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Second-Order Fade Statistics in Underwater Optical Wireless Communications in Air-Bubble and Turbidity-Impaired Environments

Pedro Salcedo-Serrano, Mohammad-Ali Khalighi, Rubén Boluda-Ruiz, José María Garrido-Balsells, Antonio García-Zambrana

arXiv:2609.07955v1eess.SP

TL;DR

Second-order fading statistics in bubble-impaired UOWC channels remain less explored than first-order statistics, despite their relevance to temporal fading. This paper develops and experimentally validates two analytical approaches to LCR and AFD across bubble sizes and water turbidity. Turbidity has little effect for small bubbles, while large-bubble fades in turbid water are shorter and more frequent; the statistics are also applied to packet outage analysis.

  • Problem

    Although first-order fading statistics are well studied in bubble-impaired UOWC, the temporal statistics LCR and AFD remain underexplored and matter for burst-error behavior and communication-system parameters.

    Source: B. Need for Second-Order Channel… · p. 2

  • Method

    The paper characterizes LCR and AFD experimentally and theoretically using Rice’s formula and a sampled-fading approach based on irradiance CDFs, then applies the statistics in a two-state Markov packet-outage model.

    Source: §VI · p. 10, §VII · pp. 11–12

  • Results

    Turbidity has negligible effect on small-bubble fade statistics, while large-bubble turbid-water scenarios have higher LCR and lower AFD; the sampled-fading approach more closely matches measurements across thresholds.

    Source: 2) Analytical LCR results · p. 8, B. Average Fade Duration Analysis · p. 9, §VII · p. 12

  • Takeaways & Limitations

    Second-order statistics can support packet-level outage evaluation, and the two-state Markov model closely matches tap-water empirical POP while its turbid-water agreement weakens beyond Tp ≈ 2 × 10^-3 s.

    Source: §VI · pp. 10–11

    Under turbid-water conditions, the Markov model progressively overestimates empirical POP beyond Tp ≈ 2 × 10^-3 s, a discrepancy attributed to deviations from its exponential good-state survival assumption.

    Source: §VI · p. 11

Abstract

from arXiv · show

The propagation of underwater optical signals through air-bubble-impaired channels induces irradiance fluctuations and partial line-of-sight blockages that degrade the performance of underwater optical wireless communication (UOWC) systems. Although the first-order fading statistics of such channels have been extensively investigated, their second-order statistics, namely the level crossing rate (LCR) and average fade duration (AFD), remain rather unexplored. This paper presents a theoretical and experimental characterization of the LCR and AFD in bubble-impaired UOWC channels under different bubble sizes and water turbidity conditions. Two analytical approaches are developed based on Rice's formula and on the cumulative distribution function (CDF) together with the bivariate CDF of the sampled irradiance, and are validated experimentally. The results show that water turbidity has a negligible effect on fade statistics under small-bubble conditions. In contrast, for large bubbles, turbid water leads to shorter but more frequent fades than tap water, as particle-induced beam spreading reduces the likelihood of deep fades and prolonged blockage events. Finally, the derived second-order statistics are applied to evaluate the packet outage probability (POP) performance using a two-state Markov channel model, demonstrating that the higher fade rate in the turbid water case results in a larger POP despite the shorter fade durations.

I. INTRODUCTION

Prior bubble-impaired UOWC studies largely characterized first-order fading, leaving LCR and AFD underexplored despite their relevance to temporal fading and burst errors. This paper measures and models those statistics across bubble sizes and turbidity conditions, then applies them to packet outage analysis.

