I do understand why the seismic velocity decreases with temperature and increases with pressure, but I am not really familiar with the relationship between the velocity with Poisson's ratio. I understand the main idea of the Poisson's ratio and how it relates to the deformation, but I do not understand how it contributes to the wave velocity and also how does it change with temperature and pressure?
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2 Answers
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Good question (and welcome!)...
I like the quantitative nature of the initial answer above (as well as the very good disclosure of isotropic and homogenous subsurface nature that is technically never true in the so-called "real world"), but for my take, I'm going to forgo presenting any equations or numbers (although, embedded links herein will do that for me!).
Read all this paragraph and the links if you want more technical info: Let's parse out some things before presenting an answer below. Remember, that temperature (usually) increases with depth (think of lithostatic pressure). Also, you say "pressure" but at this point, it's better (long term in school or professional life) to think of that aspect of subsurface conditions as stress (and how that affects strain - see last sentence of this verbose paragraph). Also, even though you've only mentioned pressure, seismic velocities, and Poisson's ratio, it's critically important to keep in mind other so-called "elastic parameters" such as Young's Modulus, Shear Modulus, and Bulk Modulus, and (arguably most important of all) Lame parameters. Also, think of how these things change (or are absent) based on P- and S-wave velocities. Lastly, I implore you to do some investigation of what rheology means in regards to this topic.
OK. So, what does Poisson's ratio basically describe? Well, it's estimating the "rigidity" of something (a rock or mineral, since we are taking about the subsurface!). What do I mean by rigidity? It's something that is technically anything (some would argue with this) but pure water (i.e. it has some form of elastic behavior).
To answer your question: simply, think that that pressure affects temperature, which then affects the geology (which is the proxy here for Poisson's ratio). So, "hotter" and/or "viscous" less rigid material (think heated rubber) will facilitate slower seismic velocities while "colder" and/or "solid" more rigid material (think cold steel) will result in higher seismic velocities.
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In terms of the seismic velocities, the Poisson's ratio can be expressed as the following, under the assumption the material is isotropic and homogeneous:
