The following definitions should be used
Mean error
where the sum of the weights
\[ M_w = \sum_{i=1}^n w_i \]
Root mean square (rms) error
Correlation coefficient between forecast and analysis anomalies
rms vector wind error
\[ rms = \sqrt {\frac{1}{M_w} \sum_{i=1}^n w_i (\vec{V}_f - \vec{V}_v)_i^2 } \]Mean absolute error
\[ MAE = \frac{1}{M_w} \sum_{i=1}^n w_i | x_f - x_v |_i \]rms anomaly
\[ rmsa = \sqrt {\frac{1}{M_w} \sum_{i=1}^n w_i (x - x_c)_i^2 } \]standard deviation of field
\[ sd = \sqrt {\frac{1}{S_w} \sum_{i=1}^n w_i (x - M_x)_i^2 } \]where
\[ M_x = \frac{1}{M_w} \sum_{i=1}^n w_i x_i \]S1 score
\[ S_1 = 100 \frac{\sum_{i=1}^n w_i (e_g)_i}{\sum_{i=1}^n w_i (G_L)_i} \]
Where:
\( x_f \)
\( x_v \)
\( x_c \)
\( M_{f,c} \)
\( M_{v,c} \)
\( \vec{V}_f \)
\( \vec{V}_v \)