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 . The interval velocity is given by


(5.13a)

In Figure 5.13a the triangle with apices at the points , , and is similar to the triangle with sides and , so we have

Substituting in equation (5.13a), we get

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 or change the slope of the diagonal and hence change and ; the error is proportional to where is either or .
  2. Changes in or has an effect similar to that in (i).
  3. When both and are small, the slope of the diagonal and are very sensitive to errors.
  4. Picking each event late by the same amount will increase by an amount proportional to

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

i) Equation (5.13a) is

Hence the error is directly proportional to either or and inversely proportional to ; it increases rapidly as approaches .

ii)

Likewise,

Since , errors in are more serious than errors in ; also, the errors increase rapidly as approaches .

iii) Let , . Then using equation (5.13a),

In general , so the error in depends mainly on the factor .

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

where is usually fairly constant over a moderate range of depths; in this case the error in is proportional to the error

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