# Digitization

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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 4 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

A continuous signal — that is, some continuous recording of data versus time — can be converted into a sequence of numbers. Each number represents the reading, or amplitude, of the signal at a specific time instant. The time points chosen are spaced equally so that the time interval between two consecutive readings of the signal is always the same. In seismic work, the value ${\displaystyle \Delta t{\rm {=4}}}$ ms is used frequently (4-ms data). The process of converting a continuous signal into a sequence of numbers at equally spaced time points is called digitization. Analog-to-digital converters automatically digitize the incoming signals when they are recorded in the field as seismic traces. Figure 1a shows a portion of a digitized signal. The dots in the figure represent the amplitudes of the signal at the indicated time points.

Instead of using the time scale appearing on the signal, it is more convenient to first choose a time increment ${\displaystyle \Delta t}$ and then define the time index n selected so that time is given by ${\displaystyle t{\rm {=}}n\Delta t}$. Figure 1b shows the same signal with the new discrete time scale. The time increment is 0.004 s (4 ms) for this example. In this way, the time index n associated with any reading ${\displaystyle x_{n}}$ is a whole number or integer. For convenience, we usually refer to the time index n as simply the discrete time n.

A digital signal comprises the numerical signal readings. For example, the digital signal shown in Figure 1a and 1b is given in Table 1.

When we plot a wavelet, we see its entire life history, from its precursor (in the case of a noncausal wavelet) to the time at which it arrives to the time at which it damps out. Its origin time, or arrival time, is associated with time index 0, which serves as the reference point for the wavelet. At time index 0, we plot the amplitude of the wavelet at its origin. At time index 1, we plot the amplitude of the wavelet when it is one unit old. At time index 2, we plot the amplitude of the wavelet when it is two units old, and so on. For example, suppose that a causal wavelet has amplitude 4 at time index 0, amplitude 2 at time index 1, amplitude 1 at time index 2, and zero amplitude for all succeeding times. Table 2 describes this wavelet.

More concisely, we can summarize this causal wavelet by the numerical sequence {4, 2, 1}, where it is understood that 4 is the initial amplitude (i.e., at time index 0) so that all preceding amplitudes are zero, 2 is the amplitude at time index 1, and 1 is the amplitude at time index 2, or the final amplitude (so that all following amplitudes are zero).

Figure 1.  (a) A digitized signal with the original time scale, (b) The same digitized signal with a time-index scale.

We can write the wavelet as

 {\displaystyle {\begin{aligned}b{\ =\ }\left\{b_{0}{\rm {,\ }}b_{\rm {1}}{\rm {,\ }}b_{\rm {2}}\right\}.\end{aligned}}} (2)

This notation means that wavelet b has amplitude ${\displaystyle b_{0}}$ at time index 0, amplitude ${\displaystyle b_{\rm {1}}}$ at time index 1, and amplitude ${\displaystyle b_{\rm {2}}}$ at time index 2. For example, the relation

 {\displaystyle {\begin{aligned}\left\{b_{0}{\rm {,\ }}b_{\rm {1}}{\rm {\ ,\ }}b_{\rm {2}}\right\}{\rm {\ =\ }}\left\{{\rm {4,\ 2,\ 1}}\right\}\end{aligned}}} (3)

implies that ${\displaystyle b_{0}{\rm {=4,}}}$ ${\displaystyle b_{1}=2}$, ${\displaystyle b_{2}{\rm {=4,}}}$. The amplitudes of the wavelet also are called the coefficients of the wavelet. We call ${\displaystyle b_{0}}$ the coefficient at time index 0, and so on. Because the wavelet ${\displaystyle \left\{b_{0}{\rm {,\ }}b_{\rm {1}}{\rm {\ ,\ }}b_{\rm {2}}\right\}}$ has three coefficients, we say that it has length (or time duration) equal to 3, or that it is a three-length wavelet.

Table 1. Digital signal readings as a function of time or of an index.
Time (in seconds) Time index n Signal reading ${\displaystyle x_{n}}$
0.200 0 ${\displaystyle x_{0}{\rm {=}}{\rm {10}}}$
0.204 1 ${\displaystyle x_{\rm {1}}{\rm {=20}}}$
0.208 2 ${\displaystyle x_{\rm {2}}{\rm {=}}{\rm {10}}}$
0.212 3 ${\displaystyle x_{\rm {3}}{\rm {=0}}}$
0.216 4 ${\displaystyle x_{\rm {4}}{\rm {=}}-{\rm {10}}}$
0.220 5 ${\displaystyle x_{\rm {5}}{\rm {=0}}}$
 Time index 0 1 2 3 4 5 Wavelet 4 2 1 0 0 0