The nonrelativistic (nr) impulse approximation (NRIA) expression for Compton-scattering doubly differential cross sections (DDCS) for inelastic photon scattering is recovered from the corresponding relativistic expression (RIA) of Ribberfors [Phys. Rev. B 12, 2067 (1975)] in the limit of low momentum transfer ($q\ensuremath{\rightarrow}0$), valid even at relativistic incident photon energies ${\ensuremath{\omega}}_{1}>m$ provided that the average initial momentum of the ejected electron $\ensuremath{\langle}{p}_{i}\ensuremath{\rangle}$ is not too high, that is, $\ensuremath{\langle}{p}_{i}\ensuremath{\rangle}$ $<m$. This corresponds to a binding energy ${E}_{b}<10$ keV. This $q\ensuremath{\rightarrow}0$ nr limit is simultaneous with the approach of the scattering angle $\ensuremath{\theta}$ to ${0}^{\ifmmode^\circ\else\textdegree\fi{}}$ ($\ensuremath{\theta}\ensuremath{\rightarrow}{0}^{\ifmmode^\circ\else\textdegree\fi{}}$) around the Compton peak maximum. This explains the observation that it is possible to obtain an accurate Compton peak (CP) even when ${\ensuremath{\omega}}_{1}>m$ using nr expressions when $\ensuremath{\theta}$ is small. For example, a $1%$ accuracy can be obtained when ${\ensuremath{\omega}}_{1}=1\phantom{\rule{0.3em}{0ex}}\text{MeV}$ if $\ensuremath{\theta}<{20}^{\ifmmode^\circ\else\textdegree\fi{}}$. However as ${\ensuremath{\omega}}_{1}$ increases into the MeV range, the maximum $\ensuremath{\theta}$ at which an accurate Compton peak can be obtained from nr expressions approaches closer to zero, because the $\ensuremath{\theta}$ at which the relativistic shift of CP to higher energy is greatest, which starts at ${180}^{\ifmmode^\circ\else\textdegree\fi{}}$ when ${\ensuremath{\omega}}_{1}<300$ keV, begins to decrease, approaching zero even though the $\ensuremath{\theta}$ at which the relativistic increase in the CP magnitude remains greatest around $\ensuremath{\theta}={180}^{\ifmmode^\circ\else\textdegree\fi{}}$. The relativistic contribution to the prediction of Compton doubly differential cross sections (DDCS) is characterized in simple terms using Ribberfors further approximation to his full RIA expression. This factorable form is given by $\mathrm{DDCS}=\mathit{KJ}$, where $K$ is the kinematic factor and $J$ the Compton profile. This form makes it possible to account for the relativistic shift of CP to higher energy and the increase in the CP magnitude as being due to the dependence of $J({p}_{\mathrm{min}},{\ensuremath{\rho}}_{\mathrm{rel}})$ (where ${p}_{\mathrm{min}}$ is the relativistic version of the $z$ component of the momentum of the initial electron and ${\ensuremath{\rho}}_{\mathrm{rel}}$ is the relativistic charge density) and $K({p}_{\mathrm{min}})$ on ${p}_{\mathrm{min}}$. This characterization approach was used as a guide for making the nr QED $S$-matrix expression for the Compton peak kinematically relativistic. Such modified nr expressions can be more readily applied to large systems than the fully relativistic version.