# Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence
## 2 Materials and Methods  
### Structure Parameters  
In their widely acknowledged development of atmospheric electromagnetic wave propagation amidst fluctuations associated with the temperature, moisture, and pressure, [PERSON] and then [PERSON] advanced the concept of structure functions. In combination, this led to an analytical basis for quantifying optical turbulence, the correlation of the associated random localized small-scale fluctuations of the former physical parameters, the spatial/temporal characteristics of turbulent eddies in the atmosphere, and, subsequently, the effects on the index of refraction [7,8]. In those instances where the turbulent flow can be characterized as Kolmogorov--isotropic, homogenous, and an associated energy spectrum falling within the inertial sub-range where energy is neither created nor dissipated by the large and small eddies--the structure function is described according to the Kolmogorov-Obhukov two-third power law. For the sake of illustration, to account for temperature fluctuations, the structure function, \(D_{T}\), is expressed as follows [7]:  
\[D_{T}(r)=\left\langle\left[T(r_{2})-T(r_{1})\right]^{2}\right\rangle=C_{T}^{2} \left|r_{2}-r_{1}\right|^{2/3} \tag{1}\]  
where \(C_{T}^{2}\) is the structure parameter for temperature, \(T\), and the brackets, \(\left\langle\right\rangle\), signify an ensemble average. The underlying analytical theory presumed temperature measurements at two points separated by the distance \(r\). In practice though, such micrometeorological parameters are generally measured at a single point as a function of time, \(t\). In order to correlate these temporally varying point measurements to spatial variability in temperature, [PERSON]'s frozen turbulence hypothesis generally is invoked. [PERSON]'s hypothesis postulates the shapes of the turbulent eddies is frozen in time and move perpendicularly past a probe or observer at the convection velocity, \(\left\langle V_{\perp}\right\rangle\), which for [PERSON]'s frozen turbulence hypothesis is taken as equivalent to the local mean wind velocity. Subsequently, Equation (1) can be re-written as follows:  
\[C_{T}^{2}=\left\langle\left[T(t+\Delta t)-T(t)\right]\right\rangle^{2}(\left \langle V_{\perp}\right\rangle\Delta t)^{-2/3} \tag{2}\]  
[PERSON] and [PERSON] [9] (pp. 61-62), however, counter the tenets of this hypothesis through references to field studies that demonstrated larger eddies moved faster than their smaller counterparts. Naturally, such a variable flow and consequent non-uniform momentum, heat, and mass transfer throughout the atmosphere, especially in the mixed atmospheric boundary layer, lead one to discount assumptions of frozen eddy shapes and sizes. Nonetheless, this reference argued as convincingly that such assumptions have been validated through a large number of observations and associated time-averaged statistics. Additionally, many measurements, such as those recorded with sonic anemometers, are conducted at extremely high frequencies, wherein eddies are not likely to change over the course of the measurement period. Specific to optical turbulence and its effects on electromagnetic wave propagation, the causal index of refraction fluctuations are characterized by the structure parameter for the index of refraction, which can be expressed in a form similar to Equation (2) but with regard to each of the physical scalars influencing the atmospheric index of refraction--temperature, moisture, and pressure. In large part, though, scientists and engineers characterizing atmospheric and optical turbulence effects in the visible and near infrared are satisfied with the following straightforward expression tying \(C_{n}^{2}\) to \(C_{T}^{2}\):  
\[C_{n}^{2}=\left(79.2\times 10^{-6}\frac{P}{T^{2}}\right)^{2}C_{T}^{2} \tag{3}\]  
where \(P\) is pressure (hPa) and \(T\) is the ambient air temperature (K).