How Inverse Problems Open the Unknowable to Science
Incomplete data can reveal entire worlds.
Earth's gravitational field. Image: NASA
Consider the gravitational field of the planet Earth. This is a thing we can measure here on the surface, thanks in part to the wildly varying lithology of the planet's subterranean otherworld, which allows us to view the field in something like high-relief. This field, as measured, gives us information about the field itself, of course, but we can also use these measurements to acquire data about, say, the subsurface density of Earth.
We can take our measurements and turn to Newton's Law of Gravitation. Here, d is the localized gravitational acceleration (field), while K is the universal gravitational constant, M is the mass density of subsurface rock found r units of distance underground.
M is the tricky one because we can't directly measure that mass density very easily (or at all). But we have the above formula, so if we pile all of our gravitational field measurements into a stack of data points, we can invert the relationship, revealing, whoooosh, an inverse problem: "The (mathematical) process of predicting (or estimating) the numerical values (and associated statistics) of a set of model parameters of an assumed model based on a set of data or observations." The textbook definition.
Here is a better explanation, courtesy of Kings College physicist Roy Pike:
So: knowing the unknowable, solving the impossible with help from guesstimates.