Pulsar timing arrays (PTAs) are galactic-scale nanohertz gravitational wave observatories. Each individual arm, composed of a millisecond pulsar, a radio telescope, and a kiloparsecs-long path, differs in its properties but, together, can be used to extract low-frequency gravitational wave (GW) signals. While gravitational waves in the nanohertz band are orders of magnitude larger than signals in the ground-based and space-based regimes, the noise sources in pulsar data are also comparatively larger. Here we will describe detector characterization for PTAs, which requires simultaneously modeling the signal and noise across our astrophysical detector. Serendipitously, many of the noise processes at work in pulsar observations, considered noise when searching for nanohertz gravitational waves, contain interesting astrophysics in their own right. After a general introduction to the subject, I will highlight new time-domain analysis work that allows us to mitigate stochastic processes from the interstellar medium and intrinsic processes in the pulsars. Lastly, I will present a noise and sensitivity analysis of the NANOGrav 15-year data release.