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Statistical Studies of Fading in Underwater Wireless Optical Channels in the Presence of Air Bubble, Temperature, and Salinity Random Variations (Long Version)
Mohammad Vahid Jamali, Ali Mirani, Alireza Parsay, Bahman Abolhassani, Pooya Nabavi, Ata Chizari, Pirazh Khorramshahi, Sajjad Abdollahramezani, Jawad A. Salehi
TL;DR
Underwater optical channels suffer absorption, scattering, and fading, but comprehensive statistical characterization of turbulence-induced fading remains limited. The paper experimentally induces air bubbles and random temperature and salinity variations, compares distribution fits across weak-to-strong turbulence, and measures coherence time. Generalized Gamma and exponentiated Weibull distributions fit single-lobe data well with beam collimation and/or aperture averaging, while coherence times of 10^-3 seconds and larger indicate slow fading.
Problem
Comprehensive statistical evidence for turbulence-induced fading distributions in UWOC channels remains limited across air-bubble, temperature, salinity, and weak-to-strong turbulence conditions.
Method
The paper experimentally induces air bubbles and random temperature and salinity variations, evaluates distribution goodness of fit, and measures channel coherence time.
Results
Generalized Gamma and exponentiated Weibull distributions excellently match single-lobe histograms, while air-bubble-dominated moderate-to-strong fading may require a two-lobe distribution.
Takeaways & Limitations
10^-3 seconds and larger coherence times indicate slow fading, and transmitter beam-collimation and/or receiver aperture averaging suits single-lobe statistical distributions.
Abstract
from arXiv · showhide
Optical signal propagation through underwater channels is affected by three main degrading phenomena, namely absorption, scattering, and fading. In this paper, we experimentally study the statistical distribution of intensity fluctuations in underwater wireless optical channels with random temperature and salinity variations as well as the presence of air bubbles. In particular, we define different scenarios to produce random fluctuations on the water refractive index across the propagation path, and then examine the accuracy of various statistical distributions in terms of their goodness of fit to the experimental data. We also obtain the channel coherence time to address the average period of fading temporal variations. The scenarios under consideration cover a wide range of scintillation index from weak to strong turbulence. Moreover, the effects of beam-collimator at the transmitter side and aperture averaging lens at the receiver side are experimentally investigated. We show that the use of a transmitter beam-collimator and/or a receiver aperture averaging lens suits single-lobe distributions such that the generalized Gamma and exponential Weibull distributions can excellently match the histograms of the acquired data. Our experimental results further reveal that the channel coherence time is on the order of $10^{-3}$ seconds and larger which implies to the slow fading turbulent channels.
I. INTRODUCTION
UWOC performance is limited by absorption, scattering, and turbulence-induced fading, while comprehensive statistical modeling of the latter remains insufficient. The paper experimentally evaluates fading distributions across varied underwater turbulence conditions and assesses fit and coherence time.
- UWOC propagation is impaired by absorption, scattering, and turbulence-induced fading, typically limiting viable communication ranges to less than 100 m.
- The literature lacks a comprehensive statistical study of turbulence-induced fading in UWOC channels despite the importance of accurate channel modeling.
- Prior temperature-gradient experiments mainly examined very weak turbulence, whereas this paper covers scintillation indices from weak to strong turbulence using air bubbles and random temperature and salinity variations.
- The study compares statistical distributions using goodness-of-fit measures and obtains channel coherence time to assess the slow-fading behavior of turbulent UWOC channels.
B. Coherence Time
The paper evaluates fading coherence time experimentally and uses distribution normalization and goodness-of-fit optimization when modeling measured intensity fluctuations. Coherence time characterizes how long the fading coefficient remains effectively constant.
- B. Coherence Time: Channel coherence time is measured as the interval over which the irradiance temporal covariance remains above a threshold such as −3 dB.
- The fading coefficient is normalized so its average power is neither amplified nor attenuated, enforcing E[˜h] = 1.
- Distribution parameters are optimized to achieve the best fit to experimental data according to the goodness-of-fit measure.
B. Gamma Distribution
This section describes Gamma and K distributions as statistical models for optical turbulence, relating their parameters to normalized fading and scintillation strength. Weibull distributions are also identified as candidates for a wide scintillation range.
- B. Gamma Distribution: The Gamma distribution uses shape parameter k and scale parameter θ, with normalization kθ = 1 and θ = 1/k = σ2_I.
- The K distribution is used in the FSO literature for strong atmospheric turbulence and already satisfies the normalization condition E[˜h] = 1.
- Weibull and exponentiated Weibull distributions have described atmospheric turbulence across a wide range of scintillation indices and are evaluated for turbulent UWOC fading.
E. Exponentiated Weibull Distribution
The paper introduces exponentiated Weibull and Gamma-Gamma models for optical fading and derives their moments and scintillation-index relationships. These formulations support parameter selection under fading normalization.
- E. Exponentiated Weibull Distribution: The exponentiated Weibull distribution is specified by parameters α, β, and η through its probability density function.
