A Risk-Contour-Based Ship Domain Model Using Riemann-Sphere Projection and AIS Validation

Conventional ship-domain models often prescribe boundary geometries empirically or infer them statistically from observed traffic, which may limit their ability to represent the directional variation in modeled encounter risk. This study develops a risk-contour-based ship-domain model using Riemann-sphere projection. Relative position and motion are mapped from the Euclidean encounter plane into a bounded spherical space, where projected relative distance and signed projected closing speed are combined to construct a composite encounter-risk index. A sigmoid transformation is used for the closing-speed component, and softmax-based weights adaptively combine the distance- and speed-related components. For head-on, starboard-side crossing, port-side crossing, and overtaking scenarios, the first positions at which the composite risk reaches a prescribed threshold are extracted for each relative bearing. These scenario-specific first-threshold-crossing points are fused using a pointwise outer-envelope rule, followed by periodic interpolation and smoothing, to reconstruct a directional ship-domain boundary. The reconstructed boundary is evaluated using AIS-derived encounter trajectories from the Florida Straits. The evaluation examines the temporal relationship among predicted domain entry, observed domain entry, and detectable changes in course or speed under a constant-course and constant-speed extrapolation assumption. Under the adopted encounter-screening and high-risk classification criteria, all identified high-risk samples generated a predicted-entry warning, while the predicted-entry warning rate for routine encounters ranged from 30.4% to 40.5%. Among the analyzed encounter trajectories, 17.4% entered the reconstructed ship domain. These results indicate that the proposed procedure can generate a directional and fore-and-aft asymmetric model-based warning contour. However, the reported warning performance is conditional on the selected projection scale, risk parameters, threshold, motion-extrapolation rule, AIS processing procedure, and high-risk labeling criteria. The reconstructed ship domain should therefore be interpreted as a model-based warning contour rather than as a direct measurement of mariners’ psychological perception or a direct indicator of collision probability.