# 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 |

## Contents

## 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**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ^{\prime}}**
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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_2}**
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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta_c = (V_1/V_2)}**
and is known, we can find

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

**(**)

## Problem 4.24c

Show that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_2}**
is given approximately by either of the following equations, the latter being less accurate:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{1}{V_2}\approx \frac{1}{2}\left(\frac{1}{V_d}+\frac{1}{V_u}\right),\quad V_2\approx \frac{1}{2}(V_d+V_u). \end{align} }****(**)

### Solution

Expanding equation (4.24d) we have

Adding the two equations, we get

Because is usually small, we set **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos \xi = 1}**
. Since **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sin\theta_c = V_1/V_2}**
, we get the first result in equation (4.24f).

Returning to equation (4.24d), we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_B(m)}
| |||||||
---|---|---|---|---|---|---|---|

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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_u}**
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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_d}**
and , and finally . From equation (4.24b)

## Continue reading

Previous section | Next section |
---|---|

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

Previous chapter | Next chapter |

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