Physics-Informed Deep Learning-Based Inversion of Marine Atmospheric Boundary-Layer Optical Duct Parameters from Free-Space Optical Link Measurements

1. Introduction

Free-space optical (FSO) communication over the ocean has attracted growing attention for its high bandwidth, license-free spectrum, and low interception probability compared to radio-frequency links [1,2]. However, the marine atmospheric boundary layer introduces refractive phenomena that can severely degrade or, conversely, enhance FSO link performance. When warm, dry air advects over a cooler sea surface, a strong temperature inversion develops in the lowest tens of meters. If the inversion gradient exceeds a critical threshold (~0.13 K/m at optical wavelengths), the modified refractivity gradient turns negative, creating an optical duct that traps near-horizontal rays and guides them along the surface—much like a submarine periscope seeing over the geometric horizon [3,4].

Optical ducting has two profound effects on maritime FSO links. First, trapped rays propagate with anomalously low loss, enabling over-the-horizon reception at distances far beyond the line-of-sight limit. Second, the duct supports multiple propagation paths (multipath) between transmitter and receiver, producing strong interference patterns and angle-of-arrival (AOA) spreading that can severely degrade link quality through intersymbol interference and scintillation [5,6]. Accurate prediction of these effects requires knowledge of the duct parameters—the inversion gradient, the layer-top height, and the surface temperature—which are traditionally obtained from radiosonde measurements or buoy-based sensors. These methods are sparse, intermittent, and ill-suited to the dynamic coastal environment.

The problem of inferring atmospheric refractivity parameters from propagation measurements—an inverse problem—has been extensively studied in the microwave regime. Techniques based on radar sea clutter [7,8], GPS radio occultation [9], and wave-theory inversion [10] have been developed to estimate evaporation duct height and surface refractivity from radio-frequency observations. In recent years, machine learning approaches have been applied to microwave duct inversion, demonstrating improved accuracy and generalization over classical methods [11,12].

At optical wavelengths, however, the problem is fundamentally different. The optical refractivity formula lacks a humidity term (water vapor affects radio refractivity but is negligible at optical frequencies), so the dominant ducting mechanism shifts from the evaporation duct (humidity-gradient driven) to the temperature inversion duct. The observable quantities also differ: optical links provide angle-of-arrival and received-power measurements rather than radar clutter or propagation loss. Despite these distinctions, no published work has attempted to invert optical duct parameters directly from FSO channel observations.

Physics-informed deep learning has emerged as a powerful paradigm for solving inverse problems in science and engineering [13,14]. By embedding physical constraints—such as governing equations, conservation laws, or constitutive relations—into the neural network loss function, these methods regularize the solution space, improve generalization from limited data, and ensure physical consistency of predictions. Applications span fluid mechanics [13], heat transfer [15], and materials science [16], but to our knowledge, physics-informed neural networks have not been applied to atmospheric refractivity inversion.

This paper addresses the gap by proposing a physics-informed deep learning framework that inverts marine boundary-layer optical duct parameters from multi-wavelength FSO link measurements. The main contributions are:

• First inversion framework for optical ducts. We formulate, for the first time, the inversion of marine boundary-layer optical duct parameters from FSO channel observations. While prior work has focused exclusively on microwave evaporation ducts, we exploit the wavelength dependence of optical refractivity and the multipath structure of ducted propagation to sense the temperature-inversion duct directly from optical measurements.
• Systematic identifiability analysis. We demonstrate that the three profile parameters have fundamentally different identifiability: the inversion gradient γ is strongly identifiable (R² = 0.843, MAE = 0.033 K/m), the layer-top height z_i is weakly identifiable (R² = 0.22), and the surface temperature T₀ is physically unobservable from link measurements—ray bending is effectively invariant under a uniform vertical offset of the refractivity profile. This analysis motivates a practical inversion strategy in which T₀ is supplied by a co-located ground meteorological station, as is standard in deployed FSO systems.
• Dual-wavelength forward model with diffraction analysis. A Bouguer ray-tracing forward model in a spherically stratified atmosphere is developed for 850/1550 nm propagation over 5–50 km maritime links. We establish the diffraction-negligible regime analytically (Rayleigh distance ~148 km >> link range) and confirm it with split-step Fourier parabolic-equation (SSFT-PE) analysis, which shows that the grid resolution required for optical-wavelength PE (dz < 0.43 mm) renders it computationally intractable, validating geometric optics as the appropriate forward model.
• Physics-guided network with ablation validation. A residual-learning neural network with a trapping-consistency physics constraint in the loss function is trained on 25,000 synthetic samples. Ablation experiments quantify the contribution of dual-wavelength dispersion features (R² improvement of +0.024, consistency improvement of +3.4 percentage points) and the physics constraint (consistency improvement of +2.5 pp). A noise robustness study demonstrates sub-linear error degradation under elevated measurement noise.

