\max \bigl\{h_{1}w_1(
{\mathbf{s}}),\ldots,h_{L}w_L({\mathbf{s}}) \bigr\}.$$ This amounts to truncating the de Haan spectral representation. If $h_l\sim  \operatorname{GEV}(1,1,1)$, then $X({\mathbf{s}})$ is max-stable with joint distribution $$\label{joint2} \mathrm{P} \bigl[X({\mathbf{s}}_1)<c_1,\ldots,X(
{\mathbf{s}}_n)<c_n \bigr] = \exp \Biggl[-\sum
_{l=1}^L\max_i \biggl\{
\frac{w_l({\mathbf{s}}_i)}{c_i} \biggr\} \Biggr],$$ which implies that the marginal distributions are unit ${\mbox{Fr\'{e}chet}}$. For equally-spaced knots, this distribution converges weakly to the full GEVP distribution function (\[joint1\]) as $L$ increases. We note that this finite approximation could be applied to other max-stable models such as those in @schlather-2002a and @kabluchko-2009a by allowing the functions $K$ to be suitably scaled Gaussian processes, unlike the current approach where $K$ is a kernel function.

Using the model described by (\[joint2\]) directly is problematic because it may not yield a proper likelihood. The process (\[Smith3\]) at $n$ locations $\{X({\mathbf{s}}_1),\ldots,\break X({\mathbf{s}}_n)\}$ is completely determined by the intensities $\{h_1,\ldots,h_L\}$. Therefore, the likelihood for $\{X({\mathbf{s}}_1),\ldots,X({\mathbf{s}}_n)\}$ requires a map from $\{
X({\mathbf{s}}_1),\ldots,X({\mathbf{s}}_n)\}$ to $\{h_1,\ldots,h_L\}$. This map may not exist, for example, if $L<n$, and generally does not have a closed form. This is common in dimension reduction methods for Gaussian process models \[e.g., @higdon-1998, @banerjee-2008a, and @cressie-2008a\].

As with the Gaussian process dimension reduction methods, the model in Section \[smodel\] includes both a spatial process ($\theta$) 