It took me a while to get it (and I'm not a geologist or petroleum engineer or anything, just someone who knows basic physics and has been reading a geology textbook), but I think I understand Erik's answer: It's based on buoyancy.
Consider a column of permeable rock and soil that has a zone of aeration above a zone of saturation. (I'm gonna talk about water,...
I think Camilo Rada's answer is the one to follow if you want to do anything serious with the data.
However, to answer the question a bit more directly, if you want to use actual equations then an alternative that should work just fine if you are looking at just the Holocene (maybe all of the Quaternary?) is to use the original tables from André Berger
Drilling mud is only used when drilling for fluids that are pressurized such as oil, gas and sometimes water. It is not used when drilling for minerals such as gold, iron ore, copper.
Exploratory drilling minerals will use a combination of:
Rotary air blast drilling (RAB) - where compressed air is forced
through the drill steel to flush drill cutting out of ...
The keyword that you are looking for is a spectrogram, this is the plot of a frequency time analysis. If you search for that term for can find many examples of what you are after e.g. https://fairyonice.github.io/implement-the-spectrogram-from-scratch-in-python.html.
On a related note if you are interested in ambient noise tomography you might want to look ...
The simplest method is the spectral ratio method where the frequency spectra at two different depth levels in the VSP are computed and divided i.e. the ratio of the spectra. IN theory the slope of this spectral ratio is related to the attenaution factor Q. However, this method is not very robust and the data needs careful preconditioning to remove ...
This is the paper
GEOPHYSICS Volume 49, Issue 8
Seismic versus sonic velocities: A vertical seismic profiling study
Authors: Robert R. Stewart, Phil D. Huddleston and Tze Kong Kan