Recently we proposed the exponentiated Weibull (EW) probability density function (PDF) as a useful distribution to model the received power fluctuations in an optical link, through the atmospheric turbulence, under aperture averaging conditions [1].

The EW PDF is given by Eq. (7) in [1] and has three free parameters, namely, two shape parameters α and β, and one scale parameter η.

This work has provoked critical comments by H. T. Yura and T. S. Rose, in which they set several reasons by which, in their opinion, the proposed EW distribution has little, if any, usefulness to model the power fluctuations in an atmospheric optical link. We will try here to briefly defend the proposed model from this criticism.

We understand that the main concerns of the authors are twofold.

Firstly they focus the attention on the fact that there it exists an inconsistency between the values of the EW parameters obtained from Eqs. (10)–(12) in [1] and the ones derived from the direct fitting over the experimental reported data. Yura and Rose show a graphical comparison of the distributions obtained in both cases in their comment to support this statement. Second, they argue that there is not a physical justification supporting the EW model and, consequently, from a pure mathematical point of view, one can build other kind of distributions with three, four or more free parameters to obtain even a better fits to data than is obtained with the EW distribution.

On the former point, we have to agree with them. As we explained in [1] we were very aware of the limitations of such equations, which are only a first approximation to relate the distribution free parameters with those defining the link and the atmospheric conditions. However, it is noteworthy to say that we derive this set of equations from the simulations results, and there the consistency is clearly better, as shown in Fig. 1 of this reply. From the figure below is readily seen that Eqs. (10)–(12) in [1] are in close agreement to the simulation data and the best fit curve (green solid line), except for the receiving aperture diameter D = 3 mm. This situation is expected, and has been addressed accordingly in [1], as the expressions derived for the EW parameters do not have the expected performance for point-like apertures (see [1, Fig. 3]).

 

Fig. 1 Simulation data results from [1, Fig. 2]. The Gamma-Gamma and the Weibull fit curves are dropped from the original figure. A curve for the EW model using [1, Eqs. (10)–(12)] is added with a magenta solid line and labeled as EW*.

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Certainly it exists apparent problems regarding the congruence of the values of the distribution parameters derived from experimental data. However, we think the problems do not come necessarily from the proposed analytical equations, but they rely also on the own nature of the available data. From a set of experimental data it is possible to obtain an average value of the refractive-index structure parameter $C_n^2$ and scintillation index ${\sigma }_I^2$, and then, also for the Rytov variance ${\sigma }_R^2$. However these data are not free of unexpected variations, as the $C_n^2$ could be not constant over the whole path and, even, a certain degree of non-stationarity will be ever present. Having these problems in mind, it is a more secure procedure to work with data obtained from simulations, where all the observable parameters are controlled. In this respect the set of equations provided in [1] has a much better behavior with these kind of data, with the exception of the smaller aperture, as it is already mentioned in the paper. Moreover, this situation is also found in [2] where the discrepancies between experimental data and those obtained from simulations arises up to a 25 % in α and up to a 93 % in β.

Although, of course, this is not enough justification we also have to say that currently we are using a slightly different set of equations, improved with respect the first version reported in [1], which will be published soon [3]. Moreover, the discrepancy between the values of the parameters obtained from some analytical expressions and the values derived from direct fitting is not only a problem for the EW distribution: other researchers have found the same problem with the well-accepted Gamma-Gamma distribution. In this regard the following quote, from Vetelino et al. [2], offers additional insight on this issue: “However, the scintillation index calculated directly from the simulation data does not match predictions made by the scintillation theory. Hence, the ${\sigma }_{\text{ln}I}^2$ parameter in the LN distribution was determined by using Eq. (2) with the scintillation index calculated directly from the simulation data. The α and β parameters in the GG distribution were obtained by doing a best fit to the simulation data...”. Here it becomes evident that, even for well accepted and established models in the literature, the expressions to derive the model’s parameters—supported by the theory— do not always offer good predictions. Other authors have reported this issue as well [4].

With respect to the second concern pointed above, we have to say that certainly at the time of the publication of [1], we had not an enough developed physical model to explain the appearance of the EW distribution at the time. We have been working in that essential point and a justification will be published soon [3]. However the last comment made by Yura and Rose seems not be right, from our personal experience. They argue that the fact of having three free parameters in the EW distribution makes much easier to find a fit with experimental or simulation results. However in the previous studies before the publication of [1], we tested a not small number of three-parameter distributions and any of them, with the notable exception of the EW, was able to reproduce our available data. Among the distribution tested are the exponentiated exponential [5] and exponentiated Gumbel distribution [6]. Following the strategy of the exponentiated distribution, we also tried to fit the irradiance data with the exponentiated Log-normal and exponentiated Gamma—here we used its two-parameter version— distributions. Additionally, a modified Weibull distribution was tested [7].

Finally, regarding the Yura and Rose concern about the EW model, defined in Eq. (7) in [1], not becoming the negative exponential, when using Eqs. (10)–(12) in [1], it is noteworthy to recall that in Sec. 5 of [1] we annotate this discrepancy. Nevertheless, if one is certain that the saturation regime will occur the equations for the classical Weibull distribution indeed degenerate to the negative exponential model. A similar scenario can be found in the seminal paper of Al-Habash et al. [8], where the GG model approaches the negative exponential distribution in the saturation regime when α → ∞ and β → 1. Analyzing the data for the saturation regime in Figs. (3)–(5) in [8], where ${\sigma }_R^2=25$, the β parameter tends to unity, however the α parameter does not become unbounded. Furthermore the PDF plots clearly do not correspond to a negative exponential distribution.

We like to finish this reply mentioning that we understand that this is still, of course, an open field, where other better mathematical functions and with a solid physical foundation may surge.