Question: 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?

Answer: 
Gravity can be used to investigate the lower crust and upper mantle (see for example [Fullea et al, 2014](http://dx.doi.org/10.1016%2Fj.jag.2014.02.003)). Satellite measurements of gravity could even be used to investigate deeper structures in the mantle, like subducting slabs ([Panet, 2014](http://dx.doi.org/10.1038%2Fngeo2063)). However, I couldn't find any use of gravity data to probe deeper, into the core for example.

The [magnetotelluric method](http://en.wikipedia.org/wiki/Magnetotellurics) is sometimes used for deep crustal structure. And features of the geomagnetic field, like [secular variation](http://en.wikipedia.org/wiki/Geomagnetic_secular_variation), can be inverted to investigate as far down as the outer core ([Gubbins, 1996](http://dx.doi.org/10.1016/S0031-9201(96)03187-1)).

However, the most common and well developed method to image the lower mantle and core is still through seismic waves.
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.