Average Individual Risk: Exposed & Total Population

Calculation methodology for Average Individual Risk (IR_av) using CCPS Equations 4.4.6 and 4.4.7 — exposed population weighting, total population averaging, and ALARP classification

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1. Purpose#

This document describes the complete calculation methodology for the Average Individual Risk ( $IR_{av}$ ) as implemented in TekRisk. It covers:

How individual risk $IR(x,y)$ is computed at each geographic point
How the exposed population is estimated across the risk contours
How the two $IR_{av}$ metrics are calculated and what they represent
How the result is classified against international tolerance criteria

2. Key Concepts#

IR(x,y)

Individual risk at geographic location $(x,y)$ . Probability per year that a person permanently located at that point would die as a consequence of an industrial accident. Unit: $\text{yr}^{-1}$ .

IR_av

Average individual risk. The population-weighted average of $IR(x,y)$ over all locations where people are present.

P_T (Total Population)

Total predetermined population for averaging risk. May include people located outside the risk contours.

ALARP

As Low As Reasonably Practicable. The zone between tolerable and intolerable risk thresholds.

Term	Definition
$f_i$	Annual frequency of accident scenario $i$ . Unit: events/yr.
$P_{f,i}(x,y)$	Probability of fatality at location $(x,y)$ given that scenario $i$ occurs. Range: 0 to 1.
$P_{xy}$	Number of people at location $(x,y)$ . Derived from receivers or background density.
Contour level	A specific IR value (e.g., $10^{-5}\ \text{yr}^{-1}$ ). The contour line encloses all locations where $IR \geq$ that value.

3. Calculation Chain Overview#

The complete calculation follows this sequence:

Define accident scenarios — source location, frequency, consequence model
Calculate $IR(x,y)$ at every point on a geographic grid:
$IR(x,y) = \sum_{i} f_i \times P_{f,i}(x,y)$
Extract contour polygons from the grid (lines of equal individual risk)
Assign population to grid cells (receivers + background density)
Calculate $IR_{av}$ (Exposed Population) — Eq. 4.4.6:
$IR_{av} = \frac{\sum(IR_{xy} \times P_{xy})}{\sum(P_{xy})}$
Calculate $IR_{av}$ (Total Population) — Eq. 4.4.7:
$IR_{av} = \frac{\sum(IR_{xy} \times P_{xy})}{P_T}$
Classify the result against tolerance criteria (Acceptable / ALARP / Intolerable)

4. Accident Scenarios#

Each scenario represents a specific accident that could occur at an industrial source. The following data is required:

Parameter	Description	Example
Source coordinates	Geographic location (latitude, longitude)	19.4326 N, -99.1332 W
Frequency ( $f_i$ )	How often this accident is expected to occur	$1.5 \times 10^{-4}$ events/yr
Model type	Type of consequence model	Pool fire, jet fire, VCE, flash fire, fireball
Fatality profile	Distance vs. fatality probability	See Section 4.1

4.1 Fatality Profile#

For thermal models (fireball, pool fire, jet fire) and explosions (VCE), the fatality profile is a table of radial distance vs. fatality probability:

Distance (m)	Fatality (%)
50	100
100	80
200	40
300	10
400	0

This profile is derived from probit equations applied to the thermal radiation or overpressure results for each scenario.

5. Individual Risk at Each Point: IR(x,y)#

5.1 Formula#

IR(x,y) = \sum_{i=1}^{n} f_i \times P_{f,i}(x,y)

Where:

$n$ = number of accident scenarios
$f_i$ = frequency of scenario $i$ ( $\text{yr}^{-1}$ )
$P_{f,i}(x,y)$ = probability of fatality at $(x,y)$ for scenario $i$

This means: sum the risk contributions from all scenarios at each point.

5.2 Calculation Grid#

The calculation is performed on a uniform rectangular grid:

Parameter	Description	Typical value
Center	Centroid of all source locations	Automatic
Resolution	Distance between grid points	25 meters
Extent	Half-width of the grid from center	500 to 6,000 meters (automatic)

Grid size example: Resolution = 25 m, extent = 2,500 m produces 201 $\times$ 201 = 40,401 calculation points.

