Wyrobek’s refraction interpretation method

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Problem 11.11

Sources $ C $, $ D $, $ E $, 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/":): F , 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/":): G in Figure 11.11a are 5 km a part. The data in Table 11.11a are for three profiles 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/":): CE , 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/":): DF , 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/":): EG with sources at $ C $, 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/":): 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/":): E , no data being recorded for offsets less than 3 km. For profiles from $ F $ 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/":): G the intercepts were 1.52 and 1.60 s, respectively. Use Wyrobek’s method (Wyrobek, 1956) to interpret the data.

Background

Wyrobek’s method is based on a series of unreversed profiles such as those shown in Figure 11.11a. The steps in the interpretation are as follows:

Figure 11.11a.  Unreversed refraction profiles.
  1. The traveltimes are measured, corrected, and plotted, and apparent velocities and intercepts are measured. If 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} cannot be measured, $ \theta _{c} $ is calculated from an assumed value.
  2. The total delay 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/":): \delta are calculated [see equation (11.8b)] for each geophone location for each profile. The curves for the different profiles are displaced up or down to obtaina composite curve covering the entire range.
  3. The half-intercept times are plotted at the source locations and a curve drawn through them. This curve is compared with the composite curve in (d); if the curves are not sufficiently parallel, 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} is adjusted to achieve parallelism. The composite delay-time curve is also used to interpolate or extrapolate the half-intercept curve to cover the complete range. Delay times are now converted into depths using equation (11.9a), i.e., by multiplying half-intercept times 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/":): V_{1} /\left(\cos \theta _{c} \right) .
Table 11.11a. Time-offset data for three refraction profiles.
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 (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_{CE} (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_{DF} (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_{EG} (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 (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_{CE} (s) $ t_{DF} $ (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_{EG} (s)
3.00 1.18 1.20 1.19 6.60 1.90 2.12 2.49
3.20 1.22 1.29 1.28 6.80 1.94 2.16 2.54
3.40 1.24 1.38 1.35 7.00 1.97 2.20 2.57
3.60 1.28 1.45 1.43 7.20 2.01 2.25 2.60
3.80 1.35 1.54 1.50 7.40 2.06 2.30 2.65
4.00 1.38 1.60 1.58 7.60 2.10 2.33 2.68
4.20 1.41 1.70 1.68 7.80 2.14 2.37 2.71
4.40 1.47 1.74 1.76 8.00 2.17 2.41 2.74
4.60 1.51 1.77 1.82 8.20 2.20 2.45 2.77
4.80 1.53 1.80 1.89 8.40 2.24 2.47 2.82
5.0 1.58 1.82 2.00 8.60 2.30 2.52 2.85
5.20 1.63 1.85 2.06 8.80 2.32 2.55 2.89
5.40 1.65 1.91 2.15 9.00 2.35 2.61 2.93
5.60 1.69 1.95 2.21 9.20 2.38 2.64 2.97
5.80 1.74 1.97 2.29 9.40 2.44 2.68 3.00
6.00 1.78 1.99 2.38 9.60 2.47 2.73 3.04
6.20 1.82 2.03 2.43 9.80 2.50 2.78 3.07
6.40 1.87 2.08 2.46 10.00 2.54 2.82 3.10

Solution

The traveltimes in Table 11.11a are plotted in the upper part of Figure 11.11b. The 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/":): V_{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/":): V_{2} have different accuracies since different numbers of points are used for each value, so we obtain weighted averages using as weights the horizontal extent of the data for each value. 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} V_{1} &= \left(2.50\times 1+2.52\times 3\right)/4=2.52\ {\rm km/s}, \\ V_{2} &= \left(5.13\times 7+5.08\times 6+5.59\times 4\right)/17=5.22\ {\rm km/s}, \\ \hbox{so}\quad \quad \theta _{c} =\sin ^{-1} \left(2.52/5.22\right)=28.9^{\circ}. \end{align}

