Uncertainty modeling and error reduction for pathline computation in time-varying flow fields

When the spatial and temporal resolutions of a time-varying simulation become very high, it is not possible to process or store data from every time step due to the high computation and storage cost. Although using uniformly down-sampled data for visualization is a common practice, important information in the un-stored data can be lost. Currently, linear interpolation is a popular method used to approximate data between the stored time steps. For pathline computation, however, errors from the interpolated velocity in the time dimension can accumulate quickly and make the trajectories rather unreliable. To inform the scientist the error involved in the visualization, it is important to quantify and display the uncertainty, and more importantly, to reduce the error whenever possible. In this paper, we present an algorithm to model temporal interpolation error, and an error reduction scheme to improve the data accuracy for temporally down-sampled data. We show that it is possible to compute polynomial regression and measure the interpolation errors incrementally with one sequential scan of the time-varying flow field. We also show empirically that when the data sequence is fitted with least-squares regression, the errors can be approximated with a Gaussian distribution. With the end positions of particle traces stored, we show that our error modeling scheme can better estimate the intermediate particle trajectories between the stored time steps based on a maximum likelihood method that utilizes forward and backward particle traces.