  Merging supermassive black hole binaries produce low-frequency gravitational
waves, which pulsar timing experiments are searching for. Much of the current
theory is developed within the plane-wave formalism, and here we develop the
more general Fresnel formalism. We show that Fresnel corrections to
gravitational wave timing residual models allow novel measurements to be made,
such as direct measurements of the source distance from the timing residual
phase and frequency, as well as direct measurements of chirp mass from a
monochromatic source. Probing the Fresnel corrections in these models will
require future pulsar timing arrays with more distant pulsars across our Galaxy
(ideally at the distance of the Magellanic Clouds), timed with precisions less
than $100$ ns, with distance uncertainties reduced to the order of the
gravitational wavelength. We find that sources with chirp mass of order $10^9 \
\mathrm{M}_\odot$ and orbital frequency $\omega_0 > 10$ nHz are good candidates
for probing Fresnel corrections. With these conditions met, the measured source
distance uncertainty can be made less than 10 per cent of the distance to the
source for sources out to $\sim 100$ Mpc, source sky localization can be
reduced to sub-arcminute precision, and source volume localization can be made
to less than $1 \ \text{Mpc}^3$ for sources out to 1-Gpc distances.