- Its nth moment is computed using a rapidly convergent series, for which the first 10 terms provide an acceptable approximation.
- The exponentiated Weibull model uses its first and second moments to satisfy fading normalization and calculate the scintillation index.
- The Gamma-Gamma model factorizes irradiance into two independent Gamma-distributed random processes and has scintillation index σ2_I = 1/α + 1/β + 1/αβ.
G. Generalized Gamma Distribution
The paper evaluates the generalized Gamma distribution as a flexible model for underwater turbulence-induced fading. Its parameterization encompasses several important statistical distributions, with parameters optimized for goodness of fit.
- The generalized Gamma distribution is evaluated to simultaneously cover features of several statistical distributions used for optical turbulence.
- When d = p, the generalized Gamma distribution reduces to Weibull; when p = 1, it becomes Gamma; and when d = p = 1, exponential.
- Its moments are E[h~] = aΓ(d/p+1/p)/Γ(d/p) and E[h~^2] = a^2Γ(d/p+2/p)/Γ(d/p).
- The parameters a, d, and p are optimized to enforce E[h~] = 1 while obtaining the best goodness of fit.
V. EXPERIMENTAL SETUP
The experiments use a controlled water-tank setup to collect received-intensity samples under air-bubble, temperature, and salinity variations. Transmitter collimation and receiver aperture averaging are varied across scenarios.
- The experimental setup uses a black-colored water tank measuring 30 × 40 × 200 cm^3 for underwater optical-channel characterization.
- A 532 nm green laser diode with maximum output power of 50 mW provides constant optical irradiance at the transmitter.
- A plano-concave lens with f = −30 mm followed by a plano-convex lens with f = 200 mm forms the transmitter beam-collimator.
- The air-bubble scenario samples received intensity with and without the transmitter beam-collimator and receiver aperture averaging lens.
- Salinity experiments insert three flows of extremely salty water through independently tunable water droppers, including tests with random air bubbles.
VI. EXPERIMENTAL RESULTS
The results section evaluates statistical fading models across three experimentally generated channel scenarios and a broad turbulence range. Experiments are selected to include both average and highest attainable scintillation conditions.
- All three experimental scenarios are used to evaluate statistical distributions for turbulence-induced fading across varied underwater channel conditions.
- The transmitter laser power is adjusted in each scenario so considerable optical power reaches the receiver.
- The temporal covariance coefficient of received irradiance is numerically calculated to characterize temporal channel behavior.
- Experiments vary air-bubble populations and hot- or salty-water flow rates to cover a wide range of optical turbulence.
- Selected results include scintillation-index values near the averages and the highest values obtained at maximum air-bubble or hot/salty-water flow rates.
A. Intensity Fluctuation due to the Random Presence of Air Bubbles
Experiments with randomly varying air bubbles examine fading across weak-to-strong conditions and evaluate how transmitter beam-collimation and receiver aperture averaging affect statistical modeling. Aperture averaging and combined transmitter-receiver diversity reduce fading and make single-lobe distributions, especially generalized Gamma and exponentiated Weibull, better fits.
- Experimental configurations: The experiments cover bubbly UWOC channels across a wide range of fading strengths, with signal fluctuations, histograms, fitted distributions, and GoF values evaluated.The channel configurations include B, BAAL, and BBC AAL links, distinguished by transmitter beam-collimator and receiver aperture-averaging lens use.
- Bubbly links without diversity: Without transmitter BC or receiver AAL, air bubbles drive received intensity mainly toward large or small values, requiring a two-lobe distribution such as mixed exponential-lognormal.The configuration is highly sensitive to beam scattering caused by randomly present bubbles, limiting the suitability of typical single-lobe models.
- Receiver aperture averaging: Receiver AAL reduces fading strength and makes bubbly-channel fluctuations predictable by single-lobe distributions; generalized Gamma and exponentiated Weibull provide the best GoF.The study reports that all seven considered distributions can acceptably model BAAL fading, while these two distributions consistently give the best accordance.
- Diversity mechanisms: AAL averages irradiance fluctuations over a larger receiver aperture, while transmitter BC sends an expanded beam through partially independent spatial paths.Together, the components implement receiver and transmitter diversity in a manner compared with spatial-diversity MIMO.
- Combined diversity: A transmitter BC can increase absorption and scattering losses because the expanded beam creates more photon interactions with suspended particles and molecules.The added received-power loss may be negligible under some channel conditions, such as highly multiple-scattering turbid harbor links.
- Combined diversity: With both transmitter BC and receiver AAL, the highest bubble concentration has scintillation index below one and samples concentrate around the normalized mean.BBC AAL links are appropriately characterized by single-lobe distributions, with generalized Gamma and exponentiated Weibull again yielding the best GoF.
B. Turbulence-Induced Fading due to the Temperature Random Variations
The experiments examine temperature-induced fading across multiple UWOC configurations and scintillation regimes, showing that aperture averaging and beam collimation affect both fading strength and distributional fit.
