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It is well-known that we can learn a lot about the structure of the lower crust, mantle, and core by observing the ways in which they refract different kinds of seismic waves.
Do we have any other ways of imaging the deeper parts of the Earth, though?
Gravity can be used to investigate the lower crust and upper mantle (see for example Fullea et al, 2014). Satellite measurements of gravity could even be used to investigate deeper structures in the mantle, like subducting slabs (Panet, 2014). However, I couldn't find any use of gravity data to probe deeper, into the core for example.
The magnetotelluric method is sometimes used for deep crustal structure. And features of the geomagnetic field, like secular variation, can be inverted to investigate as far down as the outer core (Gubbins, 1996).
However, the most common and well developed method to image the lower mantle and core is still through seismic waves.
One interesting and relatively new technique is by the detection of geoneutrinos. These particles are produced by radioactive decay in the Earth's interior. They are uncommonly suitable for probing the deep Earth because -- unlike most particles and waves -- they can travel through thousands of kilometres of rock with very little absorption. Of course, this very same characteristic makes actually detecting them something of a challenge, and detectors tend to be rather large (in the thousands of cubic metres). Araki et al. (2005) gives some early results -- but as the Wikipedia article shows, there are more and bigger detectors on the drawing boards, so we should expect to see more geoneutrino results in the coming years and decades. This abstract gives the best elevator pitch I've found so far for geoneutrino research:
Radioactive decay of U and Th gives off ghost-like, neutrino particles that can be detected by 1000 ton detectors built a mile underground, where they are shielded from the cosmic rays that rain down on the Earth. Collaborations between physicists and geologists are detecting these "geo-neutrinos". Future underwater detectors, deployed at different points on the ocean floor, will create a neutrino tomographic image of mantle structures sited at the base of the mantle above the core.
Inverse problems are some of the most important mathematical problems in science and mathematics. Inverse problems arise in many branches of geophysics, medical imaging, remote sensing, ocean acoustic tomography, nondestructive testing, astronomy, physics and many other fields.
Geophysicists remotely measure the seismic (acoustic), gravity, and electromagnetic fields of the earth and then treat the inverse problem to constrain the properties of the earth's interior.
