<p>The Fourier transform of a Gaussian is a Gaussian.</p>

<p>If your signal is given by </p>

<p><span class="math-container">$$g(t)=\frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left( -\frac{(t-t_0)^2}{2 \sigma^2}\right)$$</span></p>

<p>then your frequency spectrum is</p>

<p><span class="math-container">$$G(f) = \exp(-2 \pi^2 \sigma^2 f^2),$$</span></p>

<p>where f is the frequency (<span class="math-container">$2 \pi \omega).$</span></p>

<p>I'l let you do the substitution with your particular numbers. Enjoy!</p>

<ul>
<li><a href="https://warwick.ac.uk/fac/sci/mathsys/courses/msc/ma934/resources/notes8.pdf" rel="noreferrer">https://warwick.ac.uk/fac/sci/mathsys/courses/msc/ma934/resources/notes8.pdf</a></li>
<li><a href="http://www.cse.yorku.ca/~kosta/CompVis_Notes/fourier_transform_Gaussian.pdf" rel="noreferrer">http://www.cse.yorku.ca/~kosta/CompVis_Notes/fourier_transform_Gaussian.pdf</a></li>
</ul>

CC BY-SA 4.0
2019-07-13T12:47:00.793Z
2019-07-13T12:47:00.793Z
uhoh
<p>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.</p>
<p>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.</p>
<p>Let's assume my source time function in time domain looks like this:</p>
<pre><code>source = 1/(2* np.pi * 17.**2) * np.exp( - (t-100)**2 / (2* 17.**2))
</code></pre>
<p>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?</p>

CC BY-SA 4.0
2019-07-14T06:57:03.177Z
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 ) ) .)
Erik
<seismic>
wavelength of seismic wave with a gaussian source
guest101
# wavelength of seismic wave with a gaussian source

(Asked by: username_0 on 2019-07-13)

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?

**Tags:** seismic

> **Comments:**
> **username_1**: 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 ) ) .)  

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## 1 answers

### Answer by username_2 on 2019-07-13. Score: 4

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!

* <https://warwick.ac.uk/fac/sci/mathsys/courses/msc/ma934/resources/notes8.pdf>
* <http://www.cse.yorku.ca/~kosta/CompVis_Notes/fourier_transform_Gaussian.pdf>


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