Apparent-velocity filtering

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Problem 9.25a

On a north-south line, the noise arriving from the south is mainly in the band and the noise arriving from the north in the band , where and are apparent velocities. Given that , sketch an (or ) plot.

Background

Apparent velocity is defined in problem 4.2d by the equation

where is the angle of approach. Using the relation , where the angular frequency equals and the angular wavenumber is . we obtain the relation


(9.25a)

being the apparent wavelength and is the apparent wavenumber. Writing this equation as , we have


(9.25b)

The slope of a line on an plot gives the apparent velocity.

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

The two-dimensional Fourier transform is defined by the equation


(9.25c)

the inverse transform being


(9.25d)

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


(9.25e)

In digital form, equation (9.25e) becomes


(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 in time sampling [see equation (9.4c)], spatial sampling involves a spatial Nyquist “frequency” = Nyquist wave-number = . Using the symbols and for the time and spatial sampling intervals, the Nyquist frequencies are

and

Figure 9.25a.  An plot.

Apparent-velocity filters can be designed to remove pie-sliced portions, e.g., a filter in Figure 9.25a to remove would remove the noise between the 6 km/s line and the -axis.

Solution

Taking north as the positive direction, on an plot, km/s is a straight line through the origin with slope km/s and the noise from the south is mainly between this line and the -axis. Similarly the noise from the north lies between the -axis and a straight line with slope –3 km/s extending from the origin. The plot is shown in Figure 9.25a for the Nyquist wavenumber , or .

If Figure 9.25a were rolled into a vertical cylinder by matching with , it can be seen that the alias slopes are simply extensions of the apparent velocity lines.

Problem 9.25b

Repeat for .

Solution

The only change from part (a) is that the Nyquist frequency is now double that in (a), that is, . 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 whose transform is defined by the equations

and being the Nyquist wavenumber and Nyquist frequency. Transforming to the domain gives

The recorded data are real, hence 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

We integrate first with respect to and obtain

where sinc .

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