• Allen, M. R., , and S. F. B. Tett, 1999: Checking for model consistency in optimal fingerprinting. Climate Dyn., 15, 419434, doi:10.1007/s003820050291.

    • Search Google Scholar
    • Export Citation
  • Belkin, M., , and P. Niyogi, 2003: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput., 15, 13731396, doi:10.1162/089976603321780317.

    • Search Google Scholar
    • Export Citation
  • Chen, M., , P. Xie, , J. E. Janowiak, , and P. A. Arkin, 2002: Global land precipitation: A 50-year monthly analysis based on gauge observations. J. Hydrometeor., 3, 249266, doi:10.1175/1525-7541(2002)003<0249:GLPAYM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Coifman, R. R., , and S. Lafon, 2006: Diffusion maps. Appl. Comput. Harmonic Anal., 21, 530, doi:10.1016/j.acha.2006.04.006.

  • Courant, R., , and D. Hilbert, 1962: Methods of Mathematical Physics. Vol. 1. John Wiley and Sons, 560 pp.

  • Fan, Y., , and H. van den Dool, 2008: A global monthly land surface air temperature analysis for 1948–present. J. Geophys. Res., 113, D01103, doi:10.1029/2007JD008470.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., , and P. J. Ioannou, 2001: Accurate low-dimensional approximation of the linear dynamics of fluid flow. J. Atmos. Sci., 58, 27712789, doi:10.1175/1520-0469(2001)058<2771:ALDAOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Flato, G., and et al. , 2013: Evaluation of climate models. Climate Change 2013: The Physical Science Basis, T. F. Stocker et al., Eds., Cambridge University Press, 741–866.

  • Fogelson, A., , and J. Keener, 2001: Immersed interface methods for Neumann and related problems in two and three dimensions. SIAM J. Sci. Comput., 22, 16301654, doi:10.1137/S1064827597327541.

    • Search Google Scholar
    • Export Citation
  • Giannakis, D., , and A. J. Majda, 2012: Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability. Proc. Natl. Acad. Sci. USA, 109, 22222227, doi:10.1073/pnas.1118984109.

    • Search Google Scholar
    • Export Citation
  • Grebenkov, D. S., , and B.-T. Nguyen, 2013: Geometrical structure of Laplacian eigenfunctions. SIAM Rev., 55, 601667, doi:10.1137/120880173.

    • Search Google Scholar
    • Export Citation
  • Lawley, D. N., 1956: Tests of significance for the latent roots of covariance and correlation matrices. Biometrika, 43, 128136, doi:10.1093/biomet/43.1-2.128.

    • Search Google Scholar
    • Export Citation
  • Mardia, K. V., , J. T. Kent, , and J. M. Bibby, 1979: Multivariate Analysis. Academic Press, 518 pp.

  • Nadler, B., , S. Lafon, , R. R. Coifman, , and I. G. Kevrekidis, 2006: Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmonic Anal., 21, 113127, doi:10.1016/j.acha.2005.07.004.

    • Search Google Scholar
    • Export Citation
  • Ropelewski, C., , and M. Halpert, 1987: Global and regional scale precipitation patterns associated with the El Niño/Southern Oscillation. Mon. Wea. Rev., 115, 16061626, doi:10.1175/1520-0493(1987)115<1606:GARSPP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Saito, N., 2008: Data analysis and representation on a general domain using eigenfunctions of Laplacian. Appl. Comput. Harmonic Anal., 25, 6897, doi:10.1016/j.acha.2007.09.005.

    • Search Google Scholar
    • Export Citation
  • Smith, T. M., , R. W. Reynolds, , T. C. Peterson, , and J. Lawrimore, 2008: Improvements to NOAA’s historical merged land–ocean surface temperature analysis (1880–2006). J. Climate, 21, 22832296, doi:10.1175/2007JCLI2100.1.

    • Search Google Scholar
    • Export Citation
  • Taylor, K. E., , R. J. Stouffer, , and G. A. Meehl, 2012: An overview of CMIP5 and the experimental design. Bull. Amer. Meteor. Soc., 93, 485498, doi:10.1175/BAMS-D-11-00094.1.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 200 200 36
PDF Downloads 165 165 40

Laplacian Eigenfunctions for Climate Analysis

View More View Less
  • 1 George Mason University, Fairfax, Virginia, and Center for Ocean–Land–Atmosphere Studies, Calverton, Maryland
  • | 2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York, and Center of Excellence for Climate Change Research, Department of Meteorology, King Abdulaziz University, Jeddah, Saudi Arabia
© Get Permissions
Restricted access

Abstract

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.

Corresponding author address: Timothy DelSole, 112 Research Hall, MS 2B3, George Mason University, 4400 University Drive, Fairfax, VA 22030. E-mail: delsole@cola.iges.org

Abstract

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.

Corresponding author address: Timothy DelSole, 112 Research Hall, MS 2B3, George Mason University, 4400 University Drive, Fairfax, VA 22030. E-mail: delsole@cola.iges.org
Save