Analyzing the Diurnal Cycle by Bayesian Interpolation on a Sphere for Mapping GNSS Radio Occultation Data

Bayesian interpolation has previously been proposed as a strategy to construct maps of radio occultation (RO) data, but that proposition did not consider the diurnal dimension of RO data. In this work, the basis functions of Bayesian interpolation are extended into the domain of the diurnal cycle, thus enabling monthly mapping of radio occultation data in synoptic time and analysis of the atmospheric tides. The basis functions are spherical harmonics multiplied by sinusoids in the diurnal cycle up to arbitrary spherical harmonic degree and diurnal cycle harmonic. Bayesian interpolation requires a regularizer to impose smoothness on the fits it produces, thereby preventing the overfitting of data. In this work, a formulation for the regularizer is proposed and the most probable values of the parameters of the regularizer determined. Special care is required when obvious gaps in the sampling of the diurnal cycle are known to occur in order to prevent the false detection of statistically significant high-degree harmonics of the diurnal cycle in the atmosphere. Finally, this work probes the ability of Bayesian interpolation to generate a valid uncertainty analysis of the fit. The postfit residuals of Bayesian interpolation are dominated not by measurement noise but by unresolved variability in the atmosphere, which is statistically nonuniform across the globe, thus violating the central assumption of Bayesian interpolation. The problem is ameliorated by constructing maps of RO data using Bayesian interpolation that partially resolve the temporal variability of the atmosphere, constructing maps for approximately every 3 days of RO data.