Theory (what is calculated)

This is the theory track – a short umbrella that tells you where to find each accident type’s derivation and the few global conventions shared by all of them. For the call-tree companion (the “how” track), see Code Flow: From “Run Model” to Results.

The IWRAP framework in one equation

Every accident frequency OMRAT reports has the form

\[F_\mathrm{accident} = N_A \cdot P_C\]

  • \(N_A\) is the geometric candidate count – how often an accident could happen from geometry + traffic alone.

  • \(P_C\) is the causation factor – the conditional probability that a candidate becomes an actual accident (the crew fails to avoid, the machinery fails, etc.).

The accident-type chapters derive \(N_A\) from first principles and give sources for \(P_C\):

Chapter

Covers

Drifting Risk Calculations

Drifting grounding / allision / anchoring. The largest chapter, with five worked examples in drifting/debug/level_1 … level_5.

Ship-Ship Collision Calculations

Head-on, overtaking, crossing, bend collisions (Hansen eq. 4.2-4.4, Pedersen).

Powered Grounding and Allision

IWRAP Category II powered grounding + allision (\(N_{II} = P_c Q m \exp(-d/(a_i V))\)).

Default causation factors

OMRAT ships the IALA default table. These are the values most published studies use unless there’s local data.

Default causation factors (IALA / Fujii / Pedersen)

Accident type

IALA default

Fujii (1974)

Notes

Head-on collision

\(4.9 \times 10^{-5}\)

\(4.9 \times 10^{-5}\)

TSS present helps; higher in narrow lanes.

Overtaking collision

\(1.1 \times 10^{-4}\)

\(1.1 \times 10^{-4}\)

Crossing collision

\(1.3 \times 10^{-4}\)

\(1.2 \times 10^{-4}\)

Pedersen value.

Bend collision

\(1.3 \times 10^{-4}\)

–

Pedersen value.

Powered grounding

\(1.6 \times 10^{-4}\)

\(1.6 \times 10^{-4}\)

Allision (structure)

\(1.9 \times 10^{-4}\)

\(1.9 \times 10^{-4}\)

Drifting

\(1.0\)

–

No avoidance – the ship is powerless.

Local adjustment factors are typically applied on top:

  • Ferry / passenger vessels – divide by 20 (two navigators, well-known route).

  • Pilot on board – divide by 3 (COWIconsult).

  • Poor visibility (3-10 %) – multiply by 2.

  • Poor visibility (10-30 %) – multiply by 8.

Adjust these in Settings -> Causation Factors if your project needs them.

Lateral traffic distribution

Ship positions across a lane are modelled as a mixture of up to three Gaussian components and one uniform component:

\[f(z) = \sum_{i=1}^{3} w_i \; \phi\!\left(\frac{z - \mu_i}{\sigma_i}\right) + w_u \; U(z; a, b)\]

where

  • \(z\) is the lateral distance from the leg centreline (m),

  • \(w_i\), \(\mu_i\), \(\sigma_i\) are the weight, mean, and standard deviation of Gaussian \(i\),

  • \(w_u\) is the weight of the uniform component on \([a, b]\),

  • weights are normalised so \(\sum w_i + w_u = 1\).

You can fit the mixture from AIS data or set it by hand. Values are stored per segment per direction (mean1_1, std1_1, weight1_1, u_min1, u_max1, u_p1, etc.).

Implemented in compute.data_preparation.get_distribution().

Coordinate systems

  • WGS84 (EPSG:4326) – all stored geometry, user inputs, and the .omrat JSON file. Lon/lat.

  • UTM – used internally for metric calculations. OMRAT picks the zone from the study-area centroid and projects once per run.

The drifting model lives in UTM end-to-end. The powered model uses a cheaper per-project equirectangular projection centred on the first leg’s start point (SimpleProjector), which is fine at the per-leg length scale where rays travel tens of kilometres.

Compass convention

OMRAT uses standard nautical bearings everywhere: 0 = N, 90 = E, 180 = S, 270 = W, measured clockwise from north. The wind rose, stored segment bearings, and drift directions all follow this convention.

Internal math uses the standard math convention (0 = E, counter-clockwise). The conversion is:

\[\theta_\mathrm{math} = (90^\circ - \theta_\mathrm{compass}) \bmod 360^\circ\]

Canonical implementation: drifting/engine.py:compass_to_math_deg. Callers that need an (x, y) step-vector directly use geometries/drift/coordinates.py:compass_to_vector.

Direction -> compass bearing

Direction

Bearing

Vector (+X=East, +Y=North)

N

0

(0, +d)

NE

45

(+d/sqrt(2), +d/sqrt(2))

E

90

(+d, 0)

SE

135

(+d/sqrt(2), -d/sqrt(2))

S

180

(0, -d)

SW

225

(-d/sqrt(2), -d/sqrt(2))

W

270

(-d, 0)

NW

315

(-d/sqrt(2), +d/sqrt(2))

References

  • Friis-Hansen, P. (2008). IWRAP MK II - Basic Modelling Principles for Prediction of Collision and Grounding Frequencies. Technical University of Denmark.

  • Pedersen, P.T. (1995). Collision and Grounding Mechanics. WEMT’95.

  • Fujii, Y. et al. (1974). Some factors affecting the frequency of accidents in marine traffic. Journal of Navigation, 27.

  • Talavera, A. et al. (2013). Application of Dempster-Shafer theory for the quantification and propagation of the uncertainty caused by the use of AIS data. Reliability Engineering and System Safety, 111, 95-105.

  • Engberg, P.C. (2017). IWRAP Mk2 v5.3.0 Manual. GateHouse A/S.