Error saturation level

Forecast errors do not grow indefinitely but asymptotically approach a maximum, the  “Error Saturation Level” (ESL).  

Figure 66 :  The error growth in a state-of-the-art NWP forecast system will at some stage display larger errors than a climatological average used as forecast and will, as do the errors of persistence forecasts and guesses, asymptotically approach an error level 41% above that of a forecast based on a climatological average

For extended forecast ranges, with decreasing correspondence between forecast and observed anomalies, the covariance term approaches zero. For A_f=A_a  this yields an ESL at

RMSE = E_saturation= A_a

which is 41% larger than E_climate, the error when a climatological average is used as a forecast (see Figure 66). The value A_a  is also the ESL for persistence fore­casts or guesses based on climatological distributions.