Gaussian process models for interpolation and prediction from stationary spacetime processes require careful specification of the covariance kernel. Employing Bochner's theorem, which, loosely speaking, states that stationary covariance kernels have a spectral representation, one can equivalently manipulate a model for the spectral matrix in lieu of the covariance matrix. Previous authors have employed this representation together with an expansion of the spectrum in Gaussian basis functions in order to learn the kernel. In this work, we employ two different expansions for the kernel, in wavelets and in terms of bandlimited (discrete prolate spheroidal wave) functions and compare these in terms of ease of use, interpretability, and computational complexity. The results are compared on both simulated data and vertical profile Doppler lidar wind speed measurements having one space and one time dimension.