ReVAR: A Data-Driven Algorithm for Generating Aero-Optic Phase Screens

Abstract

The propagation of light through a turbulent flow field around an aircraft results in optical distortions commonly known as aero-optic effects. The development of methods to mitigate these effects requires large amounts of realistic aero-optic data. However, methods for obtaining this data, including experiment, computational fluid dynamics, and simple phase screen algorithms (e.g., boiling flow), each have significant drawbacks such as high cost, high computation, limited quantity, and/or inaccurate statistics. More recently, data-driven algorithms have been proposed that are computationally efficient and can synthesize aero-optic data to match the statistics of measured data, but these approaches still have drawbacks including limited quality, inaccurate statistics, and the use of complicated algorithms. In this paper, we introduce ReVAR (Re-whitened Vector AutoRegression), a data-driven algorithm for generating synthetic aero-optic data that matches the statistics of measured data. A key contribution in this algorithm is Long-Range AutoRegression, a linear predictive model that combines a standard autoregression with a set of low-pass filters of the data to fit both short-range and long-range temporal statistics. ReVAR uses Long-Range AR together with a spatial re-whitening step to convert measured aero-optic data to temporally and spatially un-correlated white noise. ReVAR can then generate synthetic aero-optic data by reversing this process using white noise input. Using two measured turbulent boundary layer data sets, we demonstrate that ReVAR better matches the measured data's temporal power spectrum and other key metrics than do two conventional phase screen generation methods and an existing single time-lag autoregressive model.

Figures (7)

• Figure 1: Parameter estimation from measured aero-optic data. This process estimates the parameters $\theta$ of a multivariate Gaussian random process. After normalizing by the time-averaged statistics through Eq. \ref{['eq: Normalizing']} and computing a spatial Principal Component Analysis (PCA) through Eqs. (\ref{['eq: First Spatial PCA']}-\ref{['eq: Top Principal Coefficients']}), ReVAR trains a linear predictive model defined through Eq. \ref{['eq: Long-Range AR Model']}, which we call Long-Range AutoRegression (AR); Long-Range AR uses the long-range predictor of Eq. \ref{['eq: Long-Range Predictor']} (illustrated in Fig. \ref{['fig: Long-Range Predictor']}). Finally, ReVAR takes the residuals of the Long-Range AR model and computes a second spatial PCA using Eq. \ref{['eq: Second Spatial PCA']}. Importantly, this converts the residuals to white noise through Eq. \ref{['eq: Spatial Whitening']}, so we can reverse this process with white noise input to generate synthetic aero-optic data.
• Figure 2: Long-range predictor corresponding to Eq. \ref{['eq: Long-Range Predictor']}. This predictor takes in a time history of data and extracts two components: i) the previous $N_L$ time-steps and ii) two low-pass filters of the entire time series, computed through Eq. \ref{['eq: Low-Pass Filter']}. The predictor then takes a linear combination of these two components with weights $\bm{A}$ to estimate the next vector. We use the low-pass filters to introduce long-range (equivalently, low-frequency) temporal correlations in data generated by this predictor without increasing the number of time-lags $N_L$ to arbitrarily large values. We use two distinct low-pass filters to increase the low-frequency resolution of this predictor.
• Figure 3: Diagram of ReVAR synthesis. This procedure samples from a multivariate Gaussian random process with the parameters $\hat{\theta}$ from Eq. \ref{['eq: ReVAR Parameters']} by taking white noise input and applying the inverse of the transformation illustrated in Fig. \ref{['fig: Parameter Estimation']} through Eqs. (\ref{['eq: Spatially-Correlated Noise']}, \ref{['eq: Low-Pass Filter']}, \ref{['eq: Long-Range Predictor']}, \ref{['eq: Long-Range AR Model']}, and \ref{['eq: Generating Synthetic Data']}). The synthetic aero-optic phase screen data matches the statistics of the measured data.
• Figure 4: Results from each method for data set F06. First and second rows: Comparisons of the slopes and OPD TPS (respectively) obtained from the measured data (green line with circular markers) and synthetic data (orange line with triangular markers), where OPD has units of microns. Boiling flow and Vogel's method both show a mismatch from the measured slopes and OPD TPS at frequencies below 30 kHz and 5 kHz (respectively), while ReVAR (red box) closely matches both measured TPS at all frequencies. Third Row: Comparison of the 2D structure functions obtained from measured data (leftmost column) and synthetic data (second-fifth columns). Isotropic boiling flow does not match the anisotropic statistics of the measured structure function and anisotropic boiling flow only roughly matches its contours, while Vogel's method and ReVAR (red box) both closely match the measured structure function.
• Figure 5: Results analogous to Fig. \ref{['fig: F06 Results']} but for data set F12.