Uncertainty amplification due to density/refractive-index gradients in volumetric PTV and BOS experiments

TL;DR

This paper provides a theoretical analysis of how density and refractive-index gradients amplify measurement uncertainties in volumetric PTV and BOS experiments, offering a model to quantify and optimize experimental accuracy.

Contribution

It introduces a model for the diffraction-limited image of particles through density gradients, deriving the CRLB for centroid estimation and proposing a methodology to estimate uncertainty amplification.

Findings

Density gradients cause image blurring, increasing centroid estimation error.

The Amplification Ratio (AR) quantifies uncertainty increase due to density gradients.

Experimental validation shows high AR regions correspond to high density gradients.

Abstract

We theoretically analyze the effect of density/refractive-index gradients on the measurement precision of Volumetric Particle Tracking Velocimetry (V-PTV) and Background Oriented Schlieren (BOS) experiments by deriving the Cramer-Rao lower bound (CRLB) for the 2D centroid estimation process. A model is derived for the diffraction limited image of a particle or dot viewed through a medium containing density gradients that includes the effect of various experimental parameters such as the particle depth, viewing direction and f-number. Using the model we show that non-linearities in the density gradient field lead to blurring of the particle/dot image. This blurring amplifies the effect of image noise on the centroid estimation process, leading to an increase in the CRLB and a decrease in the measurement precision. The ratio of position uncertainties of a dot in the reference and gradient images is a function of the ratio of the dot diameters and dot intensities. We term this parameter the Amplification Ratio (AR), and we propose a methodology for estimating the position uncertainties in tracking-based BOS measurements. The theoretical predictions of the particle/dot position estimation variance from the CRLB are compared to ray tracing simulations with good agreement. The uncertainty amplification is also demonstrated on experimental BOS images of flow induced by a spark discharge, where we show that regions of high amplification ratio correspond to regions of density gradients. This analysis elucidates the dependence of the position error on density and refractive-index gradient induced distortion parameters, provides a methodology for accounting its effect on uncertainty quantification and provides a framework for optimizing experiment design.