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Research Collection

Educational Material

Reflection seismic 1 script

Author(s):

Kruk, Jan van der Publication Date:

2001

Permanent Link:

https://doi.org/10.3929/ethz-a-004363847

Rights / License:

In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use.

ETH Library

• Temporal Fourier (t Þf) transformation

• Spatial Fourier (x Þk_x) transformation applications

Þ f-k_x transformation

• Radon (τ-p_x) transformation

– Linear Radon transform – Parabolic Radon transform

t domain f domain

Time domain Frequency domain

Fourier transformation

t-x domain f-k_x domain

Time-Place Frequency-Wavenumber

τ-p_x domain

(Intercept - slowness)

1-D-Transformation

2-D-Transformation

Time-Place

t-x domain

2-D-Fourier

τ-p-Transformation

Transformation domains

Temporal Fourier transformation

Fourier Transformation: ^G( )^{f ∞ g}( )^{t e−i2πftdt}

∞

−ò

=

Inverse Fourier

Transformation: ^g( )^{t =} ₋^∞ò_∞^G( )^{f ei2πftdf}

f_N=1/(2 ∆t)

Sampling will preserve all frequencies up to the Nyquist frequency:

• frequency

• amplitude

• phase f₁

f₂

Sum:

f₁

f₂

Sum:

Spatial Fourier transformation

Fourier Transformation: ^G(^kx ^f ) ^g( )^{x f ei kxxdx}

, 2π

, ⁼ ₋ò^∞_∞

Inverse Fourier

Transformation: ^g( )^{x f G}(^kx ^f )^{e i kxxdk}

π

, 2

, ⁻

∞

∞

−ò

=

Spatial Fourier transformation is discussed for one horizontal (x) direction, but can be carried out in the two horizontal directions.

Temporal versus Spatial 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 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)

Temporal Fourier transformation

Spatial Fourier transformation

α

Apparent velocity:

sinα v^app = v

f v^app

app =

λ

Apparent wavelength:

∞

= vapp

∞

=

λapp

= v vapp

f

= v λapp

The phase velocity which a wavefront appears to have along a line of geophones

ÞNumber of waves per unit distance perpendicular to a wavefront

α v_app = v / sin α

Horizontal Wave

Incoming Wave

v

v

Wavefront

α=90°

v_app = v

app app

app

Apparent wavenumber k_app

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: _max

min

min 2

1

2 f

T v

t x

app

=

∆ <

=

∆

Two spatial samples for one apparent wavelength

∆x

∆t₁

∆t₂

∆x

∆t₃

T/2 α₁

α₃ α₂

α₁ < α₂ < α₃

∆x

min 1

vapp ^> v_app^min₂ > v_app^min₃ ^sinα v_app= v

2

min 1

t < T

∆ 2

min 2

t = T

∆ 2

min 3

t > T

∆ v^appmin

t = ∆x

∆

f

k_a (apparent wavenumber)

f = v_a k Slope v ^a

f-k-Spectrum

a

Yilmaz, 1987

Yilmaz, 1987

Yilmaz, 1987

Yilmaz, 1987

Yilmaz, 1987

And its k-f amplitude spectrum

Composite walk-away noise test

Yilmaz, 1987

Rejection ground roll energy

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

corresponding f-k spectra

CMP gathers from a shallow marine survey

Before and after f-k dip filtering to remove coherent linear noise Yilmaz, 1987

Synthetic CMP gathers containing multiples

primaries Water-bottom multiples

+ = VM velocity multiples

VP velocity primaries

Yilmaz, 1987

NMO correction using primary velocity function

Yilmaz, 1987

After NMO correction

Zero-ing in f-k quadrant

Inverse

NMO Yilmaz, 1987

To suppress aliased energy

Yilmaz, 1987

VM1= slow (water-bottom) multiples VM2= fast (peg-leg) multiples

NMO corrected data using primary velocities

Yilmaz, 1987

Yilmaz, 1987

• Velocity filter

• Suppression of multiples

• Interpolation of traces

• Analysis of guided waves

τ-p transformation for various arrivals

Yilmaz, 1987

(1/p)

P1 and P2 are primaries W is water bottom which

results in multiples

Yilmaz, 1987

p(s/km)

1/18 1/1.5

Reducing source-generated noise

in shallow seismic data using linear and hyperbolic τ−p transformations

Roman Spitzer, Frank Nitsche and Alan G. Green

48 receivers 5 m. interval

Source location:

5 m from first geophone 3m depth

(a) Raw shot gather (b) Time and offset

varying gain

(c) Spectral balancing (80-250 Hz)

Result of

filtering Difference

Between (a) And (c)

Linear τ-p transformation

Hyperbolic τ-p transformation

Amplitude of each sample

is squared Hyperbolic τ-p

transformation Inverse hyperbolic τ-p transformation

Shot gather along a high-resolution seismic line in northern Switzerland

(a) Raw shot gather (b) Time and offset

varying gain

(c) Spectral balancing (80-250 Hz)

Linear τ-p transformation

Result of

filtering Difference

Between (c) And (c)

Hyperbolic τ-p transformation

Amplitude of each sample

is squared Hyperbolic τ-p

transformation

Inverse hyperbolic τ-p transformation

Processing:

• CMP sorting

• NMO corrections

•NMO stretch mute

• Stacking

Reflections were found to extend to shallower depths and more continuous