Quick-look velocity analysis and effects of errors

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{1} are more serious than errors in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{2} ; also, the errors increase rapidly as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{1} approaches Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{2} .

iii) Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{2} =t_{1} +\Delta t , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t\approx 0 . Then using equation (5.13a),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

In general Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{1} \gg \Delta t , so the error in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{i} depends mainly on the factor $ t_{1}/\Delta t $.

iv) We assume that both events are late by the same amount Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta\tau . Then equation (5.13a) becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan \zeta is usually fairly constant over a moderate range of depths; in this case the error in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{i} is proportional to the error Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta\tau.

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