Engineering Hydrology · Precipitation

Thiessen Polygon Method for Average Rainfall: Interactive Example, Isohyetal & Arithmetic Mean Compared

Rain gauges measure rainfall at points, but floods, water budgets and hydrological models need the average over an area. This guide explains the three classical methods, lets you drag gauges on a live catchment and watch the Thiessen polygons and the average update, works through full examples, and shows how to calculate the optimum number of rain gauges.

Live Thiessen polygonsWorked examplesIsohyetal calculatorOptimum gauges N = (Cv/ε)²

By Bimal Ghimire (PhD researcher, precipitation remote sensing) · Published October 28, 2026 · 16 min read

Background

Why We Need Average (Areal) Rainfall

A standard 8-inch (203 mm) rain gauge samples about 0.03 m² of ground. A catchment may cover hundreds of km². Runoff calculations such as the rational method, the SCS curve number method and unit hydrograph analysis all need a single representative depth for the whole area, called the mean areal precipitation (MAP).

Because rainfall varies in space, especially in convective storms and mountains, simply averaging the gauges can be badly biased when gauges are clustered. The classical methods differ in how they weight each gauge.

Overview

The Three Classical Methods at a Glance

Method	Weight of each gauge	Best for	Main weakness
Arithmetic mean	Equal (1/n)	Flat areas, evenly spread gauges, small variation	Biased by clustered gauges
Thiessen polygon	Area closer to that gauge than to any other	Most catchments; objective and repeatable	Ignores terrain; step changes at polygon edges
Isohyetal	Area between contours of equal rainfall	Hilly terrain, dense networks, storm analysis	Subjective; needs skill and many gauges

Method 1

Arithmetic Mean Method

$$\bar P=\frac{P_1+P_2+\dots+P_n}{n}=\frac{1}{n}\sum_{i=1}^{n}P_i$$

Use only gauges inside the catchment. A common textbook rule of thumb is that the method is acceptable when the gauges are uniformly distributed over flat terrain and individual readings differ from the mean by less than about 10%. It is quick and needs no map, which is why it is still used for first estimates.

Method 2

Thiessen Polygon Method: Steps and Formula

Proposed by A. H. Thiessen (1911), the method assumes the rainfall at any point equals the rainfall at the nearest gauge. Each gauge therefore represents the region closer to it than to any other gauge.

Construction (by hand or in GIS)

Plot the catchment boundary and all gauges, including those just outside the boundary.
Join neighbouring gauges with straight lines to form a network of triangles (a Delaunay triangulation).
Draw perpendicular bisectors of every triangle side. The bisectors meet and form a polygon around each gauge.
Clip the polygons to the catchment boundary and measure each area $A_i$ (planimeter, grid counting, or GIS).
Weight each gauge by its area fraction and sum.

Thiessen average

$$\bar P=\frac{A_1P_1+A_2P_2+\dots+A_nP_n}{A_1+A_2+\dots+A_n}=\sum_{i=1}^{n}w_iP_i,\qquad w_i=\frac{A_i}{A}$$

The weights $w_i$ depend only on gauge positions, so they are computed once and reused for every storm, as long as all gauges report.

GIS shortcut: QGIS → Vector → Geometry Tools → Voronoi Polygons, then Clip with the catchment and compute areas with the field calculator. ArcGIS Pro: Create Thiessen Polygons. In Python, scipy.spatial.Voronoi plus shapely clipping.

Interactive Thiessen Polygon Calculator

Drag a gauge to move it

Colour by 1 grid unit =km

Practice

Worked Example: Thiessen vs Arithmetic Mean

The catchment in the interactive tool (198.5 km², with 1 grid unit = 1 km) has five gauges inside and one gauge, F, just outside its north-east edge. Gauges A, B and E are clustered in the drier west; C sits alone in the wetter east. Storm totals (mm) are shown below with the Thiessen areas computed by clipping each Voronoi cell to the boundary.

