Relation between average and rms velocities

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	4
Pages	79 - 140
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 4.13a

Both the average velocity ${\bar {V}}$ and the root-mean-square velocity $V_{\rm {rms}}$ can be regarded as weighted averages along a particular raypath. Discuss the difference between the two velocities from this point of view.

Background

For a bed of constant velocity $V$ extending from the surface down to a horizontal reflector at a depth $h$, the two-way traveltime is $t=2h/V$. If the velocity is not constant and we assume vertical raypaths, we obtain an average velocity ${\bar {V}}$ by dividing $2h$ by the two-way traveltime, in effect replacing the actual section by a single constant-velocity layer. When the section consists of a series of horizontal constant-velocity layers, we find ${\bar {V}}$ by summing the traveltimes through each layer, and dividing the total thickness by this sum.

Dix (1955) showed that better results could be obtained by using the root-mean-square velocity $V_{\rm {rms}}$ calculated according to the formula,

${\begin{aligned}V_{\rm {rms}}^{2}=\sum _{i=1}^{n}V_{i}^{2}\Delta t_{i}/\sum _{i=1}^{n}\Delta t.\end{aligned}}$

(4.13a)

This formula applies to horizontal velocity layering as the offset goes to zero.

Fermat’s principle of stationary time states that a wave travels from one point to another along a path for which the traveltime is stationary, usually a minimum, compared to that along adjacent paths. This principle can be used to derive laws involving the geometrical aspects of wave travel (see problem 6.3).

Solution

For horizontal layering, ${\bar {V}}$ for a given depth is found by dividing the depth by the total one-way traveltime for a ray traveling vertically down to the required depth. This can be expressed as

${\begin{aligned}{\bar {V}}=h/t=(\Sigma \Delta h_{i})/\Sigma (\Delta h_{i}/V_{i}),\quad \mathrm {or} \quad h/{\bar {V}}=\Sigma (\Delta h_{i}/V_{i}),\end{aligned}}$

(4.13b)

where $\Delta h_{i}$ and $V_{i}$ are the thickness and velocity of the the $i^{\rm {th}}$ th bed and $h$ is the total thickness. Thus, $1/{\bar {V}}$ is the weighted average of the reciprocal of velocity when the weights are the layer vertical thicknesses.

Since the denominator of equation (4.13a) is the total one-way traveltime, we can write the equation in the form

${\begin{aligned}V_{\rm {rms}}^{2}t=\sum \limits _{i=1}^{n}V_{t}^{2}\Delta t_{i},\end{aligned}}$

from which we see that $V_{\rm {rms}}^{2}$ is the weighted average of $V_{i}^{2}$, the weights being the one-way vertical traveltimes through each bed.

Problem 4.13b

Calculate ${\bar {V}}$ and $V_{\rm {rms}}$ down to each of the interfaces in Figure 4.13a. Why do they differ (give a geometrical explanation)?

Figure 4.13a.  Model for the calculation of ${\bar {V}}$ and $V_{\rm {rms}}$.

Solution

${\bar {V}}$ is found by calculating the one-way time to pass through each layer, summing the time increments to a given depth and dividing the sum into the depth; $V_{\rm {rms}}$ is found from equation (4.13a). The values are shown in Figure 4.13a.

The average velocity ${\bar {V}}$ is calculated for a straight-line vertical path and is obtained from the weighted average of $1/V_{i}$, the weights being the layer thicknesses, whereas $V_{\rm {rms}}$ gives more weight to high-velocity beds. This is consistent with the fact that the least-time path between two points involves changes in path direction that increase travel in the high-velocity layers. Thus, using $V_{\rm {rms}}$ somewhat compensates for the failure of straight-line paths to account for ray bending at interfaces.

Problem 4.13c

Plot ${\bar {V}}$ and $V_{\rm {rms}}$ versus depth and versus traveltime and determine the best-fit straight lines for the four cases. What are the main problems in approximating data with functional fits?

Figure 4.13b.  Linear fits of velocity-depth and velocity-time data.

Solution

Figure 4.13b is the plot of ${\bar {V}}$ and $V_{\rm {rms}}$ versus depth and two-way traveltime. Least-squares fits (see Sheriff and Geldart, 1995, problem 15.13) are also tabulated on Figure 4.13b.

Use of any functional form involves approximations. The main problems in approximating data with functional fits are the following:

1. We must select a functional form to fit the data and the form selected may be either too simple or too complex. A simple form is easy to determine and use but it may not represent the data with sufficient precision. Use of a more complex form increases the accuracy but at the expense of increased computation time.
2. Single points far from the line of the function have a disproportionate effect on the determination of the line.
3. When the data are not evenly spaced, data bunched together have less influence on the best-fit line than more widely spaced data.
4. The accuracy of functional fits changes with offset because the relative contributions of high-velocity members increases with offset.

Table 4.14a. Depth calculations using various velocity functions.

Function	$t_{0}\to$	1.00 s	2.00 s	2.10 s	3.10 s
i)	${\bar {V}}$	2.00 km/s	2.50 km/s	2.67 km/s	3.10 km/s
	$z$	1.00 km	2.50 km	2.80 km	4.80 km
ii)	$V_{\rm {rms}}$	2.00 km/s	2.55 km/s	2.81 km/s	3.24 km/s
	$z$	1.00 km	2.55 km	2.95 km	5.02 km
iii)	${\bar {V}}$	2.02 km/s	2.54 km/s	2.59 km/s	3.12 km/s
	$z$	1.01 km	2.54 km	2.72 km	4.84 km
iv)	$V_{\rm {rms}}$	2.03 km/s	2.62 km/s	2.68 km/s	3.27 km/s
	$z$	1.02 km	2.62 km	2.81 km	5.07 km

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Table of Contents (book)

Also in this chapter

• Accuracy of normal-moveout calculations
• Dip, cross-dip, and angle of approach
• Relationship for a dipping bed
• Reflector dip in terms of traveltimes squared
• Second approximation for dip moveout
• Calculation of reflector depths and dips
• Plotting raypaths for primary and multiple reflections
• Effect of migration on plotted reflector locations
• Resolution of cross-dip
• Cross-dip
• Variation of reflection point with offset
• Functional fits for velocity-depth data
• Relation between average and rms velocities
• Vertical depth calculations using velocity functions
• Depth and dip calculations using velocity functions
• Weathering corrections and dip/depth calculations
• Using a velocity function linear with depth
• Head waves (refractions) and effect of hidden layer
• Interpretation of sonobuoy data
• Diving waves
• Linear increase in velocity above a refractor
• Time-distance curves for various situations
• Locating the bottom of a borehole
• Two-layer refraction problem

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