Game textures often use this kind of noise for terrain heights, smoke, etc.

A similar kind of noise known as blue noise can be generated by taking the Fourier transform and not applying a 1/f filter but a high-pass filter instead. You end up with noise that only has high frequencies in it, and not low frequencies. Thus the noise does not have large-scale features, which is ideal for use in dithering.

Blue noise (and its DFT) look like this: https://demofox2.files.wordpress.com/2018/08/vc.png

And an example of dithering with white and blue noise: https://demofox2.files.wordpress.com/2019/06/randomvsblue.jp...

There are scientific applications for this kind of procedure. If an experiment has a noise source with a known frequency distribution, you can simulate the experiment by generating many thousands of realizations of noise superimposed with your (expected) signal. The variance in your measurement introduced by the noise can be used to assess the systematic uncertainty of the experiment.

For example, in ground-based experiments that measure the cosmic microwave background radiation, there is a substantial foreground noise from the atmosphere that can be modeled as a 1/f distribution. And actually the observations themselves are subject to a random variance (see cosmic variance) due to the fact that we get to observe the early universe from only one point in space. So you can use a similar trick to sample many random realizations of the CMB for given physical constants, and decide if our one-off observation is compatible with the theory.

You’d never do a DFT for generating a fBM, but the same technique using a different noise spectrum is how we’ve been generating ocean waves in the VFX industry since forever: https://people.cs.clemson.edu/~jtessen/reports/papers_files/...