Adachi’s method

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Problem 11.5

Given the data in Table 11.5a for a reversed refraction profile with sources $ A $ and $ B $, use Adachi’s method to find velocities, depths, and dips.

Background

Adachi (1954; see also Johnson, 1976) derived equations for reversed refraction profiles similar to equations (4.18b,d) but with two important differences: he used angles of incidence measured relative to the vertical ($ \alpha _{i} $ and $ \beta _{i} $ in Figure 11.5a) and vertical depths. The equations are valid for a series of refractors of different dips but with the same strike. Derivation of his equations is lengthy but not difficult (see Sheriff and Geldart, 1995, Section 11.3.2); we quote the final results without proof.

The notation is illustrated in Figure 11.5a where $ \alpha _{i} $ and $ \beta _{i} $ are angles of incidence relative to the vertical at the $ i^{\rm {th}} $ interface for the downgoing rays from sources $ A $ and $ B $, respectively (these are angles of approach at the surface for $ i=1 $), $ a_{i} $ and $ a_{i}^{'} $ are the angles of incidence and refraction for the downgoing ray at interface $ i $, $ b_{i} $ 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/":): b_{i}^{'} are the same for the upcoming ray, 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 _{i+1} is the dip of the 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/":): i^{\rm th} interface, 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_{i} is the vertical thickness of the bed below this interface below the downdip source.

The traveltime 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_{n} for the refraction along the top of the 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^{\rm th} layer is given by


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_{n} = \frac{x\sin \beta _{1} }{V_{1} } + \sum\limits_{i=1}^{n-1} \frac{h_{i} }{V_{i} } \left(\cos \alpha _{i} +\cos \beta _{i} \right). \end{align} (11.5a)

If we set $ x=0 $, 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_{n} becomes the intercept time 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_{in} at the downdip source; thus,


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_{in} = \sum\limits_{i=1}^{n-1} \frac{h_{i} }{V_{i} } \left(\cos \alpha _{i} +\cos \beta _{i} \right). \end{align} (11.5b)
Table 11.5a. Reversed refraction times.
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\to 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 (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/":): t_A\to 0.00 0.25 0.50 0.74 0.98 1.24 1.50 1.70 1.81 1.91 2.02 (s)
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_B\to 3.00 2.90 2.80 2.68 2.52 2.41 2.31 2.20 2.07 1.91 1.80 (s)
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\to 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 (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/":): t_A\to 2.16 2.28 2.38 2.44 2.56 2.64 2.72 2.80 2.89 3.00 (s)
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_B\to 1.65 1.50 1.40 1.25 1.12 1.00 0.75 0.49 0.23 0.00 (s)
Figure 11.5a.  Notation used in Adachi’s equations.

The angles are related as follows:


$ {\begin{aligned}\left.{\begin{array}{l}\alpha _{i+1}=a_{i}^{'}+\xi _{i+1}=\alpha _{i+1}+\xi _{i+2},\\\beta _{i+1}=b_{i}^{'}-\xi _{i+1}=b_{i+1}-\xi _{i+2}.\end{array}}\right\}\end{aligned}} $ (11.5c)

Snell’s law [equation (3. 1a)] gives


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} \sin \alpha _{i}' =\left(V_{i+1} /V_{i} \right)\sin a_{i},\quad \sin b_{i}' =\left(V_{i+1} /V_{i} \right)\sin b_{i}. \end{align} (11.5d)

For the refraction along the 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^{\rm th} interface,


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} a_{n} =b_{n} =\theta _{cn} =\left(\alpha _{n} +\beta _{n} \right)/2,\qquad \xi _{n+1} =\left(\alpha _{n} -\beta _{n} \right)/2. \end{align} (11.5e)

The initial interpretation stage is plotting the data and determining 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_{1} and the apparent velocities 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_{un} 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/":): V_{dn} , and intercept times 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_{in} for each of the refraction events. The angles 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/":): \alpha _{1} 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/":): \beta _{1} are given by equation (4.2d). Next we use problem 4.24b to get $ \theta _{c1} $ 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/":): \xi _{2} 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/":): V_{1} , 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/":): \alpha _{1} , 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/":): \beta _{1} . The depth 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_{1} is now found using equation (11.5b).

For the next interface we find new values 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/":): \alpha _{1} 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/":): \beta _{1} using the next pair of apparent velocities. Since 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 _{2} is now known, we use equation (11.5c) to get new values 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/":): a_{1} and $ b_{1} $, after which equation (11.5d) gives 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/":): a_{1'} , 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/":): b_{1'} and equation (5.11c) gives 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/":): \alpha _{3} , 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/":): \beta _{3} . We can now 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/":): \theta _{c3} , 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 _{3} , 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_{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/":): h_{2} .

Figure 11.5b.  Plot of the data in Table 11.5a.

Solution

Figure 11.5b shows the plotted data and the measured slopes and time intercepts. The average value of the near-surface velocity 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_{1} is 2.02 km/s. Two refraction events are observed with the apparent velocities and intercept times listed below.

