: a typical path for the first type 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/":): SMNP_{1}U_{2}WP_{4} R_{4} , the reflection occurring at the shallow dipping interface, The second type, 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/":): SMNP_5 R_{6} P_{7} R_{7} , involves reflection at the surface. The first type exists between 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_{3} 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_{4} , and the curve is parallel to the head-wave curve to the right of 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_{3} . The second type exists to the right of 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_{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 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_{3} . For the first type, we find the coordinates of 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} :

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=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 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} 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 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/":): 3^{\circ} to the left and the refractor 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/":): 5^{\circ} to the left?

Figure 6.16d.  Geometry and raypaths for $ 3^{\circ } $ 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/":): 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

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 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_{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}

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