# Delay: Minimum, mixed, and maximum

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 7 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

What is a midpoint reverse of a causal finite-length wavelet? The time indices of the causal finite-length wavelet

 {\displaystyle {\begin{aligned}&b=\left(b_{0}{,\ }b_{1}{,\ }b_{2}{,\ }\dots {,\ }b_{N}\right)\end{aligned}}} (21)

run from n = 0 to n = N. To obtain the midpoint reverse, the midpoint (0 + N)/2 = N/2 is used as the point of reflection. Thus, the midpoint reverse is

 {\displaystyle {\begin{aligned}&b^{R}=\left\{b_{N}^{*},{\dots ,}b_{2}^{*}{,\ }b_{1}^{*}{,\ }b_{0}^{\star }\right\},\end{aligned}}} (22)

where now the coefficient ${\displaystyle b_{N}^{*}}$ occurs at time index 0 and the coefficient ${\displaystyle b_{0}^{*}}$ occurs at time index N. Thus, the midpoint reverse of a causal wavelet is also causal.

Unless otherwise stated, we assume that a wavelet is causal and that its reverse is its midpoint reverse, which also is causal. For example, the reverse of the wavelet (1, 0, 3) is (3, 0, 1). As another example, the reverse of the wavelet (i, 0.5) is (0.5, – i) because ${\displaystyle i^{*}=-i}$. We also see that the reverse of the reverse of a wavelet is the given wavelet.

What is minimum delay, mixed delay, and maximum delay? The concepts of minimum delay and mixed delay apply only to causal wavelets. The concept of maximum delay applies only to finite-length causal wavelets. A minimum-delay wavelet has its energy concentrated near its arrival time, as opposed to a mixed-delay wavelet, which has its energy distributed away from its arrival time. A maximum-delay wavelet is the reverse of a finite-length minimum-delay wavelet.

The concept of minimum delay is developed here for systems operating in discrete time - that is, systems in which the variables appear as a sequence of numbers at discrete, equally spaced time instants. By contrast, systems operating in continuous time have variables that are functions of continuous time - that is, the values of their variables are given at all instants of time. See Robinson (1962[1]) for a development of the concept of minimum delay for continuous-time systems.

## References

1. Robinson, E. A., 1962, Random wavelets and cybernetic systems: Charles Griffin and Co.