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A Unified Channel Model and Performance Evaluation for Underwater-to-Air Optical Links

Xing Du, Mohamed-Slim Alouini

arXiv:2610.34713v1eess.SP

TL;DR

Direct underwater-to-air optical links motivate statistical modeling, but the link performance depends on operating conditions and receiver assumptions. This paper combines conditional channel factors and evaluates BER and outage; independent checks support numerical consistency within the model, while the equivalent-beam approximation can be optimistic.

  • Problem

    Direct underwater-to-air links depend on water clarity, alignment, and the actual receiver, motivating a statistical model for the specified link conditions.

    Source: §I · p. 1

  • Method

    The framework couples interface transmission and pupil displacement through a common surface tilt and evaluates conditional gain and OOK performance with explicit support.

    Source: §I · p. 2, §V · p. 13

  • Results

    Independent component-wise Monte Carlo checks the full conditional model at six reference configurations, while the equivalent-beam pupil approximation can be appreciably optimistic near a target BER.

    Source: C. Surface Statistics and Conditional… · p. 5, §V · p. 13

  • Takeaways & Limitations

    Conditional operating-region analysis uses the circular-pupil reference, and source-traced contrasts qualify engineering conclusions by the prescribed operating conditions.

    Source: §V · p. 13

    The results establish numerical consistency within the adopted single-patch scalar model, not experimental validation of its assumptions.

    Source: §V · p. 13

Abstract

from arXiv · show

This paper develops a conditional statistical framework for nominally normal underwater-to-air optical links. It combines bulk attenuation, normalized log-normal atmospheric and Weibull underwater fluctuations, and entrance-pupil collection with surface-tilt-dependent Fresnel transmission. Conditioning transmission and beam displacement on a common tilt preserves their dependence. Product distributions with explicit support and Jacobians yield integrals valid for arbitrary positive Weibull shape. Average uncoded OOK bit error rate (BER) and outage are evaluated using consistent power, noise and SNR definitions. Independent component-wise Monte Carlo verifies the full conditional model at six reference configurations; targeted importance sampling checks selected BER values down to approximately 6 x 10^-9. A circular-pupil Gaussian integral reveals parameter-dependent errors in the equivalent-beam collection approximation, including a 36.9% BER underestimate at one tested operating point. Conditional power and distance studies therefore use the circular-pupil reference and distinguish low-error short-air-segment conditions from collection-limited conditions. Source-anchored spectral and water-type contrasts, together with illumination and receiver checks, identify the dependence of these results on the prescribed operating conditions. The framework supplies reproducible error and outage calculations within a single-patch scalar model; physical operating ranges additionally depend on the surface, propagation and receiver assumptions.

I. Introduction

The paper develops a conditional model for a nominally normal, single-patch underwater-to-air uplink, linking interface transmission and pupil collection through the same surface tilt. Its scope distinguishes entrance-pupil collection from downstream detector effects and treats bulk attenuation separately from normalized irradiance fluctuations.

B. Bulk Attenuation and Interface Transmission

  • Bulk extinction reduces the specified collected component, whereas normalized irradiance fluctuations change instantaneous strength; the Weibull choice is conditional, not a universal distance-to-turbulence mapping.Scattering in the Beer–Lambert coefficient does not model multiply scattered fields, delayed photons, or the temporal impulse response.Source: §I · p. 1, B. Bulk Attenuation and Interface… · p. 3

I. Introduction

  • The framework links Fresnel transmission and refraction-induced pupil displacement to the same surface tilt, while retaining bulk attenuation and normalized irradiance fluctuations as distinct factors.It is a conditional integration and assessment framework for a specified direct uplink, not a new fundamental fading distribution.Source: §I · pp. 1–2, A. Geometry, Reference Planes, and… · p. 3
  • The framework distinguishes source-anchored wavelength, water-type, illumination, and receiver contrasts from universal operating recommendations, which these comparisons do not establish.The comparisons identify which conclusions depend on illumination, field of view, or environmental calibration.Source: §I · p. 2

