process is $\theta({\mathbf{s}})= [\sum_{l=1}^LA_{l}w_{l}({\mathbf{s}})^{1/\alpha} ]^{\alpha}$, where $A_l$ are the basis function coefficients. To ensure max-stability and ${\mbox{Fr\'{e}chet}}$ marginal distributions, the random effects $A_l$ follow the positive stable distribution with density $p(A|\alpha)$ which has Laplace transformation $\int_0^\infty\exp(-At)p(A|\alpha)\,dA = \exp
(-t^\alpha
)$. We denote this as $A_l\sim\operatorname{PS}(\alpha)$. Although $p(A|\alpha)$ has no closed form, it possesses the essential feature that if $A_1,\ldots,A_T{\stackrel{\mathrm{i.i.d.}}{\sim}}\operatorname{PS}(\alpha)$, then $(A_1+\cdots+A_T)/T^{1/\alpha}\sim \operatorname{PS}(\alpha)$. Appendix \[appa2\] verifies that this model for $X({\mathbf{s}})$ is max-stable with unit ${\mbox{Fr\'{e}chet}}$ marginal distributions.

Marginalizing over the nugget terms $U({\mathbf{s}})$ gives the hierarchical model $$\begin{aligned}
\label{auxmodel} Y({\mathbf{s}}_i) | A_{1},\ldots,A_{L} &
{\stackrel{\mathrm{indep}}{\sim}}& \operatorname{GEV}\bigl[\mu^*({\mathbf{s}}_i),\sigma^*(
{\mathbf{s}}_i),\xi^*({\mathbf{s}}_i)\bigr],
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
A_{l} &{\stackrel{\mathrm{i.i.d.}}{\sim}}& \operatorname{PS}(\alpha),\end{aligned}$$ where $\mu^*({\mathbf{s}}) = \mu({\mathbf{s}}) + \frac{\sigma({\mathbf{s}})}{\xi({\mathbf{s}})} [\theta
({\mathbf{s}})^{\xi({\mathbf{s}})}-1 ]$, $\sigma^*({\mathbf{s}}) = \alpha\sigma({\mathbf{s}})\theta({\mathbf{s}})^{\xi({\mathbf{s}})}$, and $\xi^*({\mathbf{s}})=\alpha\xi({\mathbf{s}})$. The responses are conditionally independent given the random effects ${\mathbf{A}}$. The effect of conditioning on ${\mathbf{A}}= (A_{1},\ldots,A_{L})^T$, and thus the spatial process $\theta$, is to move spatial dependence from the residuals to a random effect in the GEV parameters. Marginalizing over the random effects induces spatial dependence. The joint distribution function of the residual process $X$ at $n$ locations ${\mathbf{s}}_1, \ldots, {\mathbf{s}}_n$ is $$\label{ASL} \mathrm{P}\bigl(X({\mathbf{s}}_i)<c_i, i=1,\ldots,n
\bigr) = \exp \Biggl\{-\sum_{l=1}^L \Biggl[
\sum_{i=1}^n \biggl(\frac{w_l({\mathbf{s}}_i)}{c_i}
\biggr)^{1/\alpha} \Biggr]^{\alpha
} \Biggr\}.$$ Therefore, although this is a process 