Two-layer refraction problem

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Problem 4.24a

Show that time-distance curves for dipping refractors take the form


$ {\begin{aligned}t_{d}&=(x/V_{1})\sin(\theta _{c}+\xi )+t_{1d},\\t_{u}&=(x/V_{1})\sin(\theta _{c}-\xi )+t_{1u},\end{aligned}} $ (4.24a)

where


$ {\begin{aligned}t_{1d}&=(2h_{d}/V_{1})\cos \theta _{c},\\t_{1u}&=(2h_{u}/V_{1})\cos \theta _{c},\end{aligned}} $ (4.24b)

$ t_{d} $ and $ t_{u} $ being the traveltimes when shooting downdip and updip, respectively (see Figure 4.24a), and $ h_{d} $, $ h_{u} $, and $ t_{id} $, $ t_{iu} $, the corresponding slant depths and sourcepoint intercepts times.

Background

Traveltime curves for horizontal refractors are discussed in problem 4.18.

Figure 4.24a.  Traveltime curves and raypaths for dipping refractor.

Solution

For the downdip case, we take O as the sourcepoint and O$ ^{\prime } $ as the receiver. Following the procedure used in problem 4.18, we have


$ {\begin{aligned}t_{d}&=(OM+PO^{\prime })/V_{1}+MP/V_{2}\\&={\frac {h_{d}+h_{u}}{V_{1}\cos \theta _{c}}}+{\frac {OO^{\prime }-(h_{d}+h_{u})\tan \theta _{c}}{V_{2}}}\\&={\frac {x}{V_{2}}}+\left({\frac {h_{d}+h_{u}}{V_{1}\cos \theta _{1}}}\right)\left[1-{\frac {V_{1}}{V_{2}}}\sin \theta _{c}\right]\\&={\frac {x}{V_{2}}}+(h_{d}+h_{u})\cos \theta _{c}/V_{1}.\end{aligned}} $ (4.24c)

But $ h_{u}=h_{d}+x\sin \xi $, so we can express equation (4.24c) in terms of $ h_{d} $:

$ {\begin{aligned}t_{d}&=(x/V_{1})(\sin \theta _{c}\cos \xi +\cos \theta _{c}\sin \xi )+(2h_{d}/V_{1})\cos \theta _{c}\\&=(x/V_{1})\sin(\theta _{c}+\xi )+t_{id}.\end{aligned}} $

Expressing equation (4.24c) in terms of $ h_{u} $, we obtain

$ {\begin{aligned}t_{u}=(x/V_{1})\sin(\theta _{c}-\xi )+t_{1u}.\end{aligned}} $

The slopes of the two traveltime curves are sin ($ \theta _{c}\pm \xi )/V_{1} $, the reciprocals being the apparent velocities, $ V_{d} $ and $ V_{u} $ (see problem 4.2d), where


$ {\begin{aligned}V_{d}=V_{1}/\sin(\theta _{c}+\xi ),\quad V_{u}=V_{1}/\sin(\theta _{c}-\xi ).\end{aligned}} $ (4.24d)

Problem 4.24b

Show how to find $ V_{2} $ and $ \xi $ from the observed data.

Solution

We obtain $ V_{1} $, $ V_{d} $, and $ V_{u} $ from the slopes of the time-distance curves. From equation (4.24d), we get

$ {\begin{aligned}\theta _{c}+\xi =\sin ^{-1}(V_{1}/V_{d}),\\\theta _{c}-\xi =\sin ^{-1}(V_{1}/V_{u}).\end{aligned}} $

Adding and subtracting the two equations gives $ \theta _{c} $ and $ \xi $. Since sin $ \theta _{c}=(V_{1}/V_{2}) $ and $ V_{1} $ is known, we can find $ V_{2} $

The dip $ \xi $ can also be found (usually more accurately) from the relation


$ {\begin{aligned}\tan \xi =(h_{d}-h_{u})/x.\end{aligned}} $ (4.24e)

Problem 4.24c

Show that $ V_{2} $ is given approximately by either of the following equations, the latter being less accurate:


$ {\begin{aligned}{\frac {1}{V_{2}}}\approx {\frac {1}{2}}\left({\frac {1}{V_{d}}}+{\frac {1}{V_{u}}}\right),\quad V_{2}\approx {\frac {1}{2}}(V_{d}+V_{u}).\end{aligned}} $ (4.24f)

Solution

Expanding equation (4.24d) we have

$ {\begin{aligned}\sin \theta _{c}\cos \xi +\cos \theta _{c}\sin \xi =V_{1}/V_{d},\\\sin \theta _{c}\cos \xi -\sin \theta _{c}\cos \xi =V_{1}/V_{u}.\end{aligned}} $

Adding the two equations, we get

$ {\begin{aligned}2\sin \theta _{c}\cos \xi =(1/V_{d}+1/V_{u}).\end{aligned}} $

Because $ \xi $ is usually small, we set $ \cos \xi =1 $. Since $ \sin \theta _{c}=V_{1}/V_{2} $, we get the first result in equation (4.24f).

Returning to equation (4.24d), we write

$ {\begin{aligned}V_{d}=V_{1}/\sin(\theta _{c}+\xi )=(V_{1}/\sin \theta _{c})(\cos \xi +\cot \theta _{c}\sin \xi )^{-1}.\end{aligned}} $

Table 4.24a. Data from refraction profile.
$ x_{A}(m) $ $ t_{A}(ms) $ $ t_{B}(ms) $ $ x_{B}(m) $ $ x_{A}(m) $ $ t_{A}(ms) $ $ t_{B}(ms) $ $ x_{B}(m) $
0 0 98 225 120 70 52 105
15 10 92 210 135 73 46 90
30 21 87 195 150 78 43 75
45 30 81 180 165 81 37 60
60 41 73 165 180 85 31 45
75 50 71 150 195 89 21 30
90 59 63 135 210 94 10 15
105 65 60 120 225 98 0 0

Setting $ \cos \xi =1 $ and expanding by the binomial theorem [see equation (4.1b)], we obtain the result

$ {\begin{aligned}V_{d}=V_{2}(1-\cot \theta _{c}\sin \xi ).\end{aligned}} $

Following the same procedure for $ V_{u} $ and adding the two expansions gives the second result in equation (4.24f). This result is less accurate than the first because we approximated the binomial expansion and also set $ \cos \xi =1 $.

Problem 4.24d

Sources A and B are located at the ends of a 225-m spread of 16 geophones. Using the data in Table 4.24a, find the velocities, dip, and depth to the refractor.

Solution

The data in Table 4.24a are plotted in Figure 4.24b and straight-line curves drawn through the data points. The slopes of these lines give the direct-wave velocity and the apparent updip and downdip velocities, and the intercepts with the $ t $-axes give $ t_{1d} $ and $ t_{1u} $. We ignore the value of $ V_{1} $ obtained on the downdip profile because it is poorly defined. The measured velocities and intercepts are now

Figure 4.24b.  Two-layer arrival times.

$ {\begin{aligned}V_{1}&=1.48\,\mathrm {km/s} ,\ V_{d}=2.72\,\mathrm {km/s} ,\ V_{u}=3.75\,\mathrm {km/s} ,\\t_{1d}&=15\,\mathrm {ms} ,\quad t_{1u}=38\,\mathrm {ms} .\end{aligned}} $

From these data, we calculate first 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/":): V_2 , 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/":): \theta_c, \xi, h_d , 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/":): h_u . The two equations (4.24f) give

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} \frac{1}{V_2}&=\frac{1}{2}\left(\frac{1}{V_d}+\frac{1}{V_u}\right)=\frac{1}{2}\left(\frac{1}{2.72}+\frac{1}{3.72}\right),\quad V_2=3.15\,\mathrm{km/s};\\ V_2&=\frac{1}{2}(V_d+V_u)=\frac{1}{2}(2.72+3.75),\quad V_2=3.23\,\mathrm{km/s}, \end{align}

the first being more accurate.

Next, 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_c = \sin^{-1}(V_1/V_2) = \sin^{-1}(1.48/3.15) = 28.0^{\circ} . Now we find 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/":): h_d 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/":): h_u , and finally 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/":): \xi . From equation (4.24b)

$ {\begin{aligned}h_{d}&={\frac {1}{2}}V_{1}t_{id}/\cos \theta _{c}={\frac {1.48\times 0.015}{2\times 0.883}}=13\,\mathrm {m} ,\quad h_{u}={\frac {1.48\times 0.038}{2\times 0.883}}=32\,\mathrm {m} ,\\\xi &=\tan ^{-1}[(32-13)/225]=4.8^{\circ }.\end{aligned}} $

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