# Sampling geophysical data

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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

Seismic data are recorded digitally and then processed on computers (Ulrych et al., 1999[1]). Digital recording requires that we sample continuous analog signals at discrete time points, quantize the samples in distinct amplitude ranges, and encode the results. After the signals have been processed, they must be restored to a form that can be displayed on various output devices.

Recording discrete data samples is carried out as follows: The output signal is sampled with a device that connects at each sampling instant a sample-and-hold amplifier to the amplifier output terminals. The sampling instant itself can be of the order of less than a microsecond. Next, the voltage stored on the sample-and-hold amplifier is compared with a reference voltage in an analog-to-digital converter. This reference voltage changes in a stepwise manner. As a result, the specific voltage interval, or quantum, within which the sample voltage falls can be determined and transmitted in digital form to a recording system. This process is repeated at each discrete time point — say, every 0.004 s (i.e., every 4 ms of data). At intermediate times, the value of the signal voltage is ignored. Of course, representation of a continuous signal by discrete data is an approximation. The closeness of the approximation depends on the sampling time interval as well as on the size of the quantum.

Next, we look at the process of reconstructing a continuous signal from sampled data. An antialias filter is applied before the sampling process — it should remove the high-frequency components that the sampling process otherwise would alias. That is, an antialias filter effectively should remove all frequencies above the Nyquist frequency. As an example, let us suppose that the sampling interval is 0.004 s. The Nyquist frequency is 125 Hz for this time interval. The original continuous signal must be low-pass-filtered by an antialias filter that effectively destroys all harmonic components above 125 Hz. In this way, these high frequencies will not appear as aliases in the Nyquist range. The basic theorem on sampling theory then states that the original filtered signal can be reconstructed from the sampled data.

In yet another method of signal reconstruction carried out by interpolation, each sample point is replaced by an interpolation function whose peak equals the sample’s height. The form of the interpolation function is the “sin x over x” function, often called the sinc function:

 {\displaystyle {\begin{aligned}{\text{sinc}}\;x\;{\text{ = }}\;{\frac {\sin \;\pi x}{\pi x}}.\end{aligned}}} (18)

The reconstructed signal is found by adding the contribution of all the interpolation functions. Perfect reconstruction never can be achieved completely. The reason is that each interpolation function is of infinite length, so that the reconstructed signal is made up of contributions from an infinite number of sample points, even those occurring very much earlier or very much later in time. However, this type of interpolation can be performed to any reasonable degree of accuracy. In practice, the errors related to the quantizing steps remain, and thus there is a limit to the meaningful refinement of the interpolation process.

In the example, our signal was low-pass-filtered with a cutoff frequency of 125 Hz, and we chose a sampling interval of 0.004 s. That is, we chose our sampling rate (250 samples/s) as twice the cutoff frequency (125 cycles/s). In other words, there are only two samples per cycle at the cutoff frequency. This sampling rate is the lowest allowable for the sampling theorem to be applicable. Any lower sampling rate, for example, 125 samples/s, or a sampling interval of 0.008 s, would not be adequate for signal construction as required by the sampling theorem.

We can look at this problem from yet another viewpoint. Consider a 125-Hz sine wave sampled 125 times per second. There will be one sample per cycle. Depending on the time instant at which the first sample occurs, the first sample can have any value between –1 and +1 (i.e., the minimum and maximum values of the sine wave). Because the sampling rate is the same as the frequency of the sine wave, the sampling is precisely in step with the sine wave, and all other samples will have the same value as the first sample. Thus, the sampled data are flat and represent a zero-frequency signal. The sampling process with this rate of sampling has caused a 125-Hz signal to appear to be a 0-Hz signal.

As we mentioned above, this phenomenon is called aliasing. Recall that for a given sampling rate, there exists the so-called Nyquist frequency. The Nyquist frequency is defined as one-half the sampling rate. In our case, the sampling interval is 0.004 s, so the sampling rate ${\displaystyle f_{s}}$ is 1/0.004 = 250 samples/s. Hence, the Nyquist frequency is 250/2 = 125 Hz. Any signal above the Nyquist frequency takes on a disguise, or alias, because of the sampling. For example, a sine wave of 250 Hz has the alias ${\displaystyle f_{a}{\ =\ }f-f_{s}{\ =\ }{\rm {250}}-{\rm {250=0\ Hz}}}$ in the sampled signal.

As another example, a sine wave of 125 Hz has the alias of 125 – 250 = –125 Hz in the sampled signal. Because we are dealing with a real-valued signal, the negative frequency –125 Hz is the same as the positive frequency +125 Hz. Thus the Nyquist frequency does not alias, as we would expect, because it lies in the Nyquist range. A sine wave of 135 Hz has the alias of 135 –250 = –115 Hz in the sample signal. Because we are dealing with a real-valued signal, the frequency –115 Hz is the same as the frequency +115 Hz. Thus, the frequency +135 Hz aliases to the frequency 115 Hz, which we see lies in the Nyquist range.

## References

1. Ulrych, T. J., M. D. Sacchi, and J. M. Graul, 1999, Signal and noise separation: Art and science: Geophysics, 64, 1648-1672.