# Ship-Ship Collision Calculations

This chapter describes OMRAT’s calculations for ship-ship collision risk. Four types of collision encounters are modelled: head-on, overtaking, crossing, and bend collisions, all based on the equations in Hansen (2008).

## Overview

Ship-ship collisions occur when two vessels occupy the same space at the same time. The IWRAP methodology models this by calculating the geometric number of collision candidates – the expected number of close encounters per year assuming no evasive action – and then multiplying by a causation factor to get the actual accident frequency.

The general formula is:

\[F_{\text{collision}} = N_G \times P_C\]

Where:

- \(N_G\) = geometric collision candidates per year

- \(P_C\) = causation probability (IALA default values)

Pipeline orchestrator: `compute/ship_collision_model.py:526` – run_ship_collision_model()

## Head-On Collisions

Head-on collisions occur between vessels travelling in opposite directions on the same leg. This is the classical encounter type for narrow shipping lanes.

### Geometry

Two streams of traffic travel in opposite directions. The collision probability depends on the lateral overlap of their position distributions.

### Equations (Hansen Eq. 4.2–4.4)

The geometric number of collision candidates is:

\[N_G = \frac{Q_1}{V_1} \times \frac{Q_2}{V_2} \times V_{ij} \times P_G \times L_w \times \frac{1}{\text{sec/year}}\]

Where:

- \(Q_1, Q_2\) = traffic volume in each direction (ships/year)

- \(V_1, V_2\) = vessel speeds (m/s)

- \(V_{ij} = V_1 + V_2\) = relative closing speed (m/s)

- \(L_w\) = leg segment length (m)

- \(P_G\) = geometric collision probability

The term \(Q/V\) converts from traffic frequency (ships/year) to traffic density (ships/m), accounting for the time ships spend on the leg.

The geometric collision probability is:

\[P_G = \Phi\!\left(\frac{\mu_{ij} + B_{ij}}{\sigma_{ij}}\right) - \Phi\!\left(\frac{\mu_{ij} - B_{ij}}{\sigma_{ij}}\right)\]

Where:

- \(\mu_{ij} = \mu_1 + \mu_2\) – combined mean lateral distance (head-on: positions add because ships face opposite directions)

- \(\sigma_{ij} = \sqrt{\sigma_1^2 + \sigma_2^2}\) – combined standard deviation

- \(B_{ij} = (B_1 + B_2) / 2\) – average vessel breadth (collision width)

- \(\Phi\) = standard normal CDF

`compute/basic_equations.py:46` – get_head_on_collision_candidates()

Head-on collision candidate calculation (compute/basic_equations.py)

    # Matching them lets us bypass scipy's frozen-distribution dispatch (~2 ms
    # per call) and use C-level ``ndtr`` / a direct Weibull formula.
    _NORM_FUNC_RE = re.compile(
        r"\.norm\(loc\s*=\s*(?P<loc>-?\d+(?:\.\d+)?)\s*,\s*scale\s*=\s*(?P<scale>-?\d+(?:\.\d+)?)\s*\)\s*\.cdf\(x\)"
    )
    _WEIBULL_FUNC_RE = re.compile(
        r"\.weibull_min\(c\s*=\s*(?P<c>-?\d+(?:\.\d+)?)\s*,\s*loc\s*=\s*(?P<loc>-?\d+(?:\.\d+)?)\s*,\s*scale\s*=\s*(?P<scale>-?\d+(?:\.\d+)?)\s*\)\s*\.cdf\(x\)"
    )

    def _repair_fn(data: dict, drift_speed: float):
        """Return a callable ``f(distance) -> P(not repaired)`` for *data*.

