}]^{\alpha}$, and then sampling $Y({\mathbf{s}}^*)$ from the independent GEV in (\[auxmodel\]). Repeating this at every MCMC iteration gives samples from the posterior predictive distribution of $Y({\mathbf{s}}^*)$.

Kernel and knot selection {#knot}
-------------------------

Although other kernels are possible, we use a scaled version of the Gaussian kernel $$\label{K} K({\mathbf{s}}|{\mathbf{v}}_l,\tau) = \frac{1}{2\pi\tau^2}\exp \biggl[-
\frac
{1}{2\tau^2}({\mathbf{s}}-{\mathbf{v}}_l)^T({\mathbf{s}}-{\mathbf{v}}_l)
\biggr],$$ where ${\mathbf{v}}_1,\ldots,{\mathbf{v}}_L\in{\mathcal{R}}^2$ are spatial knots and $\tau>0$ is the kernel bandwidth. To ensure that the kernels sum to one at each location, the kernels are scaled as $$\label{w} w_l({\mathbf{s}}) = \frac{K({\mathbf{s}}|{\mathbf{v}}_{l},\tau)}{\sum_{j=1}^L K({\mathbf{s}}|{\mathbf{v}}_{j},\tau)}.$$ The knots are taken as a fixed and regularly-spaced grid of points covering the spatial domain. Section \[ssmith\] shows that this choice of kernel function and knot distribution gives the GEVP as a limiting case. Even with a regular grid of knots, the extremal coefficient is nonstationary, that is, $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j)$ is not simply a function of $\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert $. For example, $w({\mathbf{s}}_i)$ may not equal $w({\mathbf{s}}_j)$ if ${\mathbf{s}}_i$ is close to a knot and ${\mathbf{s}}_j$ is not. This discretization artifact dissipates for large $L$.

While the extremal coefficient does not fully characterize spatial dependence, it is useful for guiding knot selection. Knot selection poses a trade-off between computational burden with too many knots and 