In most geometry inverse problem, information like the temperature measurement on the surface is use to detect the shape and location of a defect. My question is: is it possible to measure the temperature distribution or temperature gradient of a material up to certain depth? What kind of device is used for this kind of problem? Can you recommend a study that deals with such kind of problem?
Yes, in material engineering they call it Thermography. By Radiating heat on the surface or infrared radiation we can map the temperature distribution.
The heat transport throughout the material is affected by defects (holes, fracture ...). Also the heat that produce during the fracture of material can be detected in this technic.
Usually a laser puls is used to send the energy throughout the material and a thermographic camera to visualise the temperature gradient.
This technic is very popular to find delimitation in fiber re-infocred plastics, or defects in welding connections between metals and composites.
I had a brief course in college, named fracture mechanics, which deals with this kind of problem but not concentrated. I think material engineering and science is the right field. Also the basic knowledge of heat transport and thermodynamic is indispensable to understand what is happening.