Evaluation of uncertainties in the CRCM-simulated North American climate

Abstract

This work is a first step in the analysis of uncertainty sources in the RCM-simulated climate over North America. Three main sets of sensitivity studies were carried out: the first estimates the magnitude of internal variability, which is needed to evaluate the significance of changes in the simulated climate induced by any model modification. The second is devoted to the role of CRCM configuration as a source of uncertainty, in particular the sensitivity to nesting technique, domain size, and driving reanalysis. The third study aims to assess the relative importance of the previously estimated sensitivities by performing two additional sensitivity experiments: one, in which the reanalysis driving data is replaced by data generated by the second generation Coupled Global Climate Model (CGCM2), and another, in which a different CRCM version is used. Results show that the internal variability, triggered by differences in initial conditions, is much smaller than the sensitivity to any other source. Results also show that levels of uncertainty originating from liberty of choices in the definition of configuration parameters are comparable among themselves and are smaller than those due to the choice of CGCM or CRCM version used. These results suggest that uncertainty originated by the CRCM configuration latitude (freedom of choice among domain sizes, nesting techniques and reanalysis dataset), although important, does not seem to be a major obstacle to climate downscaling. Finally, with the aim of evaluating the combined effect of the different uncertainties, the ensemble spread is estimated for a subset of the analysed simulations. Results show that downscaled surface temperature is in general more uncertain in the northern regions, while precipitation is more uncertain in the central and eastern US.

Appendix

1.1 Root-mean-square difference as a function of sample length

In this appendix we develop a theoretical expression for the root-mean-squared difference between two average fields as a function of sample length. Usually, the variance estimator S ² can be written as

$$ S^{{\text{2}}} = \frac{1} {{M - 1}}{\sum\limits_{i = {\text{1}}}^M {{\left( {z_{i} - \overline{z} } \right)}^{{\text{2}}} } }, $$

(2)

where M is the number of elements z, and $\ifmmode\expandafter\bar\else\expandafter\=\fi{z}$ the mean value. When M = 2, it can be rewritten as

$$ S^{{\text{2}}} = {\left( {z_{{\text{1}}} - \overline{z} } \right)}^{{\text{2}}} + {\left( {z_{{\text{2}}} - \overline{z} } \right)}^{{\text{2}}} , $$

(3)

and, using the standard definition of average value, as

$$ S^{{\text{2}}} = \frac{{\text{1}}} {{\text{2}}}{\left( {z_{{\text{1}}} - z_{{\text{2}}} } \right)}^{{\text{2}}} . $$

(4)

If z ₁ and z ₂ are two-dimensional fields, we may estimate the domain variance by averaging all grid points in both i and j directions. Then, the estimated domain variance can be written as

$$ S_{\text{D}}^{2}= \frac{1}{D}{\sum\limits_{ij} {S_{ij}^{2}} } = \frac{1}{2}\frac{1}{D}{\sum\limits_{ij} {\left( {z_{1}^{ij} - z_{2}^{ij}} \right)}^{2}}=\frac{1}{2}{\text{MSD}}, $$

(5)

where D is the number of grid points in the domain, and MSD is the mean-square difference in its standard definition. From here we can see that

$$ {\text{RMSD}} = {\sqrt {\text{2}} }S_{\text{D}} , $$

(6)

where S _D is the estimated standard deviation and RMSD the root-mean-square difference. A better estimation of S _D from this relation can be obtained when variability is uniformly distributed spatially and spatial correlation is low.

As discussed by von Storch and Zwiers (2001), the variance of a sample mean $\ifmmode\expandafter\bar\else\expandafter\=\fi{z}$ of a collection of independent and identically distributed random variables z, can be expressed as

$$ S^{{\text{2}}} _{{\ifmmode\expandafter\bar\else\expandafter\=\fi{z}}} = \tfrac{{\text{1}}} {N}S^{{\text{2}}} _{z} , $$

(7)

where N is the number of members in the sample, and S ² the variance of z. This can be interpreted as stating that the sample mean, as an estimator of the population mean, has an uncertainty that is proportional to the population variance and inversely proportional to the size of the sample.

From (6) and (7) we obtain that

$$ {\text{RMSD}}_{{\overline{z} }} = {\sqrt {\text{2}} }S_{{\overline{z} }} \approx \frac{{{\sqrt {\text{2}} }}} {{{\sqrt N }}}S_{z} $$

(8)

which gives us a relation for the RMSD between two average fields drawn from different samples and the length of the samples. Since the RMSD is a crude estimation of the standard deviation, a departure from the function on the right-hand side is expected (particularly if the conditions of decorrelation in time and space and uniform spatial variability are far from satisfied).