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Geophysics #### Analog vs. Digital Signal in seismic

Most of the processing operations are now carried out on digital computers due to availability of powerful digital workstations,however if analog filters are used in the field recording systems ,a phase shift is introduced by analog filters.This phase shift should be removed by processing.

Digital seismic signal is discretely represented seismic signal.As seismic signal is a continuous time signal,Representing continues time seismic signal discretely introduces error in data.For analog to digital conversion,Analog signal is sampled at regular intervals and values of the analog signal in that  particular part of waveform is represented by a discrete value.Accuracy of representing the data is controlled by 1.Sampling frequency as defined by Nyquist-Shannon sampling theorem,sampling frequency controls spatial accuracy of digital signal and 2.Number of bits used to obtain digital signal ,Number of bits define how accurately signal magnitude is represented.

For analog to digital conversion , the analog signal is sampled at a fixed rate in time, called the sampling interval (or the sampling rate). Typical values of sampling interval ranges from 1 to 4 ms for most of reflection seismic work. Some of the high-resolution studies require sampling intervals as small as 0.25 milliseconds. Discrete time function is called a time series.Insufficient sampling leads to loss of information.

Nyquist-Shannon sampling theorem is a fundamental theorem to convert  continuous time analog signals into discrete-time digital signals without loosing any information. It establishes a condition for sampling rate to make sure no information is lost while converting analog to digital signal.The theorem states that, the sampling frequency must be greater than twice the highest frequency of the input signal.This will make sure proper reconstruction of analog signal into digital signal. For the given the sampling interval Δt, Nyquist frequency is the highest frequency that can be restored accurately ,it is half the sampling frequency  mathematically defined as FNyq = 1/2Δt.

Nyquist–Shannon sampling theorem got its name after the name of scientist Harry Nyquist and Claude Shannon.

Aliasing: It is a phenomenon in signal sampling and in design of experiments, when several continuous signals that are different become indistinguishable (or aliases of one another) when sampled. Consequently the signal cannot be uniquely reconstructed from the sampled signal. The term is used often in Digital Imaging, where anti-aliasing is used to limit clipping artifacts around contrasted edges (like in screen-fonts) and in poorly sampled raster images. Anti-aliasing in digital signal processing is the technique of minimizing aliasing when representing a high-resolution signal at a lower resolution. Therefore, an analog low-pass filter is typically applied before sampling to ensure that no components with frequencies greater than half the sample frequency remain. Figure 2.7:-Two different sinusoids that fit the same set of samples.one waveform is sampled at frequency above half the sampling rate