Gauge	Rainfall P (mm)	Thiessen area A (km²)	Weight A/ΣA	Weighted P (mm)
A	62	24.78	0.1248	7.74
B	58	30.63	0.1543	8.95
C	104	51.73	0.2606	27.10
D	74	43.98	0.2216	16.40
E	66	28.77	0.1449	9.56
F (outside)	118	18.61	0.0938	11.06
Total		198.50	1.0000	80.82

Thiessen average: $\bar P = 80.8$ mm
Arithmetic mean of the five gauges inside: $(62+58+104+74+66)/5 = 72.8$ mm

The arithmetic mean is about 10% lower because three of the five inside gauges sit in the dry west, while gauge C represents more than a quarter of the area and outside gauge F covers the wet north-east corner. Move gauge C toward the cluster in the tool above and watch both averages change.

Method 3

Isohyetal Method

An isohyet is a contour of equal rainfall. The analyst interpolates between gauge values (using knowledge of terrain and storm direction), draws isohyets, measures the area between each pair of successive isohyets inside the catchment, and assigns that strip the mean of the two isohyet values:

$$\bar P=\frac{\sum_{j} a_j\,\dfrac{P_j+P_{j+1}}{2}}{\sum_j a_j}$$

$a_j$ = catchment area between isohyets $P_j$ and $P_{j+1}$. For the strip next to the boundary, use an estimated boundary value or the isohyet value itself.

Worked example

Isohyets (mm)	Mean P (mm)	Area (km²)	Area × P
100 – 90	95.0	18	1,710
90 – 80	85.0	46	3,910
80 – 70	75.0	62	4,650
70 – 60	65.0	40	2,600
60 – 55 (boundary)	57.5	14	805
Total		180	13,675

$\bar P = 13{,}675/180 = 76.0$ mm. Use the calculator below for your own isohyet map.

Isohyetal Average Calculator

Upper isohyet (mm)Lower isohyet (mm)Area between (km²)

Network Design

Optimum Number of Rain Gauges

How many gauges are enough? If the existing $m$ gauges give a coefficient of variation $C_v$ and you accept an error $\varepsilon$ (%) in the mean, the required number is:

$$N=\left(\frac{C_v}{\varepsilon}\right)^2,\qquad C_v=\frac{100\,\sigma_{m-1}}{\bar P},\qquad \sigma_{m-1}=\sqrt{\frac{\sum_{i=1}^{m}(P_i-\bar P)^2}{m-1}}$$

ε is commonly taken as 10%. This is the standard formula in Indian and South Asian engineering hydrology texts (e.g. Subramanya). It assumes gauges are spread so that each is equally representative.

Example: six gauges recorded 82.6, 102.9, 180.3, 110.3, 98.8 and 136.7 mm. Mean = 118.6 mm, σ_m−1 = 35.04 mm, so C_v = 29.54%. With ε = 10%, $N = (29.54/10)^2 = 8.73 \Rightarrow 9$ gauges, so 3 more are needed.

Optimum Rain Gauge Calculator

Rainfall at existing gauges (mm), separated by commas or spaces

Allowable error ε (%)

Current Practice

Beyond Thiessen: Interpolation and Satellite Rainfall

Thiessen polygons are a special case of spatial interpolation (nearest neighbour). Modern studies typically use:

Inverse distance weighting (IDW)

Each grid cell gets a weighted mean of gauges with weights ∝ 1/d^p (p ≈ 2). Smooth, simple, no terrain information.

Kriging and co-kriging

Geostatistical interpolation using the variogram; co-kriging or regression-kriging adds elevation to capture orographic gradients and gives an uncertainty estimate.

Weather radar

High spatial and temporal resolution; needs gauge adjustment because the reflectivity–rain-rate relation varies by storm type.

Satellite and reanalysis products

NASA GPM IMERG (0.1°, half-hourly), CHIRPS (0.05°, daily, 50°S–50°N, from 1981) and ERA5-Land (≈9 km, hourly) give areal rainfall where gauges are sparse, but carry biases, especially for orographic and light rain. Validate against local gauges before use.

