In order to characterise the differences between the model and observations it is important to take into account the correspondence in both the patterns and the variances of the two fields. We define the centred pattern RMSD as:

CRMSD =

Taylor (2001) noticed that a simple relationship exists between the correlation coefficient, the centred pattern RMS difference, and the variances of the fields in question. The relationship is given by:

which takes the same form as the law of cosines (c2 = a2 + b2 - 2ab cos (y)). This relationship can be used to plot the information about R, CRMSD and the variances in the model and observations as a point on a single diagram. In order to make it possible to compare fields with different units, the statistics can be non-dimensionalised by normalising each variable in Eq. (22.8) by the standard deviation in the observed field, which leaves the correlation coefficient unchanged. A schematic Taylor diagram is shown in Fig. 22.1. If the model exactly reproduced the observations, it would lie at the point indicated by the black circle. The distance between this black circle and the actual model point (the blue diamond in this example) represents the CRMSD and the dotted arcs on the diagram represent lines of constant CRMSD. The correlation coefficient is represented on the outer arc of the diagram with increasing correlation with the angle from the y-axis. The normalised standard deviation is represented as the distance to the origin, with a ratio of one denoted by the dashed arc (if the point is closer to the origin the model has lower

Fig. 22.1 Schematic description of a Taylor diagram

Fig. 22.1 Schematic description of a Taylor diagram

variance than the observations). The power of the Taylor diagram lies in the ability to plot numerous model runs on a single diagram and to compare these various aspects of the models' performance.

One drawback of the Taylor diagram is that the mean error of the models is not accounted for. The so-called Target diagram (Jolliff et al. 2009) can be used to represent complementary information about the statistical performance of models. In this case, the relationship between the total mean square difference and the unbiased MSD and bias, RMSD2 = MD2 + CRMSD2, is plotted on a diagram where the x-axis represents CRMSD and the y-axis represents the bias. Since CRMSD is a positive quantity by definition, the negative x-axis can be utilised to include information about the standard deviation difference by multiplying the CRMSD by the sign of the standard deviation difference.

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