Geophysical signal processing

Geophysical signal processing is a method that through the use of computers aims to manipulate the acquired (raw) signal through the application of filters, algorithms, and transforms to make the wanted signal clearer in both the time and frequency domains. The two main goals of geophysical signal processing are: improvement of the signal-to-noise ratio, and results representation in a convenient manner to facilitate geological and geophysical interpretation.

Figure 1- Geophysical signal processing general workflow

Discretization of the signal

The workflow of the geophysical signal processing varies depending on the type of data that we are working with. However, since the processing of the signals is done in a digital computer the discretization of the signal (waveform) is the first step. Discretization of the signal is the process in which we convert analog data to digital data. In this case, the analog data is a continuous signal with an infinite number of samples, and the digital data is a discrete signal with finite number of samples. For a signal to be discretize, it should be a band-limited signal which means that the Fourier Transform is applied to the signal, the amplitude spectrum of the signal is non-zero for a specific range of frequencies. The discretize samples are equal intervals of time where each interval is represented by a single amplitude measurement. The discretization of the signal facilitates and makes the processing of the signal more efficient and effective.

Figure 2: Discretization of the signal. The top panel shows the continuous "analog" signal of a sine wave. The bottom panel show the discretized signal "digital" for the same sine wave.

Sampling Theory

Sampling theorem behaves as a bridge between continuous signal and discrete signals. The theory let us know what information we can recover from the discrete data. However, sampling operation is generally not invertible. The sampling theorem states that “A continuous time signal can be represented in samples and can be exactly reconstructed when sampling frequency (f_s) is greater than or equal to twice the maximum frequency component (f_m) present in the signal” (Mallat, 1999).

Important Parameters

N = number of total samples

dt = time-sampling interval

T = sampling rate

f_s= sampling frequency

f_m= maximun frequency component

f_n = Nyquist frequency

df = frequency-sampling interval

Re-sampling: over sampling and under sampling

This happens when there is either a greater number of samples (over sampling) or a smaller number of samples (under sampling) in the discrete signal. In under sampling, dt becomes larger whereas in over sampling dt becomes smaller. Over sampling increases the amount of memory and time required to complete the processing. In the other hand, down sampling can represent a significant memory save at the time of processing. Down sampling behaves as a low pass filter, it eliminates random data. It is possible that a lot of valuable information may be lost at the of re-sampling. However, the major pitfall of down sampling is that it can cause aliasing.

Figure 3: Resampling - oversampling, perfect sampling and under sampling.

Aliasing

Aliasing is the overlapping of two or more signals, it occurs as a consequence of under sampling. Aliasing can also mean the contamination of the signal by frequencies beyond the Nyquist frequency. Aliasing could be avoided by applying a band pass filter before the resampling of the signal.

Correlation

Correlation is an operation in geophysical signal processing to measure the similarities between two signals. The convolution is given by the equation given in figure #. The amount of correlation between two signals is given by the correlation coefficient, which varies between -1 and 1. The correlation coefficient is given by the equation in figure #. If the correlation coefficient is 1, it means that the signals are the same, if the correlation coefficient is -1 it means that the two signals have reverse polarity, and if the correlation coefficient is 0 it means that there is no correlation at all. There are two types of correlation: Auto-correlation and Cross-correlation.

Figure 4 - Correlation equation. The top equation works for the analog signal and the bottom equation works for the discretize signal.

Auto-correlation

Auto-correlation is defined as the correlation of the signal with itself. The correlation is calculated by the multiplication of the signal with its conjugate. It is useful to understand the behavior of the signal. Autocorrelation is symmetric

Cross-correlation

Cross-correlation is the correlation between two different signals. Cross-correlation is not symmetric.

Convolution

Convolution is an operation in geophysical signal processing to analyze the relation between the input and output of a LTI (Linear and time invariant) system. The word convolution means folding, where the input signal is convolved with the impulse response to generate and output (Robison et al., 1986). There are two main types of convolution: Continuous convolution and Discrete convolution. 

Figure 5 - Convolution Equation. The top equation works for the analog signal and the bottom equation works for the discretize signal.

Continuous convolution

It is given by the equation of figure #. The convolution between the input and output are in the time domain, and in an analog signal.

Discrete convolution

The convolution between the input and output in a discrete signal. The discrete convolution is given by equation in figure #.

Deconvolution

Deconvolution it is used to reverse the process of convolution on a signal. It represents the inverse filtering operation performed on the output of the LTI system to obtain information about the input. The word deconvolution means unfolding (Robison and Durrani, 1986). The equation for deconvolution is given by the equation in figure #.

Figure 6 - Deconvolution Equation. The top equation works for the analog signal and the bottom equation works for the discretize signal.

Discrete Time Filters

Filters are used to remove incoherent signals like noise and other undesired signals in the frequency range of interest. Some of the most common filters are: low-pass filter, high-pass filter and band-pass filter.

Low-pass filter

When a low-pass filter is applied only frequencies below the cutoff frequency remain (passband) and the frequencies above the cutoff frequency are eliminated or attenuated (stopband). By the removing of the frequencies, the low pass filter generates a smoothing effect. The low pass filtering technique is often used to remove noise, clean up signals, and perform data averaging. The most common designs of low-pass filters are the moving average filter, Butterworth filter and the Chebyshev filter.

High-pass filter

When a high-pass filter is applied signals below the cutoff frequency (stopband) are eliminated or attenuated and signal above the cutoff frequency (passband) are kept. The high-pass filters technique is often used to clean up low-frequency noise, and to highly the high-frequency trends. The most common designs of low pass filter are the Butterworth filter and the Chebyshev filter.

Bandpass filter

When a band-pass filter is applied, only frequencies within the bandwidth are maintain without distortion of the input signal. The bandwidth of the filter is defined as the difference between the higher cut off frequency and the lower cut off frequency. The band-pass filter is considered as a second order filter and it is usually used for noise cancelation.

Filter applications

Filters are used in many fields and task, the following are some of the most common tasks: Imaging (Seismic, EM, InSAR, GPR, Earthquake Sources, etc.), Similarity Search, Inversion, Sensor response correction, and Improvement of resolution.

References

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Mallat, Stéphane. A wavelet tour of signal processing. Elsevier, 1999.