Apparent-velocity filtering

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	9
Pages	295 - 366
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 9.25a

On a north-south line, the noise arriving from the south is mainly in the band Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{S} \le 6\ {\hbox{km/s}}} and the noise arriving from the north in the band Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{N} \le 3\ {\hbox{km/s}}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{S}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{N}} are apparent velocities. Given that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta x=50\ {\hbox {m}}} , sketch an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f-k} (or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega -\kappa} ) plot.

Background

Apparent velocity is defined in problem 4.2d by the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} V_{a} =V/\sin\alpha , \end{align} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha} is the angle of approach. Using the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V=f\lambda =\omega /\kappa} , where the angular frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega} equals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\pi f} and the angular wavenumber Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\pi /\lambda} . we obtain the relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} V_{a} =f\lambda _{a} =2\pi f/\kappa_{a} =\omega /\kappa_{a}, \end{align} }

(9.25a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda _{a}} being the apparent wavelength and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa_{a} =2\pi /\lambda _{a}} is the apparent wavenumber. Writing this equation as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega =V_{a} {\kappa_{a}}} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{{\hbox{d}}\omega}{{\hbox{d}}{\kappa_{a}}} =\frac{{\hbox{d}}f}{{\hbox{d}}k_{a}} =V_{a}. \end{align} }

(9.25b)

The slope of a line on an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f-k} plot gives the apparent velocity.

The one-dimensional Fourier transform relation was defined in equation (9.3c.d) as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} G(\omega)={\int}_{-\infty}^{+\infty} g(t)e^{-{\hbox {j}}\omega t} {\hbox{d}}t,\; g(t)=(1/2\pi){\int}_{-\infty}^{+\infty} G(\omega)e^{\hbox {j}\omega t} {\hbox{d}}\omega . \end{align} }

The two-dimensional Fourier transform is defined by the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} G(\kappa,\; \omega)=\int\int_{-\infty}^{+\infty} g(x,\; t) e^{-{\hbox {j}}(\kappa x+\omega t)} {\hbox{d}}x\ {\hbox{d}}t, \end{align} }

(9.25c)

the inverse transform being

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} g\; (x,\; t)=(1/2\pi)^{2} \int\int_{-\infty}^{+\infty} G(\kappa,\; \omega)\; e^{\hbox {j}(\kappa x+\omega t)} {\hbox{d}}{\kappa}\ {\hbox{d}}\omega. \end{align} }

(9.25d)

One-dimensional convolution [equation (9.2a)] becomes in two dimensions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} h(x,\; t)=g(x,\; t)*f(x,\; t)=\int{\int}_{-\infty}^{+\infty} g(\sigma ,\; \tau)f(x-\sigma ,\; t-\tau)\,\mathrm{d}\sigma \,\mathrm{d}{\tau}. \end{align} }

(9.25e)

In digital form, equation (9.25e) becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} h_{xt} =g_{xt} *f_{xt} =\mathop{\sum}\limits_{m}^{} \mathop{\sum}\limits_{n}^{} g_{mn} \ f_{x-m,t-n}. \end{align} }

(9.25f)

In applying apparent-velocity filtering, we deal with data that are sampled in both time and space. Corresponding to the temporal Nyquist frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{N} =1/2\Delta} in time sampling [see equation (9.4c)], spatial sampling involves a spatial Nyquist “frequency” = Nyquist wave-number = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/(2\Delta x)} . Using the symbols Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta _{t}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta _{x}} for the time and spatial sampling intervals, the Nyquist frequencies are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{N} =1/2\Delta _{t} ,\; \omega _{N} =\pi /\Delta _{t}, \end{align} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \kappa_{N} =\pi /\Delta _{x}. \end{align} }

Figure 9.25a.  An Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f-k} plot.

Apparent-velocity filters can be designed to remove pie-sliced portions, e.g., a filter in Figure 9.25a to remove Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle < 6\ {\hbox{km/s}}} would remove the noise between the 6 km/s line and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} -axis.

Solution

Taking north as the positive direction, on an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f-k} plot, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{a} =6} km/s is a straight line through the origin with slope Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle df/dk_{a} =6} km/s and the noise from the south is mainly between this line and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +\kappa_{a}} -axis. Similarly the noise from the north lies between the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\kappa_{a}} -axis and a straight line with slope –3 km/s extending from the origin. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f-k} plot is shown in Figure 9.25a for the Nyquist wavenumber Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{N} =(1/2\times 50){\hbox{m}}^{-1} =10/{\hbox{km}}} , or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa_{N} =\pi /50\ {\hbox{m}}^{-1} =62.8/{\hbox{km}}} .

If Figure 9.25a were rolled into a vertical cylinder by matching Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\kappa_{N}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +\kappa_{N}} , it can be seen that the alias slopes are simply extensions of the apparent velocity lines.

Problem 9.25b

Repeat for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta x=25\ {\hbox{m}}} .

