Effect of timing errors on stacking velocity, depth, and dip

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Problem 5.17a

Given that the trace spacing in Figure 5.17a is 50 m, determine the stacking velocity, dip, and depth at approximately 0.5, 1.0, 1.5, 2.0, and 2.4 s.

Solution

Using an enlarged version of Figure 5.17a, we measured arrival times on the center traces and on traces symmetrically located left and right of the center, limiting the offsets to where we felt we could pick the events with confidence. The measurements give us $ t_{o} $, $ \Delta t_{\rm {NMO}} $, 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 t_{d} /\Delta x and we must 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/":): z 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 ; for this we need the 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/":): \bar{V} . We can 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/":): V_{s} from equation (5.12a) and, since the dip is small, Failed to parse (SVG (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} \approx \bar{V} and we can therefore use equation (4.2b) to get an approximate 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/":): \xi . The equations are

Figure 5.17a.  A split-dip record (courtesy Grant Geophysical).
Table 5.17a. Measurements and calculations for five events.
Failed to parse (SVG (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_{o} (s) 0.533 0.975 1.575 2.008 2.417
offset (m) 575 m 1125 1425 1625 2375
Failed to parse (SVG (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_{\rm left} (s) 0.633 1.150 1.658 2.058 2.533
Failed to parse (SVG (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_{\rm right} (s) 0.608 1.142 1.758 2.150 2.575
Failed to parse (SVG (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_{\rm NMO} (s) 0.0875 0.171 0.133 0.096 0.137
Failed to parse (SVG (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) 1885 1950 2200 2615 2920
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z (m) 502 950 1730 2625 3530
Failed to parse (SVG (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_{d} /\Delta x (ms/m) 0.0435 0.0071 0.0702 0.0566 0.0177
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin \xi 0.0410 0.0070 0.0772 0.0740 0.0258
Failed to parse (SVG (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2.3^{\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/":): 0.4^{\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/":): 4.4^{\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/":): 4.2^{\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/":): 1.5^{\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/":): \begin{align} \Delta t_{\rm NMO} = (t_{\rm left} +t_{\rm right})/2-t_{o} \\ V_{s} =x/(2t_{o} \Delta t_{\rm NMO})^{1/2} ,\\ \sin \xi = (V_{s} /2)(\Delta t_{d} /\Delta x)\\ z=V_{s} t_{o} /2. \end{align}

The calculated results for five reflections are shown in Table 5.17a.

Problem 5.17b

What problems or ambiguities do you have in picking these events? How much uncertainty is there in your ability to pick times and how much uncertainty does this introduce into the velocity, depth, and dip calculations?

Solution

There are clearly different families of events interfering with each other on this record, which we have not attempted to sort out. The axes of symmetry of some of the data shift to the left with depth, indicating dip to the right. The event at 2.417 s may be a multiple. Clearly many more events could be picked.

We timed the centers of the black peaks, and this involves 5-10 ms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (\approx 0.5-2\%) uncertainty in this case. At a work station where a best-fit curve can be used to smooth-out noise, uncertainty can be reduced appreciably, and measurements can be accurate to 1 ms. Measurements 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_{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/":): \xi are based on time differences and their errors are probably about 5%. If the offsets had been longer, measured differences would have been larger, giving better accuracy, but then uncertainties in event continuity and interference with other events might have increased the errors. In calculating depths, the onset of reflections should be measured so time measurements are probably 20 ms (1 to 4%) late. This may introduce 1% error 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/":): V_{s} , but other errors involved 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/":): V_{s} are probably more important, including the assumption that it is the correct velocity to use. Dips are almost certainly underestimated by the use 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_{s} , which does not allow for the fact that the velocity at the reflector is usually larger 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/":): V_{s} because of the usual increase of velocity with depth.

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