Individual Risk Contours

Technical methodology for calculating Individual Risk (IR) contour maps using TekRisk — grid-based IR aggregation, Marching Squares contouring, and GeoJSON output for QRA visualization

1. Executive Summary

Individual Risk (IR) contours are iso-risk lines drawn on a map that connect all geographic points sharing the same annual probability of fatality due to industrial hazards. They are the cornerstone of Quantitative Risk Assessment (QRA) in the chemical and oil & gas industries.

TekRisk computes IR contours by:

Aggregating all defined risk scenarios (fires, explosions, toxic clouds) at a facility

Evaluating the fatality probability at thousands of grid points around the site

Extracting contour lines at standard risk levels (e.g., $1 \times 10^{-5}$ , $1 \times 10^{-6}$ per year)

Projecting results onto a geographic map as GeoJSON polygons

Map visualization

The result is a set of nested contour polygons — each representing a different risk level — overlaid on a Mapbox GL map, allowing engineers to visually assess whether vulnerable receptors (schools, hospitals, residential areas) fall within unacceptable risk zones.

2. Key Concepts

Individual Risk (IR)

Annual probability that a hypothetical person, continuously present and unprotected at a specific location, will be killed due to an industrial accident. Expressed as a dimensionless number per year (e.g., $IR = 1 \times 10^{-5}$ /year = 1 in 100,000 chance).

Scenario Frequency (f)

How often a particular accident scenario is expected to occur, in events/year. Derived from historical failure rate databases (OREDA, OGP/IOGP) combined with event tree analysis.

Fatality Probability (Pf)

Probability of death at a specific location given a scenario occurs. Derived from probit models (thermal/blast) or point-in-polygon tests (flash fire — $P_f = 1.0$ inside the LEL envelope, $0$ outside).

Contour Levels

Specific IR values at which iso-risk lines are drawn. TekRisk uses seven standard levels: $10^{-2}$ through $10^{-8}$ per year.

2.1 Core Formula

The Individual Risk at any point $(x, y)$ is the sum of contributions from all scenarios:

IR(x, y) = \sum_{i} f_i \times P_{f,i}(x, y)

Symbol	Description	Units
$IR(x, y)$	Individual risk at location $(x, y)$	per year
$f_i$	Frequency of scenario $i$	events/year
$P_{f,i}(x, y)$	Probability of fatality at $(x, y)$ given scenario $i$ occurs	dimensionless $[0, 1]$
$i$	Scenario index (each risk model is a scenario)	—

3. Supported Risk Models

TekRisk supports five consequence model types, each contributing to the overall IR calculation:

Model	Hazard Type	Symmetry	$P_f$ Method	Wind Dependence
Fireball	Thermal radiation	Radial	Distance-based interpolation on fatality profile	None — omni-directional
Pool Fire	Thermal radiation	Radial	Distance-based interpolation on fatality profile	None — radially symmetric
Jet Fire	Thermal radiation	Radial	Distance-based interpolation on fatality profile	None — radially symmetric
VCE	Blast overpressure	Radial	Distance-based interpolation on fatality profile	None — radially symmetric
Flash Fire	Flame engulfment	Directional	Point-in-polygon on rotated LEL envelope	Critical — uses full 16-direction wind rose

Fireball, Pool Fire, Jet Fire, VCE — These models assume the hazard effect decays with distance from the source in all directions equally. The fatality probability at any point depends only on the Euclidean distance to the source, following a pre-computed fatality profile (fatality percentage vs. distance).

4. Step-by-Step Methodology

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Step 1: Scenario Definition

Each enabled risk model defines a scenario characterized by:

Parameter	Description	Source
Frequency ( $f$ )	Annual occurrence rate	Risk model configuration
Source location	Geographic coordinates (lat, lon)	Source definition
Model type	fireball, poolfire, jetfire, vce, flashfire	Risk model type
Fatality profile	$P_f$ vs. distance curve (radial) or LEL polygon (flash fire)	Prior consequence calculation results

Fatality profiles are extracted from previously completed consequence calculations stored in RiskCalculation.results.fatalidades. Each profile consists of distance-fatality percentage pairs:

Distance (m)	Fatality (%)
0	100
25	95
50	70
100	30
200	5
350	0

Flash Fire profiles

For flash fire models, the fatality "profile" is replaced by a LEL polygon — the geographic outline of the flammable cloud at the Lower Explosive Limit concentration for a single wind direction.

