Motivated by the wide applications of multivariate Gaussian random fields in spatial modeling, we study the tail probability of the extremes, the inference of fractal indices and large covariance modeling of multivariate Gaussian random fields. First, we establish the precise asymptotics for the extremes of bivariate Gaussian random fields by applying the double sum method. The main results can be applied to bivariate Mat\'{e}rn fields. Second, we study the joint asymptotic properties of estimating the fractal indices of bivariate Gaussian random processes under infill asymptotics, which indicates that the estimators are asymptotically independent of the cross correlation in most cases. Third, we define a framework to couple high-dimensional and spatially indexed LiDAR signals with forest variables using a fully Bayesian functional spatial data analysis, which is able to capture within and among LiDAR signal/forest variables association within and across locations. The proposed modeling framework is illustrated by a simulated study and by analyzing LiDARand spatially coinciding forest inventory data collected on thePenobscot Experimental Forest, Maine.
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