  When analyzing the spatio-temporal dependence in most environmental and earth
sciences variables such as pollutant concentrations at different levels of the
atmosphere, a special property is observed: the covariances and
cross-covariances are stronger in certain directions. This property is
attributed to the presence of natural forces, such as wind, which cause the
transport and dispersion of these variables. This spatio-temporal dynamics
prompted the use of the Lagrangian reference frame alongside any Gaussian
spatio-temporal geostatistical model. Under this modeling framework, a whole
new class was birthed and was known as the class of spatio-temporal covariance
functions under the Lagrangian framework, with several developments already
established in the univariate setting, in both stationary and nonstationary
formulations, but less so in the multivariate case. Despite the many advances
in this modeling approach, efforts have yet to be directed to probing the case
for the use of multiple advections, especially when several variables are
involved. Accounting for multiple advections would make the Lagrangian
framework a more viable approach in modeling realistic multivariate transport
scenarios. In this work, we establish a class of Lagrangian spatio-temporal
cross-covariance functions with multiple advections, study its properties, and
demonstrate its use on a bivariate pollutant dataset of particulate matter in
Saudi Arabia.
