# Stacking velocity versus rms and average velocities

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 5.12a

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 ${\displaystyle 0^{\circ }}$, ${\displaystyle 10^{\circ }}$, ${\displaystyle 20^{\circ }}$, and ${\displaystyle 30^{\circ }}$. Calculate stacking velocity for each angle and compare with the average velocity ${\displaystyle {\bar {V}}}$ and the rms velocity ${\displaystyle V_{\rm {rms}}}$.

### Background

Average velocity ${\displaystyle ({\bar {V}})}$ and rms velocity ${\displaystyle (V_{\rm {rms}})}$ 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 ${\displaystyle V_{s}}$. If we use equation (4.1c) to remove the NMO, that is, if we assume a single horizontal constant-velocity layer, the velocity ${\displaystyle V}$ in equations (4.1a,c) becomes ${\displaystyle V_{s}}$. The ${\displaystyle x^{2}-t^{2}}$ plot of equation (4.1a) is a straight line with slope ${\displaystyle 1/V_{s}^{2}}$; thus,

 {\displaystyle {\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 ${\displaystyle t_{o}}$ and ${\displaystyle t}$ are the two-way traveltimes at the origin and at offset ${\displaystyle x}$ while ${\displaystyle \Delta t=t-t_{o}}$. When the velocity changes with depth, the ${\displaystyle x^{2}-t^{2}}$ 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, ${\displaystyle V_{s}\approx V_{\rm {rms}}}$.

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

### Solution

We use Snell’s law to calculate the raypath angles ${\displaystyle \theta _{i}}$ in each layer. The two-way time in a layer is ${\displaystyle t_{i}=600/V_{i}\cos \theta _{i}}$ and the offset in a layer is ${\displaystyle x_{i}=600\tan \theta _{i}}$. 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 ${\displaystyle {\bar {V}}}$ 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 ${\displaystyle V_{s}}$. A general rule for ${\displaystyle V_{s}}$ calculations, that the offset should be comparable to the depth, is not reached for any of these situations.

