# F-k filtering

F-k filtering is when seismic data, that is traditionally in the time and displacement domain, is converted into the frequency and wave number (F-k) domain as seen in figure 1 below. The seismic data is then filtered to remove unwanted frequencies higher and/or lower than the seismic signal band, and then converted back to the time-displacement domain. The process is a two dimensional Fourier transformation and must be sampled according to the Nyquist criterion to avoid aliasing. Figure 1: Surface and reflected waves in the time–space domain before and after phase-shift processing in the FKXKY domain: (a) surface and reflected waves in the time–space domain; (b) phase of the surface and reflected waves in the FKXKY domain; (c) phase of the surface and reflected waves after phase-shift processing in the FKXKY domain

## Background

Acoustic signals that are not reflections appear in shot records as noise. These signals have a constant “apparent velocity” as they travel along the receiver cable. This simple organization allows them to be isolated from the reflection signal and to be removed from the record. To accomplish this you can use the F-k, or pie slice, filter in which the range of apparent velocities can be selected and removed to eliminate this linear noise.

## Process Figure 2: FFTxy( ) denotes the two-dimensional Fourier transform with x, y as variables, eiθd( f, kx, ky) , eiθs( f, kx, ky) , and eiθg( f, kx, ky) denote the two-dimensional Fourier transform of eiθd( f, kx, ky) ,eiθs( f, kx, ky) and eiθg( f, kx, ky) with x, y as variables.

The first step is to take a common shot gather, CMP gather, or CMP stack and perform a 2-D Fourier Transform on it. Then select the amplitudes that are to be removed and zero-out the transform within the reject zone. NextAfter that perform a 2-D inverse Fourier trareturnfo go back to the original spectrum.

Without considering random noise the seismic record is noted as d(t,x,y) = s(t,x,y) + g(t,x,y) where d is the seismic record and is a sum of the reflected wave (s) and the surface wave (g). In the F-k domain this equation becomes D(ƒ,x,y)eiΘd(ƒ,x,y) = S(ƒ,x,y)eiΘs(ƒ,x,y) + G(ƒ,x,y)eiΘg(ƒ,x,y). To obtain the results in the F-k domain a two-dimensional Fourier transformation is done along the x–y plane using the equation in figure 2.

According to the dispersion relation the equation ƒ = vk must be satisfied

## Applications Figure 3: Seismograms of shot 19: (a) raw data; (b) after mute and trace edition. F-k filter After editing and mute of trace amplitudes, commonly in seismic processing flow chart, it recommends some kind of filtering. The first filter used was f-k filter, which is a two- dimensional filter defined in the frequency domain obtained by Double Fourier Transform: 

F-k filtering is used for things like eliminating the aliasing surface-wave energy and maintain the low frequency information of the reflected waves, and attenuating the residual weak energy of ground roll. The main application for F-k filtering is to eliminate coherent noise in seismic data as exemplified by figure 3.

## Issues

Traditional F-k filtering cannot eliminate all of spatial aliasing of surface waves. 

A filter that covers too wide of a range can remove too great an amount of information and make interpreting correctly difficult.