A. Geometry, Reference Planes, and Assumptions

  • The model represents a nominally normal uplink through one locally planar patch, with the receiver entrance pupil centered on the nominal transmitted axis.Oblique nominal incidence, surface shadowing, foam, and multiple interface encounters are outside this single-patch model.Source: A. Geometry, Reference Planes, and… · p. 2
  • Beam displacement, pupil radius, beam radius, and residual-displacement variance are defined at the entrance-pupil plane, while focal-plane detector loss requires separate specifications.The model assumes the downstream detector collects all power passed by the pupil.Source: A. Geometry, Reference Planes, and… · p. 2

C. Surface Statistics and Conditional Pointing Loss

Surface statistics determine tilt, while residual displacement is modeled conditionally on that tilt. This preserves the shared dependence of interface transmission and pupil collection; the reduced constant-transmission model is only a comparison case.

C. Surface Statistics and Conditional Pointing Loss

  • A common surface tilt determines both interface transmission and refraction-induced pupil displacement, so the factors are not treated as independent.Residual displacement is modeled as an independent Gaussian vector in the pupil plane, with the conditional displacement described by a Rice density.Source: C. Surface Statistics and Conditional… · p. 4
  • The equivalent-beam collection approximation is used for the prescribed Gaussian pupil-plane envelope, but an exact circular-aperture Gaussian integral is retained as a reference.The approximation is not an exact diffraction calculation, and the beam radius must follow a consistent propagation or optical-design specification.Source: C. Surface Statistics and Conditional… · pp. 4–5
  • The reduced model is a comparison case: replacing tilt-dependent transmission with a constant discards its dependence on pointing loss, and the algebra alone does not establish the error.The optional first-order reduction also approximates transmission, air-path attenuation, refraction angle, and tilt statistics.Source: C. Surface Statistics and Conditional… · p. 5

D. Normalized Atmospheric and Underwater Fluctuations

Atmospheric and underwater irradiance fluctuations are modeled as normalized random factors, separate from mean bulk attenuation. Their distribution parameters are specified or fitted rather than assigned a universal distance relation.

D. Normalized Atmospheric and Underwater Fluctuations

  • The atmospheric fluctuation uses a log-normal model under a weak-fluctuation assumption, with a plane-wave estimate distinguished from beam- or aperture-averaged values.A Gaussian-beam or aperture-averaged value should be obtained for the actual beam and pupil.Source: D. Normalized Atmospheric and… · p. 5
  • The underwater fluctuation uses a Weibull family whose scale enforces unit mean, separating mean bulk attenuation from turbulence strength.The shape is specified or fitted to irradiance measurements; no universal shape–distance relation is assumed.Source: D. Normalized Atmospheric and… · p. 5

III. Performance Analysis of Cross-Medium Communication Systems

The conditional product law forms gain from atmospheric fluctuations, interface transmission, and pupil collection at a fixed surface tilt. It retains the total-internal-reflection atom and distinguishes the coupled model from an independent reduced model.

A. Scaled Atmospheric Gain and Conditional Product Law

  • At fixed transmitted tilt, H2 combines atmospheric gain, Fresnel transmission, and pupil collection through a positive conditional density with support and Jacobian tied to pupil collection.The positive-density part is not normalized to one; the full law also accounts for total internal reflection.Source: A. Scaled Atmospheric Gain and… · pp. 5–6
  • The full law assigns zero gain to the total-internal-reflection event, while transmission-conditioned performance requires renormalizing the slope density and excludes that atom.The critical-angle tail follows from the adopted untruncated slope model, not from a measured blockage probability.Source: A. Scaled Atmospheric Gain and… · p. 6
  • The independent reduced model recovers a conventional product formula using the reduced gain scale and normal-incidence pupil collection.A marginal pupil-collection density cannot simply be multiplied by random marginal interface transmittance in the coupled model.Source: A. Scaled Atmospheric Gain and… · p. 6
  • The reduced-case check uses exact expressions for the stated model, but those expressions do not apply to the tilt-coupled law.The stable implementation evaluates the normal survival function directly.Source: A. Scaled Atmospheric Gain and… · p. 6

B. Composite Distribution for Arbitrary Weibull Shape

The composite gain distribution retains any zero-gain mass and gives a CDF construction valid for every positive Weibull shape. Numerical evaluation must preserve endpoint contributions and handle truncation and quadrature carefully.