5.3 Fatality Probability: Thermal and Explosion Models#

For a grid point at distance $d$ from the source:

Calculate distance: $d = \sqrt{(x - x_{\text{source}})^2 + (y - y_{\text{source}})^2}$
Interpolate the fatality profile:
- If $d \leq$ minimum profile distance: $P_f =$ first profile value / 100
- If $d \geq$ maximum profile distance: $P_f = 0$
- Otherwise: linear interpolation between the two nearest profile points
Contribution to IR: $f_i \times P_f$

5.4 Fatality Probability: Flash Fire (Directional)#

Flash fire is direction-dependent. The LEL cloud polygon is rotated to each possible wind direction:

P_{f,\text{ff}}(x,y) = \sum_{\theta} P(\text{wind} = \theta) \times \mathbb{1}\{(x,y) \in \text{LEL}_{\theta}\}

Where:

$\theta$ = wind direction (16 directions from wind rose: N, NNE, NE, ..., NNW)
$P(\text{wind} = \theta)$ = probability that wind blows FROM direction $\theta$
$\mathbb{1}()$ = 1 if the point is inside the rotated LEL polygon, 0 otherwise

5.5 Combining All Scenarios#

At each grid point, the IR contributions from ALL scenarios are summed:

IR(x_i, y_j) = \sum_{k=1}^{n} f_k \times P_{f,k}(x_i, y_j)

6. Contour Extraction#

6.1 Contour Levels#

Level ( $\text{yr}^{-1}$ )	Meaning
$10^{-2}$	1 in 100 per year
$10^{-3}$	1 in 1,000 per year
$10^{-4}$	1 in 10,000 per year
$10^{-5}$	1 in 100,000 per year
$10^{-6}$	1 in 1,000,000 per year
$10^{-7}$	1 in 10,000,000 per year
$10^{-8}$	1 in 100,000,000 per year

6.2 Extraction Method#

Contour polygons are extracted using the marching squares algorithm:

For each contour level, classify every grid cell as above or below
Interpolate exact crossing points on cell edges
Connect all crossing points into closed polygons
Convert to geographic coordinates

6.3 Area Calculation#

The enclosed area of each contour is calculated using the shoelace formula applied to the polygon vertices. Results in $\text{m}^2$ and hectares.

7. Population Assignment#

Population is assigned to each grid cell from three sources, in priority order:

Polygon receivers (e.g., residential zones, school campuses):

P_{\text{cell}} = P_{\text{receiver}} \times \frac{A_{\text{cell}}}{A_{\text{receiver}}}

The receiver's population is distributed uniformly across its geographic area.

8. IR_av: Exposed Population (Eq. 4.4.6)#

8.1 Formula#

IR_{av}^{(\text{exp})} = \frac{\sum(IR_{xy} \times P_{xy})}{\sum(P_{xy})} \qquad \text{(CCPS Eq. 4.4.6)}

Variable	Description
$IR_{xy}$	Individual risk at location $(x,y)$ — the actual grid value ( $\text{yr}^{-1}$ )
$P_{xy}$	Number of people at location $(x,y)$
$\sum(P_{xy})$	Total exposed population — only people within risk contours

8.2 Meaning#

This metric answers: "What is the average annual fatality risk for a person in the exposed population?"

The "exposed population" consists only of people who are within the risk contours — i.e., people who face some non-zero level of risk from the facility.

8.3 Calculation Procedure#

totalWeightedRisk = 0     (numerator)
totalPopulation = 0       (denominator)

For each grid cell (i, j) where IR > 0:
  IR_cell = grid value at (i, j)

  Determine P_cell:
    - If cell is inside a polygon receiver: proportional population
    - Else if cell matches a point receiver: receiver population
    - Else if project density > 0: density × cell_area
    - Else: skip (no people at this location)

  If P_cell > 0:
    totalWeightedRisk += IR_cell × P_cell
    totalPopulation += P_cell

IR_av(exposed) = totalWeightedRisk / totalPopulation

If totalPopulation = 0, the result is null (no exposed population found).

8.4 Audit Breakdown#

For traceability, each grid cell is classified into a contour band:

Band	Risk Level
$\geq 10^{-2}$	Innermost, highest risk
$10^{-3}$ to $10^{-2}$
$10^{-4}$ to $10^{-3}$
$10^{-5}$ to $10^{-4}$
$10^{-6}$ to $10^{-5}$
$10^{-7}$ to $10^{-6}$
$10^{-8}$ to $10^{-7}$	Outermost, lowest risk

For each band, the report shows:

Number of grid cells
Area ( $\text{m}^2$ )
Population
Representative IR (weighted average of grid values in that band)
Weighted risk contribution ( $IR \times P$ )
Percentage of total contribution

This breakdown is available in both the application interface and the PDF report.

