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Downscaled Climate Projections

In a probabilistic approach, however, one has flexibility to control the correlation between two stations. For example, consider a small scale variable at two stations, $s_1(t)$ and $s_2(t)$, together with the corresponding large-scale predictors, $L_1(t)$ and $L_2(t)$, where all variables are a function of time, $t$. For simplicity, we assume that linear regression is a good approximation so that the downscaled $s_j(t)$ is related to $L_j(t)$ by \begin{equation*} \tilde{s}_j(t) = a_j L_j(t) + b_j + c_j n_j(t), \end{equation*} where $a_j$ and $b_j$ are the ordinary least squares constants, $n_j(t)$ is noise with zero mean and unit variance, $c_j$ is the square root of the "unexplained" variance, $\tilde{s}(t)$ is the downscaled $s(t)$, and $j = 1$ or $2$. One can show that the correlation between the downscaled variables is: \begin{equation} \label{corr} \mathrm{corr} (s_1,s_2) = \frac{ a_1 a_2 \mathrm{cov} (L_1,L_2) + c_1 c_2 \mathrm{corr}(n_1,n_2) }{ \sqrt{(a_1^2 \sigma_1^2 + c_1^2)(a_2^2 \sigma_2^2 + c_2^2)} }, \end{equation} where $\sigma_j$ is the standard deviation of $L_j(t)$. In a deterministic approach $n_j(t) = 0$ and the correlation between $s_1$ and $s_2$ \eqref{corr} simplifies to \begin{equation*} \mathrm{corr} (s_1,s_2) = \mathrm{corr} (L_1,L_2). \end{equation*} In a probabilistic approach, on the other hand, one has the freedom to choose the cross-correlation between $n_1(t)$ and $n_2(t)$ and this in turn determines the correlation between $s_1$ and $s_2$. In this simple example, one can solve \eqref{corr} for $\mathrm{corr}(n_1,n_2)$ and easily determine the noise cross-correlation required to correctly reproduce $\mathrm{corr}(s_1,s_2)$.