# Two-layer refraction problem

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 4.24a

Show that time-distance curves for dipping refractors take the form

**(**)

**where **

**(**)

** and being the traveltimes when shooting downdip and updip, respectively (see Figure 4.24a), and , , and , , the corresponding slant depths and sourcepoint intercepts times.**

### Background

Traveltime curves for horizontal refractors are discussed in problem 4.18.

### Solution

For the downdip case, we take O as the sourcepoint and O as the receiver. Following the procedure used in problem 4.18, we have

**(**)

But , so we can express equation (4.24c) in terms of :

Expressing equation (4.24c) in terms of , we obtain

The slopes of the two traveltime curves are sin (, the reciprocals being the apparent velocities, and (see problem 4.2d), where

**(**)

## Problem 4.24b

Show how to find and from the observed data.

### Solution

We obtain , , and from the slopes of the time-distance curves. From equation (4.24d), we get

Adding and subtracting the two equations gives and . Since sin and is known, we can find

The dip can also be found (usually more accurately) from the relation

**(**)

## Problem 4.24c

Show that is given approximately by either of the following equations, the latter being less accurate:

**(**)

### Solution

Expanding equation (4.24d) we have

Adding the two equations, we get

Because is usually small, we set . Since , we get the first result in equation (4.24f).

Returning to equation (4.24d), we write

0 | 0 | 98 | 225 | 120 | 70 | 52 | 105 |

15 | 10 | 92 | 210 | 135 | 73 | 46 | 90 |

30 | 21 | 87 | 195 | 150 | 78 | 43 | 75 |

45 | 30 | 81 | 180 | 165 | 81 | 37 | 60 |

60 | 41 | 73 | 165 | 180 | 85 | 31 | 45 |

75 | 50 | 71 | 150 | 195 | 89 | 21 | 30 |

90 | 59 | 63 | 135 | 210 | 94 | 10 | 15 |

105 | 65 | 60 | 120 | 225 | 98 | 0 | 0 |

Setting and expanding by the binomial theorem [see equation (4.1b)], we obtain the result

Following the same procedure for and adding the two expansions gives the second result in equation (4.24f). This result is less accurate than the first because we approximated the binomial expansion and also set .

## Problem 4.24d

Sources A and B are located at the ends of a 225-m spread of 16 geophones. Using the data in Table 4.24a, find the velocities, dip, and depth to the refractor.

### Solution

The data in Table 4.24a are plotted in Figure 4.24b and straight-line curves drawn through the data points. The slopes of these lines give the direct-wave velocity and the apparent updip and downdip velocities, and the intercepts with the -axes give and . We ignore the value of obtained on the downdip profile because it is poorly defined. The measured velocities and intercepts are now

From these data, we calculate first , then , and . The two equations (4.24f) give

the first being more accurate.

Next, . Now we find and , and finally . From equation (4.24b)

## Continue reading

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Locating the bottom of a borehole | Maximum porosity versus depth |

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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