# Quick-look velocity analysis and effects of errors

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 5.13a

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 ${\displaystyle V_{m}}$. The interval velocity is given by

 {\displaystyle {\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 ${\displaystyle V_{m}}$, ${\displaystyle V_{1}}$, and ${\displaystyle A}$ is similar to the triangle with sides ${\displaystyle \Delta V}$ and ${\displaystyle \Delta t}$, so we have

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

Substituting in equation (5.13a), we get

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

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

i) Equation (5.13a) is

{\displaystyle {\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 ${\displaystyle t_{1}}$ or ${\displaystyle t_{2}}$ and inversely proportional to ${\displaystyle (t_{2}-t_{1})}$; it increases rapidly as ${\displaystyle t_{1}}$ approaches ${\displaystyle t_{2}}$.

ii) {\displaystyle {\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,

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

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

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

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

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