        Caches the compiled scalar distribution callable, so the hot per-edge /
        per-ship call sites don't re-parse strings or re-create distribution
        objects.  For the two dominant shapes (``norm.cdf(x)`` /
        ``weibull_min.cdf(x)``) we bypass scipy's frozen-distribution machinery
        and call ``scipy.special.ndtr`` or a direct Weibull formula -- scipy's
        generic ``.cdf`` has ~2 ms of Python-level dispatch per call, and
        ``get_not_repaired`` runs tens of thousands of times per cascade.
        """
        key: tuple
        if data.get('use_lognormal') == 1:
            key = ('lognorm', float(data['std']), float(data['loc']),
                   float(data['scale']), float(drift_speed))
        else:
            key = ('func', str(data.get('func', '')), float(drift_speed))

        fn = _REPAIR_FN_CACHE.get(key)
        if fn is not None:
            return fn

        if data.get('use_lognormal') == 1:
            # lognorm.cdf(x; s, loc, scale) = ndtr(log((x-loc)/scale) / s) for x>loc
            s = float(data['std'])
            loc = float(data['loc'])
            scale = float(data['scale'])

            def fn(distance: float, _s=s, _loc=loc, _scale=scale, _spd=drift_speed) -> float:
                drift_time = distance / _spd / 3600.0
                t = drift_time - _loc
                if t <= 0 or _scale <= 0:
                    return 1.0

### Default causation factor

\[P_C = 4.9 \times 10^{-5} \quad \text{(Fujii et al. 1974)}\]

## Overtaking Collisions

Overtaking collisions occur between vessels travelling in the same direction at different speeds. The faster vessel catches up to the slower one.

### Geometry

Both vessels travel in the same direction. The collision probability depends on the speed difference (overtaking is only possible if \(V_{\text{fast}} > V_{\text{slow}}\)) and the lateral overlap of their distributions.

### Equations (Hansen Eq. 4.5)

The formula is structurally identical to head-on, but with different relative speed and lateral offset:

\[V_{ij} = V_{\text{fast}} - V_{\text{slow}}\]
\[\mu_{ij} = \mu_{\text{fast}} - \mu_{\text{slow}}\]

Note

If \(V_{\text{fast}} \leq V_{\text{slow}}\), overtaking is impossible and the result is zero.

The geometric probability \(P_G\), combined standard deviation \(\sigma_{ij}\), and breadth \(B_{ij}\) are calculated identically to the head-on case.

`compute/basic_equations.py:147` – get_overtaking_collision_candidates()

### Default causation factor

\[P_C = 1.1 \times 10^{-4} \quad \text{(Fujii et al. 1974)}\]

## Crossing Collisions

Crossing collisions occur at intersections where two legs cross at an angle \(\theta\). This is common at traffic separation scheme junctions, port approaches, and areas where multiple routes converge.

### Geometry

Two traffic streams cross at angle \(\theta\). The collision zone is an area around the intersection point whose size depends on vessel dimensions and the crossing angle.

### Equations (Hansen Eq. 4.6)

\[N_G = \frac{Q_1 \times Q_2 \times D_{ij}} {V_1 \times V_2 \times |\sin\theta| \times \text{sec/year}}\]

Where:

- \(\theta\) = crossing angle (radians)

- \(D_{ij}\) = collision diameter (metres)

- \(V_1, V_2\) = vessel speeds on each leg

The relative speed uses the law of cosines:

\[V_{ij} = \sqrt{V_1^2 + V_2^2 - 2 V_1 V_2 \cos\theta}\]

The collision diameter accounts for the projected area of both vessels at the crossing angle:

\[D_{ij} = (L_1 + L_2) \times |\sin\theta| + (B_1 + B_2) \times |\cos\theta|\]

Where \(L_1, L_2\) are ship lengths and \(B_1, B_2\) are ship beams.

Note

When \(\theta \to 0\) or \(\theta \to \pi\) (parallel or anti-parallel courses), \(\sin\theta \to 0\) and the crossing formula is not applicable. Use head-on or overtaking formulas instead.