The remainder of this paper is organized as follows. Section 2 develops the physical model—refractivity, profile parameterization, ray tracing, and the diffraction analysis. Section 3 describes the dataset generation pipeline. Section 4 presents the physics-informed neural network architecture and training protocol. Section 5 reports the numerical results, including the ablation study, identifiability analysis, and noise robustness. Section 6 discusses the findings, limitations, and engineering implications. Section 7 concludes.

2. Physical Model

2.1. Optical Refractivity in the Marine Boundary Layer

At radio frequencies, atmospheric refractivity is conventionally expressed as [3,17]:

$$N_{\mathrm{RF}}=77.6{\displaystyle \frac{P}{T}}+3.73\times {10}^5{\displaystyle \frac{e}{T^2}}$$

(1)

where P is total pressure (hPa), T is temperature (K), and e is water vapor pressure (hPa). The second (humidity) term drives evaporation duct formation over the ocean.

At optical wavelengths, the refractivity formula takes a different form. Following the Barrell–Sears parameterization [18,19,20], the group refractivity for visible and near-infrared light is:

$$N(\lambda ,P,T)=77.6(1+{\displaystyle \frac{7.52\times {10}^{-3}}{{\lambda }^2}}){\displaystyle \frac{P}{T}}$$

(2)

where λ is the vacuum wavelength in micrometers (μm). The critical distinction from the radio formula is the absence of a humidity term—water vapor dispersion at optical frequencies is negligible compared to the dry-air contribution. Consequently, the dominant mechanism for optical ducting is temperature inversion (dT/dz > 0), not the humidity gradient that drives evaporation ducts at radio frequencies.

The wavelength-dependent coefficient c(λ) = 1 + 7.52 × 10⁻³/λ² (with λ in µm) produces a refractivity difference of approximately 2 N-units between 850 nm and 1550 nm at standard sea-level conditions (P = 1013.25 hPa, T = 290 K). This chromatic dispersion, while small in absolute terms, is detectable through differential angle-of-arrival measurements and constitutes a key observable for the inversion framework (Section 2.3).

To describe ducting, we use the modified refractivity [3,17]:

$$M\left(z\right)=N\left(z\right)+{\displaystyle \frac{z}{a}}\times {10}^6$$

(3)

where a = 6,371 km is the Earth's radius and z is altitude. The ducting condition is:

$${\displaystyle \frac{\mathit{dM}}{\mathit{dz}}}<0$$

(4)

which means the refractivity decreases fast enough with altitude to overcome the Earth-curvature term and bend rays downward. Setting dM/dz = 0 and solving for the temperature gradient yields the critical inversion gradient γc—the minimum temperature increase per meter required for ducting:

$${\gamma }_c=({\displaystyle \frac{{10}^6/a}{77.6c\left(\lambda \right)P}}-{\displaystyle \frac{1}{\mathit{HT}}})T^2$$

(5)

where H = 8,434 m is the pressure scale height. At 850 nm, 1013.25 hPa, and 290 K: $\gamma$c = 0.1318 K/m; at 1550 nm: $\gamma$c = 0.1330 K/m. These values are achievable in marine boundary-layer inversions, where gradients of 0.1–0.5 K/m are commonly observed in the lowest 10–80 m [4].

2.2. Inversion Profile Parameterization

The temperature inversion is modeled as a bilinear profile with a smooth transition at the inversion top (Figure 1):

T(z) = T_a(z) + [T_b(z) − T_a(z)] · S(z) (6)

where:

• T_a(z) = T₀ + γ · z is the inversion-layer extension (temperature increasing with height at rate γ),
• T_b(z) = T₀ + γ · z_i + β · (z − z_i) is the above-inversion extension (standard lapse rate β = −0.0065 K/m),
• $S\left(z\right)={\displaystyle \frac{1}{2}}[1+\tanh ({\displaystyle \frac{z-z_i}{w}}\left)\right]$ is a smooth step function with transition half-width w = 3 m.

The three free parameters defining the profile are summarized in Table 1:

The corresponding modified refractivity profile M(z) is obtained by substituting T(z) and the hydrostatic pressure profile P(z) = P₀ exp(−z/H) into the optical refractivity formula. A duct forms whenever dM/dz < 0 within the inversion layer, i.e., when γ exceeds γc.

