Fitting weather forecast uncentainty with the famous tanh-model

Weather forecast uncentainty data

d = [[t_0 E_0], [t_1, E_1], ..., [t_{n-1}, E_{n-1}]]     

is fitted with the model

E(t) = A tanh(a t  + b ) + B            for norm=false
E(t) = 1 + c[tanh(a t + b) − tanh(b)]    for norm=true

where t is the time and E is the root-mean-square (r.m.s.) error. We consider two types of data:

normalized data with E(0)=1 (norm = true) and un-normalized/raw data (norm = false)

The routine fit_model in errorgrowth.py returns:

[a, b, c]         for norm=true
[a, b, A, B]      for norm=false

References:

• Lorenz, Edward (1996). "Predictability – A problem partly solved" Seminar on Predictability, Vol. I, ECMWF. link

• Nedjeljka Žagar, Martin Horvat, Žiga Zaplotnik & Linus Magnusson "Scale-dependent estimates of the growth of forecast uncertainties in a global prediction system" Tellus A: Dynamic Meteorology and Oceanography Vol. 69 , Iss. 1,2017 link