Multiquadric Equations and Optimal Areal Rainfall Estimation

Numerical surface-fitting methods such as kriging, reciprocal distance, and multiquadric equations have often been employed for interpolation of discrete spatial data such as rainfall depth observations. Such methods have been used occasionally for estimation of average areal rainfall depth. When these or similar surface-fitting techniques are used for areal rainfall depth estimation, a weighted average of the discrete rainfall depth observations results. The rainfall depth estimates are unbiased if the numerical surface can fit a nonzero uniform depth exactly everywhere in the region. The unbiased conic multiquadric surface yields optimal gauge weighting coefficients for rainfall data correlated by a linear covariance function. The unbiased conic multiquadric method is a robust estimator of average areal rainfall that does not require extensive rainfall data with which to characterize the stochastic nature of the rainfall process. Unbiased multiquadric analysis appears to reduce the problem of negative gauge weights attributed to the phenomenon of "multicolinearity" by some investigators. A simple method of calculating the multiquadric gauge-weighting coefficients by using 3D CAD is demonstrated.