Interval velocity: v int Corresponding to any of the velocity averages, an interval velocity can be defined that is simply the

2.7 Interval velocity: v int Corresponding to any of the velocity averages, an interval velocity can be defined that is simply the

particular average applied across a small interval rather than from the surface (z = 0) to depth z.

36 CHAPTER 2. VELOCITY

0.6 time (sec)

velocity (m/s)

Figure 2.8: v rms and v ave are compared with v ins for the case of the linear increase of v ins with z.

For example, the average and rms velocities across an interval defined by τ 1 and τ 2 are simply

These quantities are functions of both the upper and lower bounds of the integrals. In the limit, as the interval shrinks so small that v ins (τ )can be considered constant, both interval velocities approach the instantaneous velocity.

It follows directly from the definition of average velocity (equation (2.10)) that the average velocity across a depth interval is just the ratio of the depth interval to the vertical traveltime across the interval. That is

Thus, if the range from 0 to z is divided into n finite intervals defined by z 0 ,z 1 ,z 2 ,...,z n−1 ,z n (where z 0 = 0 and z n = z) then

]v ave (τ k ,τ ) (2.23)

k−1

k=1

where τ k = τ (z k ). Defining ∆τ k =τ k −τ k−1 then gives

where v n k ≡v ave (τ k ,τ k−1 ) and τ = k=1 ∆τ k . Comparing equation (2.24) with equation (2.11) suggests that the former is just a discrete version of the latter. However, the velocity v k in equation

37 (2.24) is the average velocity of the k th finite interval and is thus the time average of v ins across the

2.7. INTERVAL VELOCITY: V IN T

interval. Of course, if v ins (τ ) is constant across each interval then v k is an instantaneous velocity and this distinction vanishes.

Equation (2.24) can be used in a number of ways. Most obviously, it shows how to combine a set of local average velocities to obtain the macro average velocity across a larger interval. Also, it can be used to estimate a local average velocity given two macro-average velocities. Suppose the

average velocities v ave1 and v ave2 from z = 0 to depths z 1 and z 2 are known. Then an expression for the average velocity, from z 1 →z 2 or equivalently from τ 1 →τ 2 , follows from equation (2.24) and is

v ave (τ 2 ,τ 1 )=

[τ 2 v ave2 −τ 1 v ave1 ].

The two macro averages are each weighted by their time intervals and then the shallower average is subtracted from the deeper. This difference is then divided by the time interval of the local average.

If v ave1 ,τ 1 ,v ave2 , andτ 2 are all measured without error, then this process (i.e. equation 2.25) works perfectly. However, in a real case, errors in the measurements can lead to wildly incorrect estimates of v ave (τ 2 ,τ 1 ). With noisy data, there is no guarantee that the term [τ 2 v ave2 −τ 1 v ave1 ] will always be positive leading to the possibility of negative velocity estimates. Also, the division by τ 2 −τ 1 can be unstable if the two times are very close together.

The discussion so far has been only for average velocities across an interval. The entire derivation above can be repeated for rms velocities with similar results but a few more subtleties arise in interpretation. Rather than repeating the derivation just given with ‘rms’ in place of ‘ave’, it is

c b instructive to consider an alternative approach. It is a property of integrals that c

a = a + b where

a < b < c. Given τ 0 <τ 1 <τ 2 and applying this rule to equation (2.17) results in v 2 1 τ 1 τ rms 2 (τ ,τ 0 )= v 2 2 ins (˜ τ )d˜ τ+ v 2 τ .

Recognizing the integrals in [ . . . ] as rms interval velocities squared multiplied by their interval τ 1 times (i.e.

v 2 (˜ τ )d˜ τ = [τ 1 −τ 0 ]v τ 2 0 ins rms (τ 1 ,τ 0 ) and similarly for the other integral) leads to

For the case of n subdivisions between τ 2 and τ 0 this generalizes to

where v n k =v rms (τ k ,τ k−1 ) and, as before, ∆τ k =τ k −τ k−1 and τ = k=1 ∆τ k . This is the rms equivalent to equation (2.24) and all of the comments made previously about v ave apply here for

v rms . In particular, equation (2.28) should be thought of a combining interval rms velocities into

a macro rms velocity. Only in the case when v ins does not vary significantly across an interval can the v k in equation (2.28) be considered to be instantaneous velocities.

Equation (2.27) is the addition rule for rms velocities. To combine rms velocities, the squared ve- locities are added and each must be weighted by its time interval. Equation (2.27) can be rearranged

38 CHAPTER 2. VELOCITY to give an expression for estimating a local rms velocity from two macro velocities

rms (τ 2 ,τ 1 )=

2 −τ 0 )v 2 rms (τ 2 ,τ 0 ) − (τ 1 −τ 0 )v 2 rms (τ 1 ,τ 0 ) (2.29)

or, with simplified notation

rms (τ 2 ,τ 1 )=

2 v 2 rms2 −τ 1 v 2 rms1

In this expression τ 0 has been set to 0 and v rms2 =v rms (τ 2 ,τ 0 ) and similarly for v rms1 . Equation (2.30) is often called the Dix equation for interval rms velocities because it was C.H. Dix (Dix, 1955) who first recognized the connection between stacking velocities and rms velocities and showed how to calculate an interval velocity from two stacking velocity measurements.

The application of equation (2.30) in practice requires the measurement of stacking velocities and vertical traveltimes to two closely spaced reflectors. Under the assumption that stacking velocities are well approximated by rms velocities (Dix (1955) or Taner and Koehler (1969)), the rms velocity of the interval is then estimated with equation (2.30). However, as with average velocities, errors in measurement lead to problems with the estimation of v

2 rms (τ 2 ,τ 1 ). If the term 2 v 2 rms2 −τ 1 v rms1 becomes negative then imaginary interval velocity estimates result. Thus one essential condition for the use of this technique is that

v 2 rms2 τ >v 2 rms1 1 .

Since τ 1 /τ 2 < 1, this is a constraint upon how fast v rms estimates can decrease with increasing time. There is no mathematical basis to constrain the rate at which v rms can increase; however, it is reasonable to formulate a constraint on physical grounds. Since P-wave seismic velocities are not expected to exceed some v max (say 7000 m/s) a constraint would be

+v 2 τ 2 −τ rms1 1 max .

rms2 <v

Exercise 2.7.1. Show that the right-hand-side of inequality (2.32) is always greater than v rms1 provided that v max >v rms1 so that this is a constraint on the rate of increase of v rms .

Exercise 2.7.2. Suppose that the times and depths to two reflectors are known, say τ 1 ,τ 2 ,z 1 , and z 2 , and that the rms velocities to the reflectors are also known, say v rms1 and v rms2 . Consider the interval velocities defined by

Under what condition(s) will these interval velocityestimates be similar? If the interval between the reflectors is highlyheterogeneous, which estimate will be larger. Why?