The wavelength dependent model of extinction in fog and haze for free space optical communication

Martin Grabner; Vaclav Kvicera

doi:10.1364/OE.19.003379

1. Introduction

Free-space optical (FSO) communication systems are influenced by the propagation impairments due to adverse weather conditions. Particularly, the scattering on the atmospheric aerosols/hydrometeors may cause significant extinction of an optical signal resulting in system outage.

Relation between atmospheric visibility and optical extinction in atmospheric hydrometeors is often applied during link margin analysis to estimate the availability performance of FSO communication systems [1–3]. This relation is dependent on the physical characteristics of hydrometeors as well as on the wavelength. In the sections 1.1 and 1.2 below, several available models of the relationship between extinction and atmospheric visibility are summarized. Some models are based merely on the empirical data [4,5], the other ones were obtained using some theoretical considerations [6,7]. After analyzing these models however, there seems to be no general agreement on the wavelength dependence of optical attenuation due to scattering in hydrometeors such as haze, mist and fog. Furthermore the models do not offer any suitable means to adapt their behavior to local specific climate conditions.

It was pointed out by several authors, e. g., [6], that it is mainly the different radius of scatterers that is responsible for different wavelength dependence of light extinction in dense fogs in comparison with the extinction in haze conditions. Nevertheless no attempts to date are known to authors that would reveal this difference explicitly in the form of relatively simple model equations. Therefore the wavelength dependence is analyzed using Mie scattering theory and a new model that takes the radius of scatterers into account is proposed for a limited range of wavelengths.

1.1 Visibility and Kruse model

The atmospheric visibility is defined as a distance where the 550 nm collimated light beam is attenuated to a fraction (5% or 2%) of an original power. Several models relate an extinction coefficient γ (1/km) and visibility V (km). The specific optical attenuation is obtained from the extinction coefficient by A (dB/km) = 10 log(e) γ.

The model derived in [4] has served a long time to the infrared communication community [8] as the only model providing a wavelength dependent relation between the atmospheric visibility and the extinction coefficient. For visibilities lower than 6 km it reads as follows:

γ (λ) = \frac{3.91}{V} {(\frac{λ}{0.55})}^{- q} where: q = 0.585 V^{1 / 3} for V < 6 km .

Here γ (1/km) is the extinction coefficient, λ (µm) is the wavelength and V (km) is the visibility. Beside the wavelength dependence term (λ/0.55)^-q, the basic relation between γ and V is expressed via an inverse proportional term 3.91/V. This can be obtained directly from the 2% definition of visibility:

\frac{I}{I_{0}} = 0.02 = e^{- γ (0.55 μm) V} \Rightarrow γ (0.55 μm) = \frac{- \ln 0.02}{V}

where I/I₀ is the fraction of the attenuated power intensity to the original intensity. It is clear from the context of the section 5.6 in [4] where the model was introduced that the model is assumed to apply for wavelengths at least in the range from 0.55 to 6 um. However, as pointed out by several authors [6,9], the wavelength dependence of the Kruse model was derived from data gathered during haze conditions and probably not during fog. Therefore, for the lowest visibilities (V < 0.5 km), the formula (1) is less reliable.

1.2 Other models

Kim et al. argued in [6] that the recent empirical data suggests no wavelength dependence of optical attenuation during low visibility conditions at least for wavelengths in the range from 0.55 to 1.55 µm. The modification of the Kruse model was proposed such that a wavelength dependence is removed altogether (q = 0) at the lowest visibilities (V < 0.5 km).

Al Naboulsi et al. introduced in [7] a slightly different approach to fog attenuation modeling. They considered two specific types of fog – advection and convection (also called radiation) fog. Then using FASCOD software, they derived interpolation formulas applicable at the wavelengths from 0.69 µm to 1.55 µm for visibilities between 0.05 and 1 km. Al Naboulsi models were compared with experimental data measured on the site “La Turbie” at Nice, France where the transmission of light with the wavelengths 0.85 and 0.95 µm through dense fog was investigated.

Ferdinandov et al. recently introduced in [5] a model that is claimed to be valid for the visibilities 0.1 < V(km) < 50 and in the range of wavelengths between 0.3 and 1.1 µm. It was derived by fitting the experimental data published in literature. It seems however that the used experimental data was not obtained during dense fog conditions.

Nebuloni analyzed in [10] extinction data available in three infrared regions to obtain the coefficients of a power law model γ = aV^b fitting best the measured data at 1.2, 3.7 and 10.6 microns.

The general trend of the wavelength dependence is different for different models. The Kruse and Ferdinandov models describe decreasing trend for all visibilities, the Al Naboulsi models describe increasing attenuation with wavelength for visibilities up to 1 km and the Kim model gives no wavelength dependence for the lowest visibilities V < 0.5 km and it is similar to the Kruse model for higher visibilities V > 1 km.