Step 2: Wind Rose Integration (Flash Fire)

Flash fire scenarios require integration with a Pasquill-Gifford wind rose to account for the directional nature of vapor cloud propagation.

Wind Rose Structure

The wind rose divides the compass into 16 equi-spaced directions at 22.5° intervals. Each direction carries a probability (fraction of time the wind blows from that direction), split into day and night periods:

Period	Hours	Stability Classes
Day	06:00 – 18:00	A, B, C, D
Night	18:00 – 06:00	D, E, F

Polygon Rotation

For each of the 16 wind directions, the base LEL polygon is rotated around the source point:

x' = x \cos(\theta) - y \sin(\theta)

y' = x \sin(\theta) + y \cos(\theta)

Where $(x, y)$ is the original polygon vertex relative to source, $\theta$ is the rotation angle, and $(x', y')$ is the rotated vertex.

The result is 16 rotated LEL polygons, each associated with the wind probability for its direction.

Flash Fire $P_f$ Formula

P_{f,\text{ff}}(x, y) = \min\!\left(1.0,\; \sum_{j} p(\theta_j) \cdot \mathbb{1}\{(x,y) \in \text{Polygon}(\theta_j)\}\right)

Where:

Symbol	Description
$\theta_j$	Wind direction $j \in \{0°, 22.5°, \ldots, 337.5°\}$
$p(\theta_j)$	Probability of wind from direction $\theta_j$ $(0$ to $1)$
$\text{Polygon}(\theta_j)$	LEL polygon rotated to direction $\theta_j$
$\mathbb{1}\{\cdot\}$	Indicator function: 1 if point is inside polygon, 0 otherwise

Probability cap

The sum is capped at 1.0 because the maximum fatality probability cannot exceed certainty.

Step 3: Grid Generation

A uniform 2D rectangular grid is generated to evaluate IR across the study area.

Parameter	Description	Default
Center	Centroid of all source coordinates	Auto-calculated
Resolution	Spacing between adjacent grid points	25 m
Extent	Half-size of the grid from center	Auto-calculated

Available resolutions: 1, 5, 10, 25, 50, 100 meters.

Auto-Calculation of Extent

Find the maximum fatality profile distance across all scenarios

Multiply by 1.3 (30% padding factor)

Round up to the nearest 100 m

Grid Dimensions

\text{Points per side} = \frac{2 \times \text{extent}}{\text{resolution}} + 1

\text{Total grid points} = (\text{Points per side})^2

Example: With extent = 2,000 m and resolution = 25 m → 161 points per side → 25,921 total points.

Coordinate System

The grid uses a local Cartesian coordinate system in meters:

Origin: Grid center (centroid of sources)
X-axis: West → East (positive eastward)
Y-axis: South → North (positive northward)

Flat-earth approximation

This approximation is valid for distances up to ~50 km from the center — adequate for typical industrial sites.

Step 4: IR Calculation at Each Grid Point

For every point in the grid, the system accumulates IR contributions from all enabled scenarios.

Fireball, Pool Fire, Jet Fire, VCE — For each grid point $(x_i, y_j)$ and each radial scenario $s$ :

Step 1 — Euclidean distance: $d = \sqrt{(x_i - x_s)^2 + (y_j - y_s)^2}$

Step 2 — Quick reject: If $d > d_{\max}$ → skip ( $P_f = 0$ )

Step 3 — Interpolate fatality profile: If $d \leq d_0$ : $P_f = P_0 / 100$ . Otherwise find interval $[d_k, d_{k+1}]$ containing $d$ :

t = \frac{d - d_k}{d_{k+1} - d_k}, \qquad P_f = \frac{P_k + t \cdot (P_{k+1} - P_k)}{100}

Step 4 — Accumulate: $IR(x_i, y_j) \mathrel{+}= f_s \times P_f$

Step 5: Contour Extraction (Marching Squares)

Once the IR grid is fully populated, contour lines at each target level are extracted using the Marching Squares algorithm.

Step 6: Geographic Conversion

Contour polygons are converted from local meter coordinates back to geographic coordinates for map display.

Conversion Formulas

\text{lat} = \text{lat}_{\text{center}} + \frac{y}{111{,}320}

\text{lon} = \text{lon}_{\text{center}} + \frac{x}{111{,}320 \times \cos(\text{lat}_{\text{center}} \times \pi / 180)}

Where $111{,}320$ is the approximate meters per degree of latitude (WGS84).