9. IR_av: Total Population (Eq. 4.4.7)#

9.1 Formula#

IR_{av}^{(\text{tot})} = \frac{\sum(IR_{xy} \times P_{xy})}{P_T} \qquad \text{(CCPS Eq. 4.4.7)}

Variable	Description
$\sum(IR_{xy} \times P_{xy})$	Same numerator as Eq. 4.4.6 — identical calculation
$P_T$	Total predetermined population — entered by the user

9.2 Meaning#

This metric answers: "What is the average annual fatality risk across the entire surrounding population, including those who face zero risk?"

$P_T$ is the entire population of interest. For example:

The population of the town surrounding the plant
All workers inside an industrial complex
The population within a defined radius from the facility

9.3 Key Difference from Eq. 4.4.6#

Aspect	Exposed Population (Eq. 4.4.6)	Total Population (Eq. 4.4.7)
Numerator	$\sum(IR_{xy} \times P_{xy})$	$\sum(IR_{xy} \times P_{xy})$ — identical
Denominator	$\sum(P_{xy})$ — exposed only	$P_T$ — all people, including unexposed
What it includes	Only people within risk contours	All people, even those with zero risk
Typical result	Higher value	Lower value
When to use	Default metric for risk assessment	When regulations require total population basis

9.4 Calculation#

Since the numerator is already computed in Step 5:

IR_{av}^{(\text{tot})} = \frac{\text{totalWeightedRisk}}{P_T}

Where $P_T$ is entered directly in the IR Average tab of the application.

9.5 CCPS Warning#

10. Risk Level Classification#

10.1 ALARP Framework#

Zone	Condition	Required Action
Intolerable	$IR_{av} \geq$ intolerable threshold	Risk is unacceptable. Reduction required regardless of cost.
ALARP	tolerable $\leq IR_{av} <$ intolerable	Risk should be reduced unless cost is grossly disproportionate to the benefit.
Acceptable	$IR_{av} <$ tolerable threshold	Risk is broadly acceptable. No additional measures required.

10.2 Available Criteria#

Country / Standard	Target	Intolerable ( $\text{yr}^{-1}$ )	Tolerable ( $\text{yr}^{-1}$ )	Reference
UK HSE	Workers	$10^{-3}$	$10^{-6}$	R2P2
UK HSE	Public	$10^{-4}$	$10^{-6}$	R2P2
Mexico ASEA	Public	$10^{-3}$	$10^{-6}$	ASEA Guidelines
Netherlands RIVM	Public	$10^{-5}$	$10^{-8}$	Dutch Risk Criteria
Hong Kong	Public	$10^{-5}$	$10^{-6}$	HKSAR Guidelines
Australia (HIPAP)	Public	$10^{-5}$	$10^{-6}$	HIPAP No. 4
USA EPA	Public	$10^{-4}$	$10^{-6}$	EPA Guidelines
Custom	—	User-defined	User-defined	—

11. Worked Numerical Example#

11.1 Project Data#

Grid resolution: 25 m (cell area = 625 $\text{m}^2$ )
Population density: 100 p/km² = 0.0001 p/m²
$P_T$ (total population of nearby town): 15,000 inhabitants
Two accident scenarios:
- Scenario A: Pool fire, $f = 5 \times 10^{-4}\ \text{yr}^{-1}$
- Scenario B: VCE, $f = 2 \times 10^{-5}\ \text{yr}^{-1}$

11.2 IR at a Sample Point (200 m East of Scenario A)#

IR_A = f_A \times P_{f,A}(200\ \text{m}) = 5 \times 10^{-4} \times 0.40 = 2.00 \times 10^{-4}

IR_B = f_B \times P_{f,B}(200\ \text{m}) = 2 \times 10^{-5} \times 0.15 = 3.00 \times 10^{-6}

IR(200\ \text{m east}) = 2.00 \times 10^{-4} + 3.00 \times 10^{-6} = 2.03 \times 10^{-4}\ \text{yr}^{-1}

11.3 Receiver: "North Colony" (Polygon)#

Population: 200 people, area: 30,000 $\text{m}^2$
48 grid cells inside (48 $\times$ 625 = 30,000 $\text{m}^2$ )
Population per cell: $200 \times (625 / 30{,}000) = 4.17$ persons

Band	Cells	Avg IR	Pop. per cell	$IR \times P$ per band
$10^{-5}$ to $10^{-4}$	3	$4.2 \times 10^{-5}$	4.17	$5.25 \times 10^{-4}$
$10^{-6}$ to $10^{-5}$	12	$5.1 \times 10^{-6}$	4.17	$2.55 \times 10^{-4}$
Below $10^{-6}$	33	$< 10^{-6}$	4.17	negligible
Subtotal	48		200	$\approx 7.80 \times 10^{-4}$

