Stacking velocity versus rms and average velocities

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	5
Pages	141 - 180
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	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 $0^{\circ }$, $10^{\circ }$, $20^{\circ }$, 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
$\theta _{i}$	$5.0^{\circ }$	$6.0^{\circ }$	$7.0^{\circ }$	$8.0^{\circ }$	$9.0^{\circ }$	$10.0^{\circ }$
$t_{i}\ (s)$	0.402	0.325	0.288	0.252	0.225	0.203
$\Sigma t_{i}\ (s)$	0.402	0.737	1.025	1.277	1.502	1.705
$x_{i}\ (m)$	52	63	73	84	95	106
$\Sigma x\ (m)$	52	105	189	273	368	473
$\Delta t\ (s)$	0	0.077	0.111	0.143	0.181	0.218
$V_{s-10}\ (m/s)$	*	1400*	1700*	1900*	2029	2170
${\bar {V}}_{10}\ (m/s)$	1500	1636	1767	1892	2013	2131
$\theta _{i}$	$9.8^{\circ }$	$11.8^{\circ }$	$13.9^{\circ }$	$15.9^{\circ }$	$17.9^{\circ }$	$20.0^{\circ }$
$t_{i}\ (s)$	0.406	0.340	0.294	0.260	0.224	0.213
$\Sigma t_{i}\ (s)$	0.406	0.746	1.041	1.301	1.534	1.747
$x_{i}\ (m)$	104	126	146	171	194	218
$\Sigma x\ (m)$	104	230	378	549	743	961
$\Delta t\ (s)$	0.0695	0.1386	0.2128	0.2867	0.3606	0.4388
$V_{s-20}\ (m/s)$	1496	1658	1775	1914	2060	2190
${\bar {V}}_{20}\ (m/s)$	1500	1637	1768	1894	2017	2137
$\theta _{i}$	$14.5^{\circ }$	$17.5^{\circ }$	$20.5^{\circ }$	$23.6^{\circ }$	$26.7^{\circ }$	$30.0^{\circ }$
$t_{i}\ (s)$	0.413	0.349	0.305	0.273	0.248	0.231
$\Sigma t_{i}\ (s)$	0.413	0.762	1.067	1.340	1.589	1.820
$x_{i}\ (m)$	155	189	224	262	302	346
$\Sigma x\ (m)$	155	344	568	830	1132	1478
$\Delta t\ (s)$	0.103	0.208	0.316	0.430	0.549	0.673
$V_{s-30}\ (m/s)$	1508	1652	1795	1928	2060	2196
${\bar {V}}_{30}\ (m/s)$	1500	1637	1770	1898	2024	2147

*Not enough significant figures to calculate with sufficient accuracy.

Table 5.12b. Calculated values of $V_{\rm {rms}},$ $V_{s}$, and ${\bar {V}}.$

	layer 1	layer 2	layer 3	layer 4	layer 5	layer 6
${\bar {V}}\ (m/s)$	1500	1635	1770	1890	2010	2130
$V_{\rm {rms}}\ (m/s)$	1500	1643	1780	1920	2060	2190
Stacking velocity calculations:
$V_{s-10}\ (m/s)$	* (0)	1370 (4)	1707 (6)	1924 (8)	2029 (9)	2170 (14)
$V_{s-20}\ (m/s)$	1496 (6)	1658 (13)	1775 (22)	1914 (32)	2060 (43)	2190 (56)
$V_{s-30}\ (m/s)$	1508 (13)	1652 (29)	1795 (48)	1928 (71)	2060 (98)	2196 (129)
Average velocity along raypaths:
${\bar {V}}_{10}\ (m/s)$	1500	1636	1767	1892	2013	2131
${\bar {V}}_{20}\ (m/s)$	1500	1637	1768	1894	2017	2137
${\bar {V}}_{30}\ (m/s)$	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
$\theta _{i}$	$10.0^{\circ }$	$12.0^{\circ }$	$14.1^{\circ }$	$16.1^{\circ }$	$18.2^{\circ }$	$20.0^{\circ }$
$t_{i}\ (s)$	0.400	0.333	0.286	0.250	0.222	0.200
$x_{i}\ (m)$	106	128	151	173	197	221
$\theta _{i}$	$20.0^{\circ }$	$24.2^{\circ }$	$28.6^{\circ }$	$33.2^{\circ }$	$38.0^{\circ }$	$43.1^{\circ }$
$t_{i}\ (s)$	0.426	0.366	0.326	0.303	0.288	0.274
$x_{i}\ (m)$	219	270	328	393	469	561
$\theta _{i}$	$30.0^{\circ }$	$37.0^{\circ }$	$44.5^{\circ }$	$54.0^{\circ }$	$64.1^{\circ }$	$90.0^{\circ }$
$t_{i}\ (s)$	0.462	0.417	0.401	0.424	0.511	*
$x_{i}\ (m)$	346	452	591	620	1245	*

*A head wave is generated at the base of layer 5.

Table 5.12d. Calculation of stacking velocity $V_{s}$ (m/s).

$\theta _{i}$	layer 1	layer 2	layer 3	layer 4	layer 5	layer 6
$10^{\circ }$	1500	1630	1750	1860	1960	2050
$20^{\circ }$	1500	1640	1750	1880	1990	2110
$30^{\circ }$	1500	1650	1790	1920	2100	*

* Head wave generated.

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

Solution

Figure 5.12b.  Dipping model.

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

Problem 5.12c

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

Solution

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

Figure 5.12c.  Geometry of problem 5.12c.

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

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.

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• Maximum porosity versus depth
• Relation between lithology and seismic velocities
• Porosities, velocities, and densities of rocks
• Velocities in limestone and sandstone
• Dependence of velocity-depth curves on geology
• Effect of burial history on velocity
• Determining lithology from well-velocity surveys
• Reflectivity versus water saturation
• Effect of overpressure
• Effects of weathered layer (LVL) and permafrost
• Horizontal component of head waves
• Stacking velocity versus rms and average velocities
• Quick-look velocity analysis and effects of errors
• Well-velocity survey
• Interval velocities
• Finding velocity
• Effect of timing errors on stacking velocity, depth, and dip
• Estimating lithology from stacking velocity
• Velocity versus depth from sonobuoy data
• Influence of direction on velocity analyses
• Effect of time picks, NMO stretch, and datum choice on stacking velocity

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