Probabilistic Tool for Freezing Temperatures

 

 

 

A cumulative distribution function or CDF is a function of a random variable X, given by the integral of the PDF up to a particular value of x.

 

 

                                                                                      F(x) = Pr{X <= x} = ∫_X<=x f(x)dx

 

 

Therefore the CDF specifies probabilities that the random quantity X will not exceed specific values.  So in the case of surface temperatures,

The variable x = 32F and  X = the NDFD temperature field.

 

Since we don’t know f(x) or the PDF for the temperature forecast, we assume a normalized or Gaussian distribution, where temperature (T) is the mean and the variance or standard deviation (σ) is the ensemble mean. Thus we can compute a cumulative normal distribution function.

 

                                                                                     Prob (T < 32) = 1/σ_T √2π ∫_-∞³² exp[-1/2(T – T_mean / σ_T)²] dT

 

 

The probability of temperatures reaching 32 degrees F is derived from the computed cumulative normal distribution of the 1200 UTC NDFD temperature grid data (T_mean) and the 2-meter temperature grid spread data (σ_T ) from the GFS ensemble (GEFS), and setting 32F as the limit for the integration.

 

This data from for this tool is generated once per day at approximately 1400 UTC for the entire temporal domain of the NDFD data in 6-hour increments extending out through the 156 hour forecast.

 

 

References

 

Wilks, Daniel S., Statistical Methods in the Atmospheric Sciences, 2^nd Edition, 2006.