# Locating the bottom of a borehole

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 |

## Contents

## Problem 4.23a

Barton (1929) discusses shooting into a geophone placed in a borehole (Figure 4.23a) to determine where the bottom is located. Point is vertically above the well geophone at , and are equidistant from in the cardinal directions, and the traveltimes from sources and to are equal.

Assuming straight-line travelpaths at the velocity , derive expressions for and in Figure 4.23a(ii) in terms of the traveltimes to from and and .

### Solution

Since , , and , point must be in the vertical plane through and must lie on the straight line . Then, letting , we have

**(**)

**(**)

Subtracting, we find

**(**)

Adding equations (4.23a) and (4.23b) gives

**(**)

Since all quantities on the right are known, we can find .

## Problem 4.23b

What are the values of and for m/s, m, m?

### Solution

From equation (4.23a),

## Problem 4.23c

How sensitive is the method, that is, what are / and ? For the specific situation in part (b), how much change is there in and per millisecond error in ?

### Solution

From equation (4.23c), assuming fixed, we get

**(**)

Differentiating equation (4.23a) and using equation (4.23e) gives

Using values from part (b), we obtain

For ms, m.

Also, .

For ms, m.

## Problem 4.23d

Assume a velocity of 1500 m/s for the first 500 m and 3500 m/s below 500 m. What are the traveltimes now and how would these be interpreted if straight raypaths are assumed?

### Solution

By trial and error we find that the angles should be as shown in Figure 4.23b. Then

Interpreting these results as in part (a), we get

Thus, varies only 5%, mainly because we subtract the squares of traveltimes, thus partially canceling errors. However, the change in is more than 20%.

## Continue reading

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Time-distance curves for various situations | Two-layer refraction problem |

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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