11.4 Receiver: "School" (Point)#

Population: 150 people
IR at location: $8.5 \times 10^{-7}\ \text{yr}^{-1}$
Contribution: $8.5 \times 10^{-7} \times 150 = 1.275 \times 10^{-4}$

11.5 Background Population#

520 grid cells with $IR > 0$ not inside any receiver
$P_{\text{cell}} = 0.0625$ per cell
Background population: $520 \times 0.0625 = 32.5$ persons
Background weighted risk: $6.12 \times 10^{-5}$

11.6 Results#

\text{Numerator:}\quad \sum(IR \times P) = 7.80 \times 10^{-4} + 1.275 \times 10^{-4} + 6.12 \times 10^{-5} = 9.69 \times 10^{-4}

\text{Denominator:}\quad \sum(P_{xy}) = 200 + 150 + 32.5 = 382.5

\text{Eq. 4.4.6:}\quad IR_{av}^{(\text{exp})} = \frac{9.69 \times 10^{-4}}{382.5} = 2.53 \times 10^{-6}\ \text{yr}^{-1}

\text{Eq. 4.4.7:}\quad IR_{av}^{(\text{tot})} = \frac{9.69 \times 10^{-4}}{15{,}000} = 6.46 \times 10^{-8}\ \text{yr}^{-1}

11.7 Classification (UK HSE — Public)#

Metric	Value	vs. Intolerable ( $10^{-4}$ )	vs. Tolerable ( $10^{-6}$ )	Result
$IR_{av}$ Exposed	$2.53 \times 10^{-6}$	Below	Above	ALARP
$IR_{av}$ Total	$6.46 \times 10^{-8}$	Below	Below	Acceptable

12. Interpretation Guidelines for Regulators#

12.1 Which Metric to Use#

Eq. 4.4.6 (Exposed Population) is the primary metric for most frameworks. It directly answers: "how much risk do exposed people face?"
Eq. 4.4.7 (Total Population) is a supplementary metric. Use when regulations specifically require it, or to provide context about risk dilution.

12.2 Verification Checklist#

#	Check	What to look for
1	Scenario completeness	Are all relevant accident scenarios included?
2	Frequencies	Consistent with historical data, fault trees, or event trees?
3	Fatality profiles	Reasonable distances for the substances and conditions?
4	Grid resolution	Fine enough to capture risk gradients? (Recommended: $\leq$ 25 m)
5	Population data	Receivers correctly placed? Background density matches census?
6	Contour areas	Physically reasonable given the scenario types?
7	$P_T$ value	Represents the actual population in the defined influence area?
8	Tolerance criteria	Correct country-specific criteria applied?

12.3 Common Pitfalls#

Issue	Impact	How to detect
Missing scenarios	Underestimates risk	Compare scenario list against HAZOP/PHA
Coarse grid (>50 m)	Smooths out peak risk	Check resolution in report
No receivers defined	Only background density used	Check receiver count in breakdown
$P_T$ too large	$IR_{av}^{(\text{tot})}$ artificially low	Compare $P_T$ against census data
Wrong density units	Over/underestimated population	Verify p/km² matches actual area

1. Purpose#

2. Key Concepts#

3. Calculation Chain Overview#

4. Accident Scenarios#

4.1 Fatality Profile#

5. Individual Risk at Each Point: IR(x,y)#

5.1 Formula#

5.2 Calculation Grid#

5.3 Fatality Probability: Thermal and Explosion Models#

5.4 Fatality Probability: Flash Fire (Directional)#

5.5 Combining All Scenarios#

6. Contour Extraction#

6.1 Contour Levels#

6.2 Extraction Method#

6.3 Area Calculation#

7. Population Assignment#

8. IR_av: Exposed Population (Eq. 4.4.6)#

8.1 Formula#

8.2 Meaning#

8.3 Calculation Procedure#

8.4 Audit Breakdown#

9. IR_av: Total Population (Eq. 4.4.7)#

9.1 Formula#

9.2 Meaning#

9.3 Key Difference from Eq. 4.4.6#

9.4 Calculation#

9.5 CCPS Warning#

10. Risk Level Classification#

10.1 ALARP Framework#

10.2 Available Criteria#

11. Worked Numerical Example#

11.1 Project Data#

11.2 IR at a Sample Point (200 m East of Scenario A)#

11.3 Receiver: "North Colony" (Polygon)#

11.4 Receiver: "School" (Point)#

11.5 Background Population#

11.6 Results#

11.7 Classification (UK HSE — Public)#

12. Interpretation Guidelines for Regulators#

12.1 Which Metric to Use#

12.2 Verification Checklist#

12.3 Common Pitfalls#

13. References#

View complete reference list