We introduce two novel bivariate parametric covariance models, the powered exponential (or stable) covariance model and the generalized Cauchy covariance model. Both models allow for flexible smoothness, variance, scale, and cross-correlation parameters. The smoothness parameter is in $(0, 1]$. Additionally, the bivariate generalized Cauchy model allows for distinct long range parameters. We also show that the univariate spherical model can be generalized to the bivariate case only in a trivial way. The results are based on general sufficient conditions for the positive definiteness of $2\times2$-matrix valued functions. These conditions require only computing the derivatives of a bivariate covariance function of order 2 and 3 in $\R$ and in $\R^3$, respectively, and calculating an infimum of a function of one variable. In a data example on the content of copper and lead in the top soil in a flood plain along the river Meuse we compare the bivariate powered exponential model to the tra 查看全文>>