# Analog versus digital signal

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

A seismic signal is a continuous time function. In digital recording, the continuous (analog) seismic signal is sampled at a fixed rate in time, called the sampling interval (or sampling rate). Typical values of sampling intervals range between 1 and 4 ms for most reflection seismic work. High-resolution studies require sampling intervals as small as 0.25 ms.

Figure 1.1-5 shows a continuous signal in time. The discrete samples that might actually be recorded are shown by dots. A discrete time function is called a time series. The bottom curve in Figure 1.1-5 is an attempted reconstruction of the original analog signal, which is shown as the curve on top. Note that the reconstructed signal lacks the details present in the original analog signal. These details correspond to high-frequency components that were lost by sampling. If a smaller sampling interval were chosen, then the reconstructed signal would more accurately represent the original signal. For the extreme case of a zero sampling interval, the continuous signal can be represented exactly.

Is there a measure of the restorable frequency bandwidth of the digitized data? Figure 1.1-6 shows a time series, such as a seismic trace, with a 2-ms sampling interval and the corresponding amplitude spectrum. In general, given the sampling interval Δt, the highest frequency that can be restored accurately is called the Nyquist frequency and is given by

 ${\displaystyle f_{Nyq}={\frac {1}{2\Delta t}}.}$ (1)

For the time series in Figure 1.1-6, Δt = 2 ms; therefore, the Nyquist frequency is 250 Hz. The original time series was resampled to obtain a series with 4- and 8-ms sampling intervals. The corresponding Nyquist frequencies are 125 and 62.5 Hz, respectively. Figure 1.1-6 also shows the series (as reconstructed back to 2 ms for plotting purposes) sampled at 4 and 8 ms with their amplitude spectra. Note that the coarser the sampling interval, the smoother the series. Smoothness results from a loss of high frequencies as seen in the amplitude spectra. Frequency components between 125 and 250 Hz, which are present in the time series with the 2-ms sampling interval, seem to be absent in the series resampled to 4 ms. Likewise, frequency components between 62.5 and 250 Hz seem to be absent from the series resampled to 8 ms. Can these frequencies be recovered? No. Once a continuous signal is digitized, the highest frequency that can be restored accurately is the Nyquist frequency.

We may think that when the time series sampled at 4 or 8 ms is interpolated back to a 2-ms sampling interval, those high frequencies should return. As stated earlier, the time series in Figure 1.1-6 with 4- and 8-ms sampling intervals actually were reconstructed by interpolation back to 2 ms to get the same number of samples as the original series for plotting with the same scale. Interpolation does not recover the frequencies lost by sampling; it only generates extra samples.

The implication for sampling the continuous signal in the field is an important one. If the earth signal had frequencies, say up to 150 Hz, then the 4-ms sampling interval would cause a loss of the band between 125 and 150 Hz.

While maximum recoverable signal frequency is the Nyquist frequency for a 1-D digitized data set, such as a single seismic trace, the situation can be different for two- or more dimensional data. Consider the process of moveout correction and stacking. Given the sampling rate for the stacked trace, say 4 ms, data samples are searched on each input trace in the common-midpoint (CMP) gather based on the hyperbolic moveout equation. Since each input trace also is sampled at regular intervals, say 4 ms, the computed input sample location would normally fall in between two samples on a given input trace. By interpolation, the required sample value can be computed and placed on the output sample location (normal moveout). Such an output-driven process would faithfully preserve frequencies below the Nyquist, only. Consider the alternative process in which a sample in the input trace is placed at the exact time location on the output trace [2]. The resulting output stacked trace from this input-driven process would contain all the samples from all the input traces at exact time locations with irregular intervals. Such random sampling then yields a potential Nyquist frequency greater than the Nyquist frequency associated with the input traces.

## References

1. Rothman, 1981, Larner, K. L., Chambers, R., and Rothman, D., 1981, Trace interpolation and design of 3-D surveys: Presented at the Ann. Eur. Assoc. Expl. Geophys. Mtg.
2. Wisecup, 1998, Wisecup, R. D., Unambiguous signal recovery above the Nyquist using random-sample-interval imaging: Geophysics, 63, 763–771.