Evaluating the suitability of the reference method (Javelle)

Keeping the shape parameter constant, we fit a GEV distribution to durations from 1 to 72 hours (2D+1 model). This gives individual \(\mu\) and \(\sigma\) values for each duration.

The 2D+1 model was fit with both Stan (Hamiltonian Monte Carlo) and checked with a Metropolis-Hastings sampler. The results of the two model fitting procedures were in agreement.

Looking at the posteriors of the ratio \(\mu / \sigma\) for each duration we can determine if the ratio scales with duration:

Shifting the data so that the posterior of the instantaneous duration is centered at zero allows us to look at the form of the difference of the posterior means from zero. Under this shift Javelle’s model would be a horizontal line at zero, and plotting the means of the posteriors against duration shows the difference in scaling behaviors between stations:

We can also look at the difference between each station and Javelle’s model while taking the uncertainty in the posterior into account:

Figure 3: ratio facet plot

Figure 3: ratio facet plot

Using the above behavior to select a QDF model

Figure 3 shows four stations need a model that allows \(\mu\) and \(\sigma\) to change separately over duration. Figure 4 shows both the \(\mu\) and \(\sigma\) parameter benefit from a scaling factor (i.e., we can’t just put the duration dependence on one parameter and not the other).

Figure 4: scaling on mu and sigma

Figure 4: scaling on mu and sigma

Of the three models we are considering (Javelle, Crochet, Van Vyver) only Van Vyver allows for separate scaling of the \(\mu\) and \(\sigma\). However, instead of implementing Van Vyver’s model we propose the “Double Delta” model, where we have two Deltas instead of one:

\[\begin{equation} \mu_d = \frac{\mu}{1 + d/\Delta_{\mu}} \end{equation}\]

and

\[\begin{equation} \sigma_d = \frac{\sigma}{1 + d/\Delta_{\sigma}} \end{equation}\]

This model is dimensionless while Van Vyver’s is not. Additionally we get \(\sigma_0\) as the scale parameter of the GEV when d = 0. Since hydrology is often concerned with the instantaneous peak discharge distribution it is perhaps useful to save the intuitive interpretation of \(\sigma_0\) for the d = 0 case.

Implementing Javelle and Double-Delta

The main hurdle to implementing the models is finding a prior for the \(\Delta\) parameter.

To find reasonable values for the \(\Delta\) parameter, we want to fit the Double Delta model to a set of parameters obtained through fitting the 2D+1 model to our data set. If Javelle’s model holds, the double deltas should be equal.

Traditional measures of uncertainty for the \(\Delta\) parameter are meaningless since the uncertainty can be manipulated through the number of durations calculated. To circumvent this we sample different populations from the posteriors of \(\mu\) and \(\sigma\) and re-fit the Double Delta model to each population.

The Delta value analysis reveals the model needs to account for three scenarios via the Delta priors:

  1. \(\Delta_{mu} = \Delta_{sigma}\) (f.ex. Dyrdalsvatn)
  2. \(\Delta_{mu} \neq \Delta_{sigma}\) (f.ex. Gryta)
  3. \(\Delta_{mu} = \Delta_{sigma} = +\infty\) i.e. no discernable duration dependence! (f.ex. Viksvatn)

For Double-Delta to be identifiable, it needs constraints on the Deltas; we propose constraining \(\Delta_{mu} \geq \Delta_{sigma}\) in line with Van Vyver and the results above.