I. INTRODUCTION

  • UOWC is presented as a promising option for bandwidth-intensive underwater applications, while bubbles are one of several impairments affecting its links.Other listed impairments include absorption and scattering, misalignment, turbulence, and blockage by obstacles.Source: §I · p. 1

A. Literature Review on UOWC Channel Modeling Under Air Bubble Effects

B. Need for Second-Order Channel Statistics

  • Second-order statistics remain a major gap in bubble-impaired UOWC research, although LCR and AFD describe temporal fading relevant to burst errors and communication-system parameters.The cited literature had only briefly introduced AFD experimentally, without a theoretical framework; atmospheric FSO results cannot be directly transferred because bubble-induced irradiance fluctuations have different physical mechanisms.Source: B. Need for Second-Order Channel… · p. 2

C. Paper Contributions

  • The paper experimentally and theoretically characterizes LCR and AFD across bubble sizes and turbidity conditions, and applies the results to packet outage analysis.Its contributions include experimentally validated expressions based on generalized Gamma and mixture generalized Gamma fading models, plus a two-state Markov channel model for packet outage probability.Source: C. Paper Contributions · p. 2

II. UOWC CHANNEL MODEL

The channel model distinguishes small- and large-bubble fading by their mechanisms: small bubbles produce irradiance fluctuations, while large bubbles can combine blockage with fluctuations. It represents these cases with generalized Gamma and mixture generalized Gamma distributions, respectively.

Small-bubble model

  • Small bubbles induce small-scale irradiance fluctuations and occasional line-of-sight blockage from bubble clusters, modeled with a generalized Gamma distribution.The distribution is written for normalized received irradiance, with scale and shape parameters defined in the accompanying passage.Source: §II · p. 2

Large-bubble model

  • Large bubbles cause frequent line-of-sight blockage and can also induce small-scale fluctuations through fragmentation into smaller bubbles.The model represents these distinct effects with a mixture of two generalized Gamma distributions.Source: §II · pp. 2–3
  • The large-bubble mixture assigns a weight to blockage-dominated fading and the complementary weight to fluctuation-dominated fading.Its CDF is formed from the weighted component CDFs.Source: §II · p. 3

III. SECOND-ORDER STATISTICS: LCR AND AFD

The section defines LCR and AFD as temporal fading measures and presents Rice’s formula as one analytical route. For bubble-impaired channels, that route approximates the LCR using assumptions about irradiance and its time derivative.

III. SECOND-ORDER STATISTICS: LCR AND AFD

  • LCR is the average rate of positive threshold crossings, while AFD is the average time the received signal remains below the threshold.These measures characterize temporal evolution beyond the instantaneous irradiance distribution.Source: §III · p. 3

A. Approach 1 (A1): Rice model

  • The Rice formulation uses the joint irradiance and derivative density, whose analytical derivation is challenging for bubble-induced fading.The paper notes that the available channel characterization is based on irradiance-amplitude measurements.Source: A. Approach 1 (A1): Rice model · p. 3
  • The Rice approach estimates LCR from the joint distribution of received irradiance and its time derivative, then obtains AFD from the LCR.For bubble-induced fading, deriving that joint distribution analytically is challenging because a theoretical framework for temporal spectral behavior is unavailable in the cited discussion.Source: A. Approach 1 (A1): Rice model · p. 3
  • The Rice approximation assumes irradiance and its time derivative are statistically independent, with the derivative following a zero-mean Gaussian distribution.The derivative variance is estimated from the empirical irradiance autocorrelation; the approach uses the stationary-process relationship described in the section.Source: A. Approach 1 (A1): Rice model · p. 3

B. Approach 2 (A2): Sampled Fading Channels

The sampled-fading approach expresses LCR using the irradiance CDF and bivariate CDF, explicitly accounting for discrete sampling. The AFD is then obtained from the LCR relationship.

B. Approach 2 (A2): Sampled Fading Channels

  • The sampled-fading approach expresses LCR in terms of the irradiance CDF and bivariate CDF, accounting for the discrete-time nature of measurements.The method is presented as a closed-form alternative for sampled fading processes.Source: B. Approach 2 (A2): Sampled Fading… · p. 4
  • The bivariate CDF describes two irradiance samples taken at times separated by the sampling period.The samples are denoted I1 = I(t) and I2 = I(t + TS).Source: B. Approach 2 (A2): Sampled Fading… · p. 4
  • The approach derives AFD from the LCR using the stated relationship between the two statistics.The passage introduces the AFD expression following the sampled-channel LCR formulation.Source: B. Approach 2 (A2): Sampled Fading… · p. 4

A. Experimental Setup

The experiments characterize bubble-impaired UOWC channels across bubble sizes and water turbidity, then fit irradiance distributions used in the fading analysis. Turbidity leaves small-bubble scintillation nearly unchanged but reduces it for large bubbles, where the fitted distribution also indicates less severe blockage.

