Abstract: Analyses of geostatistical data are often based on the assumption that the spatial random field is isotropic. This assumption, if erroneous, can adversely affect model predictions and statistical inferences. Today, many applications consider global data, and hence, it is necessary to check the assumption of isotropy on a sphere. This study proposes a test for spatial isotropy on a sphere. The data are first projected onto the set of spherical harmonic functions. Under isotropy, the spherical harmonic coefficients are uncorrelated, but 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 this matrix as the test statistic. Extensive simulations are conducted to assess the Type-I errors of the test under different scenarios. Our method requires temporal replication in the data and, hence, is applicable to many data sets in the Earth sciences. We show how temporal correlation affects the test and provide a method for handling such correlation. We also gauge the power of the test as we move away from isotropy. The method is applied to 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 a model component. Then, we apply the test to the isotropic component in a sequence of such models to determine how well the models capture the anisotropy in the fields.

Key words and phrases: Anisotropy, spatial statistics, spherical harmonic representation.