# Summary

In this chapter we have defined important elements that will be used throughout this inversion module. These include forward and inverse mappings, their associated vector spaces and the difference between linear and nonlinear problems. The fundamental statement that “the inverse problem is ill-posed” is illustrated by considering the relationship between the interval and rms velocities in reflection seismology. A Jupyter notebook allows readers to verify the statements made in the text and to carry out further exploration regarding the nonuniqueness and ill-conditioning that is inherent in the inverse problem. Any inversion formalism must address these two aspects and identify two major pathways by which this can be done. The first uses a Bayesian framework based in probability theory. The second, is a deterministic approach where an optimization problem is solved to provide a specific solution for the inverse problem. This is a Tikhonov approach and we explore this in the next chapter.