2. Scattering theory: extinction vs. effective radius

An extinction coefficient γ of the hydrometeor can be calculated (within the first order multiple scattering approximation) as:

γ = \int_{0}^{\infty} π r^{2} n (r) Q_{e x t} d r

where r is the particle radius, n(r) is the particle (drop) size distribution and Q_ext is the extinction efficiency factor [11]. A gamma type distribution is widely used for representing the microphysical properties of clouds and fogs [12]:

n (r) = a r^{α} e^{- b r}

where a, b and α are adjustable parameters of the distribution. It is appropriate to interpret these model parameters in terms of physical quantities. A particularly fruitful approach is to define an effective radius r_e [11] of the distribution that well characterizes the whole distribution from the point of view of an integrated geometrical cross section of all particles. The effective radius is proportional to a ratio of the total volume and the total geometrical cross section of particles in the distribution:

r_{e} = \frac{\int_{0}^{\infty} r^{3} n (r) d r}{\int_{0}^{\infty} r^{2} n (r) d r} = \frac{3}{4} \cdot \frac{total volume}{total geom
. cross section} .

Substituting (4) into (5), one can find for the effective radius that

r_{e} = (α + 3) / b

.

If r_e is fixed, the shape parameter α is inversely related to the dispersion of the distribution. Numerical experiments reveal that the extinction coefficient is not very sensitive to the shape parameter α when r_e is fixed. As an additional descriptive parameter of the distribution, a liquid water content LWC is chosen because it is proportional to the total volume of particles. For fixed LWC, the total geometrical cross section obtained from (5) is inversely proportional to the effective radius. In the small wavelength limit where λ → 0, the extinction efficiency factor Q_ext → 2 and so the extinction cross section C_ext = πr ² Q_ext is simply twice the geometrical cross section. Then the extinction coefficient is twice the total geometrical cross section:

γ = 2 \int_{0}^{\infty} π r^{2} n (r) d r = \frac{3 L W C}{2 ρ} \frac{1}{r_{e}}

where ρ stands for a water density and the second equality in (6) comes from (5) and from the definition of LWC. The consistent units are assumed in (6), the additional factor of 1000 has to be included into (6) if the quantities are expressed in practical units such as γ (1/km), r and r_e (µm), LWC (g/m³), ρ (g/cm³) and n(r) (cm⁻³µm⁻¹).

The integral in (3) has been evaluated numerically (by an adaptive Simpson rule with relative accuracy better than 10⁻³) with the extinction efficiency factor Q_ext calculated using full Mie formulae [11]. The complex refractive index of water was taken from [13]. Figure 1 shows the wavelength dependence of the extinction coefficient for fixed LWC, α and different values of r_e. In the short wavelength limit, one can check in Fig. 1 that the extinction coefficient is given by γ(km⁻¹) = 1500/r_e(µm) corresponding to (6) with LWC = ρ = 1. In the region where applicable (see Fig. 1), a simple relation between the extinction coefficient, the liquid water content and the effective radius is so obtained:

γ = 1500 \frac{L W C}{r_{e}}

where γ (1/km), LWC (g/m³) and r_e (µm). The region of sufficient accuracy of (7) with a relative error less than 10% is determined by the condition: λ < r_e/5. In order to analyze the extinction dependence on visibility, the relative extinction coefficient measured relatively to the extinction coefficient at the 0.55 μm wavelength is of interest. The relative extinction is independent of LWC provided r_e and α are fixed. It is also clear from Fig. 1 that there is no generally monotonic wavelength dependence except in some interval around 0.55 μm. It seems therefore that the exponent q in (1) may serve as some approximation only in the interval 0.2 μm < λ < 2 μm that is focused on in the following.

Fig. 1 Extinction coefficient for different effective radii, LWC = 1 g/m³, α = 5

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The wavelength dependence of the relative extinction coefficient in the mentioned interval is depicted in Fig. 2 . The whole interval was divided into two subintervals: λ < 0.55 μm and λ ≥ 0.55 μm and a linear approximation of the log-log wavelength dependence was found in both of them, see the straight lines in Fig. 2.

Fig. 2 Relative extinction and the linear approximation of its log-log wavelength dependence, α = 5

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The slope of the log-log dependence s (the exponent in (10) below) as a function of the effective radius was obtained. The s(r_e) function can be approximated by the following expression:

s = 2 (\tanh (p_{1} (w + p_{4})) - 1) + p_{2} \exp (- p_{3} {(w + p_{5})}^{2})

w = \log_{10} r_{e}

where r_e (μm) and the parameters p_i for the two wavelength subintervals are summarized in the Table 1 .

