Stacking velocity versus rms and average velocities

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Problem 5.12a

Assume six horizontal layers, each 300 m thick and with constant velocity (Figure 5.12a). The successive layers have velocities of 1.5, 1.8, 2.1, 2.4, 2.7, and 3.0 km/s. Ray-trace through the model to determine offset distances and arrival times for rays that make angles of incidence at the base of the 3.0 km/s layer of $ 0^{\circ } $, $ 10^{\circ } $, $ 20^{\circ } $, and $ 30^{\circ } $. Calculate stacking velocity for each angle and compare with the average velocity $ {\bar {V}} $ and the rms velocity $ V_{\rm {rms}} $.

Background

Average velocity $ ({\bar {V}}) $ and rms velocity $ (V_{\rm {rms}}) $ were discussed in problem 4.13 [see equations (4.13a,b)].

In the common-midpoint (CMP) technique, a number of traces are obtained with different source-geophone distances (offsets, see problem 4.1) but the same midpoint. After correcting for NMO (and for dip if necessary), they are added together (stacked), the number of traces added together being the multiplicity. The velocity used to remove the NMO is the stacking velocity $ V_{s} $. If we use equation (4.1c) to remove the NMO, that is, if we assume a single horizontal constant-velocity layer, the velocity $ V $ in equations (4.1a,c) becomes $ V_{s} $. The $ x^{2}-t^{2} $ plot of equation (4.1a) is a straight line with slope $ 1/V_{s}^{2} $; thus,


$ {\begin{aligned}V_{s}^{2}=x^{2}/\left(t^{2}-t_{o}^{2}\right)\approx x^{2}/\left(2t\Delta t\right)\;,\;V_{s}\approx x/(2t\Delta t)^{1/2},\end{aligned}} $ (5.12a)

where $ t_{o} $ and $ t $ are the two-way traveltimes at the origin and at offset $ x $ while $ \Delta t=t-t_{o} $. When the velocity changes with depth, the $ x^{2}-t^{2} $ plot is curved but the curvature is generally small enough that the best-fit straight line gives reasonably accurate results. For horizontal velocity layering and small offsets, $ V_{s}\approx V_{\rm {rms}} $.

Figure 5.12a.  Model of 300-m-thick layers.

Solution

We use Snell’s law to calculate the raypath angles $ \theta _{i} $ in each layer. The two-way time in a layer is $ t_{i}=600/V_{i}\cos \theta _{i} $ and the offset in a layer is $ x_{i}=600\tan \theta _{i} $. The values in Table 5.12a have been calculated without regard to the number of significant figures to illustrate the sensitivity of the calculations. The average velocities $ {\bar {V}} $ along the respective raypaths have also been calculated for comparisons.

The calculations for the intermediate layer boundaries assume that reflections are generated at each boundary. Traveltime differences, shown in parentheses in Table 5.12b, are very small for most of the situations, and, especially where the differences are less than 20 ms, are not very reliable for calculating $ V_{s} $. A general rule for $ V_{s} $ calculations, that the offset should be comparable to the depth, is not reached for any of these situations.

