# Linear increase in velocity above a refractor

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 4.21

If the velocity function in problem 4.17c applies above a horizontal refractor at a depth of 2.40 km where the refractor velocity is 4.25 km/s, (see Figure 4.21a), plot the traveltime-distance curve.

### Background

In Figure 4.18a, the refraction event does not exist to the left of point , where is the critical distance (see problem 4.18).

### Solution

For a headwave to be generated, the angle of incidence must reach the critical angle . At the depth km, the velocity just above the refractor is km/s, so the critical angle is

To get , we have

Thus

To get the critical distance , we note that ( in Figure 4.18a) is equivalent to when . Using equation (4.17b), we have

The time required to travel the path is, using equation (4.17c),

Thus, the coordinates of the refraction traveltime curve at the critical distance are 2.56 s, 3.24 km). The refraction curve beginning at this point has the slope s/km. The curve (plotted in Figure 4.21b) has the equation

The intercept can be found graphically by extending the refraction curve or by calculating for :

The direct wave’s traveltime curve is . However, a diving wave with () related by equation (4.20a) arrives before this wave. Inverting equation (4.20a) gives

This equation has been used to plot the direct wave in Figure 4.21b. The crossover point is at approximately 5.9 km. However, the refraction and the direct wave do not differ appreciably in slope and hence picking the crossover point is not very accurate.

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Diving waves | Time-distance curves for various situations |

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Partitioning at an interface | Seismic velocity |

## 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