Analysis and Uncertainty Quantification of Thermal Transport Measurements through Bayesian Parameter Estimation

TL;DR

This work argues that Bayesian parameter estimation (BPE) provides a rigorous, interpretable framework for both fitting FDTR-based thermal models and quantifying uncertainty, by mapping the full parameter landscape and incorporating priors. Through a gold/sapphire FDTR study, the authors compare BPE against traditional approaches (LSR, RSS, MC, MSE mapping), demonstrating that BPE naturally captures multi-parameter correlations and measurement variability, and can reveal biased externally defined inputs (e.g., layer thickness). The results show that BPE can yield tighter, more physically plausible uncertainty and can shift inferred parameters when priors or fit quality are accounted for, especially in the presence of correlated inputs. Overall, the study highlights the practical benefits of BPE for uncertainty quantification in thermal transport measurements and provides code to reproduce the results.

Abstract

The thermal transport community is increasingly interested in rigorous uncertainty quantification (UQ) of their measurements. In this work, we argue that Bayesian parameter estimation (BPE) represents a powerful framework for both analysis/fitting and UQ. We provide a detailed walkthrough of the technique (including code to duplicate our results) and example analysis based on measuring the thermal conductance of a gold/sapphire interface with FDTR. Comparisons are made against traditional analysis/UQ techniques adopted by the thermal transport community. Notable advantages of BPE include the interpretability of its results, including the capacity to indicate incorrect input assumptions, as well as a way to balance overall goodness of fit against prior knowledge of feasible parameter values. In some cases, incorporating this additional information can affect not only the magnitude of error bars but the inferred values themselves.

Figures (13)

• Figure 1: Example workflow for inferring solely the interface thermal conductance $G$ using BPE. All other model parameters are fixed at their externally defined values. In (a), the experimentally measured phase lag data (black) are compared to the overall best-fit model (solid blue), as well as 3 other possible interface conductance values chosen for demonstration purposes (dashed green, yellow, and red). In (b), a single measurement (arbitrarily chosen at 3.07 MHz) is examined to demonstrate how likelihoods of a given model are determined according to their location relative to a normal distribution, with a mean and standard deviation ($\sigma_{\text{meas,}i}$ in Eq. \ref{['eq:gauss_lkl']}) corresponding to the measurement. The actual measurement (dark blue) is shown alongside a hypothetical measurement with the same mean, but an uncertainty $3\times$ greater than what was measured (gray), for demonstration purposes. Again, three possible values of the interface conductance are shown, with decreasing likelihood as they deviate more greatly from the measured value (green, yellow, red). In (c), phase lag measurements at 25 frequencies are aggregated, the resulting MSE (solid black) corresponding to different possible values of the interface conductance is shown, and the minimum MSE point is designated (dashed green, vertical). In addition, the probability distributions generated by BPE are shown, both for the actual measurements (blue) and the hypothetical set of measurements with $3\times$ the magnitude of the uncertainties (gray).
• Figure 2: RSS uncertainty contribution analysis. The influence of each parameter's uncertainty on the measured interface thermal conductance is plotted individually. Subsequently, these individual contributions are aggregated as outlined in Eq. \ref{['eq:RSS']}, and displayed on the right, in red. Each parameter's relative contribution to the aggregated uncertainty is shown below. $r$ represents the laser radius, while $k, L,$ and $C_p$ represent the thermal conductivity, thickness, and heat capacity of a layer respectively.
• Figure 3: Resulting probabilities from the multi-parameter inference with a uniform prior. The four on-diagonal plots (a, c, f, j) represent the marginalized probability distributions of each individual parameter, with the y-axis representing the relative probability that any individual parameter value is correct. The six off-diagonal plots (b, d-e, g-i) represent probability heatmaps of each pairwise combination of parameters, showing the most likely pairwise combinations of the parameter values.
• Figure 4: Comparison of model fits obtained using uniform (a) vs. Gaussian (b) priors. Gray curves denote the associated prior used for the externally defined input parameters, while green and blue bars signify the probabilities of the resulting inferences.
• Figure 5: Resulting histogram of traditional LSR fits obtained from 25 distinct sample positions. Presented for comparison are the BPE probability distributions, associated with all three choices of prior.