Effect of density gradients on the Cramer-Rao lower bound for volumetric PIV/PTV measurements

Volumetric PIV/PTV experiments in variable density environments suffer from aero-optical effects due to refraction of light rays passing through the flow field. The density/refractive index gradients affect the Mie-scattered intensity, position and diameter of the projected particle image, and influence the accuracy and precision of the particle position estimates from the projected images. This effect varies with the viewing direction, particle position within the laser sheet, optical setup and density gradients in the flow field. We propose to theoretically study this effect by deriving the Cramer-Rao Lower Bound (CRLB) for the precision of the particle position estimation process. We model the image of a particle in the presence of density gradients, building on a previous particle image model developed for Background Oriented Schlieren (BOS) experiments, and compare theoretical predictions of the particle position estimation variance from the CRLB to ray tracing simulations. The results of this analysis can help elucidate the dependence of the position error on the experimental parameters, aid in experiment design, and provide directions in improving current reconstruction tools to account for these effects.