`compute/basic_equations.py:238` – get_crossing_collision_candidates()

Crossing collision with law of cosines (compute/basic_equations.py)

    def get_head_on_collision_candidates(
        Q1: float,           # Traffic in direction 1 (ships/year)
        Q2: float,           # Traffic in direction 2 (ships/year)
        V1: float,           # Speed direction 1 (m/s)
        V2: float,           # Speed direction 2 (m/s)
        mu1: float,          # Mean lateral position dir 1 (m)
        mu2: float,          # Mean lateral position dir 2 (m)
        sigma1: float,       # Std dev lateral position dir 1 (m)
        sigma2: float,       # Std dev lateral position dir 2 (m)
        B1: float,           # Beam of vessel type 1 (m)
        B2: float,           # Beam of vessel type 2 (m)
        L_w: float           # Leg segment length (m)
    ) -> float:
        """
        Calculate geometric number of head-on collision candidates.

        Uses Hansen Eq. 4.2-4.4:
        N_G = Q1 × Q2 × V_ij × P_G × L_w

        Where:
        - V_ij = V1 + V2 (relative closing speed for head-on)
        - P_G = Φ((μ_ij + B_ij)/σ_ij) - Φ((μ_ij - B_ij)/σ_ij)
        - μ_ij = μ1 + μ2 (mean lateral distance between vessels)
        - σ_ij = √(σ1² + σ2²) (combined standard deviation)
        - B_ij = (B1 + B2)/2 (average vessel breadth)

        Parameters
        ----------
        Q1 : float
            Traffic in direction 1 (ships/year)
        Q2 : float
            Traffic in direction 2 (ships/year)
        V1 : float
            Speed direction 1 (m/s)
        V2 : float
            Speed direction 2 (m/s)
        mu1 : float
            Mean lateral position dir 1 (m)
        mu2 : float
            Mean lateral position dir 2 (m)

### Default causation factor

\[P_C = 1.3 \times 10^{-4} \quad \text{(Pedersen 1995)}\]

## Bend Collisions

Bend collisions occur at waypoints where a route changes direction. A vessel that fails to make the turn continues on its original heading and may collide with vessels that did turn correctly.

### Geometry

At a waypoint, the route changes direction by angle \(\theta\). Most vessels turn correctly, but a small fraction \(P_0\) (typically 1%) fail to turn and continue straight.

### Equations

\[Q_{\text{no turn}} = Q \times P_0\]
\[Q_{\text{turn}} = Q \times (1 - P_0)\]

The bend collision is then modelled as a crossing collision between the non-turning traffic and the turning traffic:

\[N_{\text{bend}} = N_G^{\text{crossing}}(Q_{\text{no turn}}, Q_{\text{turn}}, \theta)\]

Where \(\theta\) is the bend angle (change in heading at the waypoint) and the crossing collision formula is applied with self-interaction (same ship type on both “legs”).

`compute/basic_equations.py:331` – get_bend_collision_candidates()

### Default causation factor

\[P_C = 1.3 \times 10^{-4} \quad \text{(Pedersen 1995)}\]

Default probability of not turning:

\[P_0 = 0.01\]

## Calculation Workflow

The collision calculation for each leg proceeds as follows:

1. Extract traffic data: For each segment, extract all ship types, frequencies, speeds, and dimensions in both directions.

2. Pair ship types: For head-on and overtaking, pair every ship type \(i\) in direction 1 with every ship type \(j\) in direction 2 (or same direction for overtaking).

3. Calculate per-pair N_G: Apply the appropriate formula based on encounter type, using the specific vessel speeds, beams, and lateral distributions.

4. Sum over all pairs: The total geometric candidates is the sum over all ship type pairs.

5. Apply causation factor: Multiply by \(P_C\) to get accident frequency.

6. Display results: Results are shown in the UI as annual accident frequencies per collision type.

## Summary of Equations

Type | Relative Speed | Lateral Offset | Key Parameter
--- | --- | --- | ---
Head-on | \(V_1 + V_2\) | \(\mu_1 + \mu_2\) | Opposite directions
Overtaking | \(V_f - V_s\) | \(\mu_f - \mu_s\) | Same direction, \(V_f > V_s\)
Crossing | \(\sqrt{V_1^2+V_2^2-2V_1V_2\cos\theta}\) | \(D_{ij}\) | Crossing angle \(\theta\)
Bend | (crossing formula) | (crossing formula) | \(P_0 = 0.01\)