Spatial-temporal modeling of vector-valued data using gradient processes: An application to wind fields

We develop a stochastic spatial-temporal model of the wind field in Houston, Texas, using wind data collected at pollution monitoring stations scattered about the area. This model is intended to be used as an input to an air pollution model, since wind is a very important part of modeling pollution transport. Since wind is naturally bivariate, in our case east and north wind components, we develop a method that assumes the wind is the gradient of some smooth, latent, univariate potential field. The mean and covariance functions of this potential field then induce the mean and covariance functions for the wind field. The advantage to this approach is that developing a scalar-valued covariance model for the potential field is much easier than developing a general, matrix-valued covariance model for the wind directly. We develop the mathematics for this gradient process model, and in particular explore the structure that arises when the underlying potential has an isotropic covariance. This structure allows us to develop variogram-like functions that can be estimated from the data in a way very similar to standard variogram estimation. For the data analysis, we propose a model of the wind following Breckling's 1989 time-series study: the wind is decomposed into a geostrophic term, a diurnal term, and an error term. The geostrophic term is constant over space, since the spatial scale we are working on is small compared to pressure systems, and captures the long-range time dependence caused by these pressure systems. The diurnal term is allowed to vary over space and time, is periodic with a period of 24 hours, and captures the sea-breeze oscillations. We adjust for errors in the data that are caused by local features, such as buildings and trees, that prevent the "true" wind from being measured. We explore the nature of the sea-breeze, and find that rotating the components of the wind allows the sea-breeze to be captured in a single component that is perpendicular to the sea coast. We fit the elements of the model using the variogram and maximum-likelihood estimation