Calculating 1D spectrum from 2D spectrum

In summary: I'll try it out and see if it works.In summary, the conversation discusses a paper presenting a method to determine attenuation and Q factors from reflection seismic data. The procedure involves computing the amplitude spectrum from the Gabor spectrum transform and transforming it into a 1D spectrum. The paper does not specify how to perform this transformation and suggests using numerical integration. The use of the Gabor transform in real case applications may require approximations. The conversation also discusses the possibility of a substitution of variables in the integration.
  • #1
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0
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?
 
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  • #2
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
Thanks, I think that you are right about the notation and the substitution.
 

1. How do you calculate a 1D spectrum from a 2D spectrum?

To calculate a 1D spectrum from a 2D spectrum, you need to integrate the values along one dimension of the 2D spectrum. This can be done by summing or averaging the values along a specific axis. The result will be a 1D spectrum with the intensity values plotted along one axis.

2. Why is it important to calculate a 1D spectrum from a 2D spectrum?

Calculating a 1D spectrum from a 2D spectrum allows for easier analysis and interpretation of the data. It reduces the complexity of the data and allows for easier comparison with other spectra. Additionally, many analytical techniques are designed to analyze 1D spectra, so converting the data to this format may be necessary for further analysis.

3. What are some common methods for calculating a 1D spectrum from a 2D spectrum?

The most common methods for calculating a 1D spectrum from a 2D spectrum include summing or averaging along a specific axis, extracting a specific line or region of interest, or using specialized software or algorithms designed for this purpose.

4. Are there any limitations to calculating a 1D spectrum from a 2D spectrum?

Yes, there are limitations to this process. Depending on the data and the method used, there may be loss of information or resolution when converting to a 1D spectrum. Additionally, the choice of which axis to integrate along can affect the resulting spectrum, so it is important to carefully consider which method is most appropriate for the data.

5. How can calculating a 1D spectrum from a 2D spectrum be used in scientific research?

Calculating a 1D spectrum from a 2D spectrum is a common technique in many scientific fields, including chemistry, physics, and biology. It can be used to analyze and identify molecules, study chemical reactions, and understand the structure and dynamics of complex systems. It is also a valuable tool in fields such as spectroscopy, microscopy, and imaging, where it allows for easier interpretation and comparison of data.

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