Preparing Gumbel probability paper

Gumbel probability paper is a plotting sheet on which data that follow Gumbel's distribution plot as a straight line. It can be prepared from ordinary arithmetic graph paper because Gumbel's equation $x_T=\bar x+K\sigma$ is linear in the reduced variate $y_T$.

Steps

1. Choose the vertical axis as an ordinary arithmetic scale for the variable (flood discharge or rainfall depth), covering the range of data and predictions.
2. For selected return periods compute the reduced variate

$$y_T=-\ln\left[\ln\frac{T}{T-1}\right]$$

3. Draw the horizontal axis with a uniform scale for $y_T$ (for example 1 cm = 1 unit of $y_T$ from −1 to 7).
4. At the positions of the computed $y_T$ values mark the return periods T (and the probability of exceedance $P=1/T$) as labels. This gives the non-uniform return period scale along the top or bottom of the sheet.
5. For plotting data: arrange in descending order, assign $T=(N+1)/m$, and plot the flood against T (that is, against its $y_T$ position).
6. Draw the best-fit straight line and extend it to read the flood for any desired T.

 Q
  |                          o
  |                    o
  |              o
  |        o
  |  o
  +--+----+----+----+----+---- y_T
    0.37  1.50 2.25 3.90 4.60 6.21
    T=2   T=5  T=10 T=50 T=100 T=500

Two points are enough to fix the straight line, since for Gumbel's distribution the line is $Q = \bar x + \sigma\,(y_T-\bar y_n)/S_n$.