In Geophysics and related fields such as Reservoir Science, we typically take inadequate, insufficient and inaccurate data, often only on the surface of earth, or in a limited number of boreholes. The Inverse Problem is to obtain not just the best model based on constraints of the data, but also the range of models that can satisfy the observations. The solutions are usually non-unique, with large associated errors, because of uncertainties in the proper formulation of the assumed model(s), as well as the lack of complete and sufficiently sensitive data. The art of the subject is to attempt to merge this difficult situation with other information such as from geology, or jointly with other data (eg. seismic and EM) to obtain a more robust solution with improved - and quantifiable - uncertainty.
Image adapted from "Quantifying the uncertainty in fault plane solutions inferred from S/P amplitude ratios" by O. Poliannikov, M. Fehler, W. Rodi.