B. Composite Distribution for Arbitrary Weibull Shape

  • The final gain H=H2Y retains the zero-gain mass p0 because the independent Weibull factor Y is positive almost surely.For positive gain, the distribution is obtained by conditioning on H2.Source: B. Composite Distribution for… · p. 6
  • The composite CDF expressions hold for every K>0, preserve the atom at zero, and approach one as gain tends to infinity.The reduced independent model uses p0=0.Source: B. Composite Distribution for… · p. 6

C. Numerical Quadrature and Endpoint Treatment

  • Quadrature requires rule-specific weight conversion, convergence checks, and bounded treatment of any truncated integration interval.The bounded CDF kernel is preferred for sharply peaked or near-zero-singular densities; the omitted contribution from restricting u is nonnegative and bounded.Source: C. Numerical Quadrature and Endpoint… · pp. 6–7
  • The zero-gain atom p0 must be retained separately because an unbounded low-gain truncation is unjustified for outage or BER calculations.The identity also notes that the CDF includes p0 under the unconditional model.Source: C. Numerical Quadrature and Endpoint… · pp. 6–7

D. OOK Signal Convention and Average BER

The BER analysis specifies average launched power, signal-independent equal-variance decision noise, and a threshold-based decision SNR. Under these conventions, strictly positive multiplicative fading has no nonzero asymptotic BER floor; a floor requires modeled zero-gain probability.

D. OOK Signal Convention and Average BER

  • For equiprobable OOK, Pt is average launched optical power and the optical levels are 0 and 2Pt, with channel gain constant over the decision interval.The noise variance is that of the decision-sample noise, assumed signal-independent and equal for both symbols; background offsets are removed and H is assumed known for the midpoint threshold.Source: D. OOK Signal Convention and Average BER · p. 7
  • The stated decision SNR is defined by threshold separation and noise variance, and the alternative electrical-signal SNR convention requires retaining its factor-of-two conversion.The numerical results use the decision-SNR convention.Source: D. OOK Signal Convention and Average BER · p. 7
  • The average BER is obtained by averaging conditional error probability, with the full CDF and its possible zero-gain mass included.A finite-node quadrature across a step CDF is not an exact AWGN check when gain is deterministic.Source: D. OOK Signal Convention and Average BER · p. 7
  • The asymptotic BER floor is zero for strictly positive multiplicative fading; a nonzero floor in the extended law requires explicitly modeled zero-gain probability.A plateau over a finite SNR sweep does not establish an asymptotic floor.Source: D. OOK Signal Convention and Average BER · p. 7

E. Outage Probability

Outage evaluation uses the unconditional CDF, including zero-gain probability when present, and converts a dB threshold to linear scale.

E. Outage Probability

  • The general-K outage calculation uses the CDF including p0 and converts a dB threshold via μth=10^(μth,dB/10); it is semi-analytical, not elementary closed form.Receiver power sensitivity in watts cannot be inserted without the defined photodetection and noise conversion.Source: E. Outage Probability · p. 7

IV. Numerical Verification and Reference Link Study

The reference study treats its numerical assignments as controlled model inputs and evaluates collection for a prescribed Gaussian envelope. It distinguishes the circular-pupil collection reference from exact wave-propagation or rough-surface solutions.

