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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 18 — Aug. 27, 2012
  • pp: 20680–20683
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Exponentiated Weibull distribution family under aperture averaging Gaussian beam waves: comment

H. T. Yura and T. S. Rose  »View Author Affiliations


Optics Express, Vol. 20, Issue 18, pp. 20680-20683 (2012)
http://dx.doi.org/10.1364/OE.20.020680


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Abstract

Recently, an exponentiated Weibull distribution model was presented for describing the effects of aperture averaging on scintillation of Gaussian beams propagating through atmospheric turbulence. The model uses three parameters that are derived from physical quantities so that in principle the model could be used to predict optical link performance. After reviewing this model, however, we find several inconsistencies that render it unusable for this purpose under any scintillation conditions.

© 2012 OSA

In a recent publication, Barrios and Dios [1

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef] [PubMed]

] 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

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef] [PubMed]

] and given here as a convenient reference has three parameters denoted by α, β, and η:
fEW(I;β,η,α)=αβη(Iη)β1exp[(Iη)β]{1exp[(Iη)β]}α1,
(7)
where I denotes the irradiance normalized to its mean. The authors relate these parameters directly to observable quantities (i.e., the receiver aperture diameter, D, the aperture averaged scintillation index, σI2, and the atmospheric coherence radius, ρ0) according to Eqs. (10)(12) of their publication as shown below:
α3.931(Dρ0)0.519,
(10)
β(ασI2)6/11,
(11)
η=1αΓ(1+1/β)g(α,β),
(12)
where g(α,β)=i=0(1)i(i+1)(1+β)/βΓ(α)i!Γ(αi), and Γ()is the gamma function. Note, in particular, when α = 1 and β = 1, Eq. (12) yields η = 1 and the EW distribution in Eq. (7) becomes a negative exponential distribution, exp[–I] (in this regard see below).

We show in this comment that the values of α, β, and η obtained by fits to the experimental data, as performed in [1

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef] [PubMed]

], 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

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef] [PubMed]

], along with the EW “fit” values for α, β, and η. The parameter values are listed in Table 1

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

table-icon
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below. From the values of σI2 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

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef] [PubMed]

], using the fit and calculated (in parentheses) values of α, β, and η are shown in our Fig. 1(a)
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).
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.

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 σI2 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, σR2. Now Eqs. (10)(12) are functions of D/ρ0, and the aperture averaged scintillation index,σI2. Although ρ0 can be expressed as a function of σR2, we are not aware of any direct relationship between σI2and σR2 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

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef] [PubMed]

], 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 σI2 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, σI2. 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.

Acknowledgment

We thank S. G. Hanson for bringing [1] to our attention. This work was supported by The Aerospace Corporation’s Strategic Initiative funding.

References and links

1.

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef] [PubMed]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: June 26, 2012
Revised Manuscript: July 20, 2012
Manuscript Accepted: July 23, 2012
Published: August 23, 2012

Citation
H. T. Yura and T. S. Rose, "Exponentiated Weibull distribution family under aperture averaging Gaussian beam waves: comment," Opt. Express 20, 20680-20683 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20680

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