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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?