A. Configuration and Computational Procedure

  • The reference configuration uses 525 nm; water extinction is not assigned a named class, and slope and irradiance variances are specified directly.These are controlled model inputs rather than measurements of a deployed link.Source: A. Configuration and Computational… · p. 8, Table III · p. 8
  • During D_w sweeps, interface illumination and atmospheric log variance, underwater shape, and slope variance stay fixed, isolating bulk underwater attenuation at a prescribed interface field.The interface field is not varied as though the transmitter optical train were otherwise unchanged.Source: A. Configuration and Computational… · p. 8
  • The equivalent-beam map verifies transformed densities, while the circular-pupil integral provides a collection reference for the same prescribed Gaussian envelope.The circular-pupil reference is not an exact wave-propagation or rough-surface solution.Source: A. Configuration and Computational… · p. 8
  • 1.67 × 10^-16 was the maximum absolute discrepancy over 30 tested radius–offset pairs in the circular-pupil identity check.The check used an independent polar integral.Source: A. Configuration and Computational… · p. 8

B. Independent Distribution and BER Verification

Independent component-wise simulations verify the conditional distribution and BER calculations, while the circular-pupil reference exposes parameter-dependent errors in the equivalent-beam approximation. The resulting design illustrations use that reference and distinguish short-air-segment low-error conditions from collection-limited conditions.

B. Independent Distribution and BER Verification

  • Across six verification cases, the largest MC–quadrature CDF difference on 43-point grids is 6.38 × 10^-4, below the per-case 95% DKW envelope.The grid comparison is not a supremum evaluation over all gain values or a joint 95% statement across cases.Source: B. Independent Distribution and BER… · p. 9, Figure 2 · p. 9
  • The CDF quadrature changes by at most 1.62 × 10^-7 between the two finest retained levels, and the largest tested positive-mass accounting error is 2.07 × 10^-14.The formal large-slope test separately checks the zero-gain atom; it is not a validated rough-sea operating condition.Source: B. Independent Distribution and BER… · p. 9
  • All 18 selected low-BER quadrature values differ from importance-sampling estimates by at most 1.11 estimated standard errors, with a largest relative standard error of 7.11%.For the circular pupil at D_a=0.5 m and γ_0,dB=52, the estimates are 5.92 × 10^-9 and (5.85 ± 0.20) × 10^-9, respectively.Source: B. Independent Distribution and BER… · p. 9

C. Aperture Approximation and Link-Error Curves

  • At D_a=1 m and γ_0,dB=60, the equivalent-beam BER is 1.0388 × 10^-6 versus 1.6471 × 10^-6 for the circular-pupil reference, a 36.9% underestimate.This is a collection-map approximation error, not a failure of the corrected product or BER identities.Source: C. Aperture Approximation and… · p. 9, Figure 4 · p. 10
  • The circular-pupil BER curves show both low-error short-air-segment and high-error conditions, but do not establish a universal water-distance cutoff or underwater bottleneck.Changing air distance also changes refraction-induced centroid displacement and prescribed beam radius, as well as bulk attenuation; interpreting the SNR sweep as available launch power requires a credible power/noise budget.Source: C. Aperture Approximation and… · p. 10, Figure 5 · p. 10
  • At D_a=2 m and D_w=5 m, outage quadrature gives 0.017553 versus 0.017580 from independent MC; at 50 m, the values are 0.956080 and 0.956061.Zero observed outages at 0.5 m provide an upper bound, not proof of zero outage probability; low-outage tails need integration or targeted rare-event estimation.Source: C. Aperture Approximation and… · p. 10, Figure 6 · p. 10

D. Conditional Budget, Sensitivities, and Design Scope

The conditional budgets quantify how water properties, illumination assumptions, and receiver inputs shape required power, while keeping source-traced contrasts distinct from the controlled verification benchmark.

