It is obvious that when you have a large mass (population) and want to get the idea about the whole population then you need to select a part of it for representation of the whole. This part is simply known as sample. It means that you must do sampling. The term sampling is described in various statistics textbooks.
From the same context in seismic geophysics, signal sampling is very important in digital signal processing and interpretation processes, since it is a good approach to deal with the interested part of the signal rather than approaching the recorded signal as a whole at once.
And the fact that most of recorded seismic signals are in continuous - time waveform in seismogram see (figure 1). Then it is important to convert a continuous -time waveform into a discrete form with the help of samples so as to extract maximum information from it.
N.B: However this should be done carefully in deciding which sampling rate to use. Since if we sample high frequency components using low sampling rate , Aliasing (See Figure 2) can be resulted and if we sample by using high sampling rate the more storage capacity would be used.
Figure 1: A simple continuous time waveform
Therefore It was so challenging to sample a continuous time signal without either distorting its quality or forming Alias until Harry Nyquist, (known as the Pioneer of sampling) contributed to his theory for determining the optimum signal sampling rate.
Aliasing
Figure 2, show the process of sampling two different signals (in yellow). Both signals are sampled with the same sampling frequency at points in red. The top signal is oversampled, i.e., its frequency is lower than half the sampling frequency, so we have more than two samples per period of this sinusoid and in this case the original signal can be perfectly reconstructed. The bottom signal, however, is undersampled. We have less than two samples per period of this sinusoid and when we try to reconstruct the signal (blue line), we are not reconstructing the original signal, but rather a much lower frequency. This effect is called Aliasing. If we are undersampled, the frequencies that are higher than the Nyquist frequency are reconstructed at lower frequencies and will add noise to the actual signal at those lower frequencies.
Nyquist Sampling Theorem
The statement of Nyquist Theorem.
The Nyquist Sampling Theorem states that:
" A bandlimited continuous-time signal can be sampled and perfectly reconstructed from its samples if the waveform is sampled over twice as fast as its highest frequency component".
In seismic geophysical signal processing, the term Nyquist Sampling Theorem is often used to describe what amount of sampling rate is required to sample a continuous - time acquired seismic signal in order to reconstruct the signal with no loss of the information it carries.
Nyquist Sampling Formula
In a Layman term, Nyquist sampling theorem is that the frequency of sampled signal must be greater or equal to twice the maximum of the frequency of the original continuous - time signal.
Such as, fs > 2fmax,
Where fs = Frequency of sampled signal (sampling rate), fmax = Maximum Frequency of the continuous - time signal.
This means that if we know the maximum frequency of the continuous - time signal accurately, Let say 33050 Hz, then according to Nyquist theorem, the optimum sampling rating of this signal should be greater or equal to 66100 Hz.
fs > 2fmax, then fs > 2 × 33050 Hz = 66100 Hz.
Key terms related to Nyquist Theorem.
> Nyquist Frequency (Folding frequency): Is defined as the maximum frequency beyond which Aliasing will occur.
Nyquist frequency = 0.5 × fs = 0.5 × 66100 = 33050 Hz
> Nyquist Rate: Is the minimum sampling rate at which the continuous signal can be accurately reconstructed.
Nyquist Rate = 2B = 2 × 33050 = 66100 Hz.
Where, B = bandwidth (highest frequency) of the signal.
> Sampling Rate: Number of samples per unit time (second). It can be obtained as a reciprocal of sample period (Ts). Sampling rate (fs) = 1/Ts
Sampling rate (fs) = 66100 Hz
Relationship between Sampling Rate and Nyquist frequency
The figure 3 below demonstrates how the relationship between the sampling rate and the underlying frequency can confuse this measurement. The underlying signal (red line) remain the same, but when the rate of observations (blue points) increased throughout the animation. The result is that the peak in the Fourier transform (bottom panel) moves around bouncing off the walls before settling down once the signal begins to be sampled at least twice per cycle. This limit is called the Nyquist frequency, and it is equal to half the sampling rate. Signals with intrinsic frequencies below the Nyquist frequency will show up at their accurate locations in the sub-Nyquist part of a Fourier transform, while higher-frequency signals will not.
Figure 3: Relationship between Sampling rate and underlying frequency.
The challenge.!
The challenge raised due to the fact that the real continuous time signals are complex curves in nature with maximum frequency that is not always known accurately.
However, through Fourier Transform a mixture of sine curves components of any complex continuous time waveform can be splitted into Frequency Spectrum and then analyzed into its frequency components.
In order to gain more insight about Nyquist sampling theorem, Let take an example the frequency spectrum of our input complex curve have three sinusoids curves components with frequencies, 25 Hz, 75 Hz, and 150 Hz for f1, f2, f3 respectively such as f1 has lowest frequency while f3 has highest frequency, with f2 is at intermediate between f1 and f3 such as f1 < f2 < f3.
According to the Nyquist sampling rule, f1 requires at least 50 Hz, samples per second (sampling rate), while f2 require at least 150 Hz and the same in f3 require 300 Hz. But because f3 component is greater that f1 and f2, then using 300 Hz as sampling rate would be an optimum sampling rate that can sample this complex signal while preserving its quality such as without (with minimal) distortion. And this is the Basic concept regarding Nyquist sampling theorem.
Always the output samples will differ in time intervals known as Sampling intervals (sample period, Ts). If we reciprocate the sample period (1/Ts) we get Sampling Rate (sampling frequency).which is equal to the number of samples per second.
Nyquist Sampling theorem gives us the idea about what optimum sampling rate to use in order to get a good representation of the signal without (with minimal) distortion. So don't breach this rule when you are doing seismic signal processing.
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References
Emiel Por, Maaike van Kooten & Vanja Sarkovic (2019): Nyquist–Shannon sampling theorem
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