## R Ca

Fig. 9.9 Multi-quadric surface fitting of a rainfall surface.

where R and C are row vectors, and A is a square matrix with elements a(xi, yj ). Inverting, given the inverse of A gives the coefficients:

The equation for R conditioned by these ci fits all the data points exactly. When the equation for the rainfall surface has been so defined, the volume of rainfall is obtained by integration over the area of the catchment and the areal rainfall results from dividing this value by the catchment area. In practice, the catchment area is subdivided into several rectangles over which the integrations are made and the volumes summed to give the total catchment volume of rainfall.

The method can be used efficiently for all time periods and the computer program can be linked to a contour package to produce machine plotted isohyetal maps of the rainfall event. Versions of multi-quadric interpolation have been used by Wood et al. (2000) to interpolate between closely spaced raingauges in the HYREX experiment on the Brue catchment in Somerset and Garcia et al. (2008) on the semi-arid Walker Branch catchment in Arizona. The multi-quadric method can also be used for rain-gauge network design by trials, varying the number of gauges and their distribution over a catchment. An alternative to this type of functional interpolation is the thin plate spline method of Hutchinson (1995) that can also incorporate other variables such as elevation in the interpolation.

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