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I want to know how to get the wavelength/frequency of a seismic wave, if I only have a Gaussian source and the velocity (c = 4000m/s) of a medium given.

e.g. for a Ricker wavelet it would be easy to get the central frequency and max frequency of the source to compute the wavelength by c/f, but I have some issues with the Gaussian and unfortunately only signals of a Gaussian source.

Let's assume my source time function in time domain looks like this:

source = 1/(2* np.pi * 17.**2) * np.exp( - (t-100)**2 / (2* 17.**2))

So a Gaussian with variance of 17 squared and shifted to the right. In the frequency domain it will be centered around zero and I don't really get a maximum frequency as well. Is there a way to find these?

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  • $\begingroup$ It is correct that the Gaussian wavelet has its peak frequency at zero frequency (see in uhohs answer below, for f=0 you get the largest G). For the maximum frequency you'll have to come up with a criterion that defines it for you. For example, the point up to which you capture 95% of the spectrum. With uhohs answer and a wavelet of type source=[scaling_factor] * exp( - (t-100)^2/(2*17^2) ), I think you can find an analytical expression for this. ( >> sqrt( log(0.05) / (-2*pi^2*17^2 ) ) .) $\endgroup$ – Erik Jul 14 '19 at 6:57
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The Fourier transform of a Gaussian is a Gaussian.

If your signal is given by

$$g(t)=\frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left( -\frac{(t-t_0)^2}{2 \sigma^2}\right)$$

then your frequency spectrum is

$$G(f) = \exp(-2 \pi^2 \sigma^2 f^2),$$

where f is the frequency ($2 \pi \omega).$

I'l let you do the substitution with your particular numbers. Enjoy!

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