## 1 Introduction  
For compensation and mitigation schemes to work effectively, it helps to know how turbulence is distributed along a path. Scintillometers have been the gold standard for measuring turbulence [1], but scintillometers provide an integrated turbulence measurement, and have limited range capability. Several irradiance-based as well as phase-based techniques to profile turbulence has been discussed in the literature. The Scintillation Detection and Ranging (SCIDAR) technique has been used by astronomers to vertically profile turbulence [2]. The technique uses correlation of scintillation due to a binary star pair at the pupil plane of the telescope. A phase-based technique known as Slope Detection and Ranging (SLODAR) profiles turbulence by measuring the cross-correlation of wavefront tilts due to a binary star pair using a Shack-Hartmann sensor [3]. [PERSON] et al. introduced another phase-based method known as Difference in Differential Tilt Variance (DDTV) that relied on differential tilt measurements between two receiver apertures due to three beacons [4]. [PERSON] et al. have also used differential tilt measurements at a single receiver due to an array of LEDs to characterize turbulence along a horizontal path [5].  
In an earlier work, a time-lapse imaging technique was introduced to measure integrated turbulence parameters remotely from a single site [6]. The technique was applied to images collected over a 7 km slant path from Fitz Hall at University of Dayton to Dayton VA Medical Center. By tracking the motion of window corners on the hospital building, path-weighted estimates of the refractive index structure constant, \(C_{n}^{2}\) were evaluated. The estimates were in good agreement with co-located scintillator measurements. Time-lapse estimates from a second imaging experiment conducted over a 1 km horizontal path close to the ground with cooperative and non-cooperative targets at the Laser Experimental Range at Wright Patterson Air Force Base agreed with scintillometer measurements as well.  
In all of the above-mentioned experiments in time-lapse, a single camera was used to capture images of the scene which enabled estimation of integrated turbulence parameters. In the present work, two spatially separated cameras were used to image an array of LEDs over a 511 m heterogeneous path, half being concrete, the rest grass. The differential tilt variances due to a pair of LEDs sensed by a single camera and between two cameras form a rich set of measurements from which turbulence distribution along the path can be extracted. The profiling results were compared to 3D sonic anemometers placed every 100 m along the path. A second experiment was conducted over a shorter grassy path with a gas heater placed in the middle of the path to generate a localized spike in turbulence. A poster board with white circles against a black background was imaged by the two cameras. Both the experiments were done over an approximately horizontal path very close to the ground. It is worth mentioning here that commercially available systems such as DELTA, developed by MZA Associates Corporation profiles turbulence along a path from time-lapse imagery of a single camera [7]. A richer set of weighting functions, and hence better prospect at profiling can be obtained by use of multiple cameras, as described in this work. The imaging approach is also a low cost, portable approach to profiling when compared to more sophisticated systems such as MZA's PROPS [8]. Methodology for the present work is described in Section 2. Theoretical expressions for the path weighting functions, which are an important component of this work have been derived here. Section 2 also provides a description of the imaging experiments. Profiles derived from time-lapse imagery measurements are compared to anemometer measurements in Section 3. A summary of the work and future research directions is discussed in Section 4.