Finding velocity

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Problem 5.16

Determine the velocity by 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/":): X^{2} -T^{2} method using the data given in Table 5.16a, 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} being for a horizontal reflector 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/":): t_{B} for a reflector dipping 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^{\circ} toward the source. (The two reflections are observed on different records.)

Background

Using equation (4.3a), we have for a dipping horizon,

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} (Vt)^{2} =(x\cos \xi )^{2} +(2h_{c})^{2} , \end{align}

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/":): x is 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/":): \xi is the dip, 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_{c} the slant depth at the midpoint between source and receiver.

Table 5.16a. 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^{2} -T^{2} data.
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_{A} (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} (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_{A} (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} (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_{A} (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} (s)
0.0 0.855 0.906 1.4 1.005 0.977 2.8 1.330 1.202
0.1 0.856 0.902 1.5 1.017 0.991 2.9 1.360 1.234
0.2 0.858 0.898 1.6 1.037 1.004 3.0 1.404 1.253
0.3 0.864 0.898 1.7 1.068 1.019 3.1 1.432 1.272
0.4 0.868 0.899 1.8 1.081 1.037 3.2 1.457 1.296
0.5 0.874 0.902 1.9 1.105 1.058 3.3 1.487 1.304
0.6 0.882 0.903 2.0 1.118 1.066 3.4 1.513 1.334
0.7 0.892 0.909 2.1 1.151 1.083 3.5 1.548 1.356
0.8 0.900 0.916 2.2 1.166 1.102 3.6 1.580 1.377
0.9 0.906 0.922 2.3 1.203 1.121 3.7 1.610 1.407
1.0 0.930 0.932 2.4 1.237 1.127 3.8 1.649 1.415
1.1 0.945 0.943 2.5 1.255 1.158 3.9 1.674 1.438
1.2 0.950 0.950 2.6 1.283 1.177 4.0 1.708 1.459
1.3 0.979 0.965 2.7 1.304 1.195
Table 5.16b. Data used in 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^{2} -T^{2} calculations.
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^{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/":): t_{A}^{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/":): x\cos \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/":): t_{B}^{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/":): x^{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/":): t_{A}^{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/":): x\cos \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/":): t_{B}^{2}
0.00 0.731 0.00 0.821 4.41 1.325 4.28 1.173
0.01 0.733 0.01 0.814 4.84 1.360 4.69 1.214
0.04 0.736 0.04 0.806 5.29 1.447 5.13 1.257
0.09 0.746 0.09 0.806 5.76 1.530 5.59 1.270
0.16 0.753 0.16 0.808 6.25 1.575 6.06 1.341
0.25 0.764 0.24 0.814 6.76 1.646 6.56 1.385
0.36 0.778 0.35 0.815 7.29 1.700 7.07 1.426
0.49 0.796 0.48 0.826 7.84 1.769 7.60 1.445
0.64 0.817 0.62 0.839 8.41 1.850 8.16 1.523
0.81 0.821 0.79 0.850 9.00 1.971 8.73 1.570
1.00 0.865 0.97 0.869 9.61 2.051 9.32 1.618
1.21 0.893 1.17 0.889 10.24 2.123 9.93 1.680
1.44 0.902 1.40 0.902 10.89 2.211 10.56 1.700
1.69 0.968 1.64 0.931 11.56 2.289 11.21 1.780
1.96 1.010 1.90 0.965 12.25 2.396 11.88 1.839
2.25 1.034 2.18 0.962 12.96 2.496 12.57 1.896
2.56 1.075 2.48 1.008 13.69 2.592 13.28 1.980
2.89 1.141 2.80 1.030 14.43 2.719 14.00 2.002
3.24 1.169 3.14 1.075 15.21 2.802 14.75 2.068
3.61 1.221 3.50 1.119 16.00 2.917 15.52 2.129
3.61 1.250 3.88 1.136
Figure 5.16a.  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^{2} -T^{2} plot.

Solution

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/":): (x^{2} ,\; t_{A}^{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/":): [(x\cos \xi )^{2} ,\; t_{B}^{2}] are tabulated in Table 5.16b and plotted in Figure 5.16a. The best-fit lines in Figure 5.16a determined by eye 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} V_{A} =2.71\ {\rm km/s}, h_{A} =1.16\ {\rm km}; V_{B} =3.35\ {\rm km/s}, h_{B} =1.49\ {\rm km}. \end{align}

Using the least-squares method (see problem 9.33), 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} V_{A} =2.71\ {\rm km/s}, h_{A} =1.15\ {\rm km}; V_{B} =3.38\ {\rm km/s}, h_{B} =1.51\ {\rm km}. \end{align}

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