Solution

The only change from part (a) is that the Nyquist frequency is now double that in (a), that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa_{N} =2\pi /2\times 25\ {\hbox{m}}^{-1} =126\ \mathrm{km}^{-1}} . This will move the alias lines upward in Figure 9.25a.

Problem 9.25c

Calculate a filter that will prevent aliasing in both the wavenumber and time domains for parts (a) and (b) [see equation (9.25c)].

Solution

We require a filter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x,\; t)} whose transform is defined by the equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} F (\kappa_{a} ,\; \omega)=+1 ,\quad |\kappa_{a} |\le \kappa_{N}\quad \mathrm{and}\quad |\omega |\le \omega _{N} ,\\ =0,\quad |\kappa_{a} |\ge \kappa_{N}\quad \mathrm{and/or}\quad |\omega |\ge \omega _{N} , \end{align} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa_{N}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega _{N}} being the Nyquist wavenumber and Nyquist frequency. Transforming $F(\kappa _{a},\;\omega )$ to the $x-t$ domain gives

${\begin{aligned}f\;(x,\;t)=(1/2\pi )^{2}\int _{-\kappa _{N}}^{+\kappa _{N}}\int _{-\omega _{N}}^{+\omega _{N}}e^{{\hbox{j}}(\kappa _{a}x+\omega t)}{\hbox{d}}{\kappa _{a}}{\hbox{d}}\omega .\end{aligned}}$

The recorded data are real, hence $f(x,\;t)$ must also be real, and therefore we can set the imaginary part in the integrand equal to zero. Using Euler’s formula (Sheriff and Geldart, 1995, problem 15.12a), we get

${\begin{aligned}f\;(x,\;t)=(1/2\pi )^{2}\int _{-\kappa _{N}}^{+\kappa _{N}}\int _{-\omega _{N}}^{+\omega _{N}}\cos(\kappa _{a}x+\omega t){\hbox{d}}{\kappa _{a}}{\hbox{d}}\omega .\end{aligned}}$

We integrate first with respect to $\omega$ and obtain

${\begin{aligned}f(x,\;t)=(1/2\pi )^{2}{\int }_{-\kappa _{N}}^{+\kappa _{N}}\left[{\frac {{\rm {\;sin\;}}(\kappa _{a}x+\omega t)}{t}}|_{-\omega _{N}}^{+\omega _{N}}\right]{\hbox{d}}{\kappa _{a}}\\=(1/4\pi ^{2}t){\int }_{-\kappa _{N}}^{+\kappa _{N}}[{\rm {\;sin\;}}(\kappa _{a}x+\omega _{N}t)-{\rm {\;sin\;}}(\kappa _{a}x-\omega _{N}t)]{\hbox{d}}{\kappa _{a}}\\=(1/4\pi ^{2}t)\left({\frac {-{\rm {\;cos\;}}(\kappa _{a}x+\omega _{N}t)}{x}}+{\frac {{\rm {\;cos\;}}(\kappa _{a}x-\omega _{N}t)}{x}}\right)|_{-\kappa _{N}}^{+\kappa _{N}}\\=(1/2\pi ^{2}xt)[{\rm {\;sin\;}}(\kappa _{a}x){\rm {\;sin\;}}(\omega _{N}t)|_{-\kappa _{N}}^{+\kappa _{N}}\\=(1/\pi ^{2}xt)[{\rm {\;sin\;}}(\kappa _{N}x){\rm {\;sin\;}}(\omega _{N}t)]\\=(\kappa _{N}\omega _{N}/\pi ^{2})\mathrm {sinc} \ \kappa _{N}x\ \mathrm {sinc} \ \omega _{N}t=(1/\Delta _{x}\Delta _{t})\ \mathrm {sinc} \ \kappa _{N}x\ \mathrm {sinc} \ \omega _{N}t_{j},\end{aligned}}$

where sinc $x={\rm {\;sin\;}}x/x$.

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Table of Contents (book)

Also in this chapter

• Fourier series
• Space-domain convolution and vibroseis acquisition
• Fourier transforms of the unit impulse and boxcar
• Extension of the sampling theorem
• Alias filters
• The convolutional model
• Water reverberation filter
• Calculating crosscorrelation and autocorrelation
• Digital calculations
• Semblance
• Convolution and correlation calculations
• Properties of minimum-phase wavelets
• Phase of composite wavelets
• Tuning and waveshape
• Making a wavelet minimum-phase
• Zero-phase filtering of a minimum-phase wavelet
• Deconvolution methods
• Calculation of inverse filters
• Inverse filter to remove ghosting and recursive filtering
• Ghosting as a notch filter
• Autocorrelation
• Wiener (least-squares) inverse filters
• Interpreting stacking velocity
• Effect of local high-velocity body
• Apparent-velocity filtering
• Complex-trace analysis
• Kirchhoff migration
• Using an upward-traveling coordinate system
• Finite-difference migration
• Effect of migration on fault interpretation
• Derivative and integral operators
• Effects of normal-moveout (NMO) removal
• Weighted least-squares

External links

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