As a practicing geophysical consultant, I would like to point out that you never mentioned "geology". In oil and gas exploration the geologic focus is on sedimentary rocks and structural configuration. In geothermal exploration, the target geology is significantly more complex.

From the standpoint of computational geophysics and the solution of inverse problems, the approach to complex (multi-physical) interpretation has been considered over the past 35 years by many authors. I advise you to pay attention to the works of I. Kukkonen, M. Verdoya, N. Balling, J.-C. Mareschal, E. Hurtig, D. Stromeyer, etc. The author of this answer also dealt with similar problems.

You are right, the direction is very interesting and they undoubtedly need to be dealt with from various positions.

Bai Lige The most advanced method of monitoring physical property changes in the subsurface is time-lapse seismic. What can be observed is changes in arrival times of seismic reflections signals, and the reflections amplitude. Both signals are sensitive to the propagation velocity of seismic waves and the bulk density.

If you want to monitor geothermal processes, you have to understand how temperature changes affect seismic properties (velocity and density). Typically geophysicist subdivide rock into the rock-frame properties and the fluid properties occupying the pore space. The rock frame properties and the fluid properties are then "averaged" using a suitable equation. The classic equation is Gassmann equation. Fluid properties (Density and bulk modulus, from which you can calculate P-wave velocities) are often calculated using Batzle and Wang equations (the classic paper is Batzle and Wang, 1992, Seismic properties of pore fluids). These equations allow calculation of the temperture-dependence of fluid properties. The effect of temperature on the rock-frame takes is slightly more complex. The major effect is probably that temperature changes causes thermal expansion/contraction of the rock, which causes stress/strain changes in the rock-frame, which in turn cause velocity changes. A good place to start reading is "The Rock Physics Handbook" by Mavko, Mukerji and Dvorkin, and the references therein.

Before you move to any statistical methods, I would suggest that you try to understand the magnitude of temperature change on both fluid properties and rock-frame properties and get a sense of the percentage velocity (and density) change. These changes will depend very much on geological properties (such as porosity and mineralogy). I concur with Ernest Berkman that some focus on the geological setting is very important. A geothermal site in a hot granite (with low porosity and permeability) will behave very differently from a geothermal site in sedimentary rock with high porosity/permeability and a slightly elevated temperature.

The challenge with multi-physics is to know which effects are first order, and which effects you can safely neglect. You can of course display the results of all the various multi-physics effects in a statistical sense with correlation matrices. It always helps to first understand the underlying physics.

Good luck with your research,

Jorg