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Essay: Effect of Trace spacing /FT on F-K Spectrum Analysis

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Abstract

In this paper we represent the effect of Trace spacing /FT on F-K Spectrum Analysis, the input data were used, 2d crooked line. We tested the effect of Trace spacing /FT (0, 0.5, 25and 50) m for 2d land data, when the value of Trace spacing /FT was smaller the F max was bigger. We made comparison of input and output to see the difference made on output after applying Pie Rejection.

 

Key word: Trace spacing; F-K Spectrum, 2d land data; Fourier transform

 

INTRODUCTION

The 2D data were used called Benjamin Creek data (.also called the Husky Canadian Foothills data) was shot in the foothills of the Canadian Rockies. Husky Oil and Talisman Energy who have released it for aiding general industry technological development own the data set. There are a total of 39,763 traces comprising 141 shots of approximately 300 traces or less in each shot. The sample rate is 4 milliseconds and there are 3 seconds of data. The surface has considerable topography (elevations) and the structure while complex is predominantly 2D.

Two-dimensional Fourier transform over time and space is called an F-K (or K-F) transform where F is the frequency (Fourier transform over time) and K refers to wave number (Fourier transform over space). The space dimension is controlled by the trace spacing and (just like when sampling a time series) must be sampled according to the Nyquist criterion to avoid spatial aliasing. F-K domain there is a two-dimensional amplitude and phase spectrum but usually only the former is displayed for clarity with color intensity used to show the amplitudes of the data at different frequency and wave-number components. Several noise types such as ground roll or seismic interference may be more readily separated in the FK amplitude domain than the time-space domain and therefore will be easier to mute before the inverse transform is applied.

Spatial aliasing is a common problem to be considered when performing data processing. One way to limit the spatial aliasing of the previous figure would be to remove frequencies above 30Hz. This would be wasteful of primary signal. The trace spacing is also important. Consider the adjacent

 

The Fourier analysis is employed to obtain the spectra amplitude and phase. In general, for an arbitrary zero mean data set, the data set in terms of ‘n’ Fourier coefficients can be expressed as:

 

F (t) = S (An*cos (w t) + Bn*sin (w t)) (1)

 

Where t = either a spatial or time series

An = real Fourier coefficients

Bn = imaginary Fourier coefficients

w = 2 p f where f is the nth frequency component

The spectra amplitude (Cn) is defined as: Cn = sort (An*An + Bn*Bn) and, the phase (f) is defined as:

F= arctan (Bn/An) (2)

Calculates the F-K spectrum of the data in the analysis window. The F-K Analysis Window display shown (Fig (1).

Theory, work method and results

The formula for determining the maximum frequency, which can be handled without spatial aliasing, is given by:

(3)

 

Fig (1) Input and F-K Spectrum with Pie Rejection defined] raw data sorted in shot order 2d crooked line.

 

Effect of trace space

 

Now we will stuty the effect of trace space on our output data ,first we will apply by trace space eqauly zero then change it to be (0.5,25,50) its only random chosed . Fig (2) (a) Input Data Example with, Pie Rejection Applied, (b) with tracing space =0, raw, (c) tracing space =0.5, (d) tracing space 25,( e) tracing space 50 ,data sorted in shot order 2d crooked line.

In Fig (3), (a) Difference between Input and Output, Output F-K Filter Display Pie Rejection Example, raw data sorted in shot order 2d crooked line, (b) Difference between Input and Output with tracing space =0.5 raw data sorted in shot order 2d crooked line, (c) Difference between Input and Output with tracing space =25m raw data sorted in shot order 2d crooked line.

Smoother trace, the value of parameters shown in table (1) .For each value of parameters trace spacing there is value of smoother trace by order.

 

 

 

Input raw data

Trace spacing /FT Smoother trace

2DLand

original 0 7

1 0.5 m 10

2 25m 50

3 50m 100

Tab (1).input F-K Filter Design Parameters with apply AGC window 250 ms.

 

 

 

 

(a) Input data (b) Tracing space =0

 

 

 

 

(c ) Tracing spcae 0.5 (d) traceing space 25

 

 

(e) Tracing space 50

 

Fig (2) (a) Input Data Example with, Pie Rejection Applied, (b) with tracing space =0, raw, (c) tracing space =0.5, (d) tracing space 25,( e) tracing space 50 ,data sorted in shot order 2d crooked line.

 

Here we made comparison of input and output to see the difference made on output after applying Pie Rejection. Fig (3), (a) Difference between Input and Output, Output F-K Filter Display Pie Rejection Example, raw data sorted in shot order 2d crooked line, (b) Difference between Input and Output with tracing space =0.5 raw data sorted in shot order 2d crooked line,( c) Difference between Input and Output with tracing space =25m raw data sorted in shot order 2d crooked line., Example with Pie Rejection Applied, (b) with Trace spacing /FT =0, (c) Trace spacing /FT = 0.5, (d) Trace spacing /FT =25 , (e) Trace spacing /FT =50, raw data sorted in short order.

 

 

 

 

(a)

 

 

(b)

 

 

(c)

 

Fig (3), (a) Difference between Input and Output, Output F-K Filter Display Pie Rejection Example, raw data sorted in shot order 2d crooked line, (b) Difference between Input and Output with tracing space =0.5 raw data sorted in shot order 2d crooked line, (c) Difference between Input and Output with tracing space =25m raw data sorted in shot order 2d crooked line.

 

 

CONCLUSIONS

We tested the effect of Trace spacing /FT (0,0.5,25and 50)m for 2d land data ,when the value of Trace spacing /FT was smaller the F max was bigger , we made comparison of input and output to see the difference made on output after applying Pie Rejection. Fig (3), (a) Difference between Input and Output, Output F-K Filter Display Pie Rejection Example, raw data sorted in shot order 2d crooked line.

 

Acknowledgements

I would like to thank CREWES Project – University of Calgary for giving me permission to use their data.

REFERENCES

[1] [Bracewell, 1978] Bracewell, R. N. (1978). The Fourier transform and its applications.

McGraw-Hill Book Company.

[Bracewell, 1978] Bracewell, R. N. (1978). The Fourier transform and its applications.

McGraw-Hill Book Company.

[2] Claerbout, J. F. (1976), Fundamentals of geophysical data processing. McGraw-Hill Book.

[3] Dobrin, M. B. (1976), Introduction to geophysical prospecting, 3rd Ed.

[4] Egbai, J. C. and Ekpekpo, A. (2003), Journal of NAMP. Vol. 7.

[5] Pieuchot, 1984 Pieuchot, M. (1984). Seismic instrumentation (Handbook of Geophysical

Exploration, section I: Seismic Exploration). Geophysical Press, Amsterdam. ISBN 0-

946631-02-6.

[6]Robinson and Treitel, 1980] Robinson, E. A. and Treitel, S. (1980). Geophysical signal

Analysis. Prentice Hall, Inc.

[7] Yilmaz, 1987] Yilmaz, O. (1987). Seismic data processing. Society of Exploration Geophysicists.

[8]Yilmaz, O. (1988), Seismic data processing Investigations in Geophysics.

 

 

 

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