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Introduction to Spectral Filtering
Spectral Filtering is a method of getting rid of noise and generating a clearer more accurate image of selected data. This method filters data according to increasing or decreasing wavelength, killing data that does not fit in with this pattern. The main objective of the spectral filtering is to remove unwanted frequencies and emphasis more on the signals with certain frequencies. In seismic data processing, it can be used to remove noises of the ambient environment.

Fourier transform and filtering
Suppose that g(t) is a response of an experiment, then the Fourier transform casts it into the frequency domain. Fourier Transform (FT) is defined by

$$g(\omega)=\int_{-\infin}^{\infin}g(t)e^{i\omega t}dt$$

where $$ i=\sqrt{-1}$$ is the imaginary unit, $$ \omega $$ is angular frequency, and $$ e^{i\omega t} $$ is the kernel of FT. A basic property of the FT is that the time span of a time series is inversely proportional to its spectral bandwidth. Other Properties of the Fourier transform can be seen elsewhere.

Selected parts of the frequency spectrum can easily undergo piecewise mathematical manipulations (attenuated or completely removed). These manipulations result into a "filtered" spectrum. By applying inverse FT to the filtered frequency spectrum, the modified signal or "filtered" signal can be obtained. Therefore, signal smoothing can be easily performed with removing completely the frequency components from a certain frequency and up, while the useful (information bearing) low frequency components are retained .

Different types of Spectral Filtering
Spectral filtering could be of the forms of band-pass, band-reject, high-pass, and low-pass filters . Below, each filter is represented.

high-pass filtering passes through the high frequencies and cuts the low frequencies.



Low-pass filter that allows the low frequencies components to pass through, but cuts off the high spatial frequencies.

Band-pass filleting is one of the mostly used types of filters since seismic data usually have low-frequency noise (such as ground roll) and also high frequency noises of the ambient environment. Purpose of the band-pass filter in to pass through certain bandwidth with little or no modifications and largely suppress the remaining part of the spectrum.



Inverse Fourier transfom of the filtered data in frequency domain introduces noise if the end abounds of the domain are sharp. Tapering the filter can reduce this noise.



Band-pass vs. Tapered Band-pass
FT of step-function and tapered step-fucntion are presented here as examples.

Step-function in time domain is:



FT of step-function in frequency domain is:



Tapered step-function in time domain is:



FT of tapered-step-function in frequency domain is:



Filtering Example
A function (exponential) is plotted in figure below

Figure below shows the inverse FT of the top figure.



Figure below shows the inverse FT of the filtered function. A unit step function was used to do the filtering in the frequency domain.



As we can see, filtering has resulted in artificial noises.

Results and discussions
SuUnix was used on a set of real data. Data consist only of S-waves. Frequency spectrum of the frequencies generates from FT of the data is plotted in figure below:

Frequency spectrum of the individual traces in '.su' files can be plotted using the following command,



Following script is used for the filtering. 'Sufilter' is the standard command in SuUnix for filtering. 'f' determines the array of filter frequencies (Hz)and 'amps' defines the array of filter amplitudes.

In the below figures, array of frequencies of the tapered band-pass filter is changed from f=3,6,100,180 to f= 3, 30,100,180. Output results of the filtering (using gain control wagc=0.6) using two different tapered band-pass filters are displayed below.