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

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 5 141 - 180 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## 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 ${\displaystyle t_{o}}$, ${\displaystyle \Delta t_{\rm {NMO}}}$, and ${\displaystyle \Delta t_{d}/\Delta x}$ and we must find ${\displaystyle z}$ and ${\displaystyle \xi }$; for this we need the velocity ${\displaystyle {\bar {V}}}$. We can find ${\displaystyle V_{s}}$ from equation (5.12a) and, since the dip is small, ${\displaystyle V_{s}\approx {\bar {V}}}$ and we can therefore use equation (4.2b) to get an approximate value of ${\displaystyle \xi }$. The equations are

Figure 5.17a.  A split-dip record (courtesy Grant Geophysical).
 ${\displaystyle t_{o}}$ (s) 0.533 0.975 1.575 2.008 2.417 offset (m) 575 m 1125 1425 1625 2375 ${\displaystyle t_{\rm {left}}}$ (s) 0.633 1.150 1.658 2.058 2.533 ${\displaystyle t_{\rm {right}}}$ (s) 0.608 1.142 1.758 2.150 2.575 ${\displaystyle t_{\rm {NMO}}}$ (s) 0.0875 0.171 0.133 0.096 0.137 ${\displaystyle V_{s}}$ (m/s) 1885 1950 2200 2615 2920 ${\displaystyle z}$ (m) 502 950 1730 2625 3530 ${\displaystyle \Delta t_{d}/\Delta x}$(ms/m) 0.0435 0.0071 0.0702 0.0566 0.0177 ${\displaystyle \sin \xi }$ 0.0410 0.0070 0.0772 0.0740 0.0258 ${\displaystyle \xi }$ ${\displaystyle 2.3^{\circ }}$ ${\displaystyle 0.4^{\circ }}$ ${\displaystyle 4.4^{\circ }}$ ${\displaystyle 4.2^{\circ }}$ ${\displaystyle 1.5^{\circ }}$

{\displaystyle {\begin{aligned}\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{aligned}}}

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 ${\displaystyle (\approx 0.5-2\%)}$ 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 ${\displaystyle V_{s}}$ and ${\displaystyle \xi }$ 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 ${\displaystyle V_{s}}$, but other errors involved in ${\displaystyle V_{s}}$ are probably more important, including the assumption that it is the correct velocity to use. Dips are almost certainly underestimated by the use of ${\displaystyle V_{s}}$, which does not allow for the fact that the velocity at the reflector is usually larger than ${\displaystyle V_{s}}$ because of the usual increase of velocity with depth.