Adachi’s method

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Problem 11.5

Given the data in Table 11.5a for a reversed refraction profile with sources and , 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 ( and 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 and are angles of incidence relative to the vertical at the interface for the downgoing rays from sources and , respectively (these are angles of approach at the surface for ), and are the angles of incidence and refraction for the downgoing ray at interface , and are the same for the upcoming ray, is the dip of the interface, is the vertical thickness of the bed below this interface below the downdip source.

The traveltime for the refraction along the top of the layer is given by


(11.5a)

If we set , becomes the intercept time at the downdip source; thus,


(11.5b)
Table 11.5a. Reversed refraction times.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 (km)
0.00 0.25 0.50 0.74 0.98 1.24 1.50 1.70 1.81 1.91 2.02 (s)
3.00 2.90 2.80 2.68 2.52 2.41 2.31 2.20 2.07 1.91 1.80 (s)
5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 (km)
2.16 2.28 2.38 2.44 2.56 2.64 2.72 2.80 2.89 3.00 (s)
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:


(11.5c)

Snell’s law [equation (3. 1a)] gives


(11.5d)

For the refraction along the interface,


(11.5e)

The initial interpretation stage is plotting the data and determining and the apparent velocities and , and intercept times for each of the refraction events. The angles and are given by equation (4.2d). Next we use problem 4.24b to get and from , , and . The depth is now found using equation (11.5b).

For the next interface we find new values of and using the next pair of apparent velocities. Since is now known, we use equation (11.5c) to get new values of and , after which equation (11.5d) gives , and equation (5.11c) gives , . We can now find , , , and .

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 is 2.02 km/s. Two refraction events are observed with the apparent velocities and intercept times listed below.

First we calculate and :

Equation (11.5c) gives , . Since this interface is the refractor, equation (11.5e) gives

{}

We find using equation (11.5b): so

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

Then equation (11.5c) gives

Using equation (11.5d), we get

From equation (11.5c) we now get

From equation (5.11e) we have

{}

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

Total vertical depth at km.

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Geologic interpretation of reflection data 3D methods

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