Table 1. Parameters of the model (8) for two wavelength subintervals

View Table

Figure 3 shows the exponent s obtained for different r_e (points) and the interpolating functions (8) (lines) with the parameter values stated in Table 1. The function (8) was chosen so that the limit of s as r_e → 0 is s = −4 which corresponds to the Rayleigh scattering. The limit of s as r_e becomes large is s = 0 which corresponds to the geometrical optics approximation. Similarly as in (1), the wavelength dependence of the extinction coefficient can be expressed as:

γ (λ) = γ (0.55 μm) {(\frac{λ}{0.55})}^{s}

where γ (1/km), λ (μm). The extinction coefficient at the wavelength of 0.55 μm, γ(0.55μm), is to be obtained from the visibility using its definition as in (2).

Fig. 3 The exponent of wavelength dependence γ ~λ^s, α = 5

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3. Relation of the effective radius and atmospheric visibility

The model expressed by Eqs. (8)-(10) is based on the underlying physics as it takes the dependence on the effective radius of the particular hydrometeor into account. Both the visibility and the effective radius are the inputs of the model. Nevertheless it may be difficult in practice to get relevant information about the size of scattering particles. It is therefore desirable to estimate the relationship between the effective radius and visibility in order to express the extinction coefficient as a function of visibility only.

For that purpose, let us consider the following facts:

a) The relation between the extinction coefficient γ and the liquid water content LWC is often (e.g [14].) empirically modeled by a power-law equation: $γ = γ_{0} {(\frac{L W C}{L W C_{0}})}^{c}$

where γ ₀ stands for the extinction coefficient corresponding to the particular liquid water content LWC ₀. The value of the exponent c is usually between 0.5 and 1, but it is wavelength dependent generally.

b) The extinction coefficient γ is linearly proportional to LWC when both the effective radius r_e and the shape parameter α are fixed. It means c = 1 in such a case.
c) For a fixed LWC (and fixed α) and λ = 0.55 μm, the effective radius r_e is inversely proportional to the extinction coefficient provided r_e > r_e _min where r_e _min is about 0.5 μm.
d) It is from the definition of visibility that γ/γ ₀ = V ₀/V for the wavelength λ = 0.55 μm where V ₀ stands for the visibility corresponding to γ ₀ and LWC ₀.

Now, let us keep α constant, since it seems to be the least sensitive parameter in our context. Then it is evident from the points a) and b) above that if the empirical exponent c ≠ 1 then r_e has to vary with LWC. Consider the effective radius r_e ₀ corresponding to the LWC ₀. The further step is to find the values of effective radius r_e corresponding to the values of LWC different from LWC ₀. Figure 4 shows that (for example) for c = 2/3 and LWC = 0.01 g/m³, the effective radius r_e < r_e ₀ (see the point c) above) and it depends on the ratio of extinction coefficients γ and γ ₁.

Fig. 4 The scheme of the relation between the effective radius and the liquid water content

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So using a), b) and c), one finds the following equation (see parameters in Fig. 4):

\frac{r_{e}}{r_{e 0}} = \frac{γ_{1}}{γ} = {(\frac{L W C}{L W C_{0}})}^{1 - c} where γ_{1} = γ_{0} {(\frac{L W C}{L W C_{0}})}^{1}

from which one has:

r_{e} = r_{e 0} {(\frac{L W C}{L W C_{0}})}^{1 - c} = r_{e 0} {({(\frac{γ}{γ_{0}})}^{1 / c})}^{1 - c} = r_{e 0} {(\frac{γ}{γ_{0}})}^{(1 / c) - 1} = r_{e 0} {(\frac{V_{0}}{V})}^{(1 / c) - 1} .

In the last equality in (13), the point d) was applied, because it was assumed that the empirical exponent c was obtained for the wavelength λ = 0.55 μm. The relation (13) gives the possibility to deduce the effective radius from the visibility if the particular r_e ₀, V ₀ and c are known from experiments. For example, consider typical values c = 2/3 (from [14,15]), r_e ₀ = 10 μm ([16]), V ₀ = 0.05 km. The Eq. (13) then reads:

r_{e} = 10 {(\frac{0.05}{V})}^{1 / 2}

and it can be substituted to (9) to calculate the exponent s as a function of visibility V.

4. Model summary

The new proposed model is applied as follows. First, the effective radius r_e is estimated from visibility using (13). Second, the exponent of wavelength dependence s is calculated using (8) and (9). Finally the extinction coefficient γ is obtained using (10).