2.3. Bouguer Ray Tracing

In a spherically stratified atmosphere where n = n(z) only, ray propagation is governed by Bouguer's invariant (the spherical form of Snell's law) [21] (Figure 2):

n(z) · (a + z) · cosθ = C = const (7)

where θ is the local elevation angle and C is determined by the launch conditions: C = n(ht) · (a + ht) · cos(θ₀). The ground-referenced horizontal distance (arc length measured along the earth's surface) accumulated as the ray traverses a monotonic altitude segment [z₁, z₂] is [3]:

$$\mathit{\Delta x}={\int }_{z_1}^{z_2}{\displaystyle \frac{\mathit{aCdz}}{n\left(z\right)r(z{)}^2\sqrt{1-{\left[\frac{C}{n\left(z\right)r\left(z\right)}\right]}^2}}}$$

(8)

where r(z) = a + z. A turning point occurs where n(z)·r(z) = C, at which the ray reverses its vertical direction. In a ducting profile, rays launched at sufficiently small elevation angles undergo repeated turning-point reflections within the duct layer, interspersed with surface reflections—this oscillatory trapping produces multipath propagation.

The forward model proceeds by a shooting method: a range of launch angles θ₀ is scanned, and for each angle the ray trajectory z(x; θ₀) is computed by piecewise integration (97-point trapezoidal rule per monotonic segment, with turning-point singularities handled by endpoint truncation). Rays that arrive at the receiver location (x = R, z = hr) are identified by sign changes in z(R; θ₀) − hr, refined by bisection. The complete multipath solution set consists of all rays satisfying the boundary condition.

Dual-wavelength chromatic dispersion. For each ray solution at 850 nm, the corresponding 1550 nm ray is found by a localized angle-space search within a ±0.3 mrad window. This windowed matching is essential because the AOA separation between the two wavelengths for the same path is only 10–40 μrad—far smaller than the coarse scan resolution (~0.2 mrad)—and independent full-range scans at each wavelength would lose the path-to-path correspondence. The dual-wavelength AOA difference (in microradians) serves as a chromatic dispersion observable that carries information about the refractivity gradient (Figure 3).

The forward model outputs the following observables for each link configuration:

• Multipath count n_paths: the number of ray solutions reaching the receiver.
• AOA vector: arrival angles of each path (mrad), sorted by descending magnitude.
• Dual-wavelength dispersion vector: AOA differences between 850 nm and 1550 nm for each matched path (μrad).
• Geometric focusing factors: placeholder values (Section 6.3 discusses limitations).

2.4. Diffraction Negligibility and Forward Model Validation

A key question is whether geometric optics (ray tracing) is adequate, or whether wave effects (diffraction) must be included. For a Gaussian beam with waist w₀, the Rayleigh distance is:

$$z_R={\displaystyle \frac{\pi w_0^2}{\lambda }}$$

(9)

For a typical FSO transmitter with w₀ = 0.20 m at λ = 850 nm: z_R ≈ 148 km, far exceeding the maximum link range of 50 km. The beam broadening at 50 km is only ~5.6%, confirming that diffraction is negligible over the link distances considered.

To validate this conclusion independently, we implemented a split-step Fourier parabolic-equation (SSFT-PE) solver with Dirichlet (reflecting) boundary at the sea surface and an absorbing window at the top boundary. The PE solver correctly reproduces free-space Gaussian-beam diffraction (validated against analytic solutions: 8.8% error at w₀ = 0.02 m due to grid undersampling, 0% error at w₀ = 0.20 m). However, at optical wavelengths the required vertical grid spacing is dz < π/(k·θ_max) < 0.43 mm for 1 mrad ray angles, demanding ~2.6 × 10⁵ grid points over a 200 m vertical domain. Combined with ~10⁴–10⁵ propagation steps for a 10 km link, the total computational cost (~3 × 10⁹ operations per link) renders SSFT-PE intractable for the dataset generation pipeline.

This analysis establishes that: (i) diffraction is physically negligible for the parameter regime of interest (zR >> link range); (ii) geometric optics is the correct and computationally efficient forward model; and (iii) SSFT-PE serves as a theoretical validation tool for the diffraction-negligible regime rather than a production forward solver.

3. Dataset Generation

3.1. Sampling Design

The three profile parameters (T₀, γ, z_i) are sampled using Latin Hypercube Sampling (LHS) over their physical ranges (Table 1). LHS ensures space-filling coverage of the 3D parameter space with a modest number of samples, avoiding the clustering and empty regions characteristic of naive random sampling. A total of 5,000 profiles are generated with a fixed random seed (20260813) for reproducibility.

Each profile is propagated over five link distances (R = 5, 10, 20, 30, 50 km) at fixed transmitter and receiver heights (ht = hr = 10 m), yielding 25,000 samples. The link distances span the transition from short-range line-of-sight to long-range duct-enhanced propagation, ensuring the dataset captures both ducting and non-ducting regimes.

3.2. Forward Simulation Pipeline

For each (profile, link distance) pair, the Bouguer ray tracer (Section 2.3) computes the complete multipath solution at 850 nm and the matched dual-wavelength solution at 1550 nm. The raw observables are:

• Multipath count n_paths (0–10 in practice; capped at 8 for feature vectorization).
• AOA vector: per-path arrival angles (mrad).
• Dual-wavelength dispersion: per-path AOA differences (μrad).
• Ducting flag: whether the profile produces a duct at 850 nm (dM/dz < 0).
• Duct height: upper boundary of the trapping region.