A. Experimental Setup

  • Measurements are sampled at 2.5 MHz for 240 seconds and low-pass filtered at 5 kHz to isolate optical fluctuations while retaining components relevant to LCR and AFD.The higher-power laser diode was introduced to improve receiver SNR and measurement accuracy.Source: A. Experimental Setup · p. 4
  • The setup uses a 3 m optical link, a 520 nm, 115 mW laser, and 16 L/min airflow; removing porous stones switches from small to large bubbles.Turbidity is produced with Mg(OH)2 and Al(OH)3 antacids, selected for a volume scattering function that closely matches natural water constituents.Source: A. Experimental Setup · p. 4

B. Path Loss and Scintillation Index

  • For large bubbles, adding antacid reduced the scintillation index from 0.168 to 0.137, while it remained virtually unchanged for small bubbles.Turbidity also raised path loss to approximately −34 dB; the authors attribute the small-bubble result to negligible LoS blockage.Source: B. Path Loss and Scintillation Index · p. 4

C. PDF and CDF Characterization

  • Approach 1 fits distribution parameters by minimizing PDF mean squared error, using generalized Gamma models for small bubbles and mixture models for large bubbles.The estimated parameters and MSE values are reported in Table II.Source: C. PDF and CDF Characterization · p. 5
  • Approach 2 fits CDFs for irradiance and consecutive-sample maxima, using generalized Gamma models for small bubbles and mixtures for large bubbles.The maximum is formed from each consecutive pair of irradiance samples; fitted parameters and MSE values are reported in Table III.Source: C. PDF and CDF Characterization · p. 5
  • Fitted PDFs match empirical data; small-bubble distributions are unimodal, whereas large-bubble distributions are bimodal with a less pronounced, mean-shifted blockage peak in turbid water.The lower peak represents LoS blockage and the higher peak fluctuations around mean received power.Source: C. PDF and CDF Characterization · pp. 5–6

V. STATISTICAL CHARACTERIZATION OF FADING

The section estimates empirical LCR and AFD from low-pass-filtered irradiance measurements over a range of thresholds. It also compares those estimates with two analytical predictions and assesses variability using non-overlapping data blocks.

V. STATISTICAL CHARACTERIZATION OF FADING

  • Empirical LCR and AFD are computed from low-pass-filtered normalized irradiance measurements across thresholds spanning the received-signal dynamic range.The positive-going LCR counts transitions from at or below a threshold to above it on the next sample.Source: §V · p. 7
  • The 6 × 10^8 samples are divided into 120 non-overlapping 2-second blocks to estimate threshold-dependent variability.Each block is much longer than the channel coherence time, reported as at most about 1 ms.Source: §V · p. 7
  • The empirical results are compared with analytical predictions from the two approaches derived in equations (11) and (15).Source: §V · p. 7

A. Level Crossing Rate Analysis

The section plots LCR against threshold for small- and large-bubble scenarios, including both tap-water and turbid-water conditions.

A. Level Crossing Rate Analysis

1) Experimental LCR results:

Large bubbles produce higher LCRs in turbid water than in tap water, while small-bubble fade statistics are similar across water conditions. Of the two analytical approaches, the sampled-irradiance CDF method matches measurements more closely across thresholds.

