Maximum likelihood filtering for particle tracking in turbulent flows

Abstract

Lagrangian particle tracking enables practitioners to study various concepts in turbulence by measuring particle positions in flows of interest. These data are subject to measurement errors; filtering techniques are applied to mitigate these errors and improve the accuracy of analyses utilizing the data. We develop a new type of position filter through use of maximum likelihood estimation by considering both measurement errors and stochastic process physics. The maximum likelihood estimation scheme we develop is general, enabling it to be applied to many different turbulent flows. We propose a process model similar to existing, complimentary work in the development of B-splines. We compare our filtering scheme to existing schemes and find that our filter out performs the scheme proposed by (Mordant et al. Physica D 193(1):245–251, 2004) considerably, and produces similar performance to spline filters, proposed by (Gesemann arXiv preprint arXiv:1510.09034, 2015). We note that the maximum likelihood treatment provides a general framework which is capable of producing different filters based on the physics of interest, whereas the spline filters are built on less specific filtering theory and are therefore more difficult to adapt across diverse use cases in fluids. We quantify the performance of each of the filtering methods using error metrics which consider both noise reduction as well as signal degradation, and together these are used to define a concept of filter efficiency. The maximum likelihood filter developed in this work is applied to simulated isotropic turbulence data from the Johns Hopkins Database.

Appendix 1 Proof of Proposition 1

Proof

$$\begin{aligned} \underset{x}{\min }\ \hspace{0.25cm} \frac{1}{2 \sigma ^2}||y - x||^2 +\frac{m^2}{2 \sigma _F^2}||\Delta A x||^2 \end{aligned}$$

is equivalent to

$$\begin{aligned} \underset{x}{\min }\ \hspace{0.25cm} \frac{1}{2}(x-y)^T(x-y) +\frac{\mu ^2}{2} x^T (\Delta A)^T \Delta A x \end{aligned}$$

with $\mu = \frac{m \sigma }{\sigma _F}$. Computing the gradient of the objective with respect to x, and setting it equal to zero, yields the optimization condition

$$\begin{aligned} (x-y) +\mu ^2 (\Delta A)^T \Delta A x = 0. \end{aligned}$$

This is simplified through standard manipulations to the expression

$$\begin{aligned} (I + \mu ^2 A^T \Delta ^T \Delta A)x = y. \end{aligned}$$

The matrix $\mu ^2 A^T \Delta ^T \Delta A$ is positive semi-definite, and therefore the matrix sum $I + \mu ^2 A^T \Delta ^T \Delta A$ is positive definite and thus permits an inverse in general. Therefore the minimizing x of the objective is generally written in terms of this inverse as

$$\begin{aligned} x = (I + \mu ^2 A^T \Delta ^T \Delta A)^{-1}y. \end{aligned}$$

$\square$