A Test for Isotropy on a Sphere using Spherical Harmonic Functions

TL;DR

This paper introduces a statistical test for isotropy on a sphere using spherical harmonic functions, which helps identify anisotropy in global geostatistical data like temperature fields.

Contribution

The paper proposes a novel test based on the correlation of spherical harmonic coefficients, including handling temporal correlation and assessing power and error rates.

Findings

Test effectively detects anisotropy in simulated data.

Application to temperature data reveals model's ability to capture anisotropy.

Method provides a diagnostic tool for geostatistical modeling on a global scale.

Abstract

Analysis of geostatistical data is often based on the assumption that the spatial random field is isotropic. This assumption, if erroneous, can adversely affect model predictions and statistical inference. Nowadays many applications consider data over the entire globe and hence it is necessary to check the assumption of isotropy on a sphere. In this paper, a test for spatial isotropy on a sphere is proposed. The data are first projected onto the set of spherical harmonic functions. Under isotropy, the spherical harmonic coefficients are uncorrelated whereas they are correlated if the underlying fields are not isotropic. This motivates a test based on the sample correlation matrix of the spherical harmonic coefficients. In particular, we use the largest eigenvalue of the sample correlation matrix as the test statistic. Extensive simulations are conducted to assess the Type I errors of the test under different scenarios. We show how temporal correlation affects the test and provide a method for handling temporal correlation. We also gauge the power of the test as we move away from isotropy. The method is applied to the near-surface air temperature data which is part of the HadCM3 model output. Although we do not expect global temperature fields to be isotropic, we propose several anisotropic models with increasing complexity, each of which has an isotropic process as model component and we apply the test to the isotropic component in a sequence of such models as a method of determining how well the models capture the anisotropy in the fields.