1. Introduction
The effect of sand storms, among other phenomena, on signal propagation has gathered notable attention from many studies as communication and distant sensing are being utilized by many civil applications. As sand storms are a common meteorological phenomenon that will interfere with the highly anticipated advantages of wireless connectivity, different authors have attempted to structure mathematical representations of path attenuation to estimate the loss of signal energy, particularly when under unique environments.
Chu [
1] developed the first known attenuation coefficient (
A) expression for propagating waves in dust or sand storms by employing related scattering theories and assuming a monodispersive particle size distribution. Following a series of assumptions, derivations, and substitutions, the author arrived at a general model containing the following parameters:
where
λ is the wavelength in meters (2π/
λ = free space wavenumber),
is the optical attenuation coefficient (dB per km),
is the particle radius in meters, and
of the parenthesized expression corresponds to the imaginary part of the refractive index of the volume of space containing particulates; and
is the complex relative permittivity of the particles given by
.
Furthermore, Chu [
1] demonstrated mathematically, before obtaining the attenuation coefficient expression,
’s relationship to total particle density (
NT), and described it as follows:
The previous relationship helps solve for
, thus Chu’s [
1] attenuation coefficient expression may alternately be expressed as:
However,
and
NT in (2) and (3) impose a challenge in terms of fulfilment either due to difficulties involved with their measurements or because they are statistically very scarce [
2]. Therefore, it has been suggested that visibility should be considered as an alternative parameter as mathematical expressions relate it to
and
NT. Chu [
1] gave the simple relation between the optical attenuation coefficient (
) and visibility (
V) in km as:
where the proportionality constant of 15 dB is simply the 10-log of 0.031, which is a measured median of the normalized difference in luminance between a mark and a reference background [
1]. Patterson and Gillette [
3], on the other hand, developed an empirical relationship between the dust mass concentration (
M) in g/m
3 and the visibility (
V) in km:
where
γ and
C (g m
−3 km) are constants, and according to Patterson and Gillette [
3], their value depends on the dust storm’s geographical area of origin and associated climatic conditions. Essentially, Equation (5) is the ratio of the mass concentration,
M, to the extinction due to particles,
, which is designated by
K. At the same time, the ratio can be related to Koschmieder’s visibility law (
) by equating
to the total extinction
–
that includes gaseous as well as particulate optical effects, which is valid for low visibility conditions (
extinction by the air molecules;
) [
4]. As a result, we obtain:
where 3.912
K is denoted by
C, leading to the relation
. Assuming a consistent particle size distribution changes with increasing or decreasing visibility, Patterson and Gillette [
3] empirically fitted the measurement data with
thus relating
M and
V as demonstrated in Equation (5).
The relationship given by Equation (5) is often used to solve for
NT, assuming a case of polydispersive particle distribution. To illustrate, the particle size distribution term [
5] is introduced into Equation (3), which becomes:
where
is the number of sand/dust particles in the radii range of
. A cubic meter of all dust particles per cubic meter of air, i.e., the total relative volume (
), is given by
At the same time,
has the following relation to the dust mass concentration (
) and the mass particle density (
) in g/m
3Using Equations (8) and (9),
may alternately be expressed as
Considering the form of Equation (8) and the relation given in Equation (10) for the size distribution in Equation (7), we get
Taking advantage of the established relationships and the equations mentioned above, the various authors that have attempted modifying Chu’s [
1] attenuation model adopted the visibility parameter, and below are some of the known attenuation models constructed by them:
Dong et al.’s [
6] attenuation model:
Goldhirsh’s [
7] attenuation model:
Ahmed et al.’s [
2] attenuation model:
Ahmed’s [
8] attenuation model:
Alhaider’s [
9] attenuation model:
Dong et al.’s [
10] attenuation model:
where
F is the frequency in GHz in (12), (15), (16), and (17),
V is the visibility in km in (12) to (17),
re is the effective particle radius in μm in (14) and (15),
r is the particle radius in μm in (16), and λ is the wavelength in meters in (13) and (14).
It is evidently clear that the attempts are very persistent in designating visibility as a critical factor in signal attenuation calculations and as indicative of sand/dust storms intensity. However, while statistical information on dust storm visibility can be retrieved from weather stations, other commonly encountered sources may offer visibility in different terms. For instance, Patterson and Gillette’s [
3] work, as briefly hinted at earlier, attempted to deduce visibility from dust concentration in the air based on an empirical relationship.
In fact, several additional empirical relationships have been established (
Table 1) to calculate visibility (
V) in km from a known dust concentration (
M; occasionally, referred to as total suspended particles (
TSP) in μg/m
3. However, there are significant accompanying uncertainties, one of which is the tendency to over- or under-estimate visibility [
11], which is this paper’s interest.
With the various assumptions made during attenuation expression derivations, it is necessary to minimize biases that may arise from the encapsulated parameters due to their associated uncertainties. Variations in M will result in different values for V and, consequently, for attenuation (A) in Equation (3) via Equations (2) and (4), or in Equation (11) via Equations (5) and (7)–(10).
For convenience, the equations in
Table 1 will be referred in discussions and some graphs in this paper as denoted under the third column, where each equation is assigned an alphanumeric character (e.g., E1 corresponds to Chepil and Woodruff’s equation).
5. Conclusions
This paper aimed to address the claims of visibility over- and under-estimation tendencies resulting from existing empirical equations. To that end, an algorithm was developed to assist in processing various data sets of TSPs and recorded visibilities, and output the best fitting results in terms of equation recurrences accompanied by their estimation percentage of error. The sources consisted of variations in terms of data collection techniques, sites of interest (urban, rural, or regional level) with particulate concentrations as low 25 ug/m3 and as high as 440 mg/m3 and corresponding visibilities as high as 56 km and as low as 0.06 km.
While the results did not yield a decisive conclusion, they did partially substantiate the Tews [
13] (average RMSE of 7.16), Shao et al. [
14] (average RMSE of 7.07), and Baddock et al. [
15] (average RMSE of 11.7) equations, which emerged as reliable estimators of visibility with reasonable accuracy compared to other empirical models (average RMSE > 29). Furthermore, by using our technique, we found that by averaging Tews [
13] and Shao et al. [
14], Tews [
13] and Baddock et al. [
15], and Shao et al. [
14] and Baddock et al.’s [
15] equations, average RMSE could be further reduced to 5.4, 7.02, and 8.52, respectively.
Finally, to evaluate the proposed approach, we provided a demonstration of employability, where a study provided signal attenuation measurements at an experimental site with vaguely defined meteorological conditions. Consequently, the conditions were converted into visibility using our findings and found to be in a good agreement with projected ranges of horizontal visibility standards set by the China Meteorological Administration (CMA).