Figure 11.11b.  Time-distance plot (top half ) and plot of delay-times and half-intercept times (bottom).
Table 11.11b. Delay times for profiles 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/":): CE , 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/":): DF , 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/":): EG .
1 2 3 4 5 6 7 8 9
5.22 6.25 7.7 5.6
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 (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/":): \delta_{CE} (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/":): \delta_{DF} (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/":): \delta_{EG} (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/":): \delta_{CE} (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/":): \delta_{DF} (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/":): \delta_{EG} (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/":): \delta_{CE} (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/":): \delta_{DF} (s)
3.0 0.61 0.70
3.2 0.61 0.71
3.4 0.59 0.70
3.6 0.59 0.70
3.8 0.62 0.74
4.0 0.61 0.74
4.2 0.61 0.90 0.74 1.03
4.4 0.63 0.90 0.77 1.04
4.6 0.63 0.89 0.77 1.03
4.8 0.61 0.88 0.76 1.03
5.0 0.62 0.86 0.78 1.02 0.93
5.2 0.63 0.85 0.80 1.02 0.92
5.4 0.62 0.88 0.79 1.05 0.95
5.6 0.62 0.88 0.79 1.05 0.95
5.8 0.63 0.86 0.81 1.04 0.93
6.0 0.63 0.84 0.82 1.03 0.92
6.2 0.63 0.84 1.24 0.83 1.04 1.44 0.92
6.4 0.64 0.85 1.23 0.85 1.06 1.44 0.94
6.6 0.64 0.86 1.23 0.84 1.06 1.43 1.04 0.94
6.8 0.64 0.86 1.24 0.85 1.07 1.45 1.06 0.95
7.0 0.63 0.86 1.23 0.85 1.08 1.45 1.06 0.95
7.2 0.63 0.87 1.22 0.86 1.10 1.45 1.07 0.96
7.4 0.64 0.88 1.23 0.88 1.12 1.47 1.10 0.98
7.6 0.64 0.87 1.22 0.88 1.11 1.46 1.11 0.97
7.8 0.65 0.88 1.22 0.89 1.12 1.46 1.13 0.98
8.0 0.64 0.88 1.21 0.89 1.13 1.46 1.13 0.98
8.2 0.63 0.88 1.20 0.89 1.14 1.46 1.14 0.99
8.4 0.63 0.86 1.21 0.90 1.13 1.48 1.15 0.97
8.6 0.65 0.87 1.20 0.92 1.14 1.47 1.18 0.98
8.8 0.63 0.86 1.20 0.91 1.14 1.48 1.18 0.98
9.0 0.63 0.89 1.21 0.91 1.17 1.49 1.18 1.00
9.2 0.62 0.88 1.21 0.91 1.17 1.50 1.19 1.00
9.4 0.64 0.88 1.20 0.94 1.18 1.50 1.22 1.00
9.6 0.63 0.89 1.20 0.93 1.19 1.50 1.22 1.02
9.8 0.62 0.90 1.19 0.93 1.21 1.50 1.23 1.03
10.0 0.62 0.90 1.18 0.94 1.22 1.50 1.24 1.03

The intercept times from the data in Table 11.11a 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/":): t_{C} =0.60 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_{D} =0.82 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_{E} = 1.31 s, and we are also given 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_{F} =1.52 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_{G} =1.60 s. Obviously the refractor is dipping down 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/":): C towards 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/":): G 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_{2} above is in fact 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_{d} . However, initially we shall ignore dip and use 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} =5.22 km/s.

The calculated delay times are listed in Table 11.11b; 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 is the offset distance from the sources for profiles 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/":): CE , 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/":): DF , and $ EG $, while 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_{CE} , 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_{DF} 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/":): \delta_{EG} are total delay times. These were obtained in the same way as 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_{AR} 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/":): \delta_{BR} in Table 11.9b using the value 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} =5.22 km/s to get columns 2, 3, and 4 in Table 11.11b.

The delay times can also be obtained by drawing straight lines through sources 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/":): C , $ 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/":): E with slopes 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/":): 1/V_{2} (the lines 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/":): HJ , 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/":): KL , 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/":): MN in Figure 11.11b) and then measuring the time differences between these lines and the observed times.

The delay times in columns 2, 3, and 4 are plotted in the lower part of Figure 11.11b using small circles (o). The half-intercept times for sources 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/":): C , 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/":): 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/":): G are also plotted (solid line at top of the lower figure) but using a different scale from that used for delay times.

The next step is to shift the delay-time values to form a continuous composite curve; we achieve this by moving 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/":): CE curve up and 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/":): EG curve down. Since this is merely a preliminary step we do not move individual values but displace the average straight lines through the points, giving the composite curve 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/":): PQ .

The delay-time curve is not parallel to the half-intercept line and, to achieve parallelism, we must change 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} to increase the delay times at large 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/":): x relative to those at small values. For profile 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/":): CE we need to change 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} so that 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/":): J moves downward about 0.2 s more than 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 ; this gives the curve 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'J' with slope equal 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/":): 1/V_{2} = 1/6.25 km/s, the other two curves becoming 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/":): K'L' 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/":): M'N' . We recalculate the delay times using 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} =6.25 km/s; the new values are given in columns 5, 6, and 7 of Table 11.11b and plotted as 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's in Figure 11.11b. The new curves do roughly parallel the half-intercept curve, and we obtain a new composite delay-time curve by moving 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_{DF} 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/":): \delta_{CE} upward by 0.2 s and 0.3 s, respectively, to join 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/":): \delta_{EG} values to form a continuous curve. The values agree exactly except for the first and last overlapping values, which differ by 2 ms; we used the average values at these two points.