The forward simulation is parallelized across profiles using a process pool (4 workers), with each profile processed independently. The complete dataset generation required approximately 3 hours on a quad-core CPU.

3.3. Noise Model and Feature Vectorization

Realistic measurement noise is added to the raw observables before feature vectorization (Table 2):

The AOA noise of 50 μrad corresponds to approximately 10 arcseconds, consistent with high-precision FSO tracking systems. The dual-wavelength noise of 5 μrad reflects the differential measurement precision achievable with coherent dual-wavelength receivers.

After noise injection, the observables are vectorized into a 28-dimensional feature vector (Table 3):

The target vector is the 3-dimensional profile parameter: Y = [$T_0$, $\gamma$, $z_i$] in physical units (K, K/m, m).

3.4. Dataset Statistics

Table 4 summarizes the key statistics of the generated dataset, and Figure 4 visualizes the marginal distributions and their correlations with the inversion gradient.

The 81.8% ducting fraction reflects the sampling range (γ ∈ [0.05, 0.50] K/m, with γc ≈ 0.13 K/m): the majority of sampled profiles produce ducting conditions. The moderate correlation between multipath count and inversion gradient (r = 0.365) confirms that n_paths carries information about γ, but the relationship is nonlinear—multipath depends on the interplay of γ, z_i, and link geometry, not γ alone. The weaker correlation between maximum dual-wavelength dispersion and γ (r = 0.104) indicates that this feature provides complementary, non-redundant information.

4. Physics-Informed Neural Network

4.1. Architecture

The inversion network is a multi-layer perceptron (MLP) with residual connections, designed to map the 28-dimensional (or 20-dimensional without dual-wavelength features) observation vector to the 3-dimensional profile parameter vector. The architecture is summarized in Table 5:

Batch normalization stabilizes training across features with heterogeneous scales (link distance in km vs. AOA in mrad vs. dispersion in μrad). Dropout provides regularization. The output is de-standardized to physical units before physics-loss computation.

4.2. Physics-Guided Loss

The total loss combines a data-fidelity term and a physics-consistency term:

$ℒ$ = $ℒ$_MSE + λ$ℒ$_phys (10)

where $ℒ$_MSE is the standard mean-squared error between predicted and true profile parameters (in standardized space), and λ = 0.5 is the physics weight.

The physics loss $ℒ$_phys enforces trapping–observation consistency: the predicted profile must be consistent with the observed multipath structure. Specifically:

• If the observation shows multipath (n_paths ≥ 2), the predicted profile must produce a duct (dM/dz < 0), because multipath arises from ray trapping.
• If the observation shows no paths (n_paths = 0), the predicted profile must not produce a duct (dM/dz ≥ 0), because the absence of received rays indicates non-trapping conditions.
• If n_paths = 1 (weak signal, ambiguous), no constraint is applied.

The modified refractivity gradient dM/dz is computed analytically from the predicted (T₀, γ, z_i) using the mid-layer approximation (T_mid = T₀ + γ·z_i/2, P_mid = P₀·exp(−z_i/2H)), making it fully differentiable for backpropagation:

$${\displaystyle \frac{\mathit{dM}}{\mathit{dz}}}{|}_{\mathrm{mid}}=77.6c{P}_{\mathrm{mid}}\left(-{\displaystyle \frac{1}{H{T}_{\mathrm{mid}}}}-{\displaystyle \frac{\gamma }{T_{\mathrm{mid}}^2}}\right)+{\displaystyle \frac{{10}^6}{a}}$$

(11)

The physics loss is then:

$${\mathcal{L}}_{\mathrm{phys}}={\displaystyle \frac{1}{\left|\mathcal{S}\right|}}{\sum }_{i\in \mathcal{S}}\{\begin{array}{cc}\mathrm{ReLU}(\mathit{dM}/d{z}_i)& \mathrm{if}{n}_{\mathrm{paths},i}\ge 2\\ \mathrm{ReLU}(-\mathit{dM}/d{z}_i)& \mathrm{if}{n}_{\mathrm{paths},i}=0\end{array}$$

(12)

where $𝒮$ is the set of samples with n_paths ≥ 2 or n_paths = 0. The ReLU activation produces zero loss when the constraint is satisfied (correct sign of dM/dz) and a positive penalty when it is violated.

4.3. Training Protocol

The 25,000 samples are split into training (80%, 20,000), validation (10%, 2,500), and test (10%, 2,500) sets with a fixed random seed (42). Standardization parameters (mean, std) are computed from the training set only and applied to all splits.

