# Improvement of signal/noise ratio by stacking

Series | Geophysical References Series |
---|---|

Title | Problems in Exploration Seismology and their Solutions |

Author | Lloyd P. Geldart and Robert E. Sheriff |

Chapter | 6 |

Pages | 181 - 220 |

DOI | http://dx.doi.org/10.1190/1.9781560801733 |

ISBN | ISBN 9781560801153 |

Store | SEG Online Store |

## Contents

## Problems

Select random numbers between to represent noise and add to each a signal . Sum 4 values of and determine the mean, standard deviation of , and values of the ratio . Repeat for 8, 16, and 32 values. Note how the mean converges toward as the number of values increases, how approaches a limiting value (which depends on the statistical properties of the noise), and how the ratio converges toward .

### Background

The standard deviation is a measure of the scatter of measured values of a quantity, large values corresponding to large variations. It is given by the equation

**(**)

where is the number of values and is the mean value.

20897 | 13007 | 95217 | 19221 | 15433 | 94882 | 23741 | 86571 | 20504 | 22169 |

20737 | 19305 | 71148 | 04035 | 03180 | 79506 | 12771 | 34806 | 37279 | 62739 |

31552 | 59282 | 16856 | 38655 | 31802 | 84283 | 08694 | 06945 | 19286 | 16924 |

60605 | 97685 | 26147 | 51379 | 39553 | 04893 | 25469 | 96469 | 57436 | 97888 |

42094 | 17446 | 27775 | 99466 | 63704 | 60957 | 55029 | 02764 | 91845 | 76174 |

54774 | 15832 | 04324 | 73597 | 42328 | 74303 | 58231 | 85798 | 16725 | 27836 |

89730 | 31886 | 34683 | 07814 | 57000 | 63721 | 43798 | 12003 | 04676 | 08367 |

12049 | 18538 | 96266 | 62439 | 81839 | 13093 | 22659 | 75018 | 31494 | 89519 |

22364 | 15913 | 51674 | 94189 | 10336 | 97801 | 21025 | 58966 | 40663 | 26197 |

80102 | 39977 | 78674 | 29634 | 38652 | 85289 | 47962 | 16594 | 50834 | 93484 |

To obtain a sequence of random numbers within a given range, adopt a rule for selecting digits from Table 6.22a; use this rule to get the units digit, then get the tens digit, etc. To determine the algebraic sign of each number, adopt another rule for fixing the sign, e.g., letting even/odd digits signify plus/minus, respectively.

### Solution

Using Table 6.22a, we get 32 values of and sums in Table 6.22b.

Writing for the mean of (), we get the results in Table 6.22c.

0 | 0 | ||||||

0 | 0 | ||||||

0 | |||||||

1st 4 | 14 | 3.5 | 4.4 | 0.57 |

1st 8 | 24 | 3.0 | 5.3 | 0.67 |

2nd 8 | 15 | 1.9 | 4.0 | 1.05 |

3rd 8 | 9 | 1.1 | 4.4 | 1.82 |

4th 8 | 14 | 1.8 | 5.5 | 1.11 |

1st 16 | 39 | 2.4 | 4.8 | 0.83 |

2nd 16 | 23 | 1.4 | 5.0 | 1.43 |

32 | 62 | 1.9 | 4.9 | 1.05 |

We see that as the number of samples increases, approaches 4.9, approaches 2, and approaches .

## Continue reading

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

Ricker wavelet relations | Radiolocation errors because of velocity variations |

Previous chapter | Next chapter |

Geometry of seismic waves | Characteristics of seismic events |

## Also in this chapter

- Characteristics of different types of events and noise
- Horizontal resolution
- Reflection and refraction laws and Fermat’s principle
- Effect of reflector curvature on a plane wave
- Diffraction traveltime curves
- Amplitude variation with offset for seafloor multiples
- Ghost amplitude and energy
- Directivity of a source plus its ghost
- Directivity of a harmonic source plus ghost
- Differential moveout between primary and multiple
- Suppressing multiples by NMO differences
- Distinguishing horizontal/vertical discontinuities
- Identification of events
- Traveltime curves for various events
- Reflections/diffractions from refractor terminations
- Refractions and refraction multiples
- Destructive and constructive interference for a wedge
- Dependence of resolvable limit on frequency
- Vertical resolution
- Causes of high-frequency losses
- Ricker wavelet relations
- Improvement of signal/noise ratio by stacking