Population Forecasting Methods for Water Supply: Arithmetic, Geometric and Incremental Increase
Population forecasting estimates how many people a town will have at the end of the design period, so that intakes, treatment plants, pipelines and sewers are not overloaded before their life is over. The three methods asked most in exams are the arithmetic increase, geometric increase and incremental increase methods. For the same census data, geometric gives the highest forecast, arithmetic the lowest, and incremental falls in between.
Formulas
| Method | Formula | Best suited to |
|---|---|---|
| Arithmetic increase | Pn = P0 + n x̄ | Old, large cities with slow, steady growth |
| Geometric increase | Pn = P0 (1 + r)n | Young, fast-growing towns (tends to over-estimate for old cities) |
| Incremental increase | Pn = P0 + n x̄ + [n(n + 1) / 2] ȳ | Average towns — a balance between the two above |
Here P0 is the latest census population, n is the number of decades ahead, x̄ is the average increase per decade, ȳ is the average of the changes in that increase, and r is the growth rate per decade (usually the geometric mean of the decade rates).
Other methods
- Decreasing rate of growth — assumes the percentage growth falls each decade as the town matures.
- Simple graphical — the census curve is extended by eye.
- Comparative graphical — the town's curve is compared with similar, larger towns that have already passed this stage.
- Logistic curve — growth is fast at first, then slows towards a saturation population.
- Master plan / zoning — population is taken from the planned density of each zone.
Exam tip: In the incremental increase method, the extra term is n(n+1)/2 × ȳ, not n × ȳ. For n = 3 decades the multiplier is 6. If the question gives four census figures you get three increases and two incremental increases — average each separately.
Worked example
Census records of a town are: 1981 — 50,000; 1991 — 62,000; 2001 — 76,000; 2011 — 92,000. Forecast the population three decades after 2011 by all three methods.
- Increases per decade: 12,000, 14,000 and 16,000. Average increase x̄ = 42,000 / 3 = 14,000.
- Incremental increases: 2,000 and 2,000. Average ȳ = 2,000.
- Arithmetic: P = 92,000 + 3 × 14,000 = 1,34,000.
- Incremental: P = 92,000 + 3 × 14,000 + (3 × 4 / 2) × 2,000 = 92,000 + 42,000 + 12,000 = 1,46,000.
- Decade growth rates: 12,000/50,000 = 0.240; 14,000/62,000 = 0.226; 16,000/76,000 = 0.211.
- Geometric mean rate r = (0.240 × 0.226 × 0.211)1/3 = 0.225 (22.5% per decade).
- Geometric: P = 92,000 × (1.225)3 = about 1,69,000.
As expected, geometric (1.69 lakh) > incremental (1.46 lakh) > arithmetic (1.34 lakh). An engineer would normally adopt the incremental value for a town like this.
From population to water demand
Once the design population is fixed, the average daily demand follows directly. At 135 litres per capita per day (the CPHEEO figure for cities with piped sewerage), 1,46,000 people need 1,46,000 × 135 = 1,97,10,000 litres a day, or about 19.7 MLD. The intake, treatment plant and mains are then sized for this demand together with the peak factors for each component.
Frequently asked questions
Which population forecasting method gives the highest value?
The geometric increase method usually gives the highest forecast, the arithmetic increase method the lowest, and the incremental increase method a value in between.
Which method is suitable for an old, large city?
The arithmetic increase method, because the growth of a large established city is slow and roughly constant per decade.
What is the design period of a water supply scheme?
A water supply scheme is commonly designed for about 30 years ahead (CPHEEO manual), with shorter periods for some components such as pumps.
What is the formula for the incremental increase method?
Pn = P0 + n·x̄ + [n(n+1)/2]·ȳ, where x̄ is the average increase per decade and ȳ is the average incremental increase.
All environmental engineering formulas — demand, treatment units, sewer design — are in the free Advanced Handbook.
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