Calculating 1D spectrum from 2D spectrum

  • Thread starter nkinar
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  • #1
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Hello---

I am reading a paper which presents a method to determine attenuation (and Q factors) from reflection seismic data (Y. Wang, "Q analysis on reflection seismic data," Geophysical Research Letters, Vol. 31, 2004).

To perform signal processing on a seismic trace, the paper describes the following procedure:

(1) From the real and complex parts of the Gabor spectrum transform, compute the (real numbered) amplitude spectrum [tex]U(t, \omega) [/tex] on a seismic trace [tex]s(t)[/tex], where [tex] t [/tex] is the time (s), [tex]\omega[/tex] is the angular frequency (1/s), and [tex] \omega = 2 \pi f[/tex], where [tex]f[/tex] is the frequency in Hz.

(2) Define [tex]\chi = t \omega[/tex] as the product of [tex]t[/tex] and [tex]\omega[/tex], and transform the 2D spectrum [tex]U(t, \omega)[/tex] into the 1D spectrum [tex]U(t\omega) = U(\chi)[/tex].

The paper does not describe how to transform [tex]U(t, \omega)[/tex] into [tex]U(\chi)[/tex].

Would numerical integration be able to do this transformation? How might I proceed?
 

Answers and Replies

  • #2
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For the sake of completeness (https://en.wikipedia.org/wiki/Gabor_transform):

1574631547672.png

In our case we have
$$
U_x(\tau \, , \,\omega)=\displaystyle{\int_{-1.9143+\tau}^{1.9143+\tau}}x(t)e^{-\pi(t-\tau)^2}e^{-i\omega t}\,dt )
$$
and I'm a bit confused whether ##t## or ##\tau## is meant, i.e. if we are talking about a simple variable substitution. I assume that the exponent ##-\omega t## is abbreviated by ##\chi## and we have such a substitution of the integral variable. The real case applications of the Gabor transform lives with approximations. So maybe the ##\alpha## in the Wikipedia article can be used to simplify the other terms of the integrand. Or - what could be as well - ##U(\chi)## is a sloppy notation for ##U(\tau,\chi)##.
 
  • #3
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Thanks, I think that you are right about the notation and the substitution.
 

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