# Introduction - Chapter 15

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

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

Author | Enders A. Robinson and Sven Treitel |

Chapter | 15 |

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

ISBN | 9781560801481 |

Store | SEG Online Store |

The fountains mingle with the river And the rivers with the ocean, The winds of heaven mix forever With a sweet emotion. —Percy Bysshe Shelley

In this chapter, we present the basic mathematical framework for the study of input-output models, along with some simple examples that show various approaches to the description and analysis of those models. The mathematical tools are derived from the general field of operational calculus, so for the most part, we are limited to the consideration of linear systems. Linear digital systems are described by linear difference equations, whereas linear analog systems are described by linear differential equations. Fortunately, many cases occur in engineering and science in which the systems either are linear or can be approximated sufficiently closely by a linear representation.

Linear methods have been applied very successfully to the analysis of geophysical systems and the processing of geophysical data. The input and output of a system are related by a difference equation (digital case) or a differential equation (analog case), the solution of which gives the output for a given input. This equation provides a complete description of the system, but often it must be converted to other forms to be useful.

Other modes of description of the system are related to the outputs produced by special types of inputs. Thus, we have

1) the

impulse responseof the system, which is the output produced by an impulse input2) the

frequency responseof the system, which relates the outputs produced by sinusoidal inputs3) the

transfer function(orsystem function), which is a generalization of the frequency-response function

These modes of description are related to one another, and each offers advantages in different types of applications. Because the characteristics of the entire system are contained in these functions, it is important to understand their properties. The essential properties of a transfer function are revealed by the location of its poles and zeros. If all the poles lie outside the unit circle (in the digital case) or within the left half-plane (in the analog case), the transfer function represents a *stable causal system*. All systems operating in real time are causal. All natural systems existing in long-run time are stable; otherwise, their output would keep increasing until finally it destroyed the system.

Man-made systems and short-run natural systems can be unstable, and finding ways to stabilize them represents an important aspect of systems analysis. If all the zeros as well as the poles lie outside the unit circle (in the digital case) or within the left half-plane (in the analog case), the transfer function represents a special kind of stable causal system called a *minimum-delay* (or *minimum-phase*) *system*. In a nutshell, a minimum-delay system is one that is invertible in real time, and that is a major reason why such systems are important.

In this chapter, we develop the properties of *autoregressive-moving-average* (ARMA) *models*. We then describe prediction systems for the deterministic impulse responses of ARMA models, and we use those prediction systems to illustrate prediction methods in general. This chapter gives a more detailed treatment than the ones found in Robinson and Treitel (1969^{[1]}, 1980^{[2]}).

## References

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## Also in this chapter

- Digital linear time-invariant systems
- Analog linear time-invariant systems
- Digital transfer functions
- Analog transfer functions
- Causality and stability of digital systems
- Causality and stability of analog systems
- Frequency response of a digital system
- Digital prediction
- Digital prediction error
- Analog prediction error