Table 5.12a. Calculation of ${\displaystyle V_{s}}$, ${\displaystyle {\bar {V}}}$, and ${\displaystyle V_{\rm {rms}}.}$
layer 1 layer 2 layer 3 layer 4 layer 5 layer 6
${\displaystyle \theta _{i}}$ ${\displaystyle 0^{\circ }}$ ${\displaystyle 0^{\circ }}$ ${\displaystyle 0^{\circ }}$ ${\displaystyle 0^{\circ }}$ ${\displaystyle 0^{\circ }}$ ${\displaystyle 0^{\circ }}$
${\displaystyle t_{i}\ (s)}$ 0.400 0.333 0.286 0.250 0.222 0.200
${\displaystyle \Sigma t_{i}\ (s)}$ 0.400 0.733 1.019 1.269 1.491 1.691
${\displaystyle x_{i}\ (m)}$ 0 0 0 0 0 0
${\displaystyle {\bar {V}}\ (m/s)}$ 1500 1640 1770 1890 2010 2130
${\displaystyle V_{\rm {rms}}\ (m/s)}$ 1500 1640 1780 1920 2060 2190
${\displaystyle t_{o}^{2}({\rm {s}}^{2})}$ 0.1600 0.5373 1.0384 1.6104 2.2231 2.8595
${\displaystyle \theta _{i}}$ ${\displaystyle 5.0^{\circ }}$ ${\displaystyle 6.0^{\circ }}$ ${\displaystyle 7.0^{\circ }}$ ${\displaystyle 8.0^{\circ }}$ ${\displaystyle 9.0^{\circ }}$ ${\displaystyle 10.0^{\circ }}$
${\displaystyle t_{i}\ (s)}$ 0.402 0.325 0.288 0.252 0.225 0.203
${\displaystyle \Sigma t_{i}\ (s)}$ 0.402 0.737 1.025 1.277 1.502 1.705
${\displaystyle x_{i}\ (m)}$ 52 63 73 84 95 106
${\displaystyle \Sigma x\ (m)}$ 52 105 189 273 368 473
${\displaystyle \Delta t\ (s)}$ 0 0.077 0.111 0.143 0.181 0.218
${\displaystyle V_{s-10}\ (m/s)}$ * 1400* 1700* 1900* 2029 2170
${\displaystyle {\bar {V}}_{10}\ (m/s)}$ 1500 1636 1767 1892 2013 2131
${\displaystyle \theta _{i}}$ ${\displaystyle 9.8^{\circ }}$ ${\displaystyle 11.8^{\circ }}$ ${\displaystyle 13.9^{\circ }}$ ${\displaystyle 15.9^{\circ }}$ ${\displaystyle 17.9^{\circ }}$ ${\displaystyle 20.0^{\circ }}$
${\displaystyle t_{i}\ (s)}$ 0.406 0.340 0.294 0.260 0.224 0.213
${\displaystyle \Sigma t_{i}\ (s)}$ 0.406 0.746 1.041 1.301 1.534 1.747
${\displaystyle x_{i}\ (m)}$ 104 126 146 171 194 218
${\displaystyle \Sigma x\ (m)}$ 104 230 378 549 743 961
${\displaystyle \Delta t\ (s)}$ 0.0695 0.1386 0.2128 0.2867 0.3606 0.4388
${\displaystyle V_{s-20}\ (m/s)}$ 1496 1658 1775 1914 2060 2190
${\displaystyle {\bar {V}}_{20}\ (m/s)}$ 1500 1637 1768 1894 2017 2137
${\displaystyle \theta _{i}}$ ${\displaystyle 14.5^{\circ }}$ ${\displaystyle 17.5^{\circ }}$ ${\displaystyle 20.5^{\circ }}$ ${\displaystyle 23.6^{\circ }}$ ${\displaystyle 26.7^{\circ }}$ ${\displaystyle 30.0^{\circ }}$
${\displaystyle t_{i}\ (s)}$ 0.413 0.349 0.305 0.273 0.248 0.231
${\displaystyle \Sigma t_{i}\ (s)}$ 0.413 0.762 1.067 1.340 1.589 1.820
${\displaystyle x_{i}\ (m)}$ 155 189 224 262 302 346
${\displaystyle \Sigma x\ (m)}$ 155 344 568 830 1132 1478
${\displaystyle \Delta t\ (s)}$ 0.103 0.208 0.316 0.430 0.549 0.673
${\displaystyle V_{s-30}\ (m/s)}$ 1508 1652 1795 1928 2060 2196
${\displaystyle {\bar {V}}_{30}\ (m/s)}$ 1500 1637 1770 1898 2024 2147
 *Not enough significant figures to calculate with sufficient accuracy.
Table 5.12b. Calculated values of ${\displaystyle V_{\rm {rms}},}$ ${\displaystyle V_{s}}$, and ${\displaystyle {\bar {V}}.}$
layer 1 layer 2 layer 3 layer 4 layer 5 layer 6
${\displaystyle {\bar {V}}\ (m/s)}$ 1500 1635 1770 1890 2010 2130
${\displaystyle V_{\rm {rms}}\ (m/s)}$ 1500 1643 1780 1920 2060 2190
Stacking velocity calculations:
${\displaystyle V_{s-10}\ (m/s)}$ * (0) 1370 (4) 1707 (6) 1924 (8) 2029 (9) 2170 (14)
${\displaystyle V_{s-20}\ (m/s)}$ 1496 (6) 1658 (13) 1775 (22) 1914 (32) 2060 (43) 2190 (56)
${\displaystyle V_{s-30}\ (m/s)}$ 1508 (13) 1652 (29) 1795 (48) 1928 (71) 2060 (98) 2196 (129)
Average velocity along raypaths:
${\displaystyle {\bar {V}}_{10}\ (m/s)}$ 1500 1636 1767 1892 2013 2131
${\displaystyle {\bar {V}}_{20}\ (m/s)}$ 1500 1637 1768 1894 2017 2137
${\displaystyle {\bar {V}}_{30}\ (m/s)}$ 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
${\displaystyle \theta _{i}}$ ${\displaystyle 10.0^{\circ }}$ ${\displaystyle 12.0^{\circ }}$ ${\displaystyle 14.1^{\circ }}$ ${\displaystyle 16.1^{\circ }}$ ${\displaystyle 18.2^{\circ }}$ ${\displaystyle 20.0^{\circ }}$
${\displaystyle t_{i}\ (s)}$ 0.400 0.333 0.286 0.250 0.222 0.200
${\displaystyle x_{i}\ (m)}$ 106 128 151 173 197 221
${\displaystyle \theta _{i}}$ ${\displaystyle 20.0^{\circ }}$ ${\displaystyle 24.2^{\circ }}$ ${\displaystyle 28.6^{\circ }}$ ${\displaystyle 33.2^{\circ }}$ ${\displaystyle 38.0^{\circ }}$ ${\displaystyle 43.1^{\circ }}$
${\displaystyle t_{i}\ (s)}$ 0.426 0.366 0.326 0.303 0.288 0.274
${\displaystyle x_{i}\ (m)}$ 219 270 328 393 469 561
${\displaystyle \theta _{i}}$ ${\displaystyle 30.0^{\circ }}$ ${\displaystyle 37.0^{\circ }}$ ${\displaystyle 44.5^{\circ }}$ ${\displaystyle 54.0^{\circ }}$ ${\displaystyle 64.1^{\circ }}$ ${\displaystyle 90.0^{\circ }}$
${\displaystyle t_{i}\ (s)}$ 0.462 0.417 0.401 0.424 0.511 *
${\displaystyle x_{i}\ (m)}$ 346 452 591 620 1245 *
 *A head wave is generated at the base of layer 5.

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

We note that the stacking velocity ${\displaystyle V_{s}}$ increases with the offset ${\displaystyle x}$. 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 ${\displaystyle 0^{\circ }}$, ${\displaystyle 10^{\circ }}$, ${\displaystyle 20^{\circ }}$ and ${\displaystyle 30^{\circ }}$.

### Solution

Figure 5.12b.  Dipping model.

The case where ${\displaystyle \theta _{1}=0^{\circ }}$ is the same as that for ${\displaystyle \theta _{6}=0^{\circ }}$ so that we need to calculate only for ${\displaystyle \theta _{1}=10^{\circ },20^{\circ },30^{\circ };}$ 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 ${\displaystyle V_{s}}$ increases with the offset ${\displaystyle x}$.

## Problem 5.12c

Assume the 300-m-thick layers dip ${\displaystyle 20^{\circ }}$ 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 ${\displaystyle 30^{\circ }}$ and is reflected at ${\displaystyle B}$.

### Solution

The raypath for a zero-offset trace makes a ${\displaystyle 20^{\circ }}$ angle in the updip direction at the surface and ${\displaystyle 0^{\circ }}$ 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 ${\displaystyle 30^{\circ }}$ in the updip direction is incident on the ${\displaystyle V_{1}/V_{2}}$ interface at ${\displaystyle 10^{\circ }}$ and thus makes the same angles with other interfaces as calculated for the ${\displaystyle 10^{\circ }}$ case in part (b). The time spent in each of the layers will also be the same as in part (b) but the ${\displaystyle x_{i}}$ 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 ${\displaystyle \cos 20^{\circ }}$ 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 ${\displaystyle V_{1}}$ layer to ${\displaystyle B=0.875\ {\rm {s}}}$, time from ${\displaystyle B}$ to the base of ${\displaystyle V_{1}}$ layer 0.672 s.

Figure 5.12c.  Geometry of problem 5.12c.

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

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.