Solenoidal filtering of volumetric velocity measurements using Gaussian process regression

Abstract

Volumetric velocity measurements of incompressible flows contain spurious divergence due to measurement noise, despite mass conservation dictating that the velocity field must be divergence-free (solenoidal). We investigate the use of Gaussian process regression to filter spurious divergence, returning analytically solenoidal velocity fields. We denote the filter solenoidal Gaussian process regression (SGPR) and formulate it within the Bayesian framework to allow a natural inclusion of measurement uncertainty. To enable efficient handling of large data sets on regular and near-regular grids, we propose a solution procedure that exploits the Toeplitz structure of the system matrix. We apply SGPR to two synthetic and two experimental test cases and compare it with two other recently proposed solenoidal filters. For the synthetic test cases, we find that SGPR consistently returns more accurate velocity, vorticity and pressure fields. From the experimental test cases, we draw two important conclusions. Firstly, it is found that including an accurate model for the local measurement uncertainty further improves the accuracy of the velocity field reconstructed with SGPR. Secondly, it is found that all solenoidal filters result in an improved reconstruction of the pressure field, as verified with microphone measurements. The results obtained with SGPR are insensitive to correlation length, demonstrating the robustness of the filter to its parameters.