Notes about off-line ’Holm-like’ calibration diagnostics with UKV NMC-like data

The following procedure is followed, with each of two options that we call “Anal” and “Fc”.
  1. Define a population of pairs of states, x1 and x2 (for total RH).
    1. For option “Anal”:
      1. x1 is an analysis, xa, and
      2. x2 is a 3-hour forecast, xf3, valid at the same time as the analysis.
    2. For option “Fc”:
      1. x1 is a 3-hour forecast, xf3, and
      2. x2 is a 6-hour forecast, xf6, valid at the same time as xf3.
  2. Construct a histogram of the joint distribution.
    1. For option “Anal”, see joint distribution here.
    2. For option “Fc”, see joint distribution here.
  3. Calculate difference fields, Δx = (x2 −  x1) ⁄ (2).
  4. Construct the following conditional PDFs for each level: Px|x1), Px|x2) and Px|x), where x = (x1 +  x2) ⁄ 2.
    1. For option “Anal”:
      1. For Px|x1) = P((xf3 −  xa) ⁄ (2)| xa) averaged over four layers of the atmosphere, see PDFs here.
      2. For Px|x2) = P((xf3 −  xa) ⁄ (2)| xf3) averaged over four layers of the atmosphere, see PDFs here.
      3. For Px|x) = P((xf3 −  xa) ⁄ (2)|( xa + xf3) ⁄ 2) averaged over four layers of the atmosphere, see PDFs here.
    2. For option “Fc”:
      1. For Px|x1) = P((xf6 −  xf3) ⁄ (2)| xf3) averaged over four layers of the atmosphere, see PDFs here.
      2. For Px|x2) = P((xf6 −  xf3) ⁄ (2)| xf6) averaged over four layers of the atmosphere, see PDFs here.
      3. For Px|x) = P((xf6 −  xf3) ⁄ (2)|( xf3 +  xf6) ⁄ 2) averaged over four layers of the atmosphere, see PDFs here.
  5. The mean, standard deviation and skewness are computed from the conditional PDFs (and the relative frequency of x1, x2 or x is also shown), each a function of x1, x2 or x, and vertical level.
    1. For option “Anal”:
      1. For Px|x1) = P((xf3 −  xa) ⁄ (2)| xa), see mean, stddev here.
      2. For Px|x2) = P((xf3 −  xa) ⁄ (2)| xf3), see mean, stddev here.
      3. For Px|x) = P((xf3 −  xa) ⁄ (2)|( xa + xf3) ⁄ 2), see mean, stddev here.
    2. For option “Fc”:
      1. For Px|x1) = P((xf6 −  xf3) ⁄ (2)| xf3), see mean, stddev here.
      2. For Px|x2) = P((xf6 −  xf3) ⁄ (2)| xf6), see mean, stddev here.
      3. For Px|x) = P((xf6 −  xf3) ⁄ (2)|( xf3 +  xf6) ⁄ 2), see mean, stddev here.