  Interactions between the lattice and charge carriers can drive the formation
of phases and ordering phenomena that give rise to conventional
superconductivity, insulator-to-metal transitions, and charge-density waves.
These couplings also play a determining role in properties that include
electric and thermal conductivity. Ultrafast electron diffuse scattering (UEDS)
has recently become a viable laboratory-scale tool to track energy flow into
and within the lattice system across the entire Brillouin zone, and deconvolves
interactions in the time domain. Here, we present a detailed quantitative
framework for the interpretation of UEDS signals, ultimately extracting the
phonon mode occupancies across the entire Brillouin zone. These transient
populations are then used to extract momentum- and mode-dependent
electron-phonon and phonon-phonon coupling constants. Results of this analysis
are presented for graphite, which provides complete information on the
phonon-branch occupations and a determination of the $A_1'$ phonon
mode-projected electron-phonon coupling strength $\langle g_{e,A_1'}^2 \rangle
= 0.035 \pm 0.001$ eV$^2$ that is in agreement with other measurement
techniques and simulations.
