Quick-look velocity analysis and effects of errors

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 5.13a

Velocity analysis usually results in a plot of stacking velocity against traveltime. Bauer (private communication) devised a “quick look” method of determining the interval velocity, assuming horizontal layering and that the stacking velocity equals the average velocity. The method is shown in Figure 5.13a. A box is formed by the two picks between which the interval velocity is to be picked; the diagonal that does not contain the two picks when extended to the velocity axis gives the interval velocity. Prove that the method is valid and discuss its limitations.

Figure 5.13a.  Interval velocity determination.

Solution

We extend the diagonal of the box as shown in Figure 5.13a, thus giving $V_{m}$. The interval velocity is given by

${\begin{aligned}V_{i}=\Delta z/\Delta t=\left(V_{2}t_{2}-V_{1}t_{1}\right)/\left(t_{2}-t_{1}\right)\\=\left[\left(V_{1}+\Delta V\right)t_{2}-V_{1}t_{1}\left]/\Delta t=\right[V_{1}\left(t_{2}-t_{1}\right)+t_{2}\Delta V\right]/\Delta t\\=\left(V_{1}\Delta t+t_{2}\Delta V\right)/\Delta t=V_{1}+t_{2}\left(\Delta V/\Delta t\right).\end{aligned}}$

(5.13a)

In Figure 5.13a the triangle with apices at the points $V_{m}$, $V_{1}$, and $A$ is similar to the triangle with sides $\Delta V$ and $\Delta t$, so we have

${\begin{aligned}\Delta V/\Delta t=\left(V_{m}-V_{1}\right)/t_{2}.\end{aligned}}$

Substituting in equation (5.13a), we get

${\begin{aligned}V_{i}=V_{1}+\left(V_{m}-V_{1}\right)=V_{m}.\end{aligned}}$

Thus the method gives the interval velocity provided the stacking velocity equals the average velocity. For horizontal velocity layering the stacking velocity is often about 2% higher than the average velocity, but the two may differ considerably if the reflectors are dipping.

Problem 5.13b

This method can be used to see the influence of measurement error. Discuss the sensitivity of interval-velocity calculations to

1. errors in picking velocity values from this graph,
2. errors in picking times,
3. picking events very close together, and
4. picking events late.

Solution

1. Errors in $V_{1}$ or $V_{2}$ change the slope of the diagonal and hence change $V_{m}$ and $V_{i}$; the error is proportional to $t/\left(t_{2}-t_{1}\right)$ where $t$ is either $t_{1}$ or $t_{2}$.
2. Changes in $t_{1}$ or $t_{2}$ has an effect similar to that in (i).
3. When both $\Delta V$ and $\Delta t$ are small, the slope of the diagonal and $V_{i}$ are very sensitive to errors.
4. Picking each event late by the same amount $\Delta \tau$ will increase $V_{i}$ by an amount proportional to $\Delta \tau .$

We now derive mathematical expressions for the changes in (i) to (iv).

i) Equation (5.13a) is

${\begin{aligned}V_{1}=(V_{2}t_{2}-V_{1}t_{1})/(t_{2}-t_{1}).\\{\rm {Thus}},\qquad \qquad \qquad \qquad {\rm {d}}V_{i}/{\rm {d}}V_{1}=-t_{1}/(t_{2}-t_{1})=-t_{1}/\Delta t,\\{\rm {and}}\qquad \qquad \qquad \qquad {\rm {d}}V_{i}/{\rm {d}}V_{2}=t_{2}/(t_{2}-t_{1})=t_{2}/\Delta t.\end{aligned}}$

Hence the error is directly proportional to either $t_{1}$ or $t_{2}$ and inversely proportional to $(t_{2}-t_{1})$; it increases rapidly as $t_{1}$ approaches $t_{2}$.

ii) ${\begin{aligned}{\frac {{\rm {d}}V_{i}}{{\rm {d}}t_{1}}}=-{\frac {V_{1}}{\left(t_{2}-t_{1}\right)}}+{\frac {\left(V_{2}t_{2}-V_{1}t_{1}\right)}{(t_{2}-t_{1})^{2}}}={\frac {-V_{1}\left(t_{2}-t_{1}\right)+\left(V_{2}t_{2}-V_{1}t_{1}\right)}{(t_{2}-t_{1})^{2}}}\\={\frac {(V_{2}-V_{1})t_{2}}{(t_{2}-t_{1})^{2}}}.\end{aligned}}$

Likewise,

${\begin{aligned}{\frac {{\rm {d}}V_{i}}{{\rm {d}}t_{2}}}={\frac {-\left(V_{2}-V_{1}\right)t_{1}}{(t_{2}-t_{1})^{2}}}.\end{aligned}}$

Since $t_{2}>t_{1}$, errors in $t_{1}$ are more serious than errors in $t_{2}$; also, the errors increase rapidly as $t_{1}$ approaches $t_{2}$.

iii) Let $t_{2}=t_{1}+\Delta t$, $\Delta t\approx 0$. Then using equation (5.13a),

${\begin{aligned}V_{t}=\left[V_{2}\left(t_{1}+\Delta t\right)-V_{1}t_{1}\right]/\Delta t=V_{2}+\left(V_{2}-V_{1}\right)\left(t_{1}/\Delta t\right).\end{aligned}}$

In general $t_{1}\gg \Delta t$, so the error in $V_{i}$ depends mainly on the factor $t_{1}/\Delta t$.

iv) We assume that both events are late by the same amount $\Delta \tau$. Then equation (5.13a) becomes

${\begin{aligned}V_{i}+\Delta V_{i}=\left[V_{2}\left(t_{2}+\Delta \tau \right)-V_{1}\left(t_{1}+\Delta \tau \right)\right]/\left(t_{2}-t_{1}\right)\\=V_{i}+\left(V_{2}-V_{1}\right)\Delta \tau /\left(t_{2}-t_{1}\right),\\{\rm {so}}\qquad \qquad \qquad \qquad \Delta V_{i}=\Delta \tau \left[\left(V_{2}-V_{1}\right)/\left(t_{2}-t_{1}\right)\right]=\Delta \tau \tan \zeta ,\end{aligned}}$

where $\tan \zeta$ is usually fairly constant over a moderate range of depths; in this case the error in $V_{i}$ is proportional to the error $\Delta \tau .$

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Also in this chapter

• 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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