Quantifying uncertainty in soot volume fraction estimates using Bayesian inference of auto-correlated laser-induced incandescence measurements

Abstract

Auto-correlated laser-induced incandescence (AC-LII) infers the soot volume fraction (SVF) of soot particles by comparing the spectral incandescence from laser-energized particles to the pyrometrically inferred peak soot temperature. This calculation requires detailed knowledge of model parameters such as the absorption function of soot, which may vary with combustion chemistry, soot age, and the internal structure of the soot. This work presents a Bayesian methodology to quantify such uncertainties. This technique treats the additional “nuisance” model parameters, including the soot absorption function, as stochastic variables and incorporates the current state of knowledge of these parameters into the inference process through maximum entropy priors. While standard AC-LII analysis provides a point estimate of the SVF, Bayesian techniques infer the posterior probability density, which will allow scientists and engineers to better assess the reliability of AC-LII inferred SVFs in the context of environmental regulations and competing diagnostics.

Appendix A: Marginalization algorithm

Here we present the pseudo-algorithm for computing the posterior density:

Discretise the expected range of soot volume fraction and temperature, $f_{\text{v}}^{1} ,f_{\text{v}}^{2} , \ldots ,f_{\text{v}}^{n}$ and T ¹, T ², …, T ^m.

Multiply the computed π(f _v, T _p|v _λ ) by a constant, so when integrated over all possible values of f _v and T _p, it integrates to unity.

In the above algorithm, the prior density of E(m _λ ) (π _pri) is a delta distribution when a fixed value is used as the absorption parameter.