fit. However, these coefficients are not all independent due to a smoothness constraint - physically, the transmission profile cannot fluctuate arbitrarily, an intuition supported by atomistic simulations [@Li:2012g; @Li:2012j]. Qualitatively, this requirement implies that each coefficient depends on the adjacent coefficients, decreasing the effective number of unknowns by a value on the order of two to three. Therefore, the number of unknowns is comparable to the number of data points.

Ample data points are a necessary but not sufficient condition to enable extraction of transmission coefficients. The last requirement is that the TDTR signal should be sensitive to the shape of the transmission coefficient profile. In the case of heat diffusion, this requirement is not satisfied: so long as the interface conductance is unchanged, the TDTR signal will not change because scattering in the substrate obscures the interfacial phonon distribution. To demonstrate this point, we simulated TDTR signals using two different transmission coefficient profiles as shown in Fig. \[fig:TDTR\_sensitivity\] (a) with the scattering rates of Si modified such that no MFP exceeds 50 nm. The chosen transmission profiles possess the same interface conductance. The amplitude and phase of the simulated TDTR signals for these two transmission coefficient profiles are shown 