Temporal Fourier (t f) transformation Spatial Fourier (x k x ) transformation applications f-k x transformation Radon(τ-p x ) transformation

Transcription

1 Temporal Fourier (t f) transformation Spatial Fourier (x k x ) transformation applications f-k x transformation Radon(τ-p x ) transformation Linear Radon transform Parabolic Radon transform

2 t domain Time domain Transformation domains 1-D-Transformation Fourier transformation f domain Frequency domain t-x domain Time-Place t-x domain Time-Place 2-D-Transformation 2-D-Fourier τ-p-transformation f-k x domain Frequency-Wavenumber τ-p x domain (Intercept - slowness)

3 Temporal Fourier transformation Fourier Transformation: ( ) ( ) i 2πft G f = g t e dt Inverse Fourier () ( ) i2πft g t = G f e df Transformation: Sampling will preserve all frequencies up to the Nyquist frequency: f N =1/(2 t)

4 frequency amplitude phase f 1 f 1 f 2 f 2 Sum: Sum:

5 Spatial Fourier transformation Fourier Transformation: ( ) ( ) i2πk xx G k f = g x, f e dx x, Inverse Fourier Transformation: ( ) ( ) i2πk xx g x, f = G k, f e dk x Spatial Fourier transformation is discussed for one horizontal (x) direction, but can be carried out in the two horizontal directions.

6 Temporal versus Spatial Fourier transformation Temporal Fourier transformation Sampling interval t sampling rate (sampling frequency) 1/ t Sampling will preserve all frequencies up to the Nyquist frequency: f N =1/(2 t) Spatial Fourier transformation Spatial sampling interval x Spatial sampling rate (sampling frequency) 1/ x Sampling will preserve all frequencies up to the Nyquist frequency: k N =1/(2 x)

7 The phase velocity which a wavefront appears to have along a line of geophones vapp λapp = v = v f α vapp λapp = = Apparent velocity: vapp = v sinα Apparent wavelength: λapp = vapp f

8 Apparent wavenumber k app Number of waves per unit distance perpendicular to a wavefront Horizontal Wave v app = v v α=90 Incoming Wave α v app = v / sin α v app app app Wavefront

9 From a practical point of view, subsequent measurements must be carried out in such a way that events on separate traces can be correlated as coming from the same horizon or reflection point in the subsurface (Yilmaz, 1987) For a given frequency component, the time delay between subsequent measurements can be at most half the period (T/2) of that frequency component to enable a correlation of two measured reflections as coming from the same horizon Max time delay: t = x v min app < min T 1 = 2 2 f max Two spatial samples for one apparent wavelength

10 x x x T/2 t 1 α 1 t 2 α 2 t 3 α 3 α 1 < α 2 < α 3 min v app1 T t 1 < 2 min min v app2 > > T t 2 = 2 min min v app3 T t 3 > 2 min v app= t = v sinα x min v app

11 f-k-spectrum f Slope v a f = v a k a k a (apparent wavenumber)

12

13

14 Yilmaz, 1987

15 Yilmaz, 1987

16 Yilmaz, 1987

17 Yilmaz, 1987

18 And its k-f amplitude spectrum Yilmaz, 1987

19 Composite walk-away noise test Rejection ground roll energy Yilmaz, 1987

20 CMP gathers from a shallow marine survey before and after F-k dip filtering to remove coherent noise with corresponding f-k spectra Yilmaz, 1987

21 CMP gathers from a shallow marine survey Before and after f-k dip filtering to remove coherent linear noise Yilmaz, 1987

22 Synthetic CMP gathers containing multiples primaries Water-bottom + = VM velocity multiples multiples VP velocity primaries Yilmaz, 1987

23 NMO correction using primary velocity function Yilmaz, 1987

24 After NMO correction Zero-ing in f-k quadrant Inverse NMO Yilmaz, 1987

25 To suppress aliased energy Yilmaz, 1987

26 VM1= slow (water-bottom) multiples VM2= fast (peg-leg) multiples NMO corrected data using primary velocities Yilmaz, 1987

27 Yilmaz, 1987

28 Velocity filter Suppression of multiples Interpolation of traces Analysis of guided waves

29

30 τ-p transformation for various arrivals

31 (1/p) P1 and P2 are primaries W is water bottom which results in multiples Yilmaz, 1987

32 p(s/km) 1/18 1/1.5 Yilmaz, 1987

33 Reducing source-generated noise in shallow seismic data using linear and hyperbolic τ p transformations Roman Spitzer, Frank Nitsche and Alan G. Green

34 48 receivers 5 m. interval Source location: 5 m from first geophone 3m depth

35 (a) Raw shot gather (b) Time and offset varying gain (c) Spectral balancing ( Hz)

36 Linear τ-p transformation Result of filtering Difference Between (a) And (c)

37 Hyperbolic τ-p transformation Hyperbolic τ-p transformation Inverse hyperbolic τ-p transformation Amplitude of each sample is squared

38 Shot gather along a high-resolution seismic line in northern Switzerland (a) Raw shot gather (b) Time and offset varying gain (c) Spectral balancing ( Hz)

39 Linear τ-p transformation Result of filtering Difference Between (c) And (c)

40 Hyperbolic τ-p transformation Hyperbolic τ-p transformation Inverse hyperbolic τ-p transformation Amplitude of each sample is squared

41 Processing: CMP sorting NMO corrections NMO stretch mute Stacking Reflections were found to extend to shallower depths and more continuous