Training uses the Adam optimizer with an initial learning rate of 10⁻³ and cosine-annealing learning-rate schedule over 200 epochs. The batch size is 256. Early stopping with patience = 20 epochs monitors the validation loss (MSE + physics loss); the best model is restored at the end of training. All experiments use the same random seed to ensure reproducibility.

4.4. Evaluation Metrics

The following metrics are computed on the held-out test set:

• Mean Absolute Error (MAE) for each parameter (T₀ in K, γ in K/m, z_i in m).
• Relative MAE: MAE divided by the mean of the true parameter, expressed as a percentage.
• Coefficient of determination R² for each parameter.
• Physical self-consistency rate: the fraction of test samples for which the predicted profile's trapping status (dM/dz < 0 or ≥ 0) is consistent with the observed multipath structure (n_paths ≥ 2 or = 0). This metric captures the benefit of the physics constraint that may not be visible in regression metrics.

The self-consistency rate is defined as:

$$\mathrm{Consistency}={\displaystyle \frac{1}{\left|\mathcal{S}\right|}}{\sum }_{i\in \mathcal{S}}1\left[\mathrm{sign}\left({\displaystyle \frac{\mathit{dM}}{\mathit{dz}}}{|}_i\right)\mathrm{consistent\; with}{n}_{\mathrm{paths},i}\right]$$

(13)

where $𝒮$ is the set of samples with n_paths ≥ 2 or n_paths = 0 (unambiguous cases), and the consistency condition is: (n_paths ≥ 2 → dM/dz < 0) and (n_paths = 0 → dM/dz ≥ 0).

5. Results

5.1. Ablation Study

To quantify the contribution of each component, four network configurations are compared, along with a mean-prediction baseline (Figure 5 and Table 6).

Key findings:

5.

Inversion gradient $\gamma$ is strongly identifiable. The full model achieves $R_{\gamma }^2$ = 0.843 and MAE_γ = 0.033 K/m, a 70.5% reduction in MAE compared to the baseline (0.112 K/m). This confirms that the multipath structure and dual-wavelength dispersion carry rich information about the inversion gradient—the parameter that directly controls ray curvature and duct formation.

6.

Surface temperature $T_0$ is effectively unidentifiable. The $\mathit{MA}{E}_{T_0}$ of the full model (4.927 K) is nearly identical to the baseline (4.961 K)—a mere 0.7% improvement. The network learns nothing about T₀ beyond the training-set mean. This is not a limitation of the architecture but a fundamental physical constraint: T₀ produces a near-uniform offset of the refractivity profile, and the residual gradient effect is exactly degenerate with γ (Section 5.2), so no link-based observation can distinguish between different T₀ values. This degeneracy is analyzed in Section 5.2.

7.

Layer-top height $z_i$ is weakly identifiable. The $R_{z_i}^2$ = 0.216 for the full model, with $\mathit{MA}{E}_{z_i}$ = 16.12 m (14.0% improvement over the baseline's 18.74 m). The moderate identifiability arises from a coupling between γ and z_i: the ray-turning height depends on both parameters, and different (γ, z_i) combinations can produce similar multipath structures.

8.

Dual-wavelength features improve both regression and consistency. Comparing "Full" vs. "No dual": $R_{\gamma }^2$ improves by +0.024 (0.843 vs. 0.819), and the consistency rate improves by +3.4 pp (82.0% vs. 78.6%). The dual-wavelength dispersion provides information that is complementary to the monochromatic AOA vector—specifically, it disambiguates profiles with similar multipath structures but different refractivity gradients, because the chromatic dispersion is directly proportional to dN/dλ and hence to the profile shape.

9.

Physics constraint improves consistency, not regression. Comparing "Full" vs. "No phys": $R_{\gamma }^2$ is essentially unchanged (0.843 vs. 0.839, Δ = +0.004), but the consistency rate improves by +2.5 pp (82.0% vs. 79.5%). This is the expected behavior: the physics loss does not add new observational information, so it cannot improve regression accuracy; instead, it regularizes predictions to be physically self-consistent, preventing the network from predicting profiles that contradict the observed multipath structure. The combined effect of both components (Full vs. No both) yields +5.7 pp in consistency, demonstrating a synergistic interaction.

10.

Consistency as a complementary metric. The regression metrics (R², MAE) for γ are similar across all four network configurations ($R_{\gamma }^2$ ∈ [0.819, 0.843]), but the consistency rate varies by 5.7 pp (76.3%–82.0%). This demonstrates that consistency captures a dimension of performance invisible to standard regression metrics: whether the predicted profile would produce the same trapping behavior as the observed data.