D. Conditional Budget, Sensitivities, and Design Scope

  • At a 5 m underwater distance, the assumed receiver requires approximately 0.407 mW for a 1 m air segment at the 10^-6 BER target.The illustrative 10 mW cap is neither a safety limit nor a proposed equipment specification.Source: D. Conditional Budget, Sensitivities,… · p. 11

E. Source-Traced Optical and Receiver Inputs

  • The source-traced contrasts keep published optical-property values separate from controlled benchmark inputs and declared receiver assumptions.The spectral comparison perturbs absorption by its published one-standard-deviation value, not as a total confidence interval.Source: E. Source-Traced Optical and Receiver… · p. 11, Table V · p. 11, Figure 7 · p. 12
  • 0.544 mW is required for the 10^-6 BER target at 525 nm with the source-traced receiver assumptions and a 5 m underwater distance.At otherwise fixed inputs, required power scales as √B_e/η_opt; the calculation does not establish a realizable hardware implementation.Source: E. Source-Traced Optical and Receiver… · pp. 11–12, Table V · p. 11
  • The shot-noise check found a largest estimated relative BER increment below 0.008% across six selected 10^-6 operating points.This supports the equal-noise approximation only at those dark-reference points, not arbitrary powers, daylight backgrounds, or an optimized unequal-noise detector.Source: E. Source-Traced Optical and Receiver… · p. 12

F. Spectral Reference and Published Water-Type Contrasts

G. Illumination and Receiver Acceptance Requirements

Illumination and receiver acceptance assumptions materially qualify the conditional performance results. The paper uses these checks to identify requirements and model boundaries, not to certify a platform or universal range.

G. Illumination and Receiver Acceptance Requirements

  • The 40 mm interface radius and 10 mrad half-divergence in the verification benchmark are not a diffraction-limited Gaussian laser specification.Treating that plane as a waist would imply M^2 ≃ 2.39 × 10^3, a consistency diagnosis rather than a measured beam-quality factor.Source: G. Illumination and Receiver… · p. 13
  • At 10 mW and a 2 m aperture, the 20 mm and 80 mm fixed-waist references give BERs of approximately 5.32×10^-2 and 6.55×10^-8.At 50 m, all four tested illumination choices remain above 0.28 under the new receiver assumptions; the comparisons separate aperture dilution from pointing tolerance.Source: G. Illumination and Receiver… · p. 13
  • For p_miss = 2 × 10^-6, the 525-nm reference requires approximately 3.92° half-FoV, or at least a 3.42-mm chief-ray radius at 50-mm focal length.This is only a necessary acceptance condition: equality consumes the entire 10^-6 error budget, so focused-spot size, residual angles, and design margin require further allowance.Source: G. Illumination and Receiver… · p. 13

V. Conclusions

  • Independent component-wise Monte Carlo checks the full conditional model at six reference configurations, while importance sampling checks selected low-BER points.The equivalent-beam pupil approximation can remain appreciably optimistic near a target BER despite agreement between its own analytical and simulated distributions; circular-pupil calculations therefore guide the operating-region analysis.Source: §V · p. 13
  • Deployment-specific predictions require measured surface and irradiance statistics, complete scattering spectra, spatial propagation, and a jointly realizable finite-detector receiver.The 50 m air case is unsupported by the tested untracked configurations, not ruled out for all designs.Source: §V · p. 13, G. Illumination and Receiver… · p. 13

Appendix A Pupil Collection and Conditional Displacement

The appendices detail the pupil-collection reference and the support-aware product and BER transformations. They establish how the conditional calculations are formed and checked.

Appendix A Pupil Collection and Conditional Displacement

  • The circular-pupil reference integral differs from the equivalent-beam approximation, so agreement must be checked for the adopted beam-to-pupil ratio.The reference integral does not add multiple scattering or a receiver-lens diffraction model.Source: Appendix A · p. 14

Appendix B Product Support, Reduced Check, and General-K

  • The product-density derivations enforce the transformed support, including the constraint x ≥ v/A_0 when V = X_1P.The Weibull-product integration retains the general exponent for arbitrary K > 0; the e^-t^2 form applies only when K = 2.Source: Appendix B · p. 14

Appendix C BER Transformation with a Possible Zero-Gain

  • The BER transformation retains a possible zero-gain mass, which contributes to the limiting error at unbounded reference SNR.The limiting expressions also provide consistency checks for numerical evaluations.Source: Appendix C · p. 14
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