5. Comparison with measured data

Figure 5 shows the example of a fog event observed in Prague, the Czech Republic. The two parameters of fog are measured [16]: liquid water content LWC (g/m³) and particulate surface area PSA (cm²/m³). The effective radius, r_e (μm) is then calculated using its definition (5) as: r_e = 30000∙LWC/PSA. It can be seen in Fig. 5 that the r_e fluctuates around about 10 microns during the developed fog phase. It is expected, however, that the typical effective radius depends on local climatic conditions [17] and so the Eq. (14) is to be modified accordingly.

Fig. 5 Liquid water content, LWC, particulate surface area, PSA, and effective radius of fog measured during a fog event observed on 20-21 January 2009 in Prague, the Czech Republic.

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Figure 6 shows a comparison of different models applied at two typical FSO wavelengths. The measured (two-year) data [18] obtained on two FSO paths located in Prague are also shown in Fig. 6 as an empirical reference. A reasonable agreement of the proposed model with the measurement is achieved at least for visibilities lower than 2 km.

Fig. 6 Specific optical attenuation vs visibility predicted by different models and measured data [18], the wavelength 1.55 μm shifted downwards (multiplied by 1/10) for better clarity.

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6. Conclusion

The wavelength dependent model of the relation between atmospheric visibility and the extinction coefficient of fog and haze was proposed based on the results of applied Mie scattering theory. The model is applicable in the interval of wavelengths 0.2 μm < λ < 2 μm and for visibilities lower than 10 km. The presented modeling approach reveals more explicitly the connection between the extinction coefficient and the microphysical parameters of fog and haze. It also provides some degree of flexibility to adapt its properties according to locally specific atmospheric conditions.

Acknowledgement

This research is supported by the Czech Science Foundation under the project No. P102/11/1376.

References and links

1. H. Willebrand, and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks (SAMS, Indianapolis, 2002), Chap. 3.

2. O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P. Favennec, Free-Space Optics, Propagation and Communication (ISTE, London, 2006), Chap. 4.

3. A. K. Majumdar, and J. C. Ricklin, eds., Free-Space Laser Communications (Springer, New York, 2008).

4. P. W. Kruse, L. D. McGlauchlin, and R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission and Detection (Jonh Wiley & Sons, New York, 1962), Chap. 5.

5. E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).

6. I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001). [CrossRef]

7. M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43(2), 319–329 (2004). [CrossRef]

8. P. Corrigan, R. Martini, E. A. Whittaker, and C. Bethea, “Quantum cascade lasers and the Kruse model in free space optical communication,” Opt. Express 17(6), 4355–4359 (2009). [CrossRef] [PubMed]

9. K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. 87(3-4), 252–258 (2008). [CrossRef]

10. R. Nebuloni, “Empirical relationships between extinction coefficient and visibility in fog,” Appl. Opt. 44(18), 3795–3804 (2005). [CrossRef] [PubMed]

11. H. C. van de Hulst, Light Scattering by Small Particles, (Dover Publications, New York, 1981).

12. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, (American Elsevier Pub. Co., New York, 1969), Chap. 4.

13. D. Segelstein, The Complex Refractive Index of Water, (University of Missouri, Kansas City, 1981).

14. R. G. Eldridge, “Haze and fog aerosol distributions,” J. Atmos. Sci. 23(5), 605–613 (1966). [CrossRef]

15. P. Chýlek, “Extinction and liquid water content of fogs and clouds,” J. Atmos. Sci. 35, 296–300 (1978).

16. M. Grabner, and V. Kvicera, “On the relation between atmospheric visibility and the drop size distribution of fog for FSO link planning,” in Proceedings of the 35th European Conference on Optical Communication (VDE VERLAG GMBH, Vienna, 2009), pp. 1–2.

17. M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011). [CrossRef]

18. M. Grabner, and V. Kvicera, “Fog attenuation dependence on atmospheric visibility at two wavelengths for FSO link planning,” in Proceedings of Loughborough Antennas & Propagation Conference (Loughborough University, Loughborough, 2010), pp. 193–196.

p_i	0.2 μm < λ < 0.55 μm	0.55 μm ≤ λ < 2 μm
p ₁	2.21888	1.94311
p ₂	0.67214	0.59076
p ₃	8.04794	6.36656
p ₄	0.8	0.45
p ₅	0.3	−0.15

The wavelength dependent model of extinction in fog and haze for free space optical communication

Abstract

1. Introduction

1.1 Visibility and Kruse model

1.2 Other models

2. Scattering theory: extinction vs. effective radius

3. Relation of the effective radius and atmospheric visibility

4. Model summary

5. Comparison with measured data

6. Conclusion

Acknowledgement

References and links

Cited By

Figures (6)

Tables (1)

Equations (14)

Optics Express