Table 5.12a. Calculation of $ V_{s} $, $ {\bar {V}} $, and $ V_{\rm {rms}}. $
layer 1 layer 2 layer 3 layer 4 layer 5 layer 6
$ \theta _{i} $ $ 0^{\circ } $ $ 0^{\circ } $ $ 0^{\circ } $ $ 0^{\circ } $ $ 0^{\circ } $ $ 0^{\circ } $
$ t_{i}\ (s) $ 0.400 0.333 0.286 0.250 0.222 0.200
$ \Sigma t_{i}\ (s) $ 0.400 0.733 1.019 1.269 1.491 1.691
$ x_{i}\ (m) $ 0 0 0 0 0 0
$ {\bar {V}}\ (m/s) $ 1500 1640 1770 1890 2010 2130
$ V_{\rm {rms}}\ (m/s) $ 1500 1640 1780 1920 2060 2190
$ t_{o}^{2}({\rm {s}}^{2}) $ 0.1600 0.5373 1.0384 1.6104 2.2231 2.8595
$ \theta _{i} $ $ 5.0^{\circ } $ $ 6.0^{\circ } $ $ 7.0^{\circ } $ 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/":): 8.0^{\circ} 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/":): 9.0^{\circ} 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/":): 10.0^{\circ}
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_{i} \ (s) 0.402 0.325 0.288 0.252 0.225 0.203
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/":): \Sigma t_{i} \ (s) 0.402 0.737 1.025 1.277 1.502 1.705
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_{i} \ (m) 52 63 73 84 95 106
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/":): \Sigma x \ (m) 52 105 189 273 368 473
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 t \ (s) 0 0.077 0.111 0.143 0.181 0.218
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_{s-10} \ (m/s) * 1400* 1700* 1900* 2029 2170
$ {\bar {V}}_{10}\ (m/s) $ 1500 1636 1767 1892 2013 2131
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_{i} 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/":): 9.8^{\circ} 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/":): 11.8^{\circ} 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/":): 13.9^{\circ} 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/":): 15.9^{\circ} 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/":): 17.9^{\circ} 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/":): 20.0^{\circ}
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_{i} \ (s) 0.406 0.340 0.294 0.260 0.224 0.213
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/":): \Sigma t_{i} \ (s) 0.406 0.746 1.041 1.301 1.534 1.747
$ x_{i}\ (m) $ 104 126 146 171 194 218
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/":): \Sigma x \ (m) 104 230 378 549 743 961
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 t \ (s) 0.0695 0.1386 0.2128 0.2867 0.3606 0.4388
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_{s-20} \ (m/s) 1496 1658 1775 1914 2060 2190
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/":): \bar{V}_{20} \ (m/s) 1500 1637 1768 1894 2017 2137
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_{i} 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/":): 14.5^{\circ} 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/":): 17.5^{\circ} 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/":): 20.5^{\circ} 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/":): 23.6^{\circ} $ 26.7^{\circ } $ 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/":): 30.0^{\circ}
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_{i} \ (s) 0.413 0.349 0.305 0.273 0.248 0.231
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/":): \Sigma t_{i} \ (s) 0.413 0.762 1.067 1.340 1.589 1.820
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_{i} \ (m) 155 189 224 262 302 346
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/":): \Sigma x \ (m) 155 344 568 830 1132 1478
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 t \ (s) 0.103 0.208 0.316 0.430 0.549 0.673
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_{s-30} \ (m/s) 1508 1652 1795 1928 2060 2196
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/":): \bar{V}_{30} \ (m/s) 1500 1637 1770 1898 2024 2147
*Not enough significant figures to calculate with sufficient accuracy.
Table 5.12b. Calculated 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_{\rm rms} , $ V_{s} $, 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/":): \bar{V}.
layer 1 layer 2 layer 3 layer 4 layer 5 layer 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/":): \bar{V} \ (m/s) 1500 1635 1770 1890 2010 2130
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_{\rm rms} \ (m/s) 1500 1643 1780 1920 2060 2190
Stacking velocity 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/":): V_{s-10} \ (m/s) * (0) 1370 (4) 1707 (6) 1924 (8) 2029 (9) 2170 (14)
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_{s-20} \ (m/s) 1496 (6) 1658 (13) 1775 (22) 1914 (32) 2060 (43) 2190 (56)
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_{s-30} \ (m/s) 1508 (13) 1652 (29) 1795 (48) 1928 (71) 2060 (98) 2196 (129)
Average velocity along raypaths:
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/":): \bar{V}_{10} \ (m/s) 1500 1636 1767 1892 2013 2131
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/":): \bar{V}_{20} \ (m/s) 1500 1637 1768 1894 2017 2137
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/":): \bar{V}_{30} \ (m/s) 1500 1637 1770 1898 2024 2147
*Not enough significant figures to calculate with sufficient accuracy.
Values in parentheses are traveltime differences.


Table 5.12c. Calculation of raypaths for dipping layers.
layer 1 layer 2 layer 3 layer 4 layer 5 layer 6
$ \theta _{i} $ 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/":): 10.0^{\circ} 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/":): 12.0^{\circ} 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/":): 14.1^{\circ} 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/":): 16.1^{\circ} 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/":): 18.2^{\circ} 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/":): 20.0^{\circ}
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_{i} \ (s) 0.400 0.333 0.286 0.250 0.222 0.200
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_{i} \ (m) 106 128 151 173 197 221
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_{i} $ 20.0^{\circ } $ 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/":): 24.2^{\circ} 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/":): 28.6^{\circ} 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/":): 33.2^{\circ} 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/":): 38.0^{\circ} 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/":): 43.1^{\circ}
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_{i} \ (s) 0.426 0.366 0.326 0.303 0.288 0.274
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_{i} \ (m) 219 270 328 393 469 561
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_{i} $ 30.0^{\circ } $ 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/":): 37.0^{\circ} 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/":): 44.5^{\circ} 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/":): 54.0^{\circ} 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/":): 64.1^{\circ} 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/":): 90.0^{\circ}
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_{i} \ (s) 0.462 0.417 0.401 0.424 0.511 *
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_{i} \ (m) 346 452 591 620 1245 *
*A head wave is generated at the base of layer 5.