Comparison of the composite delay-time curve with the half-intercept time curve shows reasonably good agreement at the two ends but significant divergence in the central part. We might assume that the intercept time at source 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/":): E is in error but the value 1.31 s would have to decrease to about 1.15 s (for a half-intercept time of about 0.58 s) to agree with the delay-time curve. Although 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/":): EG -curve is short, it is regular so that it is difficult to fit a line having an intercept of 1.15 s. A more likely source of error is variations of velocity; these could be of two kinds: (i) the actual value 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/":): V_{2} could be 6.25 at the two ends but higher than 6.25 in the range 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/":): 7<x_{c} <10 km and lower than 6.25 in the range 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/":): 10<x_{c} <15 km, (ii) velocity changes due to dip (the intercepts show an overall dip down 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/":): C 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/":): G , 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/":): V_{2} is the apparent velocity $ V_{d} $. While velocity variations due to changes in dip are the more likely explanation, we can proceed with the interpretation without deciding which velocity effect is the cause.

To reduce the gap between the two curves, we change 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} so that the difference between the 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/":): \delta_{CE} 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/":): X_{C} =10.0 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/":): x_{C} =6.6 km increases by 0.1 s. Letting 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 be the required velocity and using equation (11.8b), 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} \left(2.54-10.0/V\right)-\left(1.90-6.6/V\right)=\left(0.94-0.84\right)+0.10, \\ V=3.4/\left(0.64-0.20\right)=7.7\ {\rm km/s}. \end{align}

We also need a new velocity that will increase 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_{DF} about 0.1 s more 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/":): x_{C} =5.0 than 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/":): x_{C} =10.0 . 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} \left(1.82-5.0/V\right)-\left(2.82-10.0/V\right)=\left(1.02-1.22\right)+0.10, \\ V=5.0/0.90=5.6\ {\rm km/s}. \end{align}

These two velocities were used to calculate revised delay times in columns 8 and 9 of Table 11.11b, and the revised values are plotted in Figure 11.11b (using small squares).

The final interpreted curve is represented by inverted triangles (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/":): \nabla ) 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/":): x_{c} =3.0 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/":): x_{c} =15.0 and by crosses 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/":): \left(\times \right) 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/":): x_{c} =16.2 to 20.0. The values can be changed to depths by multiplying the half-intercept times 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/":): V_{1} /\left(\cos \theta _{c} \right) [see equation (11.9b)].

We now get approximate 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/":): \xi by finding depths 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/":): C 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/":): G using equation (11.9b); then we use 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 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_{d} to 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/":): V_{u} , 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} , 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/":): \theta _{c} which give a more accurate depth factor 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} /\cos \theta _{c} . Thus, 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} V_{1} =2.52\ {\rm km/s},\quad V_{2} =5.22\ {\rm km/s},\quad \theta _{c} =28.9^{\circ},\quad \delta_{c} =0.60/2,\quad \delta_{G} = 1.60/2. \end{align}

Using these values, the depths become

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_{C} &=0.30\times 2.52/\cos 28.9^{\circ} =0.86\ {\rm km}, \\ h_{G} &=0.80\times 2.52/\cos 28.9^{\circ} =2.30\ {\rm km}, \\ \xi &=\tan ^{-1} \left[\left(2.30-0.86\right)/20\right]=4.1^{\circ}. \end{align}

Using 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_{d} =6.25 km/s, we solve equation (4.24d) for 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_{u} , giving

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} \xi =\left(1/2\right)[\sin ^{-1} \left(2.52/6.25\right)-\sin ^{-1} (2.52/V_{u})], \\ \hbox{so}\quad \quad 4.1^{\circ} &=\left(1/2\right)[23.8^{\circ} - \sin ^{-1} (2.52/V_{u})] \\ \sin ^{-1} \left(2.52/V_{u} \right)&=23.8^{\circ} -8.2^{\circ} =15.6^{\circ},\\ \left(2.52/V_{u} \right)&=\sin 15.6^{\circ} =0.269, V_{u} =9.37\ {\rm km/s}, \\ V_{2} \approx \frac{1}{2} \left(V_{d} +V_{u} \right) &=\frac{1}{2} \left(6.25+9.37\right)=7.81\ {\rm km/s}, \\ \theta _{c} &\approx \sin ^{-1} \left(2.52/7.81\right)\approx 18.8^{\circ},\; \cos \theta _{c} \approx 0.947,\\ \hbox{depth factor} &\approx V_{1} /\cos \theta _{c} \approx 2.52/0.947\approx 2.66. \end{align}

Thus, the refractor is nearly flat over the region where we used 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} =6.25 km/s, so local dip is mainly in the places where we carried out the second revision using velocities of 7.7 and 5.6 km/s.

We shall not refine our interpretation further because of the limited acccuracy of the data.

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