Even with gridded data, the final step is the same idea as Thiessen: a catchment average is an area-weighted sum of cell values. Understanding the classical methods makes it much easier to judge when a gridded product is trustworthy. Long areal rainfall series are often then tested for change with the Mann-Kendall trend test or converted to drought indices such as SPI and SPEI.

Avoid These

Common Mistakes

Ignoring outside gauges

Nearby gauges outside the boundary still control edge areas. Include them in the construction.

Reusing weights with missing data

If a gauge did not report, its polygon must be redistributed: redraw the polygons without it.

Wrong bisectors

Bisectors must be perpendicular to the line joining two neighbouring gauges and pass through its midpoint, not just split the angle.

Map scale errors

Measured map areas must be converted with the square of the scale factor (1 cm² on a 1:50,000 map = 0.25 km²).

Keep Learning

Related Tools and Guides

FAQ

Frequently Asked Questions

What is the Thiessen polygon method?

It estimates the average rainfall over an area by giving each rain gauge a weight equal to the fraction of the catchment that is closer to it than to any other gauge. The regions are found by drawing perpendicular bisectors between neighbouring gauges, which creates polygons (Voronoi cells) clipped to the catchment boundary. Average rainfall = Σ(Ai × Pi) / A.

Which is the most accurate method for average rainfall?

Among the three classical methods, the isohyetal method is usually the most accurate when drawn by an experienced analyst with dense data, especially in hilly terrain, because it can follow orographic patterns. Thiessen is objective and repeatable and is the most widely used. The arithmetic mean is acceptable only for flat areas with evenly spread gauges and small rainfall differences.

Can gauges outside the catchment be used in the Thiessen method?

Yes. A gauge just outside the boundary can still be the nearest gauge to part of the catchment, so it receives a weight equal to that area. Ignoring nearby outside gauges biases the result, especially near the edges.

What are the limitations of the Thiessen polygon method?

It ignores topography and orographic effects, assumes rainfall is constant within each polygon, and the polygons must be redrawn whenever a gauge is added or has missing data. It also cannot represent gradients between gauges, which the isohyetal method and interpolation methods such as kriging can.

How do you calculate the optimum number of rain gauges?

Use N = (Cv / ε)², where Cv is the coefficient of variation of rainfall at the existing gauges in percent (using the standard deviation with m − 1) and ε is the allowable percentage error in the mean, commonly 10%. Round N up to the next whole number.

Is the Thiessen polygon the same as a Voronoi diagram?

Yes. Thiessen polygons are Voronoi cells, named after the meteorologist A. H. Thiessen who applied them to rainfall in 1911. In GIS software they are created with tools such as "Voronoi polygons" (QGIS) or "Create Thiessen Polygons" (ArcGIS).

Sources

References

Thiessen, A. H. (1911). Precipitation averages for large areas. Monthly Weather Review, 39(7), 1082–1084.

Chow, V. T., Maidment, D. R., & Mays, L. W. (1988). Applied Hydrology, Chapter 3. McGraw-Hill, New York.

Subramanya, K. Engineering Hydrology, Chapter 2 (Precipitation). McGraw-Hill Education, New Delhi.

World Meteorological Organization (2008). Guide to Hydrological Practices, Volume I, WMO-No. 168, 6th ed. Geneva.

Huffman, G. J., et al. (2020). Integrated Multi-satellitE Retrievals for GPM (IMERG) Technical Documentation. NASA Goddard Space Flight Center.

Funk, C., et al. (2015). The climate hazards infrared precipitation with stations—a new environmental record for monitoring extremes. Scientific Data, 2, 150066.

Muñoz-Sabater, J., et al. (2021). ERA5-Land: a state-of-the-art global reanalysis dataset for land applications. Earth System Science Data, 13, 4349–4383.