A_{1t},\ldots,A_{Lt}
&{\stackrel{\mathrm{indep}}{\sim}}& \operatorname{GEV} \bigl[\mu_t^*({\mathbf{s}}_i),
\sigma_t^*({\mathbf{s}}_i),\xi^*({\mathbf{s}}_i) \bigr],
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
A_{lt} &{\stackrel{\mathrm{i.i.d.}}{\sim}}& \operatorname{PS}(\alpha),\end{aligned}$$ where $\theta_t({\mathbf{s}})=[\sum_{l=1}^LA_{lt}w_{l}({\mathbf{s}})^{1/\alpha
}]^{\alpha
}$ is the spatial random effect for year $t$, $\mu_t^*({\mathbf{s}}) = \mu
({\mathbf{s}})
+ \frac{\sigma({\mathbf{s}})}{\xi({\mathbf{s}})} (\theta_t({\mathbf{s}})^{\xi({\mathbf{s}})}-1 )$, $\sigma_t^*({\mathbf{s}})=\alpha\sigma({\mathbf{s}})\theta_t({\mathbf{s}})^{\xi({\mathbf{s}})}$, and $\xi^*({\mathbf{s}})=\alpha\xi({\mathbf{s}})$. Note that while the GEV parameters conditioned on $\theta_t({\mathbf{s}})$ in (\[auxmodel2\]) vary by year, marginally, $Y_t({\mathbf{s}})\sim\operatorname{GEV}[\mu({\mathbf{s}}),\sigma({\mathbf{s}}),\xi({\mathbf{s}})]$ for all $t$.

Gaussian process priors are used for the GEV parameters $\mu({\mathbf{s}})$, $\gamma({\mathbf{s}}) = \log[\sigma({\mathbf{s}})]$, and $\xi({\mathbf{s}})$. The Gaussian process $\mu$ has mean $\mathbf{x}({\mathbf{s}})^T{\bolds{\beta}}_{\mu}$, where $\mathbf{x}({\mathbf{s}})$ includes the spatial covariates such as elevation. The spatial covariance of $\mu$ is ${ \mbox{Mat\'{e}rn}}$ \[@Banerjee03 [@Cressie93; @Handbook]\] with variance $\delta_{\mu}^2>0$, range $\rho_{\mu}>0$, and smoothness $\nu_{\mu}>0$. The other GEV parameters $\gamma({\mathbf{s}})$ and $\xi({\mathbf{s}})$ are modeled similarly. In some applications, it may also be desirable to allow for the GEV parameters to evolve over time, perhaps following a separate linear time trend at each location, which would be a straightforward modification of this model. The MCMC algorithm used to sample from this model is described in Appendix \[appa3\].

Connection with the Gaussian extreme value process {#ssmith}
==================================================

The GEVP of @smith-1990a is a well-known spatial max-stable process. In this section we show that the proposed positive stable random effects model in Section 