Autorregulación

every effort should be made to gather a complete view on the realistic coverage of the historical information available. Nevertheless, it is possible that no clear beginning of the length of time period covered by the historical series ℎ is available. In cases where no information on ℎ is available, it is possible to estimate the length of the historical period ℎ from the properties of the threshold exceedances.

In the case of only one threshold exceedance in the whole historical period, Strupczewski et al. (2014) suggested estimating its value taking ℎ^ = 2 ∗ 𝑡, where 𝑡 indicates the length of the period between the time of the historical record and the beginning of the systematic series. The motivation behind the proposed estimate stems from the idea that the time at which the perception threshold is exceeded is uniformly distributed on a discrete domain [0,1, … , ℎ]. Since the expected value of a uniform distribution on [0, ℎ] (𝐻 ∼ 𝑈(0, ℎ)) is equal to 𝐸[𝐻] = (ℎ − 0)/2, when only one observation is available the suggested estimate for ℎ is 2 ∗ 𝑡.

It can be shown that this estimate corresponds to using a constrained L-moment estimate for the upper limit of a uniform distribution. Hosking and Wallis (1997) gave the relationship between the first 2 L-moments (𝜆₁, 𝜆₂) and the parameters of a uniform distribution on the continuous domain [𝛼, 𝛽], 𝑈(𝛼, 𝛽) as:

𝜆₁=1

2(𝛼 + 𝛽) (equation B.21)

𝜆₂=1

6(𝛽 − 𝛼) (equation B.22)

from which the following relationships can be derived:

(𝛽 − 𝛼) = 6𝜆2 (equation B.24)

The practical case of estimating the starting year of the historical record could be reframed as the estimation of the value of 𝛽 after having fixed 𝛼 to 0, since the beginning of the gauged record is known.

If the sample of timing of threshold exceedance is composed of only one record of value 𝑡, only the first sample L-moment 𝑙1 can be computed and it corresponds to 𝑙1=

𝑡. The estimate for 𝛽 can then be taken to be 𝛽^ = 2𝑙₁= 2𝑡, which corresponds to the estimate proposed in Strupczewski et al. (2014).

If more than one threshold exceedance is recorded in the historical period, the distance between the beginning of the systematic record and the timing of the historical

exceedances is recorded in the sample (𝑡(1), … , 𝑡(𝑛)), where samples are ordered in

decreasing order from the largest value of 𝑡 (that is, the first historical record) to the smallest (that is, the most recent historical record). From the sample of time records, sample values for the first 2 L-moments 𝑙₁ and 𝑙₂ can be obtained. Keeping 𝛼 fixed at 0, an estimate for 𝛽 can be again be found as 𝛽^ = 2𝑙1. This estimate is here called L1

estimate.

However, rather than fixing the 𝛼 parameter, an estimate for the interval width 𝑤 = (𝛽 − 𝛼) could be taken to be 𝑤^ = 6𝑙₂. The width 𝑤 can then be plugged in as an estimate for the historical record length ℎ. This estimate is here called L2 estimate. Both estimation approaches suffer from the drawback that there is no formal assurance that the estimated value of ℎ is actually larger than 𝑡₍₁₎, the time of record of the first historical event. This can be resolved in practice by taking the estimate of ℎ to be the maximum between 𝑡₍₁₎ and the preferred estimate of 𝑏^. Furthermore, if more than one historical record is available, the estimated value of ℎ would belong to the continuous scale rather than the discrete scale, which can be easily fixed in practice by rounding the 𝛽^ value.

Since the sample 𝑙₁ value corresponds to the average value of a sample, the L1 estimate corresponds to twice the average distance between the time of record of the historical values and the beginning of the systematic sample. This value is relatively easy to compute and easier to communicate than the L2 estimate. Furthermore, the L2 estimate can only be calculated if at least 2 threshold exceedances are present in the historical sample. The performance of the L2 estimate has been investigated in the simulation study, but little difference from the performance of the L1 estimate was found and is not discussed further.

Once again, next to the use of L-moments, another possible approach to the statistical estimation of a parameter characterising a distribution is maximum likelihood. The maximum likelihood estimate for the left boundary of a uniform distribution corresponds to the minimum of the sample. In the practical application at hand, this corresponds to taking ℎ = 𝑡₍₁₎, or, in other words, having the period covered by historical information to start at the time of recording of the first historical record. Here this estimate is called the T1 estimate. The use of the T1 estimate is widely discouraged in the literature on the inclusion of historical data for flood frequency analysis, but it is important to

acknowledge that a statistical motivation for such estimate could be given, especially in the unlikely case of a large number of historical events.

All the approaches to the estimation of ℎ presented in this section (L1, L2 and T1 estimates) rely on the assumption that the timing of the threshold exceedances is uniformly distributed. This is a realistic assumption when the process describing the flood magnitude is stationary, that is, it is equally likely at any point in time to record a threshold exceedance. Deviations from this assumption – due to either the natural alternation between flood rich and flood poor periods, or a higher likelihood of not

recording events which happened further away in time – would undermine the performance of all estimation approaches. The use of a statistical estimation for the historical period length should only be employed when it is truly impossible to identify a sensible point in time at which it is credible that all events above a threshold are

present in the historical sample.

Figure B.31 shows the RMSE for the estimated shape parameter and Figure B.32 the estimated scale parameter as a function of the systematic record length for selected lengths of the historical records when using different approaches to estimate the value of ℎ within the likelihood framework. The RMSE values obtained when using the true value of ℎ and when using systematic data only are also shown.

The bigger differences in the performance for the different approaches to the estimation ℎ can be seen for shorter records and higher perception threshold. The historical samples in these cases is fairly small and all approaches are likely to give a poor estimate of ℎ, the T1 approach in particular. Once the historical sample size is larger, however, using an estimated value of ℎ gives a similar performance to using the true value of ℎ, with little difference between the T1 and L1 estimation.

Figure B.31 RMSE for the shape parameter as a function of the systematic record length for selected historical record lengths

Notes: Line types and shapes indicate the estimation approach used to estimate 𝒉. Each panel shows a different 𝑿𝟎 and shape parameter combination.

Figure B.32 RMSE for the scale parameter as a function of the systematic record length for selected historical record lengths

Notes: Line types and shapes indicate the estimation approach used to estimate 𝒉. Each panel shows a different 𝑿𝟎 and shape parameter combination.

Estimation method: Likelihood.