Let’s say you are standing on the outside of a hollow sphere, about the size of the Earth. There is a single rod underneath where you’re standing which varies in density d(x) along its entire length. Its mass pulls on you. You have tools which can measure precisely the gravitational pull wherever you’re standing. How many different measurements do you need to take—and where—in order to figure out where the rod is heavy and where it’s light?
The real problem of finding stuff in the ground has to be much much harder, but I just want to know the answer to this “simple” question for starters.
My first intuition would be to measure right on top of the rod, then move to two other spots so you have formed an equiangular triangle on the sphere. But still leaves ∞ degrees of freedom if you started out with an entire function’s worth of freedom, right?