1) Experimental LCR results:

  • For small bubbles, both water conditions show comparable LCRs: approximately 100 crossings/s at Ith = 0 dB and 10 crossings/s near Ith = −6 dB.The curve is asymmetric because bubble clusters produce short, consecutive deep fades, while high-irradiance states persist longer.Source: 1) Experimental LCR results · p. 8, Figure 5 · p. 8
  • For large bubbles, turbid water shows higher LCR across thresholds than tap water, with a bimodal profile peaking near 1 dB and −9 dB.Scattered photons partially restore irradiance during LoS blockage, raising the irradiance floor and concentrating crossings near the secondary peak.Source: 1) Experimental LCR results · p. 8, Figure 6 · p. 9

2) Analytical LCR results:

  • Approach 2 matches measured LCR more closely over the full threshold range, while Approach 1 overestimates LCR at low thresholds.Approach 2 remains within experimental variability bands, though its fitted CDF and bivariate CDF parameters make estimates sensitive to fitting errors.Source: 2) Analytical LCR results · pp. 8–9
  • Approach 1's low-threshold overestimation reflects missed crossings from finite sampling, high-frequency noise amplification, and errors fitting rare deep fades.The low-threshold estimates are particularly uncertain where crossing events are scarce.Source: 1) Experimental LCR results · p. 7, 2) Analytical LCR results · pp. 8–9

B. Average Fade Duration Analysis

The AFD increases quasi-exponentially with threshold for small bubbles, while large-bubble results show local minima. For large bubbles, turbid water shortens fades, including a decrease from approximately 10 ms to 1 ms at Ith = −10 dB.

B. Average Fade Duration Analysis

VI. PACKET OUTAGE PROBABILITY ANALYSIS

The two-state Markov model links packet outage to the probability that a packet starts in a good channel state and avoids a transition into a bad state. POP rises with packet duration and is consistently higher in turbid than tap water across both bubble scenarios.

VI. PACKET OUTAGE PROBABILITY ANALYSIS

  • A packet succeeds only if the channel starts in the good state and remains there throughout its duration.The threshold separates good and bad states according to whether irradiance maintains the FEC target BER of 3.8 × 10−3.Source: §VI · p. 10
  • The model assumes exponential good-state durations, and measured survival functions generally follow that prediction under the tested conditions.This supports the memoryless assumption used for the two-state continuous-time Markov model.Source: §VI · p. 10, Figure 10 · p. 11, Figure 9 · p. 11
  • Empirical POP is the fraction of packet-length sliding windows containing at least one irradiance sample below the threshold.The number of samples in a packet is m = floor(Tp/Ts).Source: §VI · p. 10
  • As packet duration increases, POP rises because the chance of a good-to-bad transition during reception increases; as Tp approaches zero, POP approaches outage probability.Thus channels with equal outage probability but different crossing rates can have different packet-level performance.Source: §VI · pp. 10–11
  • Turbid water has higher POP than tap water across the considered packet durations for both bubble scenarios, despite shorter fades.The model also captures the knee point beyond which POP rises markedly with packet duration.Source: §VI · pp. 10–11

VII. CONCLUSIONS

The study characterizes LCR and AFD in bubble-impaired UOWC channels with two experimentally validated analytical approaches, then applies those statistics to packet outage. Turbidity has little effect for small bubbles, while large-bubble turbid water has shorter, more frequent fades and higher POP.

VII. CONCLUSIONS

  • Approach 1, based on Rice’s formula, tends to overestimate LCR and underestimate AFD, especially at low threshold levels.Approach 2 uses the fading-envelope CDF and bivariate CDF to account for sampling effects and agrees more closely with experimental LCR and AFD.Source: §VII · pp. 11–12
  • For large bubbles, turbid water has higher LCR and lower AFD than tap water, while turbidity has negligible fade-statistics impact under small bubbles.The large-bubble fades are therefore shorter but more frequent; particle scattering and increased relative measurement-noise contribution may both play a role.Source: §VII · p. 12
  • Despite shorter fades, turbid water has higher POP than tap water because of its increased fade rate.The conclusions report that packet-level performance can diverge from bit-level metrics such as BER.Source: §VII · p. 12
  • The authors identify matched-SNR measurements as necessary to isolate the contributions of particle-induced scattering and measurement noise.They also propose higher-order channel models to better capture temporal dynamics and support adaptive link-layer mechanisms.Source: §VII · p. 12
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