All the information about a plasma species is encoded in its distribution function. While it would be helpful to measure the distribution function directly, it is only possible to measure its moments. If the form of the distribution function is not known a priori, it can be difficult to interpret diagnostic signals. This is particularly true in fast-ion physics, where diagnostics that nominally view the same thing, the fast-ion distribution function, give seemingly discordant measurements. The process of going from a fast-ion distribution to a measurement and the reverse process of going from a set of measurements to a fast-ion distribution are the main topics of this thesis.

Chapters 2-3 concern the modeling of fast-ion diagnostics. Here we derive functions that translate the information about a fast ion into measurable quantities, i.e. forward models. This is done for three diagnostics: the neutral particle analyzer (NPA), fast ion D-α (FIDA) spectroscopy, and neutron scintillators. Chapter 3 discusses the development of FIDASIM, the practical implementation of the forward models.

Chapter 4 deals with diagnostic velocity-space weight functions, an ansatz which is used to aid in the interpretation of experimental measurements and as an approximate forward model of the diagnostic. From the forward models discussed in Chapter 2, we derive weight functions in a full 6D generalized coordinate system, from which we also derive the velocity-space weight functions. Using an action-angle formulation, orbit-space weight functions, which can be used to exactly represent a diagnostic's forward model in a linear form, are derived.

Chapters 5-6 detail how to use weight functions to infer the fast-ion distribution function from experimental measurements. Orbit weight functions, in particular, facilitate the inference of the entire distribution function, using any fast-ion diagnostic that views the plasma. Benchmarks with synthetic data and a reconstruction of a classical distribution from experimental measurements, show that systematic errors and intrinsic biases in the inference methods are the main impediments to accurately inferring the fast-ion distribution function. However, experimental studies of the redistribution of fast ions by sawtooth crashes at ASDEX Upgrade demonstrate that the effects of systematic error and biases become less important when only considering relative changes in the distribution function.