JEOS RP ISSN03 | Page 134

J. Eur. Opt. Society-Rapid Publ. 22, 13( 2026) 127

be deliberately disabled by“ dark vessels,” or may be unavailable in remote ocean areas due to sparse receiver coverage. Consequently, there is a critical need for passive, non-cooperative surveillance systems that do not rely on the target’ s participation. Submarine cables, which already crisscross the world’ s oceans, offer an ideal, covert platform for this purpose [ 7 ]. By connecting a DAS interrogator to dark fiber strands, real-time acoustic monitoring can be achieved over vast distances with high resolution. This work presents a rigorous physical – mathematical framework for the acoustic detection of surface vessels using bottommounted optical fibers, deriving the opto-acoustic transfer function and hyperbolic moveout equations from the wave equation for a moving point source, and solving the inversion using a Differential Evolution – based stochastic optimization strategy.

2 Physical principles of phase-sensitive OTDR

To interpret the acquired data, we model the Phasesensitive Optical Time Domain Reflectometry(/-OTDR) process [ 2 ]. The fiber acts as a one-dimensional waveguide with random scattering centers. As a coherent laser pulse propagates, the backscattered electric field is formed by the superposition of contributions from scatterers within the pulse width.

When an external acoustic pressure wave P( r, t) interacts with the cable, it induces a mechanical strain on the fiber core. In DAS, we measure the differential phase shift D / between two sections of fiber separated by a gauge length L g. Assuming the acoustic wavelength is larger than the gauge length, the relationship between the recorded optical phase and the axial dynamic strain zz is linear:

/ ðz; t Þ 4pnnL g

zz ðz; tÞ; ð1Þ k

where n 0.78 is the photo-elastic scaling factor and n is the refractive index. This establishes that the recorded optical phase is a direct proxy for the strain induced by the vessel’ s acoustic emission.

3 Kinematic and acoustic wavefield modeling

We model the ocean as a semi-infinite homogeneous halfspace and the vessel as a moving point source emitting a broadband signal S( t).

3.1 Hyperbolic moveout and directional sensitivity

For a vessel moving with velocity v s along a linear trajectory at the surface, the acoustic arrival time at the fiber creates a hyperbola:

ð Þ ¼ t2 CPA þ ðx � x CPAÞ 2; ð2Þ t 2 arr x

V 2 app

where t CPA is the time at the Closest Point of Approach( CPA) and V app is the moveout velocity. This relationship

Algorithm 1. Pseudocode: Forward model simulation

1: Input: Grid ðx; tÞ, Vessel Params m 2: Output: Synthesized Data Matrix Dðx; tÞ 3: Initialize D 0 4: for each time step t j do 5: Calculate vessel position r s ðt j Þ using v s; h 0 6: for each sensor channel x i do 7: Compute range R ij ¼kr s ðt j Þ�r fib ðx i Þk 8 Compute Doppler factor f Dop and Arrival Time t

P arr sinð2pfDop t arr Þ

allows us to invert the vessel speed and impact parameter from the data curvature. Additionally, the fiber acts as a dipole sensor; its sensitivity to axial strain results in a characteristic amplitude“ null” when the acoustic wavefronts arrive perpendicular to the cable( at CPA), facilitating precise localization [ 8 ].

3.2 Spectral interference and Doppler effects

Beyond kinematics, the signal contains spectral features derived from the source depth and motion. The interaction between the direct acoustic path and the surface reflection( Lloyd’ s Mirror effect) creates frequency notches dependent on the source depth z s. Simultaneously, the broadband noise undergoes a Doppler shift due to the relative radial velocity, resulting in a sigmoidal frequency transition. These physical constraints – interference patterns and Doppler shifts – are critical for validating estimated velocities and depths [ 9 ].

3.3 Numerical implementation

To validate the model, we implemented a numerical forward model that synthesizes the DAS waterfall plot D( x, t). The simulation procedure is summarized in Algorithm 1. It generates a parametric source S( t) composed of a fundamental frequency and harmonics, modulated by Lloyd’ s Mirror interference. The source is spatially windowed by a geometric hull function to simulate the acoustic wake, scaled by geometrical spreading, and finally convolved with the gauge length operator to replicate the physical measurement process.

4 Inverse problem and optimization strategy

Inverting kinematic parameters from DAS data is a nonlinear optimization problem complicated by oscillatory