Typical inversion examples. Figure 6 illustrates the inversion behavior on three representative test profiles: a strongly ducting profile (γ = 0.30 K/m, z_i = 25 m), for which the rich multipath structure yields an accurate γ prediction; a weakly ducting profile (γ = 0.15 K/m, z_i = 50 m), for which γ and z_i are predicted with moderate accuracy; and a non-ducting profile (γ = 0.08 K/m), which is correctly classified as non-trapping. The strong-duct case benefits from abundant multipath constraints, whereas the weak-duct case exhibits fewer paths and hence weaker parameter constraints.

5.2. Identifiability Analysis

The ablation results reveal a striking hierarchy of identifiability among the three profile parameters (Figure 7). This hierarchy has a clear physical explanation.

$T_0$ unidentifiability: iso-observation degeneracy. The surface temperature T₀ sets the absolute refractivity level, N₀ = 77.6·c(λ)·P₀/T₀. Because N ∝ P/T is nonlinear in T, changing T₀ while keeping γ and z_i fixed does more than translate M(z): it also perturbs the gradient dM/dz (by ≈10% of its ducting-layer value for a ±10 K change). However, this gradient perturbation can be exactly compensated by a small adjustment of γ: since ray bending is governed by dM/dz, any (T₀, γ) pair lying on the iso-observation line γ − κ·(T₀ − 290 K) = const, with κ ≈ 2.2×10⁻³ (K/m)/K, produces essentially identical trajectories (Figure 8). The link therefore constrains only a linear combination of T₀ and γ—the parameters are degenerate along this direction—and T₀ itself is unobservable. Moreover, the degeneracy is shallow: a 10 K error in T₀ is equivalent to a 0.022 K/m shift in γ, comparable to the γ inversion error (MAE = 0.033 K/m), so the network cannot separate the two parameters.

This degeneracy is confirmed numerically: the $\mathit{MA}{E}_{T_0}$ of every network configuration is within 1% of the baseline, and $R_{T_0}^2$ is effectively zero (not shown, as it is negative—worse than predicting the mean). No amount of network capacity, training data, or feature engineering can overcome a physical degeneracy.

$\gamma$ strong identifiability: curvature control. The inversion gradient γ directly determines the modified refractivity gradient dM/dz within the ducting layer, which in turn controls the ray curvature. A larger γ produces tighter ray bending, shorter bounce distances, and more multipath paths within a given link distance. These effects are directly observable through the AOA vector and multipath count, providing strong constraints on γ. The $R_{\gamma }^2$ = 0.843 and the 70.5% MAE reduction over baseline confirm this strong identifiability.

$z_i$ weak identifiability: $\gamma$–$z_i$ coupling. The layer-top height z_i determines where the inversion gradient transitions to the standard lapse rate, setting the vertical extent of the ducting layer. However, the ray-turning height depends on both γ and z_i through the integral of dM/dz: a larger z_i with a smaller γ can produce a similar turning height and thus similar multipath structure. This coupling limits the independent information about z_i, resulting in $R_{z_i}^2$ = 0.22.

Practical strategy: A+B. Based on this analysis, the inversion strategy adopted in this work is:

• $\gamma$: inverted from FSO link measurements (R² = 0.843, MAE = 0.033 K/m).
• $z_i$: inverted from FSO link measurements, accepting moderate accuracy (R² = 0.22, MAE = 16.1 m). The trapping/non-trapping classification, validated by the 82% consistency rate, is more reliable than the continuous regression.
• $T_0$: not inverted from link measurements. Instead, it is supplied by a co-located ground meteorological station—a single temperature sensor at the link terminal. This is consistent with standard FSO deployment practice, where terminal meteorological sensors are routinely installed for link-margin monitoring.

This strategy transforms an apparent limitation (T₀ unidentifiability) into a practical advantage: by removing the unidentifiable parameter from the inversion, the network can focus its capacity on the identifiable parameters (γ, z_i), and the ground station provides T₀ at zero additional deployment cost.

5.3. Noise Robustness

To assess robustness under degraded measurement conditions, additional Gaussian noise is injected into the AOA features of the test set at multiples of the baseline noise level (σ_base = 50 μrad), and the full model (with dual-wavelength features and physics constraint) is re-evaluated (Figure 9 and Table 7). The additional noise levels are 0.5×, 1×, 2×, and 4× the baseline, corresponding to 25, 50, 100, and 200 μrad additional 1σ noise.

vs. noise multiplier. (c) Self-consistency rate vs. noise multiplier (paradoxical increase explained by conservative predictions). (d) Example predicted profiles at ×0.5 and ×4 noise levels.

Key observations:

11.

Sub-linear MAE degradation. From ×0.5 to ×4.0 noise (an 8-fold increase in additional noise), the MAE_γ degrades from 0.034 to 0.097 K/m—a factor of 2.85×. This sub-linear growth (error grows slower than noise) indicates that the network has learned robust features that do not overfit to the nominal noise level.

12.

