8.1Score definitions

The following definitions should be used
Mean error Image Removed error 

M = \frac{1}{M_w} \sum_{i=1}^n w_i (x_f - x_v)_i

where the sum of the weights

M_w = \sum_{i=1}^n w_i

Root mean square (rms) error Image Removed

rms = \sqrt {\frac{1}{M_w} \sum_{i=1}^n w_i (x_f - x_v)_i^2 }

Correlation coefficient between forecast and analysis anomalies
Image Removed
rms vector wind error Image Removed
Mean absolute error Image Removed
rms anomaly Image Removed
standard deviation of field Image Removed where Image Removed
S1 score Image Removed
Where:
Image Removed =the forecast value of the parameter in question
Image Removed =the corresponding verifying value
Image Removed =the climatological value of the parameter
Image Removed =the number of grid points or observations in the verification area
Image Removed =the mean value over the verification area of the forecast
anomalies from climate
Image Removed =the mean value over the verification area of the analysed
anomalies from climate
Image Removed =the forecast wind vector
Image Removed = Image Removed
Image Removed = Image Removed
where the differentiation is approximated by differences computed on the verification grid.
The weights Image Removed applied at each grid point or observation location are defined as
Verification against analyses: Image Removed , cosine of latitude at grid point i
Verification against observations: Image Removed , all observations have equal weight

r = \frac{\sum_{i=1}^n w_i (x_f-x_c-M_{f,c})_i (x_v-x_c-M_{v,c})_i}{\sqrt{\left(\sum_{i=1}^n w_i (x_f-x_c-M_{f,c})_i^2 \right) \left(\sum_{i=1}^n w_i (x_v-x_c-M_{v,c})_i^2 \right) }}

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

...