- Experimental scenarios: Two temperature-variation procedures—tunable water droppers and heating elements—were used to experimentally evaluate fading distributions across channel conditions.Experiments also included mixed air bubbles and temperature variations, with receiver aperture averaging used throughout this subsection.
- Heating-element variations: Heating-element scenarios produced relatively weak fading because the large receiver aperture averaging lens compensated for beam wandering.Many considered distributions consequently fitted the experimental data in these conditions.
- Tunable-dropper variations: At scintillation index σ2_I = 0.0577, only generalized Gamma excellently predicted the channel statistics; Weibull and exponentiated Weibull fairly matched, while Gamma and lognormal failed.This result corresponds to the DAAL configuration under tunable-dropper temperature variations.
- Distributional fit: For DAAL links, generalized Gamma matched the data excellently across all scintillation-index regions, while Weibull and exponentiated Weibull also performed fairly across turbulence regimes.Lognormal and Gamma could fail, especially around scintillation index 0.1 and below.
- Beam-collimator effects: Transmitter beam collimation reduced fading strength and concentrated intensity samples around their mean, but reduced received optical power because the beam expanded spatially.With receiver aperture averaging, the resulting histograms could be modeled well by relatively simple single-lobe distributions.
- Mixed bubbles and temperature variations: When air bubbles dominated under moderate-to-strong fading, single-lobe distributions were no longer capable of modeling the fading statistics.Using transmitter beam collimation with receiver aperture averaging reduced sensitivity to beam scattering and restored excellent single-lobe fits.
C. Turbulence-Induced Fading due to the Salinity Random Variations
The study experimentally examines fading under salinity variations across channel configurations and scintillation levels, finding that generalized Gamma and exponentiated Weibull distributions model the observed behavior across the tested range.
- Motivation: Salinity-induced turbulence can become dominant over temperature-induced turbulence as w approaches 0, potentially producing stronger turbulence.The parameter w represents the relative strength of temperature and salinity fluctuations, and salinity-induced turbulence increases as w approaches its maximum value.
- Experimental setup: The experiments introduce extremely salty water through three independent droppers to create salinity variations in a fresh-water tank.Results are presented for links using both a transmitter beam-collimator and receiver aperture averaging lens.
- Goodness of fit: Not all tested distributions match the experimental histograms well, particularly at small scintillation-index values.The comparison uses acquired temporal data and corresponding numerical results for the tested channel conditions.
- Goodness of fit: Generalized Gamma and exponentiated Weibull distributions perfectly model salinity-variation fading across the full scintillation-index range tested.This conclusion concerns the salinity-random-variation channel conditions examined experimentally.
- Combined effects: When all three effects coexist, the histogram tends to follow the statistical behavior of the most dominant effect.For example, a high air-bubble population tends to produce the behavior associated with bubble-dominated links.
- Combined effects: Combining generalized Gamma or exponentiated Weibull models with an exponential distribution is proposed when link geometry is highly sensitive to beam wandering.The passage frames this combination as a promising solution for such conditions.
D. Coherence Time Evaluation
The paper evaluates coherence time through the temporal covariance coefficient of irradiance across multiple channel scenarios. Coherence times are generally longer than 10^-3 seconds, indicating slow temporal fading.
- Definition: Channel coherence time is defined as the average interval over which the fading coefficient remains effectively constant.Equivalently, it is the interval during which the irradiance temporal covariance coefficient remains above a threshold.
- Results: Coherence times are usually larger than 10^-3 seconds across the different tested channel scenarios.The scenarios include B, H, D, M, and S links with or without transmitter beam-collimation and receiver aperture averaging.
- Effects of link configuration: Transmitter beam-collimation produces larger coherence times by alleviating sensitivity to beam scattering.This corresponds to more slowly varying fading.
- Effects of scintillation: Increasing the scintillation index slightly decreases coherence time for a given channel condition.The passage associates higher scintillation-index values with lower temporal covariance coefficients.
VII. CONCLUSIONS AND FUTURE DIRECTIONS
The conclusions synthesize experimental fading results across air-bubble, temperature, and salinity conditions and identify suitable statistical models for different regimes. They also outline mathematical-modeling and real-sea validation as future directions.
- Conclusions: Air-bubble-dominated channels with moderate-to-strong fading generally require two-lobe distributions such as mixed exponential-lognormal.Single-lobe distributions cannot acceptably model these conditions unless beam-collimation or aperture averaging reduces sensitivity to beam scattering.
- Conclusions: Generalized Gamma and exponentiated Weibull are the best candidates in scenarios where simple lognormal and Gamma distributions fail.The paper reports this conclusion for temperature- and salinity-induced turbulence across weak-to-strong scintillation regions.
- Future directions: The study provides extensive experimental data as benchmarks for evaluating rigorous mathematical models of UWOC fading.Future modeling may use statistical-optics methods such as the Karhunen–Loève expansion.
- Future directions: Extending the laboratory study to real sea environments with advanced transmitter–receiver pairs is identified as a useful future research direction.The proposed extension would investigate fading behavior under real underwater conditions.