# Velocity analysis and statics corrections exercises

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Exercise 3-1. Why does salt have anomalously high velocity (4.5 to 5.5 km/s)?

Exercise 3-2. Measure the traveltimes corresponding to offset values of 1 and 3 km in Figure 3.1-2. Then compute the velocity above the reflector and verify that it is 2264 m/s. Note that the zero-offset trace is not recorded; therefore, t0 normally is not a known quantity.

Exercise 3-3. The CMP gather in Figure 3.1-5a contains a hyperbola. Following NMO corrections and using 2000 (Figure 3.1-5c) and 2500 m/s (Figure 3.1-5d) velocities, are the traveltime trajectories hyperbolic?

Exercise 3-4. Make velocity picks from the velocity panels in Figures 3.2-4, 3.2-5, 3.2-6, and 3.2-7.

Exercise 3-5. Make velocity picks from the CVS panel in Figure 3.E-1.

Exercise 3-6. Consider two intersecting lines. Would you expect that velocity analyses at the intersection point yield the same velocity function?

Exercise 3-7. Which is correct: velocity analysis from datum or velocity analysis from surface?

Exercise 3-8. Fill the missing elements in the following table. Average velocity vavg, which relates vertical traveltime to depth in a horizontally layered medium, is defined as ${\displaystyle v_{avg}={\frac {\sum \nolimits _{i=1}^{N}v_{i}\Delta t_{i}}{\sum \nolimits _{i=1}^{N}\Delta t_{i}}},}$ where Δti = Δzi / vi, Δzi = layer thickness and vi = interval velocity. The rms velocity is given by equation (4).

 Layer Thickness, m Interval Velocity, m/s RMS Velocity, m/s Average Velocity, m/s 200 2,000 300 3,000 400 4,000 350 3,500 500 5,000

Exercise 3-9. Explain why the velocity for horizon A in Figure 3.2-34 behaves as shown in the HVA display below the salt dome S.

Exercise 3-10. Suppose you want to fit a set of observed traveltimes to a parabola of the form t = a + bx + cx2. The tabulated input values are given below.

 i xi Observed ${\displaystyle t'_{i}}$ 1 0 0.4 2 1 1.1 3 2 3.5 4 3 7.9 5 4 14.4

Set up the least-squares problem and solve for a, b, and c. You will have five equations and three unknowns.

Exercise 3-11. Solve the system

${\displaystyle {\begin{array}{lcr}x_{1}-2x_{2}=1\\x_{1}+4x_{2}=4\\\end{array}}}$

by the Gauss-Seidel iterative method. Verify the results by solving these equations by the method of substitution to obtain the correct solution: x1 = 2 and x2 = 0.5.

Exercise 3-12. Solve the system

${\displaystyle {\begin{array}{lcr}x_{1}+4x_{2}=4\\x_{1}-2x_{2}=1\\\end{array}}}$

by the Gauss-Seidel iterative method. Note that this is the same problem as in Exercise 3-11, except that the order of the equations is reversed. The solution should be the same. You will find that the solution cannot be obtained because the process of iteration will not converge. This demonstrates the importance of ordering the equations when solving by the Gauss-Seidel method.

Exercise 3-13. Write equation (37a) for i = 1, 2, 3, and j = 1, 2, 3. You will find that there are six unknowns, but five independent equations.

Exercise 3-14. Can you use refraction-based statics techniques in permafrost areas?

Exercise 3-15. Which of the following adversely affects the quality of velocity spectrum — long-wavelength or short-wavelength (less than a cable length) statics anomalies?

Exercise 3-16. In what way does inside muting of a CMP gather affect the velocity spectrum?

## Figures and equations

 ${\displaystyle v_{rms}^{2}={\frac {1}{t_{0}}}\sum _{i=1}^{N}v_{i}^{2}\Delta \tau _{i},}$ (4a)

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{rms}^{2}}}.}$ (4b)

 ${\displaystyle t_{ij}=s_{j}+r_{i}.}$ (37a)