# Frequency

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

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 4 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

Why is the use of sines and cosines of the time variable *t* relevant to the theory of digital filtering? The reason is that sines and cosines let us define frequency, and the concept of frequency is basic to what we mean by a filter, be it digital or analog. Let us think about a rotating wheel, as on an old-fashioned wagon or stagecoach. Consider, for example, a wheel that is rotating at a rate of 10 times a second. If we consider one spoke on this wheel to be a reference vector, this vector makes 10 complete rotations in every second. Each rotation through (which is radians) represents one cycle, so we say that the vector has a *cyclic frequency* of 10 cycles per second. The word *hertz* (Hz) stands for cycles per second, so alternatively, we say that the vector has a frequency of 10 Hz. Up to this point, we have not specified in which direction the vector is rotating. As a matter of mathematical convention, we say that the vector has a frequency of +10 Hz if it is rotating in the counterclockwise direction, whereas it has a frequency of –10 Hz if it is rotating in the clockwise direction. The frequency customarily is denoted by the symbol *f*.

We talk about negative frequencies, but do we encounter them in seismic work? In the seismic case, only real-valued signals are recorded. Therefore, a positive frequency cannot be distinguished from its negative counterpart. For example, a frequency of 10 Hz is indistinguishable from a frequency of –10 Hz. Hence, only positive frequencies need be considered in ordinary circumstances.

*Period* is a general term used to denote an interval of time, but the word *period* has special meaning in frequency analysis. By the period of a repeating process, we mean the shortest time interval for which the process exactly repeats itself. In our example of the rotating vector, the motion exactly repeats itself 10 times a second, so we say that it has a period . What does this have to do with the subject of frequency?

The period *T* and the frequency *f* are reciprocals of each other, that is, and . As we have seen, hertz is a measure of cyclic frequency, that is, a measure of the number of cycles per second. However, there is yet another type of frequency: the angular frequency , which is measured in radians per second. Angular frequency is related to cyclic frequency *f *by the fundamental relation

**(**)

In general, the word *frequency* can be used for either or *f*, so one often must determine from the context which frequency is meant. Usually, if the word *frequency* occurs by itself, cyclic frequency in units of hertz is implied. The factor in the above formula comes from the fact that there are radians (or ) in each complete rotation.

## Continue reading

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Digitization | Sinusoidal motion |

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Visualization | Filtering |

## Also in this chapter

- Time series
- The wavelet
- Digitization
- Sinusoidal motion
- Aliasing
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises