Laplacian Eigenfunctions for Climate Analysis

This paper proposes a new method for representing data in a general domain on a sphere. The method is based on the eigenfunctions of the Laplace operator, which form an orthogonal basis set that can be ordered by a measure of length scale. Representing data with Laplacian eigenfunctions is attractive if one wants to reduce the dimension of a dataset by filtering out small-scale variability. Although Laplacian eigenfunctions are ubiquitous in climate modeling, their use in arbitrary domains, such as over continents, is not common because of the numerical difficulties associated with irregular boundaries. Recent advances in machine learning and computational sciences are exploited to derive eigenfunctions of the Laplace operator over an arbitrary domain on a sphere. The eigenfunctions depend only on the geometry of the domain and hence require no training data from models or observations, a feature that is especially useful in small sample sizes. Another novel feature is that the method produces reasonable eigenfunctions even if the domain is disconnected, such as a land domain comprising isolated continents and islands. The eigenfunctions are illustrated by quantifying variability of monthly mean temperature and precipitation in climate models and observations. This analysis extends previous studies by showing that climate models have significant biases not only in global-scale spatial averages but also in global-scale dipoles and other physically important structures. MATLAB and R codes for deriving Laplacian eigenfunctions are available upon request.