TDTR signal. From the plot, one can immediately tell the transmission coefficient profiles used in the calculations are not correct as the experimental and calculation signals do not match. The correct transmission coefficient profile will reproduce the measured TDTR signal. Our procedure to identify this profile is described in the next section.

Solution of inverse problem {#subsec:TransmissionMeasurements}
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The final step is to solve the inverse problem that identifies the transmission coefficients that best explain the observed data. From the BTE model, we obtain a surface temperature decay curve as a function of time just like the one measured in the experiments. For a given sample, the actual transmission coefficient profile as a function of phonon frequency will minimize the difference between the simulation curves and experimental TDTR traces at all modulation frequencies.

We solve the inverse problem using a particle swarm optimization (PSO) method to search for the optimal profile. The goal of the PSO method is to minimize the objective function defined as $$f = \alpha|g_{ab-initio}(T_{12}(\omega)) - g_{measured}| + (1-\alpha)\int\left(\frac{d^2T_{12}}{d\omega^2}\right)^2d\omega. \label{eq:ObjectiveFunction}$$ The first part of the equation evaluates the norm of the difference between the experimentally measured and BTE-simulated TDTR signals given a transmission profile profile $T_{12}(\omega)$. The 