The Smoothing Spline: A New Approach to Standardizing Forest Interior Tree-Ring Width Series for Dendroclimatic Studies

Abstract

A new approach to removing the non-climatic variance of forest interior tree-ring width series, using the smoothing spline, is described. This method is superior to orthogonal polynomials because it makes no assumptions about the shape of the curve to be used for standardization. Also, because the spline curve can range continuously from a linear least squares fit to cubic interpolation through the data, it is far more flexible than polynomials and provides a more "natural" fit. For computing the spline, we found that specifying the Lagrange multiplier p which appears in the calculus of variation solution rather than the residual variance as suggested by Reinsche was both practical and more efficient. In effect, the smoothing spline is a one-parameter family of low-pass filters defined by p. We describe the general characteristics of these filters in the time and frequency domains and compute the response functions for several of them. The smoothing spline is an excellent tree-ring standardization method because its filtering characteristics are well defined. Its utility for dendroclimatology should be considerable since, outside of semiarid environments, sites similar to forest interiors predominate.