Table 5.12d. Calculation of stacking 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_{s} (m/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/":): \theta_{i} layer 1 layer 2 layer 3 layer 4 layer 5 layer 6
$ 10^{\circ } $ 1500 1630 1750 1860 1960 2050
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/":): 20^{\circ} 1500 1640 1750 1880 1990 2110
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/":): 30^{\circ} 1500 1650 1790 1920 2100 *
* Head wave generated.


We note that the stacking 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_{s} increases with 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/":): x . The calculated velocities are summarized in Table 5.12b.

Problem 5.12b

Repeat part (a) for the case where rays make angles of incidence at the free surface 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/":): 0^{\circ} , 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} , 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/":): 20^{\circ} 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/":): 30^{\circ} .

Solution

Figure 5.12b.  Dipping model.

The case 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/":): \theta _{1} =0^{\circ} is the same as that for $ \theta _{6}=0^{\circ } $ so that we need to calculate only 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/":): \theta _{1} =10^{\circ}, 20^{\circ}, 30^{\circ}; the results are given in Table 5.12c.

We now calculate a stacking velocity for reflections for each layer for each of the angles (Table 5.12d).

As before, we note that the stacking 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_{s} increases with 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/":): x .

Problem 5.12c

Assume the 300-m-thick layers 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/":): 20^{\circ} as shown in Figure 5.12b and determine arrival times for a zero-offset ray and one that leaves the free surface at an angle 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/":): 30^{\circ} and is reflected 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/":): B .

Solution

The raypath for a zero-offset trace makes a 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/":): 20^{\circ} angle in the updip direction at the surface 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/":): 0^{\circ} angles at all of the interfaces so that after reflection the raypath will return to the sourcepoint. The traveltimes are the same as calculated in part (a).

A ray that leaves the free surface 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/":): 30^{\circ} in the updip direction is incident on the $ V_{1}/V_{2} $ interface 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/":): 10^{\circ} and thus makes the same angles with other interfaces as calculated for 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/":): 10^{\circ} case in part (b). The time spent in each of the layers will also be the same as in part (b) but 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_{i} distances are now measured along the bedding planes. Thus, to determine the locations of the source and the emergent location, these have to be corrected 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/":): \cos 20^{\circ} The geometry is shown in Figure 5.12c. We have from part (b), e = 435 m, g = 488 m, one-way time from top 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} layer 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/":): B = 0.875\ {\rm s} , time 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/":): B to the base 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} layer 0.672 s.

Figure 5.12c.  Geometry of problem 5.12c.

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} {\rm a}+\ {\rm b}=300/\sin 20^{\circ} =877\ {\rm m},\\ {\rm c} =\ {\rm g} \left(\sin 110^{\circ} /\sin 60^{\circ} \right)=488\left(0.934/0.866\right)=526\ {\rm m},\\ {\rm k} =\ {\rm g}\left(\sin 20^{\circ} /\sin 60^{\circ} \right)=488\left(0.342/0.866\right)=193\ {\rm m},\\ \Delta t_{1} ={\rm k}/1500=0.128\ {\rm s},\\ {\rm e}+\ {\rm f}=300/\tan 20^{\circ} =300/0.364=824\ {\rm m},\\ {\rm f} =824-435=389\ {\rm m},\\ {\rm j}= {\rm f}\left(\sin 20^{\circ} /\sin 80^{\circ} \right)=389\left(0.342/0.985\right)=135\ {\rm m},\\ \Delta t_{2} ={\rm j}/1500=0.090\ {\rm s},\\ {\rm b} =877-389=488\ {\rm m},\\ {\rm traveltime} = t = 0.875+\Delta t_{1} +672+\Delta t_{2} =1.765\ {\rm s},\\ {\rm source\text{-}receiver\ distance} = {\rm b}+ {\rm c}=488+526=1014\ {\rm m}. \end{align}

The source is farther from the zero-offset location than the emergent point, so that the data are not suitable for stacking velocity calculations unless a DMO correction (Sheriff and Geldart, 1995, section 9.10.2) has been applied. Calculating arrival times for dipping reflections for split-dip situations is often done by trial and error.

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