GeoJSON Output

Each contour polygon is packaged as a GeoJSON Feature:

{
  "type": "Feature",
  "properties": {
    "level": 1e-5,
    "levelFormatted": "1e-5",
    "color": "#EA580C",
    "opacity": 0.3,
    "type": "ir_contour"
  },
  "geometry": {
    "type": "Polygon",
    "coordinates": [[[lon1, lat1], [lon2, lat2], "...", [lon1, lat1]]]
  }
}

Area Calculation

The area enclosed by each contour is computed using the shoelace formula on local meter coordinates:

A = \frac{1}{2} \left| \sum_{i} (x_i \cdot y_{i+1} - x_{i+1} \cdot y_i) \right| \quad [\text{m}^2]

Area Range	Display Unit
$< 10{,}000$ m²	m²
$10{,}000 – 1{,}000{,}000$ m²	hectares (ha)
$\geq 1{,}000{,}000$ m²	km²

5. Risk Tolerability Criteria

The following table summarizes international standards for individual risk tolerability:

Standard	Region	Intolerable Limit	Broadly Acceptable	Reference
UK HSE R2P2	United Kingdom	$1 \times 10^{-4}$ (public)	$1 \times 10^{-6}$	HSE (2001)
ASEA	Mexico	$1 \times 10^{-3}$	$1 \times 10^{-6}$	ASEA Guidelines
RIVM	Netherlands	$1 \times 10^{-5}$	$1 \times 10^{-8}$	RIVM guidelines
Hong Kong	Hong Kong	$1 \times 10^{-5}$	$1 \times 10^{-6}$	HK risk guidelines
HIPAP No. 4	Australia	$1 \times 10^{-5}$	$1 \times 10^{-6}$	NSW HIPAP No. 4
EPA / OSHA	United States	$1 \times 10^{-4}$	$1 \times 10^{-6}$	EPA RMP / OSHA PSM

ALARP Principle

ALARP — As Low As Reasonably Practicable

Between the intolerable and broadly acceptable limits lies the ALARP region. Within this zone, risk is tolerated only if further reduction is impracticable or disproportionately costly.

Zone	Risk Level	Required Action
Intolerable	$\geq 1 \times 10^{-3}$ to $1 \times 10^{-4}$ (varies by standard)	Risk cannot be justified except in extraordinary circumstances
ALARP	Between intolerable and acceptable thresholds	Tolerable only if reduction is impracticable
Broadly Acceptable	$\leq 1 \times 10^{-6}$ to $1 \times 10^{-8}$ (varies by standard)	No further action needed

6. Contour Color Code

TekRisk uses a seven-level color scheme to distinguish risk zones on the map:

Level (per year)	Color	Hex Code	Risk Interpretation
$1 \times 10^{-2}$	Indigo	`#4B0082`	Extreme — immediate action required
$1 \times 10^{-3}$	Dark Red	`#8B0000`	Intolerable — exceeds all standards
$1 \times 10^{-4}$	Red	`#DC2626`	Intolerable for public (UK HSE, US EPA)
$1 \times 10^{-5}$	Orange	`#EA580C`	Upper ALARP bound (many jurisdictions)
$1 \times 10^{-6}$	Yellow	`#EAB308`	Broadly acceptable threshold
$1 \times 10^{-7}$	Light Green	`#84CC16`	Low risk
$1 \times 10^{-8}$	Green	`#22C55E`	Negligible risk

Non-standard levels

Contour levels not in this standard set default to gray (#6B7280).

7. Comparison with CCPS Methodology

TekRisk implements the general approach described in CCPS Guidelines for Chemical Process Quantitative Risk Analysis (2nd Ed., 2000), §4.4.1.2. The core formula documented in §2.1 is literally Eq. 4.4.2 of CCPS. This section traces the implementation-to-implementation correspondence so that auditors and regulatory reviewers can map TekRisk's outputs back to the canonical reference.

7.1 Equation-to-Equation Mapping

CCPS (2000)	TekRisk Equivalent	Note
Eq. 4.4.1 — $IR_{x,y} = \sum_{i} IR_{x,y,i}$	Scenario additivity (§2.1)	Identical
Eq. 4.4.2 — $IR_{x,y,i} = f_i \cdot p_{f,i}$	Per-grid-cell term (§4 Step 4)	Shared foundation
Eq. 4.4.3 — $f_i = F_I \cdot p_{O,i} \cdot p_{OC,i}$	$f_i$ consumed from upstream `RiskCalculation.results` (§4 Step 1)	TekRisk does not recompute the event tree; it ingests the frequency already derived
Eq. 4.4.4 — directional factor $f_i \cdot (\theta_i / 360)$	Replaced by 16-direction wind rose with empirical $p(\theta_j)$ (§4 Step 2)	TekRisk refines: real directional probabilities vs. uniform wind
Eq. 4.4.5 — cumulative contour-by-contour summation	Replaced by full-grid IR evaluation + Marching Squares (§4 Step 5)	Eq. 4.4.5 only applies to the CCPS simplified approach

