Bayesian SPDE–INLA modelling of rainfall in South Africa: a structured thermodynamic spatiotemporal interaction approach

Abstract Climate variability in South Africa is increasingly characterised by rising temperatures and changing rainfall patterns, posing challenges for climate risk assessment and adaptation planning. This study develops a hierarchical Bayesian spatiotemporal framework to model monthly rainfall, combining the Stochastic Partial Differential Equation (SPDE) approach with Integrated Nested Laplace Approximation (INLA) and a Tweedie likelihood. A Type IV space-time interaction term is incorporated to capture non-separable temperature-rainfall dynamics, using maximum and minimum temperatures as covariates. The results indicated that models including non-separable interaction structures provide improved fit and predictive performance compared to nested alternatives, emphasising the relevance of capturing complex spatiotemporal dependencies. Trend analysis reveals spatially heterogeneous warming, with stronger increases in maximum temperature across northern interior regions and less consistent patterns in minimum temperature. The estimated effects suggest contrasting thermodynamic influences on rainfall, with minimum temperature positively associated and maximum temperature negatively associated with precipitation, and residual diagnostics highlighted remaining spatial dependence and over-dispersion, suggesting limitations related to the sparse observational network. The proposed Bayesian SPDE-based framework provides a flexible and robust tool for analysing regionally differentiated hydroclimatic responses, supporting improved understanding of climate variability and informing region-specific adaptation strategies.