## The Thiessen polygon

Devised by an American engineer (Thiessen, 1911), this is also an objective method of calculating a weighted average. In this case, the rainfall measurements at individual gauges are weighted by the fractions of the catchment area represented by the gauges as:

where R[ are the rainfall measurements at n rain gauges and A is the total area of the catchment. To calculate the areal weights, given a map of the catchment with the rain-gauge stations plotted, the catchment area is divided into polygons by lines that are equidistant between pairs of adjacent stations. A typical configuration for well-distributed gauges is shown in Fig. 9.7. The polygon areas, ai, corresponding to the rain-gauge stations are then measured. In the illustrated example, there are nine measurements contributing to the calculation even though gauge 1 is outside the catchment boundary. The area ai within the catchment is however nearer to gauge 1 than to the neighbouring gauges 2, 3 and 8, and is therefore better represented by measurements at gauge 1.

The area fractions ai /A are called the Thiessen coefficients and, once they have been determined for a stable rain gauge network, the areal rainfall is very quickly computed for any set of rainfall measurements. Thus the Thiessen method lends itself readily to computer processing. If there are data missing for one rain-gauge station, it can then be simpler to estimate the missing values and retain the original coefficients rather than to redraw the polygons and evaluate fresh Thiessen coefficients. If, however, a rain-gauge network is altered radically, then the Thiessen polygons have to be redrawn, the new areas measured and a new set of coefficients evaluated.

The simplicity of the Thiessen polygon method derives from the assumption that the rainfall in areas between the gauges can be interpolated linearly. This is only one possible assumption, however. Other methods involve different types of interpolation, such as the annual average rainfall weighting mentioned in the previous section and inverse distance weighting (e.g. Creutin and Obled, 1982; Dirks etal., 1998; Tomczak, 1998; Garcia et al., 2008).

Fig. 9.7 Thiessen polygon method of estimating areal rainfall.

Fig. 9.7 Thiessen polygon method of estimating areal rainfall.

The Thiessen method for determining areal rainfall is sound and objective, but it is dependent on a good network of representative rain gauges. It is not particularly good for mountainous areas, since altitudinal effects are not allowed for by the areal coefficients, nor is it useful for deriving areal rainfall from intense local storms. To overcome some of the shortcomings, investigations into the use of height-weighted polygons combining altitudinal and areal effects on the rainfall measurements have been made. This modification will lead to improved estimates of areal rainfall, relative to the methods based on distance from gauge measurements alone, where there is a strong correlation of rainfall with elevation (annual average rainfall weighting can also be used to reflect such correlations). Previous editions of Hydrology in Practice have also outlined the hypsometric method of relating rainfalls to the area in a catchment in a particular elevation band for use where there are multiple gauges at different elevations and such a correlation is particularly strong but statistical interpolation methods are now generally preferred (see Section 9.2.5 below).

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