In a recent publication, Barrios and Dios [1] claim that an exponentiated Weibull (EW) probability density function (PDF) is an excellent fit to both simulation and experimental data under all aperture averaging conditions (including point apertures) for weak and moderate scintillation conditions. The EW PDF, given in Eq. (7) of [1] and given here as a convenient reference has three parameters denoted by α, β, and η:

We show in this comment that the values of α, β, and η obtained by fits to the experimental data, as performed in [1], are inconsistent with the values obtained from Eqs. (10)–(12). The disconnect between the parameter values derived from the fit and those derived from physical quantities (Eqs. (10)–(12)) and other inconsistencies discussed below, render the EW model proposed by the authors ineffective for predicting aperture averaged scintillation statistics and optical communications performance.

The experimental results under weak to moderate turbulence conditions for four aperture diameters are shown by the authors in Figs. 4(a)–4(d) of [1], along with the EW “fit” values for α, β, and η. The parameter values are listed in Table 1 below. From the values of _{${\sigma }_I^2$} and, D/ρ₀ also shown in Figs. 4(a)–4(d), we calculate α, β, and η from Eqs. (10)–(12), and display them in parentheses in Table 1 for comparison. Plots of the PDFs generated from the EW model, expressed by Eq. (7) of [1], using the fit and calculated (in parentheses) values of α, β, and η are shown in our Fig. 1(a) –1(d), below. All four test cases clearly reveal that the PDFs generated from the “fit” and “calculated” parameter sets are substantially different. Therefore, modeling of the aperture averaging using the EW distribution is unjustified, as it has no connection to physical parameters, and cannot be used as a predictive tool for performance.

Table 1. Parameter values from conditions specified in Fig. 4 of [1] along with the corresponding values obtained from Eqs. (10)–(12) in parentheses.

View Table

 

Fig. 1 Comparison of the EW probability density functions for the fit (green curves) and theoretically predicted (blue curves) for the experiments of [1] Figs. 4(a)–4(d).

Download Full Size | PDF

Further, we note an inconsistency between “fit” parameters obtained for the simulation (Fig. 2) and the experiment data (Fig. 4). For example with a 3 mm aperture (Figs. 2(a) and 4(a)), D/ρ₀ = 0.32 for both simulation and experiment, whereas the values of _{${\sigma }_I^2$} differ by less than 2% (0.486 vs. 0.477). We would therefore expect the two sets of fit parameters,α, β, and η, to be nearly identical. This, however, is not the case as α, β, and η differs by about 69% (2.4 vs. 4.05), 12% (0.95 vs. 0.84), and 53% (0.55 vs. 0.84), respectively.

Also, in Fig. 3 Barrios and Dios present plots of the fitted and calculated (from Eqs. (10)–(12)) parameters as a function of the Rytov variance, _{${\sigma }_R^2$}. Now Eqs. (10)–(12) are functions of D/ρ₀, and the aperture averaged scintillation index,_{${\sigma }_I^2$.} Although ρ₀ can be expressed as a function of _{${\sigma }_R^2$,} we are not aware of any direct relationship between _{${\sigma }_I^2$}and _{${\sigma }_R^2$} and are at a loss as how the estimated values of α, β, and η shown in Fig. 3 were obtained. Further, as stated in the text, we find that the parameter values obtained from the fits and those obtained from Eqs. (10)–(12) are not consistent.

Additionally in the strong saturated turbulence limit, in contrast to what is stated in the last paragraph of [1], the EW PDF defined in Eq. (7), becomes the negative exponential distribution exp[-I] if and only if both α = 1 and β = 1. Further it is well known that under these conditions not only does the PDF of irradiance become a negative exponential but also _{${\sigma }_I^2$} for a point receiving aperture becomes unity, consistent with Eq. (11), for α, β = 1. From Eq. (10), however, the condition α = 1 implies that D/ρ₀ = 14.0, which by definition indicates significant aperture averaging and which must necessarily yield a non-unity value of the aperture averaged scintillation index, _{${\sigma }_I^2$}. Thus, the EW theory leads one to a physically inconsistent conclusion and is likewise inappropriate for this turbulence regime as well.

Finally, Barrios and Dios note that the EW distribution is superior to the gamma-gamma distribution. This is expected because the gamma-gamma distribution has two parameters while the EW distribution has three parameters. Mathematically, one could construct a PDF that has four or more free parameters and obtain even better fits to data than is obtained for the EW distribution. However, such distributions, as well as the EW distribution, are not physically based. In conclusion we find that although a mathematically fitted tractable formula for the PDF obtained from the data that describes a measured distribution may be useful, the EW distribution cannot be used as a predictive irradiance statistics model for other physical conditions.