\[smodel\] contains this model as a limiting case. The GEVP construction for the residual process is $$\label{Smith} X({\mathbf{s}}) = \max\bigl\{h_{1}K({\mathbf{s}}|{\mathbf{u}}_{1},
\Sigma),h_{2}K({\mathbf{s}}|{\mathbf{u}}_{2},\Sigma ),\ldots\bigr\},$$ where $\{(h_1,{\mathbf{u}}_1), (h_2,{\mathbf{u}}_2),\ldots\}$ follows a Poisson process with intensity $\lambda(h,{\mathbf{u}}) = h^{-2}I(h>0)$, and $K$ is a kernel function standardized so that $\int K({\mathbf{s}}|{\mathbf{u}},\tau)\,d{\mathbf{u}}=1$ for all ${\mathbf{s}}$. The construction (\[Smith\]) is a special case of the de Haan \[@dehaan-1984a\] spectral representation. A useful analogy is to think of $X({\mathbf{s}})$ as the maximum rainfall at site ${\mathbf{s}}$, generated as the maximum over a countably-infinite number of storms. The $k$[th]{} storm has center ${\mathbf{u}}_{k}\in{\mathcal{R}}^2$, intensity $h_{k}>0$, and spatial range given by $K({\mathbf{s}}|{\mathbf{u}}_{k},\tau)$.

Under this model, the joint distribution at locations ${\mathbf{s}}_1,\ldots,{\mathbf{s}}_n$ is $$\label{joint1}
\quad\mathrm{P} \bigl[X({\mathbf{s}}_1)<c_1,\ldots,X(
{\mathbf{s}}_n)<c_n \bigr] = \exp \biggl[-\int\max_i
\biggl\{\frac{K({\mathbf{s}}_i|{\mathbf{u}},\Sigma
)}{c_i} \biggr\}\,d{\mathbf{u}}\biggr].$$ The GEVP has extremal coefficient $$\label{ECsmith} \vartheta({\mathbf{s}}_i,{\mathbf{s}}_j) = \int\max \bigl
\{K({\mathbf{s}}_i|{\mathbf{u}},\Sigma ),K({\mathbf{s}}_j|{\mathbf{u}},\Sigma) \bigr\}\,d{\mathbf{u}},$$ which simplifies to $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j)=2\Phi (\frac
{\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert }{2\tau} )$ for the Gaussian kernel (\[K\]). This does not include a nugget effect, since $\vartheta({\mathbf{s}}_i,{\mathbf{s}}_j)=1$ if $\Vert {\mathbf{s}}_i-{\mathbf{s}}_j\Vert =0$.

The connection to the model in Section \[smodel\] is made by restricting the storm locations to the set of $L$ knot locations $\{{\mathbf{v}}_1,\ldots,{\mathbf{v}}_L\}$ and rescaling the kernels to sum to one as in (\[w\]), giving $$\label{Smith3} X({\mathbf{s}}) = 