R² degrades faster than MAE. While the MAE grows sub-linearly, $R_{\gamma }^2$ drops sharply from 0.832 to 0.064. This is because R² measures the fraction of variance explained: as noise increases, the residual variance grows, reducing R² even if the absolute error remains moderate. At ×4 noise, the $R_{\gamma }^2$ ≈ 0.06 indicates that the model's predictions are barely better than the mean—effectively, the signal-to-noise ratio has dropped below the inversion threshold. However, the MAE_γ = 0.097 K/m is still below the baseline's 0.112 K/m, meaning the model retains some predictive power.

13.

Consistency paradoxically increases with noise. The self-consistency rate rises from 81.9% at ×0.5 noise to 93.8% at ×4.0 noise. This counterintuitive trend has a simple explanation: as noise increases, the network's predictions become more conservative (closer to the training-set mean), and since 81.8% of training profiles are ducting (γ > γc), the mean prediction is ducting. For samples with n_paths ≥ 2 (which are predominantly ducting profiles), this conservative prediction is more likely to satisfy the consistency condition. This behavior underscores that the consistency rate measures physical self-consistency, not inversion accuracy—both must be considered together.

14.

Engineering threshold. The results suggest that the inversion remains useful up to approximately ×2 noise (100 μrad additional), where $R_{\gamma }^2$ = 0.604 and MAE_γ = 0.059 K/m. Beyond this point, the noise overwhelms the signal, and the inversion degrades to a near-baseline level. For reference, state-of-the-art FSO tracking systems achieve AOA measurement precision of 10–50 μrad, well within the useful range.

6. Discussion

6.1. Comparison with Microwave Duct Inversion

The optical duct inversion problem differs from its microwave counterpart in several fundamental respects:

15.

Ducting mechanism. Microwave evaporation ducts are driven by the humidity gradient (the 4810·e/T² term in radio refractivity), while optical ducts are driven by the temperature inversion gradient. The two mechanisms coexist in the same marine boundary layer but affect different frequency bands independently—a radiosonde observation that characterizes the humidity profile does not directly constrain the optical duct, and vice versa.

16.

Observable types. Microwave inversion techniques use radar sea clutter [7], propagation loss [8], or GPS occultation [9] as observables. Optical inversion uses angle-of-arrival and multipath structure—fundamentally different quantities that require a different forward model (ray tracing vs. parabolic equation) and feature representation.

17.

Wavelength diversity. The dual-wavelength (850/1550 nm) chromatic dispersion observable has no direct microwave analogue. The 2 N-unit refractivity difference between the two wavelengths produces 10–40 μrad AOA separations per path—small but measurable, and directly sensitive to the refractivity gradient. This feature adds genuine information content, as demonstrated by the +0.024 R² improvement in the ablation study.

18.

Identifiability structure. The iso-observation degeneracy of T₀ is specific to the optical regime (where refractivity depends on P/T, and a uniform temperature shift produces an almost-uniform N shift whose residual gradient effect is absorbed by γ). In the microwave regime, the humidity term breaks even this near-degeneracy because water vapor has its own vertical structure. The identifiability analysis framework, however, is transferable: the same approach (ablation + physical reasoning) can be applied to microwave duct parameters.

6.2. Engineering Implications

The proposed framework has several implications for the design of maritime FSO systems:

• Duct-aware link planning. By inverting γ from link measurements, the system can predict whether the current atmospheric conditions support ducting and adjust link-margin allocations accordingly. The 82% consistency rate ensures reliable trapping/non-trapping classification.
• Over-the-horizon detection. When ducting is detected (high γ, n_paths ≥ 2), the link may experience anomalous over-the-horizon reception. The inversion framework provides real-time awareness of this condition without requiring additional sensors beyond the FSO terminal itself and a single temperature sensor.
• Dual-wavelength deployment. The ablation study quantifies the benefit of dual-wavelength operation: the +0.024 R² improvement and +3.4 pp consistency improvement justify the added complexity of a dual-wavelength (850/1550 nm) transceiver. These two wavelengths are already standard in optical communication (silica fiber windows), making the dual-wavelength requirement a natural fit for FSO systems with fiber-optic backends.
• Ground station integration. The T₀ unidentifiability result provides a clear design directive: a single temperature sensor at the link terminal is both necessary and sufficient to supply the missing parameter. This is a minimal hardware requirement that is already met by most deployed FSO systems for environmental monitoring.

6.3. Limitations

Several limitations should be noted:

19.

Synthetic data. The training and evaluation data are generated from a forward model, not from field measurements. While the forward model is physically grounded and validated (analytic checks, Bouguer invariant conservation, diffraction-negligibility analysis), real-world atmospheric profiles may exhibit complexities not captured by the three-parameter bilinear model—e.g., multi-layer inversions, turbulence-induced scintillation, and horizontal inhomogeneity.

