Spherical Process Models for Global Spatial Statistics.

Area of Science:

• Geophysical Science
• Environmental Science
• Climate Science

Background:

• Statistical models for global data must account for the Earth's spherical geometry.
• Current models often use chordal distance, introducing distortions, instead of the natural geodesic distance.
• Covariance functions directly defined on a sphere using geodesic distance are needed.

Purpose of the Study:

• To address the limitations of existing statistical models for global spatial data.
• To introduce and evaluate covariance functions defined directly on a sphere using geodesic distance.
• To compare different methods for building Gaussian process models on spheres.

Main Methods:

• Review of current approaches: differential operator, stochastic partial differential equation, kernel convolution, and deformation.
• Illustration of Gaussian process realizations with varying covariance structures (isotropic and nonstationary).
• Application to global surface temperature data using deformations and geographical indicators.

Main Results:

• Demonstration of physically realistic distortions caused by using chordal distance.
• Comparison of methods based on log-likelihood values and prediction scores.
• Evaluation of isotropic and nonstationary covariance models for global surface temperature.

Conclusions:

• Covariance functions directly defined on a sphere using geodesic distance are crucial for accurate global data modeling.
• The choice of covariance model impacts the physical realism and predictive performance.
• Further research is needed on related problems in spherical data analysis.