Interpolating mean rainfall using thin plate smoothing splines

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Abstract
Thin plate smoothing splines provide accurate, operationally straightforward and computationally efficient solutions to the problem of the spatial interpolation of annual mean rainfall for a standard period from point data which contains many short period rainfall means. The analyses depend on developing a statistical model of the spatial variation of the observed rainfall means, considered as noisy estimates of standard period means. The error structure of this model has two components which allow separately for strong spatially correlated departures of observed short term means from standard period means and for uncorrelated deficiencies in the representation of standard period mean rainfall by a smooth function of position and elevation. Thin plate splines, with the degree of smoothing determining by minimising generalised cross validation, can estimate this smooth function in two ways. First, the spatially correlated error structure of the data can be accommodated directly by estimating the corresponding non-diagonal error covariance matrix. Secondly, spatial correlation in the data error structure can be removed by standardising the observed short term means to standard period mean estimates using linear regression. When applied to data both methods give similar interpolation accuracy, and error estimates of the fitted surfaces are in good agreement with residuals from withheld data. Simplified versions of the data error model, which require only minimal summary data at each location, are also presented. The interpolation accuracy obtained with these models is only slightly inferior to that obtained with more complete statistical models. It is shown that the incorporation of a continuous, spatially varying, dependence on appropriately scaled elevation makes a dominant contribution to surface accuracy. Incorporating dependence on aspect, as determined from a digital elevation model, makes only a marginal further improvement.