20.

Focusing factor placeholder. The geometric focusing gain is currently set to unity for all paths. In reality, the ray-tube convergence/divergence produces path-dependent intensity variations that carry additional information about the profile shape. Implementing a ray-tube-based focusing calculation is a straightforward extension that would enrich the feature vector.

21.

Single-link geometry. All links use ht = hr = 10 m. In practice, FSO terminals may be at different heights (e.g., ship-to-shore links). The network architecture accepts link parameters as input features, so heterogeneous geometries can be incorporated by expanding the training data.

22.

Turbulence not modeled. Clear-air turbulence produces scintillation and beam wander that are distinct from ducting effects. In strongly turbulent conditions, the AOA noise may exceed the levels tested in the noise sweep. Extending the noise model to include turbulence-induced fading is left for future work.

23.

$z_i$ weak identifiability. The $R_{z_i}^2$ = 0.22 limits the practical accuracy of layer-top height inversion. Future work could explore loss weighting schemes (a "focus factor" that up-weights z_i in the loss) or attention mechanisms that allocate more network capacity to z_i-sensitive features. Alternatively, multi-link configurations (different receiver heights) could provide the additional geometric diversity needed to decouple γ and z_i.

6.4. Future Work

Based on the findings and limitations, the following directions are identified:

24.

Field validation. The most critical next step is validation with real FSO link data over maritime paths. A dual-wavelength (850/1550 nm) link with AOA tracking and a co-located temperature sensor would provide all necessary inputs for the inversion framework.

25.

Focus factor for $z_i$. A loss-weighting or multi-task learning approach that allocates increased capacity to z_i-sensitive features could improve the weak identifiability of the layer-top height.

26.

Extended profile models. Generalizing beyond the three-parameter bilinear profile to multi-layer or non-parametric profiles would broaden the applicability of the framework. A neural-network-parameterized profile (rather than a fixed-form parameterization) could be jointly optimized with the inversion network.

27.

Joint optical–microwave inversion. Since the optical duct (temperature-driven) and the microwave evaporation duct (humidity-driven) share the same sea-air thermodynamic state, joint inversion of both duct types from combined optical and microwave measurements could provide tighter constraints on the boundary-layer state than either modality alone.

28.

Real-time implementation. The trained network infers profile parameters in milliseconds on a CPU, making it suitable for real-time link adaptation. Integration with an adaptive optics or power-control subsystem could demonstrate closed-loop duct-aware FSO operation.

7. Conclusions

This paper has presented a physics-informed deep learning framework for inverting marine boundary-layer optical duct parameters from multi-wavelength free-space optical link measurements. The key findings are:

29.

First optical duct inversion. The framework is the first to invert marine optical duct parameters (inversion gradient, layer-top height, surface temperature) directly from FSO channel observations, addressing a gap in the existing literature where only microwave evaporation ducts have been considered.

30.

Strong inversion of the critical parameter. The inversion gradient γ—the parameter that governs whether ducting occurs—is inverted with high accuracy (R² = 0.843, MAE = 0.033 K/m), representing a 70.5% improvement over the mean-prediction baseline. This confirms that multipath structure and dual-wavelength chromatic dispersion carry rich information about the refractivity gradient.

31.

Systematic identifiability analysis. The three profile parameters exhibit fundamentally different identifiability: γ is strongly identifiable (R² = 0.843), z_i is weakly identifiable (R² = 0.22), and T₀ is physically unobservable from link measurements due to its iso-observation degeneracy with γ (near-translation invariance of ray bending under a profile offset). This analysis motivates a practical strategy in which T₀ is supplied by a ground meteorological station—standard in deployed FSO systems—transforming a physical limitation into a minimal-cost engineering solution.

32.

Physics-guided regularization. The trapping-consistency physics constraint does not improve regression accuracy ($R_{\gamma }^2$ change of +0.004) but significantly improves physical self-consistency (+2.5 pp), demonstrating that the physics loss serves as a regularizer ensuring predictions respect observed multipath structure.

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Dual-wavelength value. The 850/1550 nm chromatic dispersion features improve $R_{\gamma }^2$ by +0.024 and consistency by +3.4 pp, justifying dual-wavelength operation for maritime FSO systems.

34.

Noise robustness. The inversion degrades sub-linearly under elevated measurement noise (MAE grows ×2.85 for ×8 noise increase), remaining useful up to approximately twice the nominal noise level.

The framework establishes a new sensing modality for maritime FSO channel prediction: by treating the atmosphere as an information channel whose multipath structure encodes the boundary-layer thermodynamic state, the FSO link itself becomes a distributed sensor of the optical duct. Combined with the identifiability-guided parameter allocation strategy, this approach offers a practical, deployable solution for duct-aware adaptive optical transmission over the ocean.