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:
\( x_f \)
= the forecast value of the parameter in question
\( x_v \)
= the corresponding verifying value
\( x_c \)
= the climatological value of the parameter
n = the number of grid points or observations in the verification area
\( M_{f,c} \)
= the mean value over the verification area of the forecast anomalies from climate
\( M_{v,c} \)
= the mean value over the verification area of the analysed anomalies from climate
\( \vec{V}_f \)
= the forecast wind vector
\( \vec{V}_v \)
= the corresponding verifying value
The differentiation is approximated by differences computed on the verification grid:
\[ e_g = \left ( \left | \frac{\partial}{\partial x}(x_f-x_v)\right | + \left | \frac{\partial}{\partial y}(x_f-x_v)\right | \right ) \] \[ G_L = max \left ( \left | \frac{\partial x_f}{\partial x}\right | , \left | \frac{\partial x_v}{\partial x}\right | \right) + max \left ( \left | \frac{\partial x_f}{\partial y}\right | , \left | \frac{\partial x_v}{\partial y}\right | \right) \]The weights w i applied at each grid point or observation location are defined as
Verification against analyses: \( w_i = \cos \theta_i \) , cosine of laltitude athe the grid point i
Verification against observations: \( w_i = 1/n \) , all observations have equal weight