7.2 General Approach (CCPS §4.4.1.2, Fig. 4.7) vs TekRisk

Aspect	CCPS General Approach	TekRisk
Geographic evaluation	"Every geographical location" (unspecified discretization)	Uniform rectangular grid (1–100 m), auto-extent from fatality profile
Effect zones	Probit model or discrete zones per incident outcome	Continuous $P_f(d)$ profile with piecewise-linear interpolation
Wind treatment	Any level of detail (manual)	16-direction Pasquill–Gifford wind rose, day/night split
Contour tracing	"Manually or by any standard graphics contouring package"	Marching Squares (16 cases + saddle-point handling)
Output format	Printed map	GeoJSON FeatureCollection + Mapbox GL visualization

7.3 Simplified Approach (CCPS §4.4.1.3, Fig. 4.8) vs TekRisk

The CCPS simplified approach bundles six conservative assumptions designed for hand calculation. Because TekRisk runs on a numerical grid, those simplifications are unnecessary and the most restrictive ones are replaced with realistic physical models, yielding contours that are more accurate, directionally correct, and less conservative than the CCPS uniform circles:

CCPS Simplified Assumption	TekRisk
Point sources	Point sources (compatible)
Uniform wind distribution (equally likely in any direction)	16-direction wind rose with empirical probabilities
Single wind speed and stability class	Day/night split across Pasquill–Gifford classes A–F
Discrete effect zones (100% inside / 0% outside)	Continuous $P_f$ vs distance profile
Uniform ignition distribution	Inherited from the scenario event tree (upstream)
Circular risk contours	Realistic contour shapes that follow the actual risk distribution, traced by Marching Squares

7.4 What TekRisk Adds Beyond CCPS

Continuous fatality profile with piecewise-linear interpolation — CCPS §4.4.1.3 contemplates only step functions or full probit models
Marching Squares contour extraction with CCW winding validation for GeoJSON compliance — CCPS specifies only "manually or by any standard graphics contouring package"
Flash fire via rotated LEL envelope across 16 directions — CCPS §8.2 only covers the pie-shaped sector simplification
Automatic local-meter ↔ lat/lon conversion plus enclosed-area computation via the shoelace formula (§4 Step 6)
Seven standardized contour levels ( $10^{-2}$ to $10^{-8}$ ) with fixed color coding (§6) — CCPS leaves level selection to the analyst

Individual Risk Contours

1. Executive Summary

2. Key Concepts

Individual Risk (IR)

Scenario Frequency (f)

Fatality Probability (Pf)

Contour Levels

2.1 Core Formula

3. Supported Risk Models

4. Step-by-Step Methodology

Step 1: Scenario Definition

Step 2: Wind Rose Integration (Flash Fire)

Wind Rose Structure

Polygon Rotation

Flash Fire $P_f$ Formula

Step 3: Grid Generation

Auto-Calculation of Extent

Grid Dimensions

Coordinate System

Step 4: IR Calculation at Each Grid Point

Step 5: Contour Extraction (Marching Squares)

Step 6: Geographic Conversion

Conversion Formulas

GeoJSON Output

Area Calculation

5. Risk Tolerability Criteria

ALARP Principle

6. Contour Color Code

7. Comparison with CCPS Methodology

7.1 Equation-to-Equation Mapping

7.2 General Approach (CCPS §4.4.1.2, Fig. 4.7) vs TekRisk

7.3 Simplified Approach (CCPS §4.4.1.3, Fig. 4.8) vs TekRisk

7.4 What TekRisk Adds Beyond CCPS

8. Limitations

9. Bibliographic References

On this page

Individual Risk Contours

Individual Risk (IR)

Scenario Frequency (f)

Fatality Probability (Pf)

Contour Levels

Algorithm Overview

Edge Interpolation

Segment Connection

Winding Order Validation

Coordinate conversion approximation

Linear interpolation

Wind rose granularity

No seasonal or hourly wind variation

No shelter or building shielding

No terrain effects

No domino effects

Grid resolution limits

Radial symmetry assumption

Uniform exposure assumption

Constant frequency

Probit model accuracy

View complete reference list

On this page