  Traveling phenomena, frequently observed in a variety of scientific
disciplines including atmospheric science, seismography, and oceanography, have
long suffered from limitations due to lack of realistic statistical modeling
tools and simulation methods. Our work primarily addresses this, introducing
more realistic and flexible models for spatio-temporal random fields. We break
away from the traditional confines of the classic frozen field by either
relaxing the assumption of a single deterministic velocity or rethinking the
hypothesis regarding the spectrum shape, thus enhancing the realism of our
models. While the proposed models stand out for their realism and flexibility,
they are also paired with simulation algorithms that are equally or less
computationally complex than the commonly used circulant embedding for Gaussian
random fields in $\mathbb{R}^{2+1}$. This combination of realistic modeling
with efficient simulation methods creates an effective solution for better
understanding traveling phenomena.
