An extended PDE-based statistical spatio-temporal model that suppresses the Gibbs phenomenon

Partial differential equation (PDE)-based spatio-temporal models are available in the literature for modeling spatio-temporal processes governed by advection-diffusion equations. The main idea is to approximate the process by a truncated Fourier series and model the temporal evolution of the spectral coefficients by a stochastic process whose parametric structure is determined by the governing PDE. However, because many spatio-temporal processes are nonperiodic with boundary discontinuities, the truncation of Fourier series leads to the well-known Gibbs phenomenon (GP) in the output generated by the existing PDE-based approaches. This article shows that the existing PDE-based approach can be extended to suppress GP. The proposed approach starts with a data flipping procedure for the process respectively along the horizontal and vertical directions, as if we were unfolding a piece of paper folded twice along the two directions. For the flipped process, this article extends the existing PDE-based spatio-temporal model by obtaining the new temporal dynamics of the spectral coefficients. Because the flipped process is spatially periodic and has a complete waveform without boundary discontinuities, GP is removed even if the Fourier series is truncated. Numerical investigations show that the extended approach improves the modeling and prediction accuracy. Computer code is made available on GitHub.