$ {\begin{aligned}V_{d2}=3.73\ {\rm {km/s}},\quad V_{u2}=4.51\ {\rm {km/s}},\quad t_{i1}=0.92\ {\rm {s}};\\V_{d3}=4.29\ {\rm {km/s}},\quad V_{u3}=5.81\ {\rm {km/s}},\quad t_{i2}=1.28\ {\rm {s}}.\end{aligned}} $

First we calculate 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/":): \alpha _{1} 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/":): \beta _{1} :

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} \alpha _{1} =\sin ^{-1} \left(V_{1} /V_{d2} \right)=32.8^{\circ},\quad \beta _{1} =\sin ^{-1} \left(V_{1} /V_{u2} \right)=26.6^{\circ}. \end{align}

Equation (11.5c) gives 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/":): \alpha _{1} =a_{1} +\xi _{2} , 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/":): \beta _{1} =b_{1} -\xi _{2} . Since this interface is the refractor, equation (11.5e) gives

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} a_{1} &=b_{1} =\theta _{c1} =\left(\alpha _{1} +\beta _{1} \right)/2=29.7^{\circ}, \\ \xi _{2} &=\left(\alpha _{1} -\beta _{1} \right)/2=3.1^{\circ}, \\ V_{2} &=V_{1} /\sin \theta _{c1} =2.02/\sin 29.7^{\circ} =4.08\ {\rm km/s}. \end{align}

{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} \hbox{Checking:}\quad \quad V_{2} &=[(1/V_{d2} +1/V_{u3} )^{2} /2]^{-1} =4.08\ {\rm km/s}. \end{align} }

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_{1} using equation (11.5b): so

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} h_{1} &= V_{1} t_{i1} /\left(\cos \alpha _{1} +\cos \beta _{1} \right)\\ &=2.02\times 0.92/\left(\cos 32.8^{\circ} +\cos 26.6^{\circ} \right)=1.07\ {\rm km}. \end{align}

For the second refractor, we calculate new angles of approach:

$ {\begin{aligned}\alpha _{1}=\sin ^{-1}\left(V_{1}/V_{d3}\right)=\sin ^{-1}\left(2.02/4.29\right)=28.1^{\circ },\\\beta _{1}=\sin ^{-1}\left(V_{1}/V_{u3}\right)=\sin ^{-1}\left(2.02/5.81\right)=20.3^{\circ }.\end{aligned}} $

Then equation (11.5c) gives

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} a_{1} = \alpha_{1} - \xi _{2} = 28.1^{\circ} - 3.1^{\circ} = 25.0^{\circ}, \\ b_{1} = \beta_{1} + \xi _{2} = 20.3^{\circ} + 3.1^{\circ} = 23.4^{\circ}. \end{align}

Using equation (11.5d), we get

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} a_{1^{'}} =\sin ^{-1} \left[\left(V_{2} /V_{1} \right)\sin a_{1} \right]=\sin ^{-1} [(4.08/2.02) \sin 25.0^{\circ} ]=58.6^{\circ}, \\ b_{1^{'}} =\sin ^{-1} \left[\left(V_{2} /V_{1} \right)\sin b_{1} \right]=\sin ^{-1} [(4.08/2.02) \sin 23.4^{\circ} ]=53.3^{\circ}. \end{align}

From equation (11.5c) we now get

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} \alpha _{2} =a_{1^{'}} + \xi _{2} =58.6^{\circ} +3.1^{\circ} =61.7^{\circ}, \\ \beta _{2} =b_{1^{'}} - \xi _{2} =53.3^{\circ} -3.1^{\circ} =50.2^{\circ}. \end{align}

From equation (5.11e) we have

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} a_{2} &= b_{2} =\theta _{c2} =\left(\alpha _{2} +\beta _{2} \right)/2=56.0^{\circ}, \\ \xi_{3} &=\left(\alpha _{2} -\beta _{2} \right)/2=5.8^{\circ}, \\ V_{3} &=V_{2} /\sin \theta _{c2} =4.08/\sin 56.0^{\circ} =4.92\ {\rm km/s}. \end{align}

{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} \hbox{Checking:}\quad \quad V_{3} &=[\left(1/V_{d3} +1/V_{u3} \right)/2]^{-1} =4.94\ {\rm km/s}. \end{align} }

Finally, we get the depth from equation (11.5b):

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_{i2} &=\left(h_{1} /V_{1} \right)\left(\cos \alpha _{1} +\cos \beta _{1} \right)+\left(h_{2} /V_{2} \right)\left(\cos \alpha _{2} +\cos \beta _{2} \right)\\ &=t_{u1} +\left(h_{2} /V_{2} \right)\left(\cos \alpha _{2} +\cos \beta _{2} \right), \\ h_{2} &=\left(1.28-0.92\right)\times 4.08/\left(\cos 61.7^{\circ} +\cos 50.2^{\circ} \right)=1.32\ {\rm km}. \end{align}

Total vertical depth 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/":): A=h_{1} +h_{2} =1.07+1.32=2.39 km.

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