# 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.^{[1]}

## Background

Acoustic signals that are not reflections appear in shot records as noise^{[3]}. 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

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.^{[5]}

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)e^{iΘd(ƒ,x,y)} = S(ƒ,x,y)e^{iΘs(ƒ,x,y)} + G(ƒ,x,y)e^{iΘg(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.^{[6]}

According to the dispersion relation the equation ƒ = vk must be satisfied^{[6]}

## Applications

F-k filtering is used for things like eliminating the aliasing surface-wave energy and maintain the low frequency information of the reflected waves,^{[6]} and attenuating the residual weak energy of ground roll^{[8]}. The main application for F-k filtering is to eliminate coherent noise in seismic data as exemplified by figure 3.^{[9]}

## Issues

Traditional F-k filtering cannot eliminate all of spatial aliasing of surface waves. ^{[6]}

A filter that covers too wide of a range can remove too great an amount of information and make interpreting correctly difficult.^{[3]}

## External Links

Domain Transformations - http://www.xsgeo.com/course/basic.htm

Filtering Techniques - https://gpg.geosci.xyz/content/seismic/seismic_reflection_filtering.html

Fourier Transformation - https://en.wikipedia.org/wiki/Fourier_transform

Dispersion Relationship - https://en.wikipedia.org/wiki/Dispersion_relation

## Sources

- ↑ “BASIC DEFINITIONS.”
*Basic Definitions*, www.xsgeo.com/course/basic.htm. - ↑ Wang, D., & Ling, Y. (2016). Phase-shift- and phase-filtering-based surface-wave suppression method.
*Applied Geophysics,**13*(4), 614-620. - ↑
^{3.0}^{3.1}Seismic processing basics. (n.d.). Retrieved from https://wiki.aapg.org/Seismic_processing_basics#f.E2.80.93k_or_apparent_velocity_filter. - ↑ Wang, D., & Ling, Y. (2016). Phase-shift- and phase-filtering-based surface-wave suppression method.
*Applied Geophysics,**13*(4), 614-620. - ↑ Developers, GeoSci.xyz. “Filtering of Seismic Data¶.”
*Filtering of Seismic Data - GPG 0.0.1 Documentation*, gpg.geosci.xyz/content/seismic/seismic_reflection_filtering.html. - ↑
^{6.0}^{6.1}^{6.2}^{6.3}Wang, D., & Ling, Y. (2016). Phase-shift- and phase-filtering-based surface-wave suppression method.*Applied Geophysics,**13*(4), 614-620. - ↑ Comparison of FK and SVD filtering in the processing of a land seismic data - Scientific Figure on ResearchGate. Available from: https://www.researchgate.net/figure/Seismograms-of-shot-19-a-raw-data-b-after-mute-and-trace-edition-F-k-filter-After_fig1_280521231 [accessed 4 Dec, 2019]
- ↑ Chen, H., Li, X., Qian, Z., & Zhao, G. (2013). Robust adaptive polarization analysis method for eliminating ground roll in 3C land seismics.
*Applied Geophysics,**10*(3), 295-304. - ↑ Herman, M., Syahmi Hashim, H., Abdul Latif, A.H., Ghosh, D.P., 2017. Application of FK Filtering for Coherent Noise Removal in High Frequency Shallow Marine Data. IOP Conference Series: Earth and Environmental Science 88, 012010.