Atmospheric turbulence parameters are estimated for an imaging path based on time-lapse imaging results. Atmospheric turbulence causes frame-to-frame shifts of the entire image as well as parts of the image. The statistics of these shifts encode information about the turbulence strength (as characterized by Cn2, the refractive index structure function constant) along the optical path. The shift variance observed is simply proportional to the variance of the tilt of the optical field averaged over the area being tracked and averaged over the camera aperture. By presuming this turbulence follows the Kolmogorov spectrum, weighting functions, which relate the turbulence strength along the path to the shifts measured, are derived. These weighting functions peak at the camera and fall to zero at the object. The larger the area observed, the more quickly the weighting function decays. One parameter we would like to estimate is r0 (the Fried parameter or atmospheric coherence diameter.) The weighting functions derived for pixel sized or larger parts of the image all fall faster than the weighting function appropriate for estimating the spherical wave r0. If we were to presume that Cn2 is constant along the path, then an estimate for r0 could be obtained for each area tracked, but since the weighting function for r0 differs substantially from that for every realizable tracked area, it can be expected that this approach would yield a poor estimate. Instead, the weighting functions for a number of different patch sizes can be combined through the Moore–Penrose pseudoinverse to create a weighting function that yields the least-squares optimal linear combination of measurements for the estimation of r0. This approach is carried out for one example and is shown to give noisy results. A modified version of this approach that creates larger patches by averaging several smaller patches together solves this noise issue. This approach can also work to estimate other atmospheric parameters.
McCrae, Jack E.; Bose-Pillai, Santasri; and Fiorino, Steven T., "Estimation of Turbulence from